Self-Assembly in Nafion Membranes Upon Hydration

Article

pubs.acs.org/JPCB

Self-Assembly in Naﬁon Membranes upon Hydration: Water Mobility and Adsorption Isotherms Aleksey Vishnyakov and Alexander V. Neimark*

Department of Chemical Engineering, Rutgers, the State University of New Jersey, 98 Brett Road, Piscataway, New Jersey 08854, United States

*S Supporting Information

ABSTRACT: By means of dissipative particle dynamics (DPD) and Monte Carlo (MC) simulations, we explored geometrical, transport, and sorption properties of hydrated Nafion-type polyelectrolyte membranes. Composed of a perfluorinated backbone with sulfonate side chains, Nafion self-assembles upon hydration and segregates into interpenetrating hydrophilic and hydrophobic subphases. This segregated morphology determines the transport properties of Nafion membranes that are widely used as compartment separators in fuel cells and other electrochemical devices, as well as permselective diffusion barriers in protective fabrics. We introduced a coarse-grained model of Nafion, which accounts explicitly for polymer rigidity and electrostatic interactions between anionic side chains and hydrated metal cations. In a series of DPD simulations with increasing content of water, a classical percolation transition from a system of isolated water clusters to a 3D network of hydrophilic channels was observed. The hydrophilic subphase connectivity and water diffusion were studied by constructing digitized replicas of self-assembled morphologies and performing random walk simulations. A non- monotonic dependence of the tracer diffusivity on the water content was found. This unexpected behavior was explained by the formation of large and mostly isolated water domains detected at high water content and high equivalent polymer weight. Using MC simulations, we calculated the chemical potential of water in the hydrated polymer and constructed the water sorption isotherms, which extended to the oversaturated conditions. We determined that the maximum diffusivity and the onset of formation of large water domains corresponded to the saturation conditions at 100% humidity. The oversaturated membrane morphologies generated in the canonical ensemble DPD simulations correspond to the metastable and unstable states of Nafion membrane that are not realized in the experiments.

I. INTRODUCTION predictions of structural and transport properties. It should be fi Polyelectrolyte membranes (PEMs) are widely used as noted that the Na on equivalent polymer weight Meq, side compartment separators in electrochemical devices.1 In chain length, and distribution of side chains along the skeleton particular, Nafion (DuPont, Figure 1), a perfluorinated linear are not precisely controlled during polymer synthesis. This uncertainty further complicates the comparison of modeling polymer with hydrophilic sulfonate side chains, is the basic ff material for proton conducting membranes of fuel cells.1 While e orts with available experimental data, even on a qualitative fi level. the acid form of Na on is most common for electrochemical fi fl applications, metal-substituted membranes are also of interest Macroscopic models typically present the Na on per uor- as permselective diffusion barriers in protective fabrics.2 Upon oalkane backbone as a continuum medium where spherical or hydration, PEMs are known to undergo nanophase segregation cylindrical water aggregates, which grow in size and coalesce as − into hydrophilic and hydrophobic subphases.1 In the case of the water activity increases (e.g., refs 4 6, review 7). The Nafion, the former contains hydrophilic side chains, counter- electrostatic interactions are modeled by double layers formed ions, and sorbed water; the latter is comprised of the at the cluster surfaces by side chains on the outer side and by perfluoroalkane backbone. At higher hydration levels, the counterions on the inner side. Such models can reasonably fi 6,8 hydrophilic subphase is continuous and provides facile water describe water sorption and ion exchange in Na on, but they fi transport. involve adjustable parameters tted directly to the experimental The transport properties of hydrated PEM depend on their results and presume a particular geometry of hydrophilic fi nanostructure, which is determined by polymer chemistry and aggregates in advance. The self-consistent eld theory allowed a solvent content. In Nafion and other similar polyelectrolytes more detailed consideration of polyelectrolyte morphology, but (e.g., Flemion, Aquivion), the segregation morphology is irregular with no particular long-range order of hydrophilic Received: May 20, 2014 aggregates.3 The irregular structure complicates both inter- Revised: August 26, 2014 pretation of SAXS and SANS results and theoretical/simulation Published: August 26, 2014

fi − Figure 1. Dissection of Na on monomer into beads (example with Meq = 944 Da shown). Beads are connected by nearest neighbor (1 2) and second neighbor (1−3) bonds to control polymer rigidity and side chain flexibility. it has been limited to linear block copolymers9,10 presented as limited size of simulated systems (up to 9 nm) is still Gaussian chains. The same applies to the mesoscale dynamic insufficient to investigate the membrane segregated morphol- density functional theory of polyelectrolytes,11,12 which was ogy. also used to model Nafion segregation.13 A more complicated A significant increase in spatial and temporal scales was approach was implemented by Galperin and Khokhlov,14 who achieved with coarse-graining and reducing the system degrees effectively divided the branched polymer into “subchains” of freedom by “lumping together” several atoms to form whose conformations may be considered independent of each quasiparticles or “beads”.43 In the most popular for modeling other. The subchain conformations were treated statistically polymeric systems coarse-grained MD (CGMD) method, using probability distributions, and the system of interacting quasiparticles interact through effective pairwise hard-core subchains was approximated by a single ideal chain in a self- potential (such as Lennard-Jones) parameters that are − consistent field. This approach was implemented on a lattice, commonly derived from atomistic modeling.13,44 47 In order producing a sponge-like irregular network of hydrophilic to draw reliable conclusions about the morphological structure aggregates.14 of self-assembled polymer, the size of the simulation box must Atomistic molecular dynamics (MD) simulations of Nafion exceed the characteristic scale of segregation by at least 1 order started with minimization of individual side chains.15 In our of magnitude. Malek et al.48 modeled a Nafion−carbon earlier works,16,17 we studied solvation of Nafion oligomers in nanocomposite using Lennard-Jones potentials between water and other solvents. These simulations confirmed quasiparticles representing different fragments of Nafion microphase segregation on the scale of a few nm and formation polymer, solvent, and carbon particles. The authors obtained of hydrophilic clusters linked by “merging” bridges, which build a self-assembled structure composed of the hydrophobic up and snap off due to thermal fluctuations. However, the size backbone and water clusters qualitatively similar to that of the simulated system was insufficient to analyze the cluster found in earlier atomistic MD simulations. However, the larger network morphology. As the atomistic and united-atom force system size allowed the evolution of segregated morphology to fields were established and computational facilities grew, MD be distinguished as hydration progressed. At low water content, simulations became able to handle systems substantially larger hydrophilic domains were roughly spherical and poorly than the size of individual hydrophilic aggregates and started to connected. At higher hydration, a sponge-like network of tackle the problems of the membrane segregation morphology roughly cylindrical aggregates of 3 nm in diameter was formed. − and water transport in perfluorinated and other ionomers.18 33 In this work, we employ the dissipative particle dynamics Generally, atomistic simulations produced a qualitatively (DPD) method,49,50 which implies soft short-range repulsion evident picture of individual water clusters that coalesce and potentials and therefore allows for much longer time steps form a continuous water cluster network as hydration increases. (compared to MD) and facilitates the system equilibration. Even with modern computers, computational expenses severely Several authors employed DPD simulations for modeling limit the spatial and temporal scales of PEM simulations. For Nafion-type membranes. The first DPD simulation of Nafion in this reason, Knox and Voth34 used in their MD simulations of acid form was conducted by Yamamoto et al.51 The Nafion the initial configurations corresponding to different conservative repulsion parameters were estimated from the morphologies suggested in the experimental literature. The mixing energy calculations conducted with atomistic modeling. large size of the system allowed for performing virtual SAXS The electrostatic interactions were implicitly mimicked by − scattering experiments with the simulated polymer config- short-range forces.51 53 The authors found irregular segrega- urations. The authors concluded that the “ionomer peak” tion morphologies, with reasonable correspondence to present in scattering results is insensitive to the segregation experimental results. Later, Dorenbos et al.54,62 and Wu et morphology, that further complicates the interpretation of al.53,55 employed the same model for studies of nanostructure SAXS and SANS results. Extensive MD simulations were and water diffusion in several perfluorinated ionomers that fl ff 56 employed to look into more subtle issues, such as the in uence di ered by Meq and side chain length. Sawada et al. accounted of equivalent weight, side chain length, and distribution along for possible cross-linking of the perfluorinated skeleton chains the skeleton on structural and transport properties of hydrated and found that this effect leads to much smaller hydrophilic − Nafion,34 37 gas adsorption,38 Nafion behavior near solid aggregates of only 1.8 nm in diameter. Eliott et al.57 combined surfaces,39 transport under electrostatic field,35 and influence of DPD results with experimental SAXS/SANS studies using a added nonvolatile solvents such as phosphoric acid and ionic novel model-independent procedure. The modeling revealed a liquids.40,41 In the most recent work, Daly et al.42 performed an multilevel membrane organization, with hydrophilic−hydro- up to 200 ns long atomistic NPT ensemble MD simulation of phobic segregation on a smaller scale and larger scale water self-diffusion in Nafion and calculated water adsorption organization of the fluorocarbon backbone. This result is isotherms with GPU based MC simulations. This work sets a consistent with previous NMR studies.58,59 Jorn and Voth60 benchmark for current atomistic simulations; however, a modeled the segregation in Nafion with DPD and calculated

11354 dx.doi.org/10.1021/jp504975u | J. Phys. Chem. B 2014, 118, 11353−11364 The Journal of Physical Chemistry B Article proton conductivity using the smoothed particle hydro- I and J. Following the standard approach to DPD simulations of 50 dynamics approach, based on local concentration and charge self-assembly, the intracomponent repulsion parameters aII densities in the resulting structures. The calculated conductiv- between beads of the same type are set equal, irrespective to the ities showed very reasonable agreement with experimental data. bead type. The beads are tightly packed with a substantial ρ 3 The DPD studies mentioned above made an important step overlap. We accepted the reduced bead packing density of Rc forward in modeling the segregation morphology of Nafion. = 3, common in DPD simulations.50 However, these models lacked several features that are critically The random and drag force also acted between overlapping important for analyses of structural and transport properties of beads along the line connecting the bead centers. Random force (R) fl PEM: polymer rigidity and explicit electrostatic interactions. Fij that accounts for thermal uctuations is taken proportional Noteworthy, parametrization of these models was based on to the conservative force that is also acting along the vector 49,50,61 (R) σ R θ θ generic DPD parameters devised for aqueous systems. In between the bead centers: Fij (rij)= w rij ij(t)rij, where ij(t) this paper, we suggest a novel DPD model of hydrated metal- is a randomly fluctuating in time variable with Gaussian fi F(D) substituted Na on membranes. The proposed model accounts statistics. The drag force is velocity-dependent: ij (rij, vij)= ff fi −γ D − for polymer chain rigidity that is di erent for the Na on w (rij)(rij*vij), where vij = vj vi and vi and vj are the current backbone and side chains and includes explicit electrostatic velocities of the particles. We assume the common relationship forces between the charged polymer fragments and dissociated between the drag and random force parameters wD(r)= R 2 − 2 D ≥ σ γ counterions. In contrast to previous works, we attempted to [w (r)] =(1 r/Rc) at r < Rc, w (r)=0atr Rc. and are customize the coarse-grained interaction parameters against the parameters that determine the level of energy fluctuation and available experimental and atomic simulation data. For the first dissipation; they are related as σ2 =2γkT, which allows a time, we not only generated self-assembled structures in constant temperature to be maintained in the course of hydrated Nafion membranes and explored the specifics of simulation via the Langevin thermostat. We assumed γ = 4.5, a nanosegregated morphologies, but we also established their common value fitted to the diffusion coefficient of water. thermodynamic properties, in particular, the relationship The polymer beads are connected by harmonic bonds (B) − between the level of hydration and humidity. Fij (rij)=Kb(rij r0)rij/rij, where Kb is the bond rigidity, which This work builds upon our earlier DPD model developed for depends on the bead type, and r0 is the equilibrium length. studies of interactions of toxic chemicals with perm-selective Following our recent papers,63,64 in addition to this nearest Nafion barriers.62 The paper is structured as follows. In section neighbor (1−2) bond, we also introduced the second neighbor II, we formulate the DPD model of coarse-grained Nafion (1−3) harmonic bonds in order to control the skeleton rigidity polymer and justify its dissection into the soft repulsive beads. and side chain flexibility. Section III discusses the DPD model parametrization that is Bead Types. We consider metal-substituted Nafion based on a combination of atomistic MD and coarse-grained oligomers of chemical structure shown in Figure 1. The MC simulations. The evolution of self-assembled morphology equivalent weight Meq of the polymer was varied from 944 (12 as the water content increases is studied in section IV. The carbon atoms between the neighboring side chains) to 1744 ff hydrophilic network connectivity and water di usion are (28 carbon atoms between the neighboring side chains); Meq explored in section V by using digitized replicas of the system values are given for the anion (the total Meq can be obtained by snapshots and modeling random walk motion of a tracer. In adding the mass of the cation). In all simulations, the side section VI, using the MC test particle insertion method, we chains are separated by equal fluorocarbon fragments, and the determine the chemical potential of adsorbed water at given side chains were of the same length. In order to explicitly hydration levels and construct the water sorption isotherms, include electrostatic interactions in the DPD model, we which allowed us to identify the conditions of saturation and to dissected the Nafion chain using four bead types. The determine the limits of stability of generated conformations. hydrophobic beads of type C, which lump together four Main conclusions and suggestions for prospective research are carbon atoms with attached fluorine atoms, represent skeleton fl fl given in section VII. per uoroalkane fragments (CFx)4. The side chain per uor- − − − − oether fragment (O CF(CF3) CF2 O ) is modeled by the II. COARSE-GRAINED MODEL OF bead of type E that is less hydrophobic than bead C. The METAL-SUBSTITUTED NAFION necessity of such detailization is caused by the role that the fl fi The proposed coarse-grained model of Nafion is constructed per uoroether fragment plays in interactions of Na on with phosphororganic chemicals.58,65 Two bead types S− and S following the classical DPD approach originated from the − seminal work of Groot and Warren.49,50,61 The system under represent negatively charged dissociated CF2CF2SO3 and consideration is presented as a multicomponent mixture of neutral non-dissociated CF2CF2SO3Msulfonategroups, beads mimicking characteristic fragments on the hydrated respectively. The latter includes an alkali metal and has to be distinguished at low hydration levels where water content is polymer, that interact via pairwise conservative soft repulsive, ffi harmonic bond, and electrostatic forces, as well as random and insu cient for dissociation of all sulfonate groups. Water velocity dependent drag forces: molecules were lumped into hydrophilic beads of type W. Hydrated alkali metal cations were represented by charged + F ()rFrFrFrFr=+++(C) () (B) () (E) () (R) () beads of type M , which in addition to the cation included ij ij ij ij ij ij ij ij ij ij several water molecules. (D) + Frvij(, ij ij ) (1) Bead Volumes. The adopted dissection of the polymer into coarse-grained beads is performed to minimize the difference in ff All beads are assigned an equal e ective diameter Rc. The soft volumes between the polymer fragments represented by (C) (C) ff ff repulsion force Fij acts between overlapping beads: Fij (rij)= di erent bead types. The e ective volumes of fragments are − (C) ≥ “ aIJ(1 rij/Rc)rij/rij at r < Rc, Fij (rij)=0atr Rc, where aIJ is estimated from the functional group volumes ( Bondi the repulsion parameter specific to the given bead pair of types tables”66,67 that are available for perfluorinated compounds68

11355 dx.doi.org/10.1021/jp504975u | J. Phys. Chem. B 2014, 118, 11353−11364 The Journal of Physical Chemistry B Article

Table 1. Coarse-Grained Model of Hydrated Naﬁon: Bead Types, Interaction Parameters, Bond Lengths, and Stiﬀness

parameterization of coarse-grained model of Naﬁon

conservative parameter, aIJRc/kT bead type fragment eﬀective volume (Å3) WK+ SS− CE

1 W (H2O)4.5 134.5 50 50 50 50 65 60 + + a 2 M [M(H2O)3.5] 122.1 50 50 50 50 65 60 − a 5 S (CF2)2SO3M 136.5 50 50 50 50 75 65 − − − 6 S (CF2)2SO3 119.3 50 50 50 50 75 65 − − 3 C (CFx)4 131.3 65 65 75 75 50 59 − − − 4 E OCF2 CFCF3O 117.1 60 60 65 65 59 50 − 2 − 2 nearest neighbor bonds U1−2(r) = 0.5K1−2(r re) second neighbor bonds U1−3(r) = 0.5K1−3(r re) 2 2 bond K1−2Rc /kT re (Å) bond K1−3Rc /kT re (Å) C−C 160 4.1 C−C 80 8.2 C−E 160 3.7 C−E 30 6.0 E−S 160 4.4 C−S 30 6.0 aEstimated for K+ counterion. but not for sulfonates), as well from the molecular volumes of group. Since each M+ bead included 3.5 molecules of water, λ = representative compounds calculated with the PQS ab initio 3.5 is the minimum hydration level at which all counterions package.69 To provide for an approximate equality with the may be considered as dissociated from their sulfonate groups volumes of the skeleton and sulfonate beads, the volume of and modeled as M+ beads. At lower hydration levels, λ < 3.5, all water bead W is chosen as 135 Å3, which corresponds to the water molecules are assigned to M+ beads; i.e., there are no effective bead size of R = 7.4 Å. With a bead density of ρR 3 = uncharged W beads. To secure electroneutrality, the number of c c − 3, the corresponding number of water molecules in the W bead charged sulfonate group S beads is equal to the number of + is nW = 4.5. Note that this noninteger number of molecules per counterions M , and the number of neutral S beads is quasiparticle is permissible in coarse-grained DPD modeling. calculated from the difference. At λ > 3.5, all sulfonate groups The bead types, respective fragments, and their volumes are are assumed to be dissociated. listed in Table 1. The underlying calculations of effective volumes are described in the Supporting Information, section I. III. PARAMETERIZATION OF INTERACTION Electrostatic Interactions. Electrostatic interactions be- POTENTIALS − tween the dissociated sulfonate beads S and hydrated Intracomponent Repulsion Parameter. We assumed the counterion M+ beads are treated using the smeared-charge “ ” 70 same intracomponent ( self-repulsion ) parameter aII =50kT/ approach of Groot. Instead of point charges interacting via R for all interactions between beads of the same type. This self- → c the Coulomb potential that diverges at r 0, each charge was repulsion parameter is chosen to approximately reflect the distributed within the sphere of the smearing diameter Re = overall compressibility of hydrated Nafion that is much higher − 1.2Rc with the charge density decreasing linearly from the bead than the pure water compressibility (e.g., compare 1.8 × 10 9 70 − 72 − − center toward the periphery. The choice of Re is rather Pa 1 for n-perfluorononane with 4 × 10 10 Pa 1 for water). It arbitrary; since the charge cloud size does not have a clear is worth noting that the chosen value of aII is lower than the physical meaning,71 its influence on the thermodynamic value of 119.8kT/Rc that would be estimated from the pure properties of electrolytes is still to be examined. We chose Re water compressibility (e.g., ref 50, Supporting Information to smaller than recommended by Groot in an attempt to improve ref 73), as was implied in previous DPD models of Nafion.51 71 calculation efficiency. The comparison between the screened This is a significant difference: solvated membrane in our and Coulomb potential created by two point charges is given in “ ” − model is much softer . the Supporting Information. The sulfonate S and counterion Perfluoroalkane Skeleton Rigidity. The Nafion skeleton M+ beads bear charges of −e and +e, respectively. Partial is modeled as a linear sequence of hydrophobic beads C, each fl ff charges on the skeleton and per uoroether fragments are representing four CFx groups. Skeleton rigidity a ects the neglected. Ewald summation was employed to account for the distance between the neighboring side chains and the overall long-range electrostatic contribution. elasticity of the matrix. In order to fit bond length and rigidities The hydrated counterion bead M+ is chosen to contain 1 that are determined by the torsional potentials rather then by cation and 3.5 water molecules and had the same parameters as the stiffness of covalent bonds, we performed MD simulations the water bead W, except for being positively charged. Thus, we of a perfluorohexadecane melt with force field from ref 74 effectively assumed that the volume of a counterion is close to similar to our prior simulations of Nafion31,32 (see the that of one water molecule, which is reasonable for larger Supporting Information, section II, for simulation details). counterions. Merging of water molecules with an alkali metal Simulations were conducted in the NPT ensemble at ion in the same bead has important implications. In our model, atmospheric pressure and a temperature of 450 °C, since the bead charges are fixed; that is, S beads were not allowed to C F freezes under ambient conditions. The MD trajectories − 16 34 dissociate into cation M+ and anion S beads in the course of were recorded to disk. Each molecule was dissected into four + − simulation and, respectively, M and anion S beads cannot fragments (four CFx groups each), and probability distributions recombine. Water content in Nafion is expressed either as water of intramolecular distances between the centers of mass of each mass per unit mass of dry polymer or by the hydration level λ fragments were calculated (Figure 2). After that, we conducted expressed as the number of water molecules per sulfonate DPD simulations of tetramers of skeleton C beads under

11356 dx.doi.org/10.1021/jp504975u | J. Phys. Chem. B 2014, 118, 11353−11364 The Journal of Physical Chemistry B Article

Figure 2. Fitting the bond rigidity in the DPD model of fl per uorohexadecane C16F34 to the results of atomistic MD simulations Figure 3. Results of MD and DPD simulations of C36F44 in water at T at T = 450 K. The distributions of distances between the chain DPD = 300 K. The distributions of distances between DPD beads separated beads separated by one, two, and three harmonic bonds are matched by three, five, and eight harmonic bonds are compared with the to the distributions of distances between the centers of mass of the distributions of distances between the centers of mass of the corresponding fragments of the atomistic chain, each of which contains corresponding fragments in the atomistic representation, each of four CF3 or CF2 groups. Reasonable agreement is obtained for beads which contains four CF3 or CF2 groups. separated by two and three bonds. hydrophilic oxygens. As a result, they are modeled as mildly similar conditions and fitted the nearest neighbor and second Δ hydrophobic aEW =9kT/Rc. The sulfonate end-groups are neighbor bond parameter paying most attention to distances Δ considered as hydrophilic ( aSW = 0) beads, whether between beads separated by two and three bonds. The resulting dissociated or not. The sulfonate groups were assumed to distributions are shown in Figure 2. The agreement between interact very unfavorably with the other fragments of the DPD and MD 1−3 and 1−4 distance distributions is very Δ polymer, and mild mismatch of aEW =9kT/Rc was assigned to reasonable. We decided not to attempt achieving a good the repulsion between C and E beads representing the skeleton agreement between MD and DPD distributions for the distance and the ether side chain fragment, respectively. between the neighboring beads in the chain and instead limit Side Chain Bonds. ff Using the short-range conservative the nearest neighbor skeleton bond sti ness to K1−2 = 160kT/ repulsion parameters described above, we obtained the nearest 2 − ff Rc . Fitting the 1 2 distance distribution requires very sti 75 neighbor and second neighbor bond parameters for the side nearest neighbor bonds, which leads to much shorter time chain from MD and DPD simulations of a single Nafion steps and drastically slows down the simulations. At the same ff ff monomer (depicted in Figure 1) in a water bath. Figure 4 time, the nearest neighbor bond sti ness hardly a ects the shows the distribution of distances between DPD beads and effective distance between side chains and therefore is of secondary importance for our purpose. Perfluoroalkane−Water Repulsion Mismatch Parame- Δ ter. aCW was obtained by performing DPD simulation of n- C36F74 that was presented as a 9-mer of C beads with the nearest neighbor and second neighbor bond parameters derived from MD simulation of perfluorohexadecane. The mismatch Δ fi parameter aCW =15kT/Rc was determined by tting to a respective atomistic MD simulation. The results are shown in Δ Figure 3. At low values of aCW, coarse-grained n-C36F74 Δ behaves as an extended chain. At aCW=15kT/Rc, the DPD simulation shows frequent transitions between the collapsed fi Δ globule and extended chain con gurations, and at aCW > fi 20kT/Rc, the globular con gurations prevail. Since the MD simulation was not long enough to estimate the time the chain spends in the coil and globule configurations, the extended and collapsed states were modeled separately. The end-to-end distances of the extended states obtained by MD and DPD are Δ in good agreement, thus justifying the choice of both aCW and bond parameters. Figure 4. Results of MD and DPD simulations of C36F44 in water at T Side Chain Repulsion Parameters. The mismatch = 300 K. The distributions of distances between S, E, and C beads are parameters for perfluoroether fragment E are estimated from fitted to the distributions of distances between the centers of mass of the water and skeleton parameter approximately taking into the corresponding fragments in the atomistic representation, shown in account that two fluorocarbon groups are replaced by Figure 2.

11357 dx.doi.org/10.1021/jp504975u | J. Phys. Chem. B 2014, 118, 11353−11364 The Journal of Physical Chemistry B Article

fi λ Figure 5. Snapshots of nanophase separation in hydrated Na on: Meq = 1144 (top) and 1744 (bottom). Hydration level = 2.25 (a, e), 6 (b, f), 9 (c, g), and 13.5 (d, h). Nafion skeleton and perfluoroether side chain beads are shown in red, sulfonate groups in dark blue, counterions in green, and water beads in light blue. respective distributions of the intramolecular distances between We considered Nafion polymer with side chains separated by fi the centers of mass of corresponding Na on fragments 12, 16, 20, and 28 skeleton beads, which corresponded to Meq = obtained in MD simulations. The presence of a side chain 944, 1144, 1344, and 1744 Da for the anion, respectively. It is substantially changes the conformations of the skeleton. worth noting that most published data that dealt with alkali − − − fi ≈ − Although distorted trans C C C C torsion angles continue metal substituted Na on membranes are of Meq 1100 1200 to dominate the skeleton conformation similarly to perfluor- Da. Experimental water sorption isotherms on these mem- oalkane chains (Figure 3), we also observed hairpin-like branes were summarized in ref 65. At 100% humidity, water conformations, where the skeleton makes a sharp bend close sorption decreases from approximately 29% wt in Li+ form to to the point of the side chain attachment. As a result, the 4−6% wt in Cs+ form. The K+ form of Nafion, which was distribution of the distances between the fragments correspond- targeted in our study, is in the middle of this range; it adsorbs ing to second neighbor fluorocarbon beads has a minor second 9.5−12% of dry polymer weight, which corresponds to λ = peak, which we could not reproduce with crude DPD models. 6.2−7.9, according to refs 65, 77, and 78. Water sorption Nevertheless, we found the agreement between MD and DPD fi decreases with the Na on equivalent weight Meq due to the results reasonable, taking into account the coarse-grained increasing fraction of hydrophobic skeleton groups.4 In the nature of the model under consideration. The bond parameters NPT DPD simulations, we considered a wide range of are presented in Table 1. hydration levels from λ = 2.25 to λ = 13.5. Figure 5 shows the snapshots of selected systems of different λ IV. DPD SIMULATIONS OF SELF-ASSEMBLED Meq and . In general, the evolution of the system morphology MORPHOLOGY OF HYDRATED NAFION is qualitatively similar to the classical scenario of the percolation 4,51,79 Using the DPD model described above, we studied the self- system formation. At low hydration levels, the water and fi counterions form small clusters around the sulfonate groups assembled phase segregation in metal-substituted Na on λ membranes by equilibrating the system in the canonical (note that in our model at < 3.5 there is not enough water to ensemble DPD simulation. We considered the polymer of dissociate all sulfonate groups). As hydration increases, a ff percolation transition occurs: isolated water clusters grow and di erent equivalent weight Meq and varied the water content to explore the evolution of membrane morphology upon coalesce, forming an irregular 3D network of worm-like channels. Further increase in hydration leads to formation of hydration. The equivalent weight Meq of the polymer is determined by the number of beads between the neighboring interconnected spheroidal clusters that grow in size. The side chains. Simulations were performed in a cubic 30 × 30 × system morphology beyond the percolation threshold resem- 3 4 30Rc cell containing 81 000 beads, starting from a random bles the classical model proposed by Gierke and Hsu, who configuration. The method of Paganabarra et al.76 was used to assumed a regular network of spherical water clusters up to 4 integrate the equations of motion with a time step of 0.02. The nm in diameter connected by cylindrical channels of balance of energy fluctuation and dissipation provided a approximately 1 nm in diameter. It is worth noting that Langevin thermostate that allowed maintaining the average narrow (2−3 water molecules in diameter) cylindrical water temperature within 1% of the required value and temperature bridges between water clusters were observed in our atomistic fluctuations below 5%. The segregation progress was simulations of Nafion.65 As the water content increases further, characterized by monitoring the number of pairs of overlapping spherical clusters grow and form large spheroidal aggregates, W beads. When this number stabilized, equilibrium was signifying macroscopic separation of the system into water and considered established. Further details of DPD simulations hydrated Nafion phases. Below, we show that these may be found in the Supporting Information, section III. morphologies correspond to supersaturated states that would

11358 dx.doi.org/10.1021/jp504975u | J. Phys. Chem. B 2014, 118, 11353−11364 The Journal of Physical Chemistry B Article not be observed under real experimental conditions with humidity limited by saturation.

V. CONNECTIVITY OF THE HYDROPHILIC SUBPHASE AND WATER DIFFUSIVITY The visual, purely qualitative observations of Nafion morphology do not allow evaluation of aqueous subphase connectivity, which determines the transport of water through PEM and is extremely important for most applications of these materials. It should be noted that the common DPD implementation employed in this work is unable to simultaneously reproduce experimental self-diffusion and viscosity in liquid systems.50 Figure 6. Digitized lattice replicas of the membrane morphology fi mapped onto a cubic lattice (hydrophilic subphase shown in light Our DPD model of Na on is designed to predict the structure λ λ of segregated polyelectrolyte, but it is not capable of directly pink): (a) Meq = 1144, = 9; (b) Meq = 1744, = 13.5. predicting the absolute values of diffusion coefficients. To location and the original location as a function of the number quantify the effect of hydrophilic subphase connectivity on of steps. water diffusion, we constructed digitized replicas of the Characteristic examples of MSD dependences are shown in simulated configurations in a fashion similar to that suggested 54 Figure 7. Line 1 is characteristic of a system with isolated water by Dorenbos et al. For each equilibrated simulation, we collected a trajectory of 400 frames. The simulated configuration was mapped onto a cubic lattice grid. The step of the grid in each dimension was 0.5Rc, which is about the size of one water molecule. Since DPD beads overlap, one lattice site may belong to several beads of different type. In order to assign a given lattice site to either a mobile hydrophilic or immobile hydrophobic subphase, we calculated the site preference as follows:

pr()l⃗ = ∑ twriil (̅ , r̅ ), i=1 2 wherewr (il̅ , r̅ )=−| (1 r il̅ − r̅|| / Rc ) at r il̅ − r̅|

<=|Rwcc,0at ril̅ − r̅|≥ R

Here, rl is the radius-vector to the center of lattice site l, ri is the ffi Figure 7. Mean square distance as a function of the number of steps radius-vector to the ith bead, and ti is a mobility coe cient for the tracer random walk within the hydrophilic subphase of related to the bead type: t = −1 for all mobile beads (that is, fi fi i hydrated Na on for several characteristic con gurations. (1) Meq = water and hydrated counterions) and ti = 1 was assigned to all 1144, λ = 2.25: water forms small isolated clusters, diffusion is λ polymer beads (C, E, and S bead types). Note that the M bead localized. (2) Meq = 1144, = 9: this level of hydration is about the mostly represents water and is a part of the water path through experimental value at 100% humidity and corresponds to fastest − ff fi the system, while S/S beads are not, despite its hydrophilic di usion obtained for given Meq. Water forms a well-de ned λ nature. The entire S bead represents a fragment of polymer and continuous subphase. (3) Meq = 1344, = 13.5: the mean square displacement reflects large water aggregates and their poor includes hydrophobic CF2 groups. Water may hydrate the S fi ff bead but does not diffuse through. p(r⃗) shows whether mobile connectivity in unstable con gurations; fast di usion at short time l and slow diffusion at longer time scale are observed. or immobile beads prevail in the close vicinity of site l.Ifp is negative, site l is assigned to a mobile (hydrophilic) subphase; otherwise, site l is assigned to the immobile subphase. Three clusters below the percolation threshold: the tracer never leaves characteristic examples of such obtained digitized lattice the cluster in which the random walk starts. Above the replicas are shown in Figure 6. percolation threshold, the MSD (lines 2 and 3) asymptotically Having created digitized lattice replicas of the membrane converges onto a straight line, which is characteristic for a morphology, we modeled water diffusion within the hydrophilic diffusion-type process along a continuous network of water subphase as a simple random walk of a tracer particle within the clusters. The slope of this line characterizes the water diffusivity replica’s mobile phase. We therefore assumed that the structural in the segregated membrane of given hydration λ relative to the evolution of the segregated polymer is much slower than the water diffusivity in the bulk water modeled as a random walk on water diffusion inside the hydrophilic subphase. Each random a simple cubic lattice. As such, the ratio of this slope to that for walk started from a randomly selected lattice site that belonged the random walk on an unrestricted lattice (all sites belong to fi to the hydrophilic subphase, and each step was an attempted the hydrophilic subphase) quanti es the ratio DW/DP of the move to one of the six sites that neighbored the current water self-diffusion coefficients in Nafion and in bulk water. ff ffi location. The move was accepted if the attempted site belonged The relative water di usion coe cients, DW/DP, calculated in to the mobile subphase. 104 random walks were performed on random walk simulations are presented in Figure 8 and Table each of 400 replicas for proper averaging. We calculated the S3 of the Supporting Information. Water diffusivity in the mean square displacement (MSD) between the current membrane is significantly reduced compared to bulk water, and

11359 dx.doi.org/10.1021/jp504975u | J. Phys. Chem. B 2014, 118, 11353−11364 The Journal of Physical Chemistry B Article

further increase of hydration, the water mobility increases, fi achieving a maximum. For Na on of Meq = 1144, the maximum is achieved around λ = 9 and corresponds to a water diffusivity 2 orders of magnitude smaller than that in the bulk. It is interesting that such a big reduction of the diffusion coefficient compares very well with the values derived by Rivin and Schneider (see ref 78) from the membrane permeation measurements80,81 for K+ substituted Nafion 112 in contact with liquid water; see Figure 8. A non-monotonic dependence of the diffusion coefficient on the saturation, which was pronounced in all systems we considered, seems to be counterintuitive, yet it is explained by ff ffi fi Figure 8. Self-di usion coe cient of water in hydrated Na on reduced the specifics of the hydrophilic subphase morphology at high to the diffusion coefficient of pure water for polymers of hydration λ hydration. Indeed, the growth of large disconnected water level for equivalent molecular weight Meq of 1144 and 1344, obtained with the random walk simulation. The green filled diamonds are clusters described in section IV (Figure 6), in which the tracer + fi ≈ gets trapped, leads to the decrease of the hydrophilic subphase experimental data for K substituted Na on-112 membranes (Meq 78 1100 Da). Similar dependences for other Meq are shown in connectivity. It is visible on the snapshots (Figure 6b) for high logarithmic scale in the Supporting Information. The error bars were Meq: large spherical clusters are not connected to each other. estimated from the diffusion coefficients for each of the individual To some extent, this “in silico” observation may be an artifact of Nafion snapshots. the static polymer matrix: in a system of large clusters, the possibility of the formation of temporal intercluster bridges shown in MD simulations16 may lead to additional pathways for DW progressively decreases with the increase of Meq. DW is water diffusion that cannot be captured with static replicas. It is smaller than DP by 1 order of magnitude for Meq = 944 and by worth noting that, as shown below, the membrane 2 orders of magnitude for Meq = 1144 and is negligibly small for conformations obtained in NPT simulations at high hydration Meq = 1344, and larger. Interestingly, such a big reduction of the diffusion coefficient compares very well with the values levels, which exhibit big and poorly connected water clusters ff ffi derived by Rivin and Schneider (see ref 78) from the and, respectively, declining di usion coe cients, correspond to membrane permeation measurements80,81 for K+ substituted oversaturated thermodynamic conditions and, thus, are not Nafion 112 in contact with liquid water; see Figure 8. dealt with in practice. At λ = 2.25, water formed a percolating network in some of Data in Figure 8 shows a dramatic dependence of the water ff the recorded configurations only in Nafion of the lowest di usion on the molecular weight Meq. The water cluster molecular weight. In other systems, the aqueous subphase network connectivity naturally declines with Meq; since the consisted of small and isolated clusters or individual counterion fraction of hydrophilic beads decreases, the distances between beads. It should be noted that the random walks were the side chains increase, thus making the water clusters more ff performed on static configurations; thus, the effects of dynamic isolated. This e ect is very pronounced: DW decreases more percolation that involves slow migration of water clusters and than an order of magnitude as Meq increases from 944 to 1344 formation of temporary bridges between water clusters found in and by another order of magnitude at Meq = 1344. For Meq = 78 λ atomistic MD simulations are not considered with this 1744, we obtained zero DW at all hydration levels above = technique. 6.75. A weaker dependence of DW on Meq was obtained in DPD A continuous mobile subphase manifested by nonzero simulations of Dorenbos et al.54 We are not aware of a detailed ff ffi − di usion coe cients was formed above the percolation experimental analysis of Dw Meq dependencies for nonacid λ fi threshold at = 4.5 in all systems but Meq = 1744. With forms of Na on, but metal-substituted Flemion membranes

Figure 9. (a) Water sorption isotherms calculated using Widom MC trial insertion into the configurations recorded from DPD simulations of Nafion of different equivalent polymer weights. Solid lines serve as guides to the eye. The configurations at a > 1 are thermodynamically unstable and correspond to oversaturated conditions; phase separation into liquid water and hydrated Nafion does not happen in simulations due to the limited system size and periodic boundary conditions. Lines are drawn as guides to the eye. The black square shows the experimental value for K+ fi ≈ 77,80 ff ffi substituted Na on with Meq 1200 Da. (b) Water activity and reduced di usion coe cient as functions of the water content for Meq = 1144 Da. The maximum of water mobility is achieved near saturation and declines in the region of oversaturation due to the formation of disconnected water domains.

11360 dx.doi.org/10.1021/jp504975u | J. Phys. Chem. B 2014, 118, 11353−11364 The Journal of Physical Chemistry B Article show an even weaker (compared to ref 54) decline of diffusivity simulations: decline in diffusion occurs at a thermodynamically with the equivalent weight.82 unstable, unphysical region of oversaturation at a > 1, which is not observed in experiment. In a homogeneous polymer VI. CALCULATION OF WATER SORPTION ISOTHERMS solution, an oversaturated condition for the solvent (that is IN HYDRATED NAFION possible as a metastable state with bulk separation prevented by The large water aggregates obtained in DPD at higher a nucleation barrier) would not cause a decline of solvent ff fi hydration levels raise a general question regarding the di usivity. Na on, however, is segregated, and the phase interpretation of the results of mesoscale modeling in the separation into hydrated polymer and pure water, when polymers that may absorb a limited amount of solvent. hydration exceeds the sorption capacity, is prevented by fi Although DPD simulations performed in the canonical arti cial periodic boundary conditions. The water cluster ensemble reveal the morphology of hydrated polymer, they network connectivity depends strongly on the number of side give no direct information on the equilibrium water activity to chains that concentrate at the interface between the hydrophilic which a particular hydration level corresponds. Furthermore, and hydrophobic subphases. The growing domains of water we do not even know whether the polymer at a given hydration drag the side chains to their surface, thus reducing the overall equilibrated as a closed system in the canonical ensemble surface area of the connecting channels and leading to the ff fi (especially at high hydration levels which are of interest for isolation of water clusters. This e ect is arti cial; it should not protective membranes) is thermodynamically stable under be observed in a macroscopic system, as it was caused by given environmental conditions, i.e., as an open system in the limited simulation time and system volume. As such, the ff grand canonical ensemble. simulated dependencies of the water di usivity on the To estimate the water isotherm and to study the system hydration level should be considered only up to the saturation stability, we calculated the chemical potential of water in the level (Figure 9). fi DPD generated hydrated membranes. For a given hydration Despite the arti cial nature of the structures with large level λ, 400 static configurations of equilibrated systems were spherical water clusters, the reason why they do not merge is “ collected. Then, the chemical potential of water beads μ in worth considering. As water bridges between the clusters dry W ” each of 400 configurations was calculated using the test particle up due to excessive surface area of the interface between the insertion method of Widom.83 A test water bead was inserted at polymer and water, large isolated clusters are formed. Each a randomly chosen location, 1 million trial insertions per frame. clustercarriesapositivechargeduetothehydrated μ counterions, which is compensated by the negative charge of As a reference, the chemical potential of pure water 0 was determined by inserting a trial water bead into a cubic box of sulfonate groups surrounding the positively cluster. Thus, two fi ρ 3 clusters experience electrostatic repulsion from each other, 10Rc in size lled with water beads at reduced density Rc =3. The saturation conditions of 100% humidity correspond to the which creates a potential barrier and prevents their fusion. equality of water chemical potentials in the membrane and in Because periodic boundary conditions suppress system the bulk water. The water humidity H that corresponds to a restructuring and the simulation time is limited, the barriers given hydration level λ is determined by the difference of these associated with the cluster coalescence never get overcome in chemical potentials. Since one bead represents 4.5 molecules of most of the simulations. Figure S4 in the Supporting λ μ λ − Information shows that the clusters experience significant water, the water activity is calculated as ln a( )=( W( ) μ fluctuations in shape but do not merge within the simulation W0)/(4.5kT). Assuming water vapor as an ideal gas, humidity is equal to the thus calculated activity. run. These calculations allowed us to construct the sorption isotherms of water in Nafion, as given by our coarse-grained VII. CONCLUSION models. The isotherms are presented in Figure 9. The We suggested a novel DPD model of hydrated Nafion isotherms have type I shape by IUPAC classification character- membranes. The proposed model accounts for polymer chain istic of microporous adsorbents with a plateau at high humidity, rigidity that is different for the Nafion backbone and side chains which extends into the oversaturation region a > 1. Water and includes explicit electrostatic forces between the charged content at a = 1 corresponds to sorption from saturated water polymer fragments and dissociated counterions. In contrast to vapor. Figure 9 shows that the higher the Meq, the higher the previous works, we attempted to customize the coarse-grained activity at the same λ, and the lower hydration level interaction parameters against the available experimental and corresponds to saturation, which is natural and agrees atomic simulation data. For the first time, we not only generate qualitatively with experimental observations. self-assembled structures in hydrated Nafion membranes with We noted that the beginning of saturation corresponds to the an experimentally informed DPD model and explore the formation of well-defined spheroidal water clusters discussed in specifics of nanosegregated morphologies, but we also establish section IV. Their growth coincides with a sharp increase of their thermodynamic properties, in particular, the relationship activity with λ. Within this regime, the chemical potential of between the level of hydration and humidity. water exceeds that in pure coarse-grained water (a > 1). After The evolution of self-assembled morphology upon hydration that, the situation changes completely and activity becomes was studied with equilibrated NPT DPD simulations performed nearly independent of hydration level. We assume that this at a series of hydration levels λ from 2.25 to 18 water molecules corresponds to the beginning of phase separation onto water per sulfonate group. A classical percolation transition from a and hydrated Nafion, signified by growth of one or two large system of isolated water clusters to a 3D network of hydrophilic clusters. These systems are typically thermodynamically channels was observed. The hydrophilic subphase connectivity unstable and a corresponding part of the isotherms have was characterized by constructing its digitized replica and negative slope dλ/da < 0, which can be noted in Figure 9. performing random walk simulations to determine the effective The adsorption isotherms from Figure 9 explain the non- water diffusivity. We found a significant decrease of water monotonic behavior of relative diffusivity found in NPT diffusivity in the membrane compared to the bulk water, e.g., by

11361 dx.doi.org/10.1021/jp504975u | J. Phys. Chem. B 2014, 118, 11353−11364 The Journal of Physical Chemistry B Article

2 orders of magnitude for Meq = 1144. These values are ■ ACKNOWLEDGMENTS comparable with the available experimental data. Noteworthy, λ This work was supported in parts by DTRA (grants HDTRA1- at high hydration levels (e.g., above = 9 for the membrane of 08-1-0042 and HDTRA1-14-1-0015) and NSF (grant equivalent molecular weight of 1144), we detected the DMR1207239). coalescence of water clusters into large domains, which were poorly connected. This effect caused the reduction of the water REFERENCES diffusivity, which showed a maximum as a function of the level ■ of hydration. Using coarse-grained MC simulations, we (1) Mauritz, K. A.; Moore, R. B. State of Understanding of Nafion. − calculated the chemical potential of water in the hydrated Chem. Rev. 2004, 104, 4535 4585. polymer and constructed the water sorption isotherms. We (2) Ramkumar, S. S.; Sata, U.; Hussein, M. 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