Self-Assembly in Nafion Membranes upon Hydration: Water Mobility

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Self-assembly in Nafion Membranes upon Hydration: Water Mobility and Adsorption Isotherms Aleksey Vishnyakov, and Alexander V. Neimark J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/jp504975u • Publication Date (Web): 26 Aug 2014 Downloaded from http://pubs.acs.org on September 6, 2014

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The Journal of Physical Chemistry

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 NJ 08854

Abstract By means of dissipative particle dynamics (DPD) and Monte Carlo (MC) simulations, we explore geometrical, transport, and sorption properties of hydrated Nafion-type polyelectrolyte membranes. Composed of perfluorinated backbone with sulfonate sidechains, Nafion selfassembles 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 introduce a coarse-grained model of Nafion, which accounts explicitly for polymer rigidity and electrostatic interactions between anionic sidechains 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 nonmonotonic dependence of the tracer diffusivity on the water content was found. This unexpected behavior was explained by 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. Keywords: Polyelectrolyte, Dissipative particle dynamics, Sorption, Diffusion *

author to whom correspondence should be addreessed; email: [email protected]

ACS Paragon Plus Environment

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I. Introduction Polyelectrolyte membranes (PEM) are widely used as compartment separators in electrochemical devices. 1 In particular, Nafion (DuPont, Figure 1), a perfluorinated linear polymer with hydrophilic sulfonate sidechains, is the basic material for proton conducting membranes of fuel cells. 1 While the acid form of Nafion is most common for electrochemical applications, metalsubstituted membranes are also of interest as permselective diffusion barriers in protective fabrics.

2

Upon hydration, PEM are known to undergo nanophase segregation into hydrophilic

and hydrophobic subphases.

1

In case of Nafion, the former contains hydrophilic sidechains,

counterions and sorbed water; the latter is comprised of the perfluoroalkane backbone. At higher hydration levels, hydrophilic subphase is continuous and provides facile water transport. The transport properties of hydrated PEM depend on their nanostructure, which is determined by polymer chemistry and solvent content. In Nafion and other similar polyelectrolytes (e.g. Flemion, Aquivion), the segregation morphology is irregular with no particular long-range order of hydrophilic aggregates. 3 The irregular structure complicates both interpretation of SAXS and SANS results and theoretical/simulation predictions of structural and transport properties. It should be noted that the Nafion equivalent polymer weight Meq, sidechain length, and distribution of sidechains along the skeleton are not precisely controlled during polymer synthesis. This uncertainty further complicates the comparison of modeling efforts with available experimental data, even on a qualitative level. Macroscopic models typically present Nafion perfluoroalkane backbone as a continuum medium where spherical or cylindrical water aggregates, which grow in size and coalesce as the water activity increases (e.g. refs 4-6, review 7). The electrostatic interactions are modeled by double layers formed at the cluster surfaces by sidechains on the outer side and by counterions


2

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on the inner side. Such models can reasonably describe water sorption and ion exchange in Nafion, 6, 8 but they involve adjustable parameters fitted directly to the experimental results and presume a particular geometry of hydrophilic aggregates in advance. The self-consistent field theory allowed a more detailed consideration of polyelectrolyte morphology, but it has been limited to linear block copolymers 9, 10 presented as Gaussian chains. The same applies to the mesoscale dynamic density functional theory of polyelectrolytes, 11, 12 which was also used to model Nafion segregation. 13 A more complicated approach was implemented by Galperin and Khokhlov, 14 who effectively divided the branched polymer into “subchains” whose conformations may be considered independent of each other. The subchain conformations were treated statistically using probability distributions, and the system of interacting subchains was approximated by a single ideal chain in a self-consistent field. This approach was implemented on a lattice, producing a sponge-like irregular network of hydrophilic aggregates. 14 Atomistic molecular dynamics (MD) simulations of Nafion started with minimization of individual sidechains.

15

In our earlier works,

16, 17

we studied solvation of Nafion oligomers in

water and other solvents. These simulations confirmed microphase segregation on the scale of few nm and formation of hydrophilic clusters linked by “merging” bridges, which build up and snap of due to thermal fluctuations. However, the size of the simulated system was insufficient to analyze the cluster network morphology. As the atomistic and united-atom forcefields were established and computational facilities grew, MD simulations became able to handle systems substantially larger than the size of individual hydrophilic aggregates and started to tackle the problems of the membrane segregation morphology and water transport in perfluorinated and other ionomers.

18-33

Generally, atomistic simulations produced qualitatively evident picture of

individual water clusters that coalesce and form a continuous water cluster network as hydration


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increases. Even with modern computers, computational expenses severely limit the spatial and temporal scales of PEM simulations. For this reason, Knox and Voth

34

used in their MD

simulations of Nafion the initial configurations corresponding to different morphologies suggested in the experimental literature. The large size of the system allowed for performing virtual SAXS scattering experiments with the simulated polymer configurations. The authors concluded that the “ionomer peak” present in scattering results is insensitive to the segregation morphology, that further complicates the interpretation of SAXS and SANS results. Extensive MD simulations were employed to look into more subtle issues, such as the influence of equivalent weight, sidechain length and distribution along the skeleton, on structural and transport properties of hydrated Nafion,

34-37

gas adsorption,

38

Nafion behavior near solid

surfaces, 39 transport under electrostatic field, 35 influence of added non-volatile solvents such as phosphoric acid and ionic liquids. 40, 41 In the most recent work, Daly et al 42 performed up to 200 ns long atomistic NPT ensemble MD simulation of water self-diffusion in Nafion and calculated water adsorption isotherms with GPU based MC simulations. This work sets a benchmark for current atomistic simulations, however, a limited size of simulated systems (up to 9 nm) is still insufficient to investigate the membrane segregated morphology. Significant increase in spatial and temporal scales was achieved with coarse-graining and reducing the system degrees of freedom by “lumping together” several atoms to form quasiparticles or “beads”. 43 In the most popular for modeling polymeric systems coarse-grained MD (CGMD) method, quasiparticles interact through effective pair-wise hard-core potentials (such as Lennard-Jones) parameters are commonly derived from atomistic modeling. 13, 44-47 In order to draw reliable conclusions about the morphological structure of self-assembled polymer, the size of the simulation box must exceed the characteristic scale of segregation by at least one


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order of magnitude. Malek et al 48 modeled a Nafion-carbon nanocomposite using Lennard-Jones potentials between quasiparticles representing different fragments of Nafion polymer, solvent, and carbon particles. The authors obtained a self-assembled structure composed of the hydrophobic backbone and water clusters qualitatively similar to that found in earlier atomistic MD simulations. However, the larger system size allowed to distinguished the evolution of segregated morphology as hydration progressed. At low water content, hydrophilic domains were roughly spherical and poorly connected. At higher hydration, a sponge-like network of roughly cylindrical aggregated of 3nm in diameter was formed. In this work, we employ the dissipative particle dynamics (DPD) method, 49, 50 which implies soft short-range repulsion potentials and therefore allows for much longer time steps (compared to MD) and facilitates the system equilibration. Several authors employed DPD simulations for modeling Nafion-type membranes. The first DPD simulation of Nafion in acid form was conducted by Yamamoto et al. 51 The conservative repulsion parameters were estimated from the mixing energy calculations conducted with atomistic modeling. The electrostatic interactions were implicitly mimicked by short-range forces. 51-53 The authors found irregular segregation morphologies, with reasonable correspondence to experimental results. Later, Dorenbos et al 54,62 and Wu et al 53, 55 employed the same model for studies of nanostructure and water diffusion in several perfluorinated ionomers that differed by Meq and sidechain length. Sawada et al 56 accounted for possible crosslinking of the perfluorinated skeleton chains and found that this effect lead to much smaller hydrophilic aggregates of only 1.8nm in diameter. Eliott et al 57 combined DPD results with experimental SAXS/SANS studies using a novel model-independent procedure. The modeling revealed a multi-level membrane organization, with hydrophilic – hydrophobic segregation on smaller scale and larger scale


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organization of the fluorocarbon backbone. This result is consistent with previous NMR studies. 58, 59

Jorn and Voth 60 modeled the segregation in Nafion with DPD and calculated proton

conductivity using the smoothed particle hydrodynamics approach, based on local concentration and charge densities in the resulting structures. The calculated conductivities showed very reasonable agreement with experimental data. The DPD studies mentioned above made an important step forward in modeling the segregation morphology of Nafion. However, these models lacked several features that are critically important for analyses of structural and transport properties of PEM: polymer rigidity and explicit electrostatic interactions. Noteworthy, parameterization of these models was based on generic DPD parameters devised for aqueous systems.49, 50, 61 In this paper, we suggest a novel DPD model of hydrated metal-substituted Nafion membranes. The proposed model accounts for polymer chain rigidity that is different for the Nafion backbone and sidechains and includes explicit electrostatic forces between the charged polymer fragments and dissociated counter-ions. In contrast to previous works, we attempted to customize the coarse-grained interaction parameters against the available experimental and atomic simulation data. For the first time, we not only generate self-assembled structures in hydrated Nafion membranes and explore the specifics of nano-segregated morphologies, but also establish their thermodynamic properties, in particular, the relationship between the level of hydration and humidity. This work builds upon our earlier DPD model developed for studies of interactions of toxic chemicals with perm-selective Nafion barriers.

62

The paper is structured as following. In

Section II, we formulate the DPD model of coarse-grained Nafion polymer and justify its dissection into the soft repulsive beads. Section III discusses the DPD model parameterization that is based on a combination of atomistic MD and coarse-grained MC simulations. The


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evolution of self-assembled morphology as the water content increases is studied in Section IV. The hydrophilic network connectivity and water diffusion are explored in Section V by using digitized replicas of the system snapshots and modeling random walk motion of a tracer. In Section VI, using the MC test particle insertion method, we determine the chemical potential of adsorbed water at given hydration levels and construct the water sorption isotherms, which allowed us to identify the conditions of saturation and to determine the limits of stability of generated conformations. Main conclusions and suggestions for prospective research are givens in Section VII.

II. Coarse-grained model of metal-substituted Nafion. The proposed coarse-grained model of Nafion is constructed following the classical DPD approach originated from the seminal work of Groot and Warren. 49, 50, 61 The system under consideration is presented as a multicomponent mixture of beads mimicking characteristic fragments on the hydrated polymer, that interact via pairwise conservative soft repulsive, harmonic bond, and electrostatic forces, as well as random and velocity dependent drag forces:

Fij (rij ) = Fij(C) (rij ) + Fij(B) (rij )+ Fij(E ) (rij ) + Fij(R) (rij ) + Fij(D) (rij , v ij )

(1)

All beads are assigned an equal effective diameter Rc. The soft repulsion force Fij( C ) acts between overlapping beads: Fij( C ) (rij ) = aIJ (1 − rij / Rc )rij / rij at r < Rc, Fij( C ) (rij ) = 0 at r ≥ Rc, where aIJ is the repulsion parameter specific to the given bead pair of types I and J. Following the standard approach to DPD simulations of self-assembly, 50 the intra-component repulsion parameters aII between beads of the same type are set equal, irrespective to the bead type. The beads are tightly packed with a substantial overlap. We accepted the reduced bead packing density of ρRc3=3, common in DPD simulations. 50


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The random and drag force also acted between overlapping beads along the line connecting the bead centers. Random force Fij( R ) that accounts for thermal fluctuations, is taken proportional to the conservative force that is also acting along the vector between the bead centers: Fij(R)(rij) = σwRrijθij(t)rij , where θij(t) is a randomly fluctuating in time variable with Gaussian statistics. The drag force is velocity-dependent: Fij(D)(rij ,vij) = −γwD(rij) (rij*vij) , where, vij = vj – vi, vi and vj are the current velocities of the particles. We assume the common relationship between the drag and random force parameters wD(r) = [wR(r)]2 = (1 − r/Rc)2 at r < Rc, wD(r) = 0 at r ≥ Rc . σ and γ are parameters that determine the level of energy fluctuation and dissipation; they are related as σ2 = 2γkT that allows to maintain constant temperature in the course of simulation via the Langevin thermostat. We assumed γ = 4.5, a common value fitted to the diffusion coefficient of water. The polymer beads are connected by harmonic bonds Fij( B ) (rij ) = K b (rij − r0 )rij / rij , where Kb is bond rigidity, which depends on the bead type, and r0 is the equilibrium length. Following our recent papers, 63, 64 in addition to this nearest neighbor (1-2) bond, we also introduced the second neighbor (1-3) harmonic bonds in order to control the skeleton rigidity and sidechain flexibility.

E

S/S-

O-CF2-CF(CF2)-O-CF2-CF2–SO3- [M+]

| -CF2-CF2-CF2-CF2-CF2-CF2-CF2–CF2–CF2-CF2-CF -CF2

C

C

C


8

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Figure 1. Dissection of Nafion 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 sidechain flexibility.

Bead types. We consider metal-substituted Nafion oligomers of chemical structure shown in Figure 1. The equivalent weight Meq of the polymer was varied from 944 (12 carbon atoms between the neighboring sidechains) to 1744 (28 carbon atoms between the neighboring sidechains); Meq values are given for the anion (total Meq can be obtained by adding the mass of the cation). In all simulations, the sidechains are separated by equal fluorocarbon fragments, and the sidechains were of the same length. In order to explicitly include electrostatic interactions into the DPD model we dissected the Nafion chain using four bead types. The hydrophobic beads of type C, which lump together four carbon atoms with attached fluorine atoms, represent skeleton perfluoroalkane fragments (CFx)4. The sidechain perfluoroether fragment (O-CF(CF3)CF2-O-) is modeled by the bead of type E that is less hydrophobic than bead C. The necessity of such detalization is caused by the role that the perfluoroether fragment plays in interactions of Nafion with phosphororganic chemicals. 58, 65 Two bead types S- and S represent negatively charged dissociated CF2CF2SO-3 and neutral non-dissociated CF2CF2SO3M sulfonate groups, respectively. The latter includes an alkali metal and has to be distinguished at low hydration levels where water content is insufficient for dissociation of all sulfonate groups. Water molecules were lumped into hydrophilic beads of type W. Hydrated alkali metal cations were represented by charged beads of type M+, which in addition to the cation included several water molecules. Bead volumes. The adopted dissection of the polymer into coarse-grained beads is performed to minimize the difference in volumes between the polymer fragments represented by


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different bead types. The effective volumes of fragments are estimated from the functional group volumes (“Bondi tables” 66, 67 that are available for perfluorinated compounds, 68 but not for sulfonates), as well from the molecular volumes of representative compounds calculated with PQS ab initio package. 69 To provide for an approximate equality with the volumes of the skeleton and sulfonate beads, the volume of water bead W is chosen as 135Å3 that corresponds to the effective bead size of Rc = 7.4Å. With bead density of ρRc3=3, the corresponding number of water molecules in W bead is nW = 4.5. Note that this non-integer number of molecules per quasiparticle is permissible in coarse-grained DPD modeling. The bead types, respective fragments, and their volumes are listed in Table 1. The underlying calculations of effective volumes are described in Supplementary Information, section S.I. Electrostatic interactions between the dissociated sulfonate beads S− and hydrated counterion M+ beads are treated using smeared-charge approach of Groot. 70 Instead of point charges interacting via the Coulomb potential that diverges at r → 0, each charge was distributed within the sphere of the smearing diameter Re=1.2Rc with the charge density decreasing linearly from the bead center towards the periphery. 70 The choice of Re is rather arbitrary, since charge cloud size does not have a clear physical meaning, 71 its influence on the thermodynamic properties of electrolytes is still to be examined. We chose Re smaller than recommended by Groot in an attempt to improve calculation efficiency. 71 The comparison between the screened and Coulomb potential created by two point charges is given in Supporting Information. The sulfonate S− and counterion M+ beads bear charges of –e and +e, respectively. Partial charges on the skeleton and perfluoroether fragments are neglected. Ewald summation was employed to account for the long-range electrostatic contribution.


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The hydrated counterion bead M+ is chosen to contain one cation and 3.5 water molecules and had the same parameters as the water bead W, except for being positively charged. Thus, we effectively assumed that a volume of counterion is close to that of one water molecule, which is reasonable for larger counterions. Merging of water molecules with an alkali metal ion in the same bead has important implications. In our model, the bead charges are fixed: that is, S beads were not allowed to dissociate into cation M+ and anion S- beads in the course of simulation and, respectively, M+ and anion S− beads cannot recombine. Water content in Nafion is expressed either as water mass per unit mass of dry polymer, or by the hydration level λ expressed as the number of water molecules per sulfonate group. Since each M+ bead included 3.5 molecules of water, λ = 3.5 is the minimum hydration level at which all counterions may be considered as dissociated from their sulfonate groups and modeled as M+ beads. At lower hydration levels, λ < 3.5, all water molecules are assigned to M+ beads, i.e. there are no uncharged W beads. To secure electroneutrality, the number of charged sulfonate group S− beads equals the number of counterions M+, and the number of neutral S beads is calculated from the difference. At λ > 3.5, all sulfonate groups are assumed to be dissociated.

III. Parameterization of interaction potentials. Intra-component repulsion parameter. We assumed the same intra-component (“selfrepulsion”) parameter aII =50kT/ Rc for all interactions between beads of the same type. This self-repulsion parameter is chosen to approximately reflect the overall compressibility of hydrated Nafion that is much higher than the pure water compressibility (e.g. compare 1.8•10-9 Pa-1 for n-perfluorononane 72 with 4•10-10 Pa-1 for water). It is worth noting that the chosen value of aII is lower than the value of 119.8kT/Rc that would be estimated from the pure water


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compressibility (e.g. ref 50, Supporting Information to ref 73), as was implied in previous DPD models of Nafion. 51 This is a significant difference: solvated membrane in our model is much “softer”.

Table 1. Coarse-grained model of hydrated Nafion: bead types, interaction parameters, and bond lengths and stiffness. Parameterization of coarse-grained model of Nafion Bead

Fragment

3

type 1

W

2

M

5

+

Conservative parameter, aIJRc/kT

Effective volume,Å

(H2O)4.5

134.5 +

a

W

K+

S

S-

C

E

50.

50.

50.

50.

65.

60.

50.

50.

50.

50.

65.

60.

[M(H2O)3.5]

122.1

S

-(CF2)2SO3K

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-

117.1

60.

60.

65.

65.

59.

50.

CFCF3ONearest neighbor bonds U1-2(r)=1/2K1-2 (r−re) 2

2

Second neighbor bonds U1-3(r)=1/2K1-3 (r−re) 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


2

12

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0.16 1-2, MD 1-3, MD 1-4, MD

0.12 normalized probability

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1-2, DPD 1-3, DPD 1-4, DPD

0.08

0.04

0 0

5

10

15

r , AA

Figure 2. Fitting the bond rigidity in the DPD model of perfluorohexadecane C16F34 to the results of atomistic MD simulations at T = 450K. The distribution of distances between the chain DPD beads separated by one, two and three harmonic bonds are matched to the distributions of distances between the centers of mass of the corresponding fragments of the atomistic chain, each of which contains four CF3 or CF2 groups. Reasonable agreement is obtained for beads separated by two and three bonds.

Perfluoroalkane skeleton rigidity. The Nafion skeleton is modeled as a linear sequence of hydrophobic beads C each representing four CFx groups. Skeleton rigidity affects the distance between the neighboring sidechains and overall elasticity of the matrix. In order to fit bond length and rigidities that are determined by the torsional potentials rather then by the stiffness of covalent bonds, we performed MD simulations of perfluorohexadecane melt with forcefield from ref 74, similar to our prior simulations of Nafion 31, 32 (see Supporting Information, section S.II for simulation details). Simulations were conducted in NPT ensemble at atmospheric pressure and temperature of 450 deg C, since C16F34 freezes at ambient conditions. The MD trajectories


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were recorded to disk. Each molecule was dissected into 4 fragments (four CFx groups each) and probability distributions of intra-molecular distances between the centers of mass of each fragments were calculated (Figure 2). After that, we conducted DPD simulations of tetramers of skeleton C beads at similar conditions and fitted the nearest neighbor and second neighbor bond parameter paying most attention to distances between beads separated by two and three bonds. The resulting distributions are shown in Figure 2. The agreement between DPD and MD 1-3 and 1-4 distance distributions is very reasonable. We decided not to attempt achieving a good agreement between MD and DPD distributions for the distance between the neighboring beads in the chain and instead limit the nearest neighbor skeleton bond stiffness to K1-2 = 160kT/Rc2. Fitting the 1-2 distance distribution requires very stiff nearest neighbor bonds, 75 which leads to much shorter timesteps and this drastically slows down the simulations. At the same time, the nearest neighbor bond stiffness hardly affects the effective distance between sidechains and therefore is of secondary importance for our purpose. Perfluoroalkane – water repulsion mismatch parameter ∆aCW was obtained by performing DPD simulation of n-C36F74 that was presented a 9-mer of C beads with the nearest neighbor and second neighbor bond parameters derived from MD simulation of perfluorohexadecane. The mismatch parameter ∆aCW = 15kT/Rc was determined by fitting to respective atomistic MD simulation. The results are shown in Figure 3. At low values of ∆aCW, coarse-grained n-C36F74 behaves as an extended chain. At ∆aCW= 15kT/Rc, the DPD simulation shows frequent transitions between the collapsed globule and extended chain configurations, and at ∆aCW > 20kT/Rc, the globular configurations 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 distance of the extended states


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obtained by MD and DPD are in good agreement, thus justifying the choice of both ∆aCW and bond parameters. Sidechain repulsion parameters. The mismatch parameters for perfluoroether fragment E are estimated from the water and skeleton parameter approximately taking in accounts that two fluorocarbon groups are replaced by hydrophilic oxygens. As a result, they are modeled as mildly hydrophobic ∆aEW = 9 kT/Rc . The sulfonate end-groups are considered as hydrophilic (∆aSW = 0) beads, whether dissociated or not. The sulfonate groups were assumed to interact very unfavorably with the other fragments of the polymer, and mild mismatch of ∆aEW = 9 kT/Rc was assigned to the repulsion between C and E beads representing the skeleton and the ether sidechain fragment, respectively. 0.16 1-4 (MD)

0.14

2-7 (MD) 1-9 (MD)

noramalized probability

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.12

1-4 (DPD) 2-7 (DPD) 1-9 (DPD)

0.1 0.08 0.06 0.04 0.02 0 0

10

20

30

40

bead-to-bead distance r , AA

Figure 3. Results of MD and DPD simulations of C36F44 in water at T = 300K. The distribution of distances between DPD beads separated by three, five, and eight harmonic bonds are compared with the distributions of distances between the centers of mass of the corresponding fragments in the atomistic representation, each of which contains four CF3 or CF2 groups.


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Sidechain bonds. Using the short-range conservative repulsion parameters described above, we obtained the nearest neighbor and second neighbor bond parameters for the sidechain from MD and DPD simulations of a single Nafion monomer (depicted in Figure 1) in water bath. Figure 4 shows the distribution of distances between DPD beads and respective distributions of the intramolecular distances between the centers of mass of corresponding Nafion fragments obtained in MD simulations. The presence of sidechain substantially changes the conformations of the skeleton. Although distorted trans C-C-C-C torsion angles continue to dominate the skeleton conformation similarly to perfluoroalkane chains (Figure 3), we also observed hairpinlike conformations, where the skeleton makes a sharp bend close to the point of the sidechain attachment. As a result, the distribution of the distances between the fragments corresponding to second neighbor fluorocarbon beads has a minor second peak, which we could not reproduce with cruse DPD models. Nevertheless, we found the agreement between MD and DPD results reasonable, taking into account the coarse-grained nature of the model under consideration. The bond parameters are presented in Table 1.


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S1-E2 (MD) S1-C3 (MD)

0.3

noramalized probability

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C4-C5 (MD) S1-E2 (DPD) S1-C3 (DPD) C4-C5 (DPD)

0.2

0.1

0 0

2

4

6

8

10

12

14

bead-to-bead distance r , AA

Figure 4. Results of MD and DPD simulations of C36F44 in water at T = 300K. The distributions of distances between S, E, and C beads are fitted to the distributions of distances between the centers of mass of the corresponding fragments in the atomistic representation, shown in Figure 2.

IV. DPD simulations of self-assembled morphology of hydrated Nafion. Using the DPD model described above, we studied the self-assembled phase segregation in metal-substituted Nafion membranes by equilibrating the system in the canonical ensemble DPD simulation. We considered the polymer of different equivalent weight Meq and varied the water content to explore the evolution of membrane morphology upon hydration. The equivalent weight Meq of the polymer is determined by the number of beads between the neighboring sidechains. Simulations were performed in a cubic 30×30×30Rc3 cell containing 81,000 beads, starting from a random configuration. The method of Paganabarra et al 76 was used to integrate the equations of motion with timestep of 0.02. The balance of energy fluctuation and dissipation provided Langevin thermostate that allowed maintaining the average temperature within 1% of the required value and temperature fluctuations below 5%. The segregation progress was


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characterized by monitoring the number of pairs of overlapping W beads. When this number stabilized, equilibrium was considered established. Further details of DPD simulations may be found in Supporting Information, section III. We considered Nafion polymer with sidechains separated by 12, 16, 20 and 28 skeleton beads, which corresponded to Meq = 944, 1144, 1344 and 1744 Da for the anion, respectively. It is worth noting that most published data dealt with alkali metals substituted Nafion membranes are of Meq ≈ 1100-1200 Da. Experimental water sorption isotherms on these membranes were summarized in refs 65. At 100% humidity, water sorption decreases from approximately 29 % wt in Li+ form to 4-6 % wt in Cs+ form. K+ form of Nafion, which was targeted in our study, is in the middle of this range; it adsorbs 9.5% to 12% of dry polymer weight, which corresponds to

λ = 6.2−7.9, according to refs. 65, 77, 78 Water sorption decreases with the Nafion equivalent weight Meq due to the increasing fraction of hydrophobic skeleton groups.4 In the NPT DPD simulations, we considered a wide range of hydration levels from λ = 2.25 το λ = 13.5.

Figure 5. Snapshots of nanophase separation in hydrated Nafion: Meq =1144 (top) and 1744 (bottom). Hydration level λ =2.25 (a,e), 6 (b,f), 9 (c,e) and 13.5 (d,h). Nafion skeleton and perfluoroether sidechain beads are shown in in red, sulfonate groups in dark blue, counterions in green, water beads in light blue.


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Figure 5 shows the snapshots of selected systems of different Meq and λ. In general, the evolution of the system morphology is qualitatively similar to the classical scenario of the percolation system formation. 4, 51, 79 At low hydration levels, the water and counterions form small clusters around the sulfonate groups (note that in our model at λ 1. Water content at a = 1 corresponds to sorption from saturated water vapor. Figure 10 shows that the higher the Meq, the higher the activity at the same λ, and the lower hydration level corresponds to saturation, which is natural and agrees qualitatively with experimental observations.


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We noted that the beginning of saturation corresponds to the formation of well-defined spheroidal water clusters discussed in Section IV. Their growth coincides with a sharp increase of activity with λ. Within this regime, the chemical potential of water exceeds that in pure coarse-grained water (a > 1). After that, the situation changes completely and activity becomes nearly independent of hydration level. We assume that this corresponds to the beginning of phase separation onto water and hydrated Nafion, signified by growth of one or two large clusters. These systems are typically thermodynamically unstable and corresponding part of the isotherms have negative slope dλ/da < 0, which can be noted in Figure 9. The adsorption isotherms from Figure 9 explain the non-monotonic behavior of relative diffusivity found in NPT simulations: decline in diffusion occurs at thermodynamically unstable, unphysical region of oversaturation at a > 1, which are not observed in experiment. In a homogeneous polymer solution, an oversaturated condition for the solvent (that is possible as a metastable state with bulk separation prevented by a nucleation barrier) would not cause a decline of solvent diffusivity. Nafion, however, is segregated, and the phase separation into hydrated polymer and pure water, when hydration exceeds the sorption capacity, is prevented by artificial periodic boundary conditions. The water cluster network connectivity depends strongly on the number of sidechains that concentrate at the interface between the hydrophilic and hydrophobic subphases. The growing domains of water drag the sidechains to their surface, thus reducing the overall surface area of the connecting channels and leading to the isolation of water clusters. This effect is artificial; it should not be observed in a macroscopic system as it was caused by limited simulation time and system volume. As such, the simulated dependencies of the water diffusivity on the hydration level should be considered only up to the saturation level (Figure 9).


27


Despite the artificial nature of the structures with large spherical water clusters, the reason why they do not merge is worth considering. As water bridges between the clusters “dry up” due to excessive surface area of the interface between the polymer and water, large isolated clusters are formed. Each cluster carries a positive charge due to the hydrated counterions, which is compensated by the negative charge of sulfonate groups surrounding the positively cluster. Thus, two clusters experience electrostatic repulsion from each other, which creates a potential barrier and prevents their fusion. Because periodic boundary conditions suppress system restructuring and the simulation time is limited, the barriers associated with the cluster coalescence never get overcome in most of the simulations. Figure S4 in supporting information shows that the clusters experience significant fluctuations in shape but do not merge within the simulation run. λ , water molecules/ sidechain

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18

1.2

0.016

aW

Meq=1144 Meq=1344

sorption diffusion

Meq=1744 12

D W/D BULK 0.012

0.8

explt Meq=1200

0.008 0.4

6

0.004

(a)

(b) 0

0 0

0.2

0.4

0.6

0.8

1

aW

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λ, water molecules/ sidechain

18

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. Black square shows experimental value for K+ substituted Nafion with Meq ≈1200 Da.

77, 80

(b)

Water activity and reduced diffusion coefficient as functions of the water content for Meq=1144 Da. Maximum of water mobility is achieved near saturation and declines in the region of oversaturation due to the formation of disconnected water domains.


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VII. Conclusion.

We suggested a novel DPD model of hydrated Nafion membranes. The proposed model accounts for polymer chain rigidity that is different for the Nafion backbone and sidechains and includes explicit electrostatic forces between the charged polymer fragments and dissociated counter-ions. In contrast to previous works, we attempted to customize the coarse-grained interaction parameters against the available experimental and atomic simulation data. For the first time, we not only generate self-assembled structures in hydrated Nafion membranes with an experimentally informed DPD model and explore the specifics of nano-segregated morphologies, but also establish their thermodynamic properties, in particular, the relationship between the level of hydration and humidity. The evolution of self-assembled morphology upon hydration was studied with equilibrated NPT DPD simulations performed at a series of hydration levels λ from 2.25 to 18 water molecules per sulfonate group. A classical percolation transition from a system of isolated water clusters to a 3D network of hydrophilic channels was observed. The hydrophilic subphase connectivity was characterized by constructing its digitized replica and performing random walk simulations to determine the effective water diffusivity. We found a significant decrease of water diffusivity in membrane compared to the bulk water, e.g. by two orders of magnitude for Meq=1144. These values are comparable with the available experimental data. Noteworthy, at high hydration levels (e.g. above λ = 9 for the membrane of equivalent molecular weight of 1144), we detected the coalescence of water clusters into large domains, which were poorly connected. This effect caused the reduction of the water diffusivity, which showed a maximum as a function of the level of hydration. Using coarse-grained MC simulations, we calculated the


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chemical potential of water in the hydrated polymer and constructed the water sorption isotherms. We determined that the maximum diffusivity and the onset of formation of large water domains corresponded to the saturation conditions at 100% humidity and concluded that the oversaturated membrane morphologies generated in the canonical ensemble DPD simulations corresponded to the metastable and unstable states that are not observed in experiments. The hydration level of λ ≈ 7 at 100% humidity established by the proposed simulation method for Meq=1144 was comparable with available experimental data. This work also reveals serious limitations of the traditional DPD models with the equal size of different types of beads and equal intra-component repulsion parameters. Firstly, this model does not take into account different compressibility of water and polymer, and thus cannot predict correctly the elasticity of the polymer matrix and the membrane swelling. Note that the membrane swelling can be taken into account in the NPT simulations with varying system volume. Secondly, the model does not distinguish between different alkali cations considered as “charged water beads”. Cations differ not only by their size, but also by the size of the solvation shell that water forms around them. This difference can be accounted by customizing the cation bead diameter, charge smearing radius Re, as well as the interactions with other beads, water beads in particular. These parameters should be fitted to thermodynamic and kinetic properties of aqueous solutions of electrolytes, but such efforts have not been reported in the literature. It is desirable to extend the proposed DPD model of the metal-substituted Nafion membrane to the acid form membranes that are used in fuel cell and other electrochemical devices. However, this extension should take into account the effects of deprotonation of sulfonate groups and proton transport within the hydrophilic subphase, including the hopping mechanism of proton diffusion. These problems will be addressed in our future work.


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Acknowledgement: This work was supported in parts by DTRA (grants HDTRA1-08-1-0042 and HDTRA1-14-1-0015) and NSF (grant DMR1207239).

Supporting information available: Supporting information includes the following sections (S.I) Details of calculation of bead volumes (S.II) Details of MD simulations for parameter fitting, (S.III) Details on DPD simulations involved in parameter fitting, including specifics of electrostatic potentials (S.IV) Tables with resulting water sorption, water activities and related diffusivities, alternative versions of selected figures in the logarithmic scale. This material is available free of charge at http://pubs.acs.org


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(69) Pulay, P.B., J.; Wolinski, K. , Pqs Ab Initio; 3.3 Ed.; Parallel Quantum Solutions (Fayetteville, AR: 2003). (70) Groot, R.D., Electrostatic Interactions in Dissipative Particle Dynamics-Simulation of Polyelectrolytes and Anionic Surfactants, J. Chem. Phys., 2003, 118, 11265-11277. (71) Warren, P.B.; Vlasov, A., Screening Properties of Four Mesoscale Smoothed Charge Models, with Application to Dissipative Particle Dynamics, J. Chem. Phys., 2014, 140, 084904. (72) Pineiro, M.M.; Bessieres, D.; Gacio, J.M.; Saint-Guirons, H.; Legido, J.L., Determination of High-Pressure Liquid Density for N-Perfluorohexane and N-Perfluorononane, Fluid Phase Equilib., 2004, 220, 127-136. (73) Vishnyakov, A.; Talaga, D.S.; Neimark, A.V., Dpd Simulation of Protein Conformations: From Alpha-Helices to Beta-Structures, J. Phys. Chem. Lett., 2012, 3, 3081-3087. (74) Cui, S.T.; Cochran, H.D.; Cummings, P.T., Vapor-Liquid Phase Coexistence of Alkane Carbon Dioxide and Perfluoroalkane Carbon Dioxide Mixtures, J. Phys. Chem. B, 1999, 103, 4485-4491. (75) Ortiz, V.; Nielsen, S.O.; Discher, D.E.; Klein, M.L.; Lipowsky, R.; Shillcock, J., Dissipative Particle Dynamics Simulations of Polymersomes, J. Phys. Chem. B, 2005, 109, 17708-17714. (76) Pagonabarraga, I.; Hagen, M.H.J.; Frenkel, D., Self-Consistent Dissipative Particle Dynamics Algorithm, Europhys. Lett., 1998, 42, 377-382. (77) Nandan, D.; Mohan, H.; Iyer, R.M., Methanol and Water-Uptake, Densities, Equivalental Volumes and Thicknesses of Several Univalent and Divalent Ionic Perfluorosulfonate Exchange Membranes (Nafion-117) and Their Methanol Water Fractionation Behavior at 298-K, J. Membr. Sci., 1992, 71, 69-80. (78) Vishnyakov, A.; Neimark, A.V., Molecular Dynamics Simulation of Microstructure and Molecular Mobilities in Swollen Nafion Membranes, J. Phys. Chem. B, 2001, 105, 9586-9594. (79) Dorenbos, G.; Morohoshi, K., Chain Architecture Dependence of Pore Morphologies and Water Diffusion in Grafted and Block Polymer Electrolyte Fuel Cell Membranes, Energy & Environ. Sci., 2010, 3, 1326-1338. (80) Schneider, N.S.; Rivin, D., Solvent Transport in Hydrocarbon and Perfluorocarbon Ionomers, Polymer, 2006, 47, 3119-3131. (81) Schneider, N.S.; Rivin, D., Steady State Analysis of Water Vapor Transport in Ionomers, Polymer, 2010, 51, 671-678. (82) Saito, M.; Hayamizu, K.; Okada, T., Temperature Dependence of Ion and Water Transport in Perfluorinated Ionomer Membranes for Fuel Cells, J. Phys. Chem. B, 2005, 109, 3112-3119. (83) Widom, B., Some Topics in Theory of Fluids, J. Chem. Phys., 1963, 39, 2808-&.


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(a)

(c) 1400

(b)

m e a n s q u a re d is p la c e m e n

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(d)

1200 1000 800 600 400 200 0 0.E+00

DH2O/DPURE 2.E+04

4.E+04

6.E+04

8.E+04

1.E+05

1.E+05

number of steps


37


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Figure 1. Dissection of Nafion 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 sidechain flexibility. 60x16mm (300 x 300 DPI)


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Figure 2. Fitting the bond rigidity in the DPD model of perfluorohexadecane C16F34 to the results of atomistic MD simulations at T = 450K. The distribution of distances between the chain DPD beads separated by one, two and three harmonic bonds are matched to the distributions of distances between the centers of mass of the corresponding fragments of the atomistic chain, each of which contains four CF3 or CF2 groups. Reasonable agreement is obtained for beads separated by two and three bonds. 92x74mm (300 x 300 DPI)



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Figure 3. Results of MD and DPD simulations of C36F44 in water at T = 300K. The distribution of distances between DPD beads separated by three, five, and eight harmonic bonds are compared with the distributions of distances between the centers of mass of the corresponding fragments in the atomistic representation, each of which contains four CF3 or CF2 groups. 81x61mm (300 x 300 DPI)


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Figure 4. Results of MD and DPD simulations of C36F44 in water at T = 300K. The distributions of distances between S, E, and C beads are fitted to the distributions of distances between the centers of mass of the corresponding fragments in the atomistic representation, shown in Figure 2. 79x58mm (300 x 300 DPI)



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Figure 5. Snapshots of nanophase separation in hydrated Nafion: Meq =1144 (top) and 1744 (bottom). Hydration level l =2.25 (a,e), 6 (b,f), 9 (c,e) and 13.5 (d,h). Nafion skeleton and perfluoroether sidechain beads are shown in in red, sulfonate groups in dark blue, counterions in green, water beads in light blue. 67x30mm (300 x 300 DPI)


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Figure 6. Digitized lattice replicas of the membrane morphology mapped onto a cubic lattice (hydrophilic subphase shown in light pink). (a) Meq = 1144, l = 9; (b) Meq=1744, l = 13.5. 51x25mm (300 x 300 DPI)



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Figure 7. Mean square distance as a function of the number of steps for the tracer random walk within the hydrophilic subphase of hydrated Nafion for several characteristic configurations: (1) Meq=1144, l = 2.25: water forms small isolated clusters, diffusion is localized. (2) Meq=1144, l = 9: this level of hydration is about the experimental value at 100% humidity and corresponds to fastest diffusion obtained for given Meq. Water forms a well-defined continuous subphase. (3) Meq=1344, l=13.5: the mean square displacement reflect large water aggregates and their poor connectivity in unstable configurations: fast diffusion at short time and slow diffusion at longer time scale are observed. 58x42mm (300 x 300 DPI)


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Figure 8. Self-diffusion coefficient of water in hydrated Nafion reduced to the diffusion coefficient of pure water for polymers of hydration level l for equivalent molecular weight Meq of 1144 and 1344, obtained with the random walk simulation. The green filled diamonds are experimental data for K+ substituted Nafion-112 membranes (Meq ≈1100 Da).78 Similar dependences for other Meq are shown in logarithmic scale in Supporting Information. The error bars were estimated from the diffusion coefficients for each of the individual Nafion snapshots. 78x45mm (300 x 300 DPI)



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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. Black square shows experimental value for K+ substituted Nafion with Meq ≈1200 Da. 77, 80 (b) Water activity and reduced diffusion coefficient as functions of the water content for Meq=1144 Da. Maximum of water mobility is achieved near saturation and declines in the region of oversaturation due to the formation of disconnected water domains. 51x16mm (300 x 300 DPI)


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none 84x78mm (300 x 300 DPI)


Self-Assembly in Nafion Membranes upon Hydration: Water Mobility

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