Morphology of Hydrated Nafion through a Quantitative Cluster

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C: Physical Processes in Nanomaterials and Nanostructures

Morphology of Hydrated Nafion through a Quantitative Cluster Analysis: A Case Study Based on Dissipative Particle Dynamics Simulations Hongjun Liu, Sara Cavaliere, Deborah J Jones, Jacques Roziere, and Stephen J. Paddison J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b01842 • Publication Date (Web): 09 May 2018 Downloaded from http://pubs.acs.org on May 15, 2018

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The Journal of Physical Chemistry

Morphology of Hydrated Nafion through a Quantitative Cluster Analysis: A Case Study Based on Dissipative Particle Dynamics Simulations Hongjun Liu,† Sara Cavaliere,‡ Deborah J. Jones,‡ Jacques Rozière,‡ and Stephen J. Paddison∗,† †Department of Chemical and Biomolecular Engineering, University of Tennessee, Knoxville, TN 37996, USA ‡Institut Charles Gerhardt de Montpellier, Agrégats, Interfaces et Matériaux pour l’Energie, UMR 5253 CNRS, Université de Montpellier, 34095 Montpellier CEDEX 5, France E-mail: [email protected] Phone: +1 865-974-2026

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Abstract The evolution of the hydrated morphology of Nafion over a range of water contents was quantified through the cluster analysis method. Our findings are in contrast with those solely based on the radial distribution functions where cluster size and separation are approximated by certain characteristics of the radial distribution function. The quantitative cluster analysis along with realistic microscopic images colored by unique IDs leads to a wealth of information on water domain size, shape, and connectivity which is essential for a mechanistic understanding of proton transport. The percolation threshold of the water domains in hydrated Nafion was found to occur at a hydration level of 5 H2 O/SO3 H. Below the threshold, isolated individual water clusters cannot contribute to the ion transport. Water clusters grow from small aggregates into larger spheres, elongated rods, and branched and twisted cylinders as the hydration level increases. Beyond the threshold, the percolating water network is conspicuously dominant in the morphology. At the higher hydration levels, a larger percentage of the water beads contribute to the polar network and the size and number of nonparticipating clusters gradually diminish. Our work emphasizes the importance of a proper quantitative tool to understand the nature of ion conducting domains.

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Introduction Nafion, a perfluorinated sulfonic acid ionomer discovered in the late 1960s, has continued to attract attention for its use in polymer electrolyte membrane (PEM) fuel cells because of its excellent chemical, electrochemical, and mechanical properties. It serves as both a proton conductor and a separator in electrochemical energy conversion devices. Fundamental understanding of water and proton transport in Nafion is crucial for the fine tuning of these materials for optimum performance. 1–6 Proton transport in Nafion is determined by the nanoscale morphology of the hydrophilic (sulfonate groups, water, protons) domains and hydrophobic (polymer backbone) domains which strongly depends on the hydration level (typically expressed as the ratio of number of water molecules to the number of sulfonic acid groups). Notwithstanding intensive and extensive experimental work, quantification of morphology with an idealistic model remains a challenge. The proposed models so far include, but not limited to: the original cluster network model of Gierke, 7 the Gebel model, 8 the early channel model of Kreuer, 9 the parallel cylindrical water channel model, 3 and locally flat ribbon-like model of Kreuer. 4 This model appears to be gaining momentum with recent experimental observations. 10–12 Complementary to experiment, molecular simulations at multiple length scales have been widely applied to understand how the molecular structure and mesoscopic morphology affect proton transport in hydrated Nafion. Ab initio molecular dynamics (AIMD), atomistic MD, and coarse-grained MD are among the most effective tools to tackle this challenge. 13–17 Dissipative particle dynamics (DPD), since its inception and further refinement, 18–20 has become a well established mesoscopic simulation technique for a variety of systems including polymer melts, polymer solutions, block copolymers, polymer brushes, bilayer membranes, and ionomers. 21–28 Employment of aggressive coarse graining and simplified soft potentials in the DPD scheme greatly expands the accessible time and length scales of simulations which enables simulation of the mesoscopic structure of Nafion feasible. 29 In this approach, Nafion and related ionomer systems enjoy great flexibility in model assignment in terms of 3



repulsion parameters. The most widely applied method is mapping onto the Flory-Huggins χ parameters, which can be estimated by solubility or mixing energies. 20,29 The parameters can also be determined by the chemical nature of each bead or be extracted as effective coarsegrained potentials from matching local structure of atomistic simulations or from electronic structure calculations. 30–33 The morphology structure emerging from accumulated DPD simulations suggests that Nafion exhibits a percolation transition from isolated hydrophilic domains to an irregular sponge-like network as hydration is increased. 15,34,35 A proper quantification of such a qualitative observation is warranted. Herein, we advocate the use of a direct cluster analysis over the conventional radial distribution function method which is shown to be inappropriate to correctly characterize the features of the proton conducting network. Our cluster analysis algorithm possesses the capability of assigning each individual water cluster a unique cluster identity so that the resultant direct 3D microscopic image of cluster size, shape, and connectivity can be readily compared with the direct imaging of atomic force microscopy (AFM) and transmission electron microscopy (TEM). 10 The cluster size distribution is not a novel concept, but has failed to obtain widespread use in characterization of the ion conducting domain in simulation studies of Nafion probably due to the complex implementation of the clustering algorithm and easy accessibility of the RDF. There are a few studies in the literature reporting a change in the cluster size distribution with hydration level from many small isolated water domains to fewer but larger domains. 14,36,37 Connolly surface analysis can also be applied to characterize the pore size as recently demonstrated by Vishnyakov et al. 38 In this contribution, a systematic cluster analysis complemented with vivid 3D microscopic images affords new physical insight into the size, shape, and connectivity of the water and ion conducting domains in hydrated Nafion.

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Simulation Methodology The DPD scheme models a polymer chain as a collection of compressible coarse-grained and connected beads following the bead-spring model to replicate polymers at the mesoscale. 18–20 DPD is a Langevin dynamics scheme, where one simulates a fluid of particles interacting through conservative, random, and dissipative forces.The net force, fi , experienced by bead i is a summation of all pairwise forces of bead i with all other beads j:

fi =

X

FijC + FijD + FijR + FijB .

j6=i

(1)

The conservative force, FijC , is given by: FijC

= aij = 0,

rij 1− Rc

rîj

if rij < Rc (2)

otherwise,

where aij is the repulsion parameter between beads i and j, vector norm rij = |rij |, unit vector rîj = rij /rij , and Rc is interaction range. The dissipative, FijD , and random, FijR , forces are defined as: FijD = −γwD (rij )(rîj · vij )rîj

(3)

ζij FijR = σwR (rij ) √ rîj , ∆t

(4)

and

where γ is the friction coefficient, σ is the noise amplitude, ∆t is the time step, and ζij is chosen from a norm Gaussian distribution with zero mean and unit variance. The pair of these two forces essentially constitutes the DPD local thermostat. To satisfy the fluctuationdissipation theorem of canonical ensemble, σ 2 = 2kB T γ. wD and wR are the weight functions written as wD (rij ) = [wD (rij ]2 = (1 − rij /Rc ) if rij < Rc 5


(5)


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and vanish for r ≥ Rc . The bonded, FijB , force provides bead connectivity between adjacent beads and is usually described by a harmonic bond. Occasionally, an additional spring angle term between two consecutive bonds is applied to improve the description of the structural properties of polymers: 32,39 1 U b = KH (rb − rb0 )2 2

(6)

U a = Ka [1 − cos(θa − θa0 )],

(7)

and

where KH is the harmonic spring constant, rb is the bond length, and rb0 the equilibrium bond length. Ka , θa , and θa0 are the angle constant, angle, and equilibrium angle, respectively. All DPD simulations were performed using the LAMMPS package. A system of reduced units is typically employed in DPD simulations. Interaction range, Rc , temperature, kB T , and bead mass, m, are chosen to be the DPD units of length, energy, and mass, respectively. The bead number density ρ is 3 and γ = 4.5. DPD length unit can be converted to real p length unit by equaling the bead volume, while DPD time unit τ = Rc m/kB T can be

estimated by matching the simulated bead diffusion constant to the experimental one of water. 40 Specifically, Rc = 3.107(ρNm )1/3

˚ A

(8)

and τ=

Nm Dsim Rc2 5/3 = 25.7Nm Dexp

ps.

(9)

The simulations were run in a cubic box of 40 DPD length units (with approximately 192000 DPD beads) with periodic boundary conditions. Each Nafion polymer chain consists of 30 or 80 repeat units. One repeat unit dictated as [AAAA(BC)] includes four fluorocarbon A beads, one side chain B bead and one sulfonic acid C bead. A revised DPD model based on the recent work of Sepehr and Paddison 33 were used to model hydrated Nafion. Parameter details and chemical identity of beads can be found in Table 1. The hydration level, λ,

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varied from 1 to 20 were simulated (corresponding to water volume fraction of 0.04 to 0.45). Table 1: DPD parameters of Nafion derived from ab initio calculations. 33 The chemical structures of A, B, C and W bead are CF2 -CF2 -CF2 -CF2 -CF2 -CF2 , OCF2 -CF(CF3 )-O-CF2 , CF2 -SO3 H+(H2 O)3 , and (H2 O)8 , respectively. One DPD length unit is about 9.0 ˚ A and DPD time unit of 820 ps. aij A B C W

A 34.6

B 32.0 28.4

bonds AA AB BC

KH 100 100 100

rb0 0.65 0.55 0.5

angles AAA AAB ABC

Ka 5 10 10

θa0 135 135 100

C 38.5 31.7 29.9

W 37.9 28.4 25.9 24.9

To quantify the morphology evolution of the water domains with increasing λ, a cluster analysis was implemented. The connectivity criterion is that two atoms/beads belong to the same cluster if their distance is smaller than a selected cutoff length (rcutof f = 1 in our case). The cutoff was dictated as the limit of the strongest Gaussian peak derived from deconvolution of the first peak of the relevant radial distribution function. 41 Identification of the first coordination limit of the connected beads is somewhat arbitrary, but definitely nontrivial. As demonstrated in our previous work, the naive application of the established criteria of the first local minimum is alarmingly misleading. 42 Cutoff length effect was also explicitly investigated. 14 Though the relative trend for cluster formation with varying hydration level will show up at the same specified cutoff length, a rigorous definition of connectivity is well appreciated and more tractable. The cluster size distribution is calculated with:

p(n) =

n

PC

Nn (i) CN i

7


(10)


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where p(n) is the probability of finding a bead in a cluster of size n, Nn (i) is the number of clusters of size n for a given configuration i, C is the total number of configurations, N is the total number of water beads which is a function of λ. Along with the morphology evolution of ion conductive channel, hydration level also affects the size and dynamics of Nafion polymers. The polymer size can be quantified by the mean end-to-end distance, Re , and radius of gyration, Rg ,:

Re2

1 = n

1 Rg2 = nN

*

*

n X i=1

(r1,i − rN,i )

2

+

n X N X (rj,i − ricom )2 i=1 j=1

(11)

+

(12)

where n is the number of polymers, each with N backbone beads, ricom is the COM of polymer i, and rj,i is the coordinates of bead j of polymer i. The Rouse ratio:

R=

Re2 6Rg2

(13)

should be unity for Rouse chains. Flory’s characteristic ratio is defined as:

CN =

Re2 (N − 1)lb2

(14)

where lb is the bond length between the monomer beads along the polymer chain. The COM self-diffusivity, D, is determined by the mean squared displacement through the Einstein relation: d 1 1 D = lim 6 t→∞ dt n

*

n X [ricom (t) − ricom (0)]2 i=1

+

(15)

The end-to-end vector autocorrelation function is a characteristic time of long-time polymer

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dynamics and is defined as:

hRe (0)Re (t)i =

n 1 X h [r1,i (t) − rN,i (t)][r1,i (0) − rN,i (0)]i. n i=1

(16)

The end-to-end vector autocorrelation function can be fitted to a stretched exponential β function exp − τt0 , then the end-to-end vector relaxation time can be obtained with: τR =

τ0 1 Γ( ). β β

(17)

Results and Discussion Before deliberation on the quantitative cluster analysis, we first demonstrate that the conventional way of quantifying water cluster size and separation between clusters based on water-water radial distribution function is inappropriate or inadequate. In our recent work, we have demonstrated that different DPD model implementations at the same hydration level (i.e. λ = 9) leads to a dramatic variation on pore sizes and separations even though they are predicted to be similar if water-water RDFs are used to derive those quantities. 43 RDFs of the water beads at different hydration levels are presented in Figure 1a. Distance at which g(r) drops below unity is usually interpreted as the cluster boundary or pore radius, and position of the second maximum of g(r) as the average separation between the water clusters or domains. The derived quantities from g(r) shown in Figure 1b seem to suggest that the pore size and separation increase with increasing hydration. Calculated water pore morphologies in terms of isosurfaces presented in Figure 2 appear to support that the water clusters keep growing and become more connected with increasing hydration, yet not the finding that water clusters become further separated with increasing water content. 44 These observations, however, are only qualitative in essence. Isosurface renderings are affected by various parameters and Figure 3 vividly demonstrates that without a proper quantitative analysis, the isosurface renderings with the misleading estimates of cluster size and separa9



tion from water-water g(r)’s may lead to a false impression of a percolating network occurring at λ = 3. (a)

2.5

gWW(r)

2 1.5 1

λ=1 2 3 4 4.5 5 6 7 8 9 10 11 13 15 20

1.25 1.2 1.15 gWW(r)

3

1.1 1.05 1 0.95 1

2

3 r

4

5

0.5 0 0.1 8

Pore radius and separation

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

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1

10

r

(b) pore radius pore separation

6

4

2

0

0

4

8

λ

12

16

Figure 1: (a) Radial distribution functions (RDFs) of the water beads at different hydration levels, λ = #H2 O/SO3 H. (b) Average pore size and separation between water clusters derived from the water-water RDF. The pore size and separation increase with increasing λ.

Characterization of the water clusters in hydrated ionomers is essential to understanding the ion/proton transport within the conducting domains. Only water in a connected percolating network can participate and facilitate the ion transport, while isolated individual

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Figure 2: Representative configuration snapshots with A, B, and C beads rendered in cyan, blue, and red spheres, respectively, and W beads in a density isosurface of 0.5. The chemical structures of A, B, C and W bead are CF2 -CF2 -CF2 -CF2 -CF2 -CF2 , O-CF2 -CF(CF3 )-O-CF2 , CF2 -SO3 H+(H2 O)3 , and (H2 O)8 , respectively. Isosurfaces were generated by the Quicksurf tool of VMD using the following parameters: resolution of 1, radius scale of 0.5, density isovalue of 0.5 and grid spacing of 0.5. (a) λ = 1, (b) λ = 2, (c) λ = 3, (d) λ = 5, (e) λ = 7, and (f) λ = 9. 11 ACS Paragon Plus Environment


Figure 3: Representative snapshot at λ = 3 with different renderings: (a) All beads as spheres: A beads (cyan), B beads (blue), C beads (red), and W beads (white). Identity of beads is given in Figure 2; (b) only A beads and W beads shown; (c) only A and W beads, isosurfaces of W beads rendered with the Quicksurf tool with resolution of 1, radius scale of 0.5, density isovalue of 0.5 and grid spacing of 0.5 denoted as (1, 0.5, 0.5, 0.5); (d) same as (c) but with a set of parameters (1, 0.5, 0.5, 1); (e) same as (c) but with a set of parameters (1, 0.5, 0.2, 0.5); and (f) same as (c) but with a set of parameters (1, 0.7, 0.5, 0.5). Without a proper quantitative analysis, the isosurface renderings along with cluster size and separation 12 estimated from water g(r) lead toACS a false impression of a percolating network. Paragon Plus Environment

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water clusters contribute very little. Therefore, proper identification of network connectivity is a principal goal in our cluster analysis. Dorenbos and coworkers developed an innovative Monte Carlo method to detect the pore connectivity based on the notion that the long range water mobility is completely governed by the connectivity of water pores and found that the percolating network exists even at a water volume fraction of 0.1. 44 This implementation seems to ignore the dynamic nature of the water domains in which water beads are in constant interaction with the environment and themselves so that the shape or connectivity of the water domains is ever-changing. Instead of implicit characterization, an explicit cluster quantification can be accomplished by the direct cluster analysis. 37,42,45 Figure 4 shows the quantitative cluster size distributions at different λ’s from the cluster analysis algorithm. Note that the resulting cluster size distribution is an ensemble average of 1000 independent configurations, in contrast to the Monte Carlo implementation which only explored a single configuration. The resulting cluster size distribution for Nafion chains of 80 repeat units is statistically indistinguishable from that for the shorter chains of 30 repeat units. Generally, the cluster size distribution of the left branch moves to the right for low λ’s ( λ < 5) and to the left for moderate and high λ’s, while that of the right branch moves to the right as λ increases. Progressive growth of the water clusters with increasing hydration corroborate the qualitative trend observed in Figure 2 that the water clusters become larger and more connected as the water content is increased. One can readily identify the percolation threshold of hydrated Nafions at λ = 5. It is evident that the cluster sizes are significantly different from those obtained from the water-water RDFs. Such a discrepancy was also acknowledged in a previous work, yet a mysterious choice of cutoff and problematic definition of cluster size as the largest distance between beads in a cluster lead to ambiguity. 37 Our cluster analysis algorithm poses a critical advantage of assigning each individual water cluster a unique cluster identity so that one may have a direct image of cluster size, shape, and connectivity. Such a manipulation implemented for several selected hydration levels will be discussed in the following paragraphs.

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0

10

λ=1 2 3 4 4.5 5 6 7 8 9 10 11 13 15 20

(a)

-1

p(n)

10

-2

10

-3

10

-4

10 1

10

100

1000

10000

1e+05

Cluster size n -1

(b)

p(n)

10

λ=3

-2

10

-3

10

-1

p(n)

10

λ=4

-2

10

-3

10

-1

p(n)

10

λ=4.5

-2

10

-3

10

-1

λ=5

10

p(n)

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

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-2

10

-3

10

1

10

100

1000

10000

Cluster size n Figure 4: (a) Quantitative cluster size distribution from the cluster analysis algorithm. (b) cluster size distributions with emphasis on approaching the percolation threshold at λ = 5. Generally, the cluster size distribution of the left branch moves to the right for low λ’s ( λ < 5) and to the left for moderate and high λ’s, while that of the right branch moves to the right as λ increases.

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Representative configuration snapshots of water clusters color coded by the unique cluster ID at water contents below the percolation threshold are shown in Figures 5 and 6. It is clearly evident that our quantitative analysis does contradict the cluster size and separation estimates derived from water-water RDFs shown in Figure 1. Water clusters keep growing from small aggregates into large spheres, elongated rods, and branched twisty cylinders as λ increases. At λ = 1, the majority of water beads exists as singlets, doublets or triplets or small aggregates with a few beads. The largest clusters have about 10 beads in aggregation forming a compact sphere (e.g. yellow cluster) or cylinder (red and blue clusters). At λ = 2, the cluster size distribution shifts to the right and the cluster size ranges from 1 to about 50 beads. The largest clusters form spherical (yellow cluster), cylindrical (cf. red and blue clusters) and an elongated and branched cylindrical (pink cluster) shapes. The cluster size distribution at λ = 3 continues to grow with as many as 140 beads. The five largest clusters are in a structure of branched cylindrical rod as conspicuously demonstrated in Figure 5. The percolation threshold is approached at λ = 4 and 4.5 and the largest clusters appear much more tortuously branched and extended, yet there is no percolating structure present (see renderings shown in Figure 6). The largest cluster at λ = 4.5 consists of ∼ 2200 water beads, while that at λ = 4 has about 1000 water beads. It appears that such a number of beads is not sufficient to form a percolating network in a simulation box of 40. A combination of informative cluster size distribution and illustrative configuration images unambiguously confirms the absence of a percolating network. Figure 7 shows representative snapshots of water clusters color coded by the unique cluster ID’s at the percolation threshold corresponding to λ = 5. The cluster size distribution shown in Figure 4 covers a broad range from 1 to 13000 corresponding to the scenario where about 40% of the total water beads constitute the largest cluster. Three independent configurations vividly demonstrate an array of possible conformations of percolating structures at this water content. In the upper panel, Configuration A, the largest (red) cluster consists of only 4410 water beads, yet it percolates along the parallel plane. Percolation

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Figure 5: Representative snapshots of water clusters color coded by the unique cluster ID’s at λ = 1, 2 and 3. Only the five largest clusters are explicitly color coded, while all the other water beads represented by white dots/spheres. The total number of water beads is denoted as Nw . At λ = 1, the red cluster has 13 water beads, blue (11), cyan (11), pink (10), and yellow (10). At λ = 2, red (50), blue (41), cyan (39), pink (33), and yellow (31). At λ = 3, red (144), blue (100), cyan (85), pink (68), and yellow (68). 16


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Figure 6: Representative snapshots of water clusters color coded by the unique cluster ID’s at λ = 4 and 4.5. Two independent configurations at each λ are shown. Only the largest five are explicitly color coded and labeled by the number of beads of each cluster, while all the others represented by white dots. No percolating cluster exists.

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can be readily identified by the iso-surface when rendered from a top view. In the middle and lower panels, the largest clusters of Configurations B and C are evidently percolating along all three directions with a larger number of constituent water beads, 13038 and 8125, respectively. Existence of water clusters inside the polymer matrix of Nafion is definitely not static but in constant dynamic equilibrium. This explains the discrete spikes at the large size side observed in the cluster size distribution curve. At certain times during the simulation, several non-percolating clusters may merge into one larger percolating cluster or conversely a large percolating cluster may decompose into a few smaller ones, none of which are percolating. The importance of proper sampling of the water clusters, and the superiority of an ensemble average over one single configuration in cluster characterization is overwhelmingly evident. The observed percolating sponge-like mesoscopic morphology is in accord with the established morphology model of Nafion where hydrophilic domains of water, hydrated protons, and sulfonate groups form an irregular three dimensional network intertwining with the hydrophobic backbone matrix. 4 Yet, the local parallel plane feature is missing. Figure 8 shows representative snapshots of water domains at hydration levels well above the percolation threshold. The percolating structures are conspicuously dominating in the snapshots. Significantly more water beads contribute to the largest cluster and the size and number of smaller clusters gradually diminish as the hydration is increased. It is expected that the macroscopic phase separation between water and polymer phases may occur at elevated λ. Figure 9 shows a structural analysis of hydrated Nafion in terms of various radial distribution functions as a function of λ. Intensity of the AA peak increases with λ. The gCW (r) and gCC (r) peaks decrease intensity with λ and move to larger separation. Similar weaker correlation and larger separation between the sulfonic acid groups (CC) and between the sulfonic acid groups and water beads (CW) with increasing hydration level were also observed in previous all-atom classical MD simulations 16,46,47 and mesoscale simulations. 37

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Figure 7: Representative snapshots of water clusters color coded by the unique cluster ID’s at percolation threshold of λ = 5. Three independent configurations are presented. The largest cluster (red) of Configuration A percolates at the parallel plane, while those of Configurations B and C percolate at all three dimensions.

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Figure 8: Representative snapshots of water clusters color coded by the unique cluster ID’s at λ = 7 and 15. Only the largest six are explicitly color coded and labeled by the number of beads of each cluster, while all the others represented by white dotes. The largest red clusters consisting of at least 90% of the total water beads are conspicuously percolating at all three dimensions.

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Side chain separation may impact the proton transport mechanism in PFSA materials. 13,48,49 Probability distribution of side chain separation is calculated as an ensemble average for the nearest neighboring side chains only. The average side chain separation slightly increases with λ, consistent with the experimental trend observed by Kreuer et al. 2,9 Size and dynamics of hydrated Nafion as a function of λ are presented in Figure 10. Hydration level does not appear to significantly alter the conformation of the polymers. Polymer size in terms of Re2 and Rg2 , bond length distribution and size ratio are intact. Increasing hydration level leads to elevated dynamics: shorter chain relaxation time, faster polymer and water diffusion. Note that the relatively mobile water and immobile Nafion polymer matrix are indicated by a large diffusivity ratio of water to polymer: DW /Dpolymer ∼ 100. It is tempting to use shorter polymer chains due to computational appeal and model tractability since the similar equilibrium mesoscopic structure was obtained even for Nafion with 8 backbone beads as that of Nafion with 20. 29,44 Our recent work suggested that sufficiently long ( about 100 constituent backbone beads) Nafion chains are required to mimic the realistic environment. 43

Conclusions In summary, we have quantified the evolution of the hydrated morphology over a broad range of hydration levels of Nafion using a cluster analysis method. Our findings are in contrast with those estimates of the widely utilized radial distribution function method. The quantitative cluster analysis along with illustrative configuration images leads to a wealth of information on water domain size, shape, and connectivity. The percolation threshold of the water domains in hydrated Nafion occurs at a hydration level of 5 H2 O/SO3 H. Below the threshold, isolated individual water clusters cannot contribute to the ion transport. Water clusters keep growing from small aggregates into larger spheres, elongated rods, and branched twisted cylinders as the hydration level increases. Above the hydration threshold,

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1.5

(a) 2

λ=1 λ=2 λ=3 λ=5 λ=7 λ=9 λ=11 λ=15

(b)

1

gCC(r)

gAA(r)

1.5

1

λ=1 λ=2 λ=3 λ=5 λ=7 λ=9 λ=11 λ=15

0.5 0.5

0

2

2

r

4

0

6

0

2

4

r

6

(d)

(c) λ=1 λ=2 λ=3 λ=5 λ=7 λ=9 λ=11 λ=15

0.4

prob(lss)

1.5

0.5

1

2.65

λ=1 λ=2 λ=3 λ=5 λ=7 λ=9 λ=11

0

gCW(r)

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

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0.3

2.6

2.55 0 2 4 6 8 10 12

λ

0.2 0.5

0

0.1 0

0

2

r

4

6

0

2

4

6

Side chain separation lss

Figure 9: Structure analysis of hydrated Nafion at different λ. (a) Radial distribution function, g(r), of the AA beads; (b) g(r) of the CC beads; (c) g(r) of the CW beads; and (d) probability distribution of side chain separation (inset: average side chain separation as a function of λ). Intensity of AA peak increases with λ. gCW (r) and gCC (r) peaks decrease intensity with λ and move to larger separation. The average side chain separation increases only slightly with λ. See Figure 2 for identity of the A, C, and W beads.

22


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(b)

(a) 2

λ=1 λ=2 λ=3 λ=5 λ=7 λ=9 λ=11

0.08

2

10

Re 2 Rg lb R CN

Probability

Polymer size

2

1

10

0

0.06

0.04

0.02

10

0

2

4

6

λ

8

10

0

12

0.4

0.8

0.6

1

1.2

Bond length 0

7000

10

(c)

(d) DW DPolymer

6500

Self diffusivity

-1

6000

τR

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


5500 5000

10

-2

10

-3

10

4500 4000

0

2

4

6

λ

8

10

12

0

2

4

6

λ

8

10

12

Figure 10: Polymer size and dynamics characterization of hydrated Nafion at different λ: (a) polymer conformation; (b) bond length distribution; (c) polymer end-to-end vector relaxation time; and (d) self diffusivity of polymers and water beads. Hydration level hardly affects the polymer conformation. Polymer size in terms of Re2 and Rg2 , bond length distribution and size ratio are intact. Increasing hydration leads to elevated dynamics: shorter chain relaxation time, and faster polymer and water diffusion.

23



the percolating water networks are conspicuously dominating in the configurations. More water beads contribute to those networks and the size and number of smaller clusters gradually diminish with increasing water content. Our work emphasizes the importance of a proper quantification tool in understanding the conducting domains in Nafion. We hope our contribution may bring due attention to this essential issue.

Acknowledgement Funding from the European Research Council under the European Union’s Seventh Framework Program (FP/2007-2013) / ERC Grant Agreement n. 306682 is acknowledged. Computing resource is provided through XSEDE allocation DMR130078.

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Graphical TOC Entry

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Morphology of Hydrated Nafion through a Quantitative Cluster

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