Impact of Nano- and Mesoscales on Macroscopic Cation Conductivity in Perfluorinated-Sulfonic-Acid Membranes

A mean-field local-density theory is outlined for ion transport in perfluorinated-sulfonic-acid (PFSA) membranes. A theory of molecular-level interactions predict nanodomain and macroscale conductivity. The effects of solvation, dielectric saturation, dispersion forces, image charge, finite size, and confinement are included in a physically consistent 3D-model domain geometry. Probability-distribution profiles of aqueous cation concentration at the domain-scale are in agreement with atomistic simulations using no explicit fitting parameters. Measured conductivities of lithium-, sodium-, and proton-form membranes with equivalent weights of 1100, 1000, and 825 g/mol(SO3) validate the macroscale predictions using a single-value mesoscopic fitting parameter. Cation electrostatic interactions with pendant sulfonate groups are the largest source of migration resistance at the domain-scale. Tortuosity of ionically conductive domains is the largest source of migration resistance at the macroscale. Our proposed transport model is consistent across multiple lengthscales. We provide a compelling methodology to guide material design and optimize performance in energy-conversion applications of PFSA membranes.


Introduction
Ion transport in cation-exchange membranes is fundamentally linked to the performance of a variety of burgeoning clean-energy technologies such as polymer-electrolyte fuel cells (PEFC). 1 A prototypical PEFC membrane consists of a phase-separated polymer with interconnected conductive, nanoscale, aqueous domains embedded in a nonconductive matrix that provides structural integrity and durability. [2][3][4] Interactions between appended charged polymer groups and aqueous counterions cause ion-transport behavior in the aqueous domains to differ from that in bulk aqueous solution. [3][4] To understand how molecular interactions among polymer, water, and ions at the nanoscale mediate transport at the macroscale, we formulate a multiscale mechanistic model for ion transport in fuel-cell membranes.
Perfluorinated-sulfonic-acid (PFSA) copolymers are the prototypical PEFC membrane material. PFSAs consist of a fluorocarbon backbone with perfluoroether sidechains that terminate in negatively charged sulfonate groups. 2 The sulfonate anion is charge compensated by an aqueous cation, such as a proton.
Unfavorable interactions between the hydrophilic sulfonate moiety and the hydrophobic backbone cause the polymer to phase separate into solid polymer bundles and an interconnected network of ionically conductive, hydrophilic domains or "pores." [2][3][4][5] Because the ionic conductivity of PFSA membranes increases drastically with water content, PEFC membranes typically operate under humidified conditions. 1 A wet environment leads to water absorption into the hydrophilic domains of the membrane with the subsequent water content described as the molar ratio of water per sulfonate site, λ (mole H 2 O/mole SO 3  ). 4 Because the sulfonate anions are immobilized by covalent bonds to the polymer matrix, electrolyte conduction through the membrane is accomplished by movement of aqueous cations. 3 The amount of absorbed water controls the degree to which the cation and the sulfonate group dissociate. 3,6 Figure 1a depicts a completely dry PFSA domain in which the sulfonate group and cation form an ionically bound ion pair. 3,[6][7] The proton exists as a hydronium cation since desorption of the constituent water molecule occurs only at extreme temperatures ( >200 °C). 4 Figure 1b depicts water solvating the bound ions. At low water contents, there is not enough water to separate the ions; they remain as bound contact pairs. 3, 6-7 Ions forming salt complexes or contact pairs are immobile and do not facilitate conduction. 7 Figure 1c depicts water completely solvating the ions at higher water contents allowing complete ion dissociation. 3, 6-7 Figure 1. Depiction of the cation center of charge (+) and water dipole (⇸) distributions around a pendant sulfonate group (). Solid lines denote the hard-sphere radius. Dotted lines denote the first solvation shell of the sulfonate group. Grey region denote PFSA polymer. a) In completely dry conditions, the sulfonate ions and cations are tightly bound as salt complex. b) In low-water conditions, ions form contact pairs. c) In high-water conditions some of the ion pairs dissociate.
Molecular-dynamics (MD) and ab-initio simulations provide invaluable understanding of intermolecular interactions among polymer, solvent, and ions in the nanodomains, but often do not describe transport processes. 9-10, 12, 25-31 Conversely, microcontinuum models provide crucial insights into transport processes in the nanodomains, but current models have selective applicability to PEFC systems 4 because they do not examine varying hydration [32][33] or exclude relevant nanoscale interactions, such as solvation energies. 8,23,[34][35][36][37] Both approaches often focus on nanoscale properties without connection to macroscopic observables. 4 Our model is grounded in physical descriptions provided by microcontinuum theories and atomistic simulations but goes beyond previous work by giving a consistent mechanistic description within the nanodomains and at the mesoscale as a function of hydration.  38 To ensure physical veracity, realistic geometric parameters are adopted from direct imaging of the PFSA membrane pore structure. The presence of mobile coins and multiple couterions is not considered, but the conductivity of fully ion-exchanged sodium and lithium-form membranes 39 are studied in addition to proton transport.

Physical Model
In PFSAs, the hydrophobic phase surrounds hydrophilic domains consisting of immobilized sulfonate groups, counterions, and absorbed water. 2 In the fully hydrated state ( λ ≈ 20), the hydrophilic domains are locally flat, ribbon-like channels with an average (mean) thickness of 0.85 nm and a width of 2.5 nm. 5, [40][41] Assuming a realistic physical representation of the aqueous nanodomains is imperative to provide a useful mathematical model. Figure 2 provides this representation. Solvent regions are completely phase 5 separated from the polymer backbone and sidechains to form lamellar channels with appended ionized sulfonate groups. Hydrophilic sulfur and oxygen atoms of the sulfonate groups are coarse grained as hemispheres. Because neighboring sulfonate groups need not be attached to the same PFSA chain, the amount of backbone between sidechains does not dictate the spacing between sulfonates along the channel. Rather, the anion groups are uniformly distributed along the walls of the channel such that the distance between groups is maximal; electrostatic repulsion between sulfonate groups is minimized.
Consequently, water swells the domain isotropically. This representation reduces the hydrophilic lamellar channel into repeating periodic cubic unit cells of dimension l .  Water molecules and cations are internal to the system; sulfonate moieties and polymer matrix are where ⟨ κ ⟩ is the integrated conductivity of the unit cell. The average current density is obtained by averaging of the local current density i over a surface of the unit cell normal to the direction of transport. Because the sulfonate anion is immobile, the local current density is due only to the cation where +¿ z ¿ is the valence of the cation, e is elementary charge, and underbars denote vectors. At constant pressure and temperature and negligible convection, the local flux +¿ J ¿ of cations in the hydrophilic channels is driven by a gradient in the cation electrochemical potential where r is the position vector inside the pore, +¿ ρ ¿ is the local molecular concentration (i.e. time averaged probability density) of the cation, and u is the anisotropic, diagonal cation mobility tensor.
u deviates from the scalar mobility in bulk solution, +¿ ∞ u ¿ , due to hydrodynamic-drag tensor 8 against the domain walls, β , [45][46][47] and increased viscosity of the liquid phase, η , around sulfonate groups due to dielectric friction (i.e. resistance of dipole rotation in an electrostatic field) 48 so that where η ∞ is the viscosity of the pure solvent. Appendix A: Cation Mobility discusses calculation of β . Einstein's law in the ideal dilute-solution limit (i.e. +¿ → 0 ρ ¿ ) relates cation mobility in bulk aqueous solution to conductivity, Boundary conditions at the upstream ( x=0 ) and downstream ( x=l ¿ boundaries are Dirichlet conditions of a fixed potential drop where +¿ us μ ¿ is a reference upstream electrochemical potential. ΔΦ is set to an applied potential of 10 -8 V (equivalent to an electric field of 116 V m -1 for a unit cell with λ=20 ), which is small enough to ensure linearity of the flux with respect to the applied potential but large enough for numerical precision.

Thermodynamics
Numerous molecular interactions in the PFSA aqueous domains dictate the distribution of the cation throughout the channel, including electrostatic interactions between the sulfonate and cation, solvation forces, dispersion and image-charge forces at the interface between the solvent and hydrophobic polymer walls and thermal entropy. Interactions are expressed through the electrochemical potential of the cation, where +¿ 0 μ ¿ is the reference electrochemical potential of the cation, k B is the Boltzmann constant, T is absolute temperature, Φ is ionic potential, and μ fs , μ solv , μ dsp , and μ img are the excess chemical potentials (i.e. excess free energies) due to ion finite-size, solvation, dispersion, and image charge, respectively. The first two terms in Equation (9) describe ideal-solution behavior, whereas the third term characterizes electrostatics. The final four terms account for ion non-idealities. Each term, except the reference chemical potential, is a function of position inside the pore.
μ fs accounts for the entropy loss by excluding water from regions with high ion concentrations. It is expressed with the widely used local-density Bickermann equation [58][59] +¿ (r ) where a is a finite-size parameter. Equation (10 is valid for lattice systems in which the solvent and ion have equal radii. When the ion and solvent are of different size, the choice of a is unclear. The results are relatively insensitive to the choice of a , and here a is set equal to the radius of a water molecule. Cation-solvation excess chemical potential, μ solv , accounts for the change in solvent potential energy due to dipoles orienting around the cation. Relative permittivity, ε , gauges the amount that dipoles can orient around a cation. Permittivity is extremely heterogeneous across a PFSA membrane nanodomain. Water solvating the sulfonate groups is dielectrically saturated ( ε =1.8 ) but water separated from the ions has a permittivity of bulk water ( ε =78.3 ). 8,33 Relative permittivity of the polymer backbone is 2.1. 60 The dependence of solvent permittivity on r is discussed in the next section. Solvation excess chemical potential at r is equal to the work to discharge a cation in a reference dielectric plus the work of charging the cation in a dielectric at r 52 where V is volume, D is the displacement field of the cation ( is the relative permittivity of bulk water, which is the reference dielectric. For a medium in which the permittivity varies slowly over space, the integral with respect to the displacement field was accurately approximated by Bontha and Pintauro as 33 where θ and ψ are spherical angular coordinates, +¿ r ¿ is the distance to the center of the cation, +¿ R ¿ is the radius of the cation, and A is a constant. The value of A is determined by interpolating between the solvation free energy in bulk solution, Δ G ∞ , and the solvation free energy in a dielectrically saturated solution, Δ G sat , where ε sat is the relative permittivity of a dielectrically saturated solvent. Values of Δ G ∞ and Δ G sat for different cations are in Table 1. The forms of Equations (12) and (13) are the same as those given by Bontha and Pintauro 33 except that we explicitly integrate over the volume rather than assume permittivity varies slowly over space. Equation (12) reduces to the solvation excess chemical potential given by Bontha and Pintauro 33 in the limit of ∇ ε=0 . To avoid simultaneously solving Equations (3), (6), (9), and (12), which is a set of coupled integral-differential equations, we approximate μ solv as the linear superposition of the solvation excess chemical potentials due to interactions with the PTFE floor and ceiling of the unit cell, which is only a function of the distance to the boundaries, d y , and interactions with the sulfonate group, which is only a function of distance to the groups, as discussed in Error: Reference source not found.
Solvation free energies account for polarization of permanent dipoles whereas van der Waals forces account for induced oscillating polarization of atoms. The dispersion force on a cation is the difference in van der Waals forces acting on a cation from water and from the PTFE polymer walls. The excess chemical potential accounting for dispersion forces, μ dsp , is thus 61 where and α , I and ρ are the diamagnetic polarizability, first ionization potential, and molecular density. Subscripts +¿ , w , and T denote the cation, water, and PTFE, respectively. Values of α and I for studied cations are in Table 1. Equation (14) superimposes dispersion forces arising from the unit-cell floor and ceiling (first and second terms, respectively).
μ img accounts for electrostatic interactions felt by an ion near the interface between two media with different dielectric constants. For the case of an aqueous ion adjacent to a water/PTFE interface, the method of image charges and Coulomb's law gives μ img as 62  Poisson's equation is necessary to close the above system of equations The strong electric field due to the charged sulfonate groups 63 and disruption of the hydrogen-bond network of water due to wall proximity 64 creates variations in ε across the unit-cell domain.
Booth's equation describes how the relative permittivity saturates with increased electric field 63, n is the bulk refractive index of water, p is the water dipole moment, and ε con is the relative water permittivity before an electric field is applied. ε con is reduced from bulk-water permittivity because the polymer walls disrupt the hydrogen-bonding network of water that causes the high permittivity of bulk water. 64 Water coordinates with neighboring water molecules forming a cluster of radius R cluster ∞ . 66 At a phase boundary, water coordination is disrupted reducing the cluster radius to R cluster . 66 Decreased cluster size reduces permittivity. 66 Lamm et al. 64 show that at 298 K the effect of water confinement on relative permittivity is well approximated as where f =( R cluster / R cluster ∞ ) 1 3 . Error: Reference source not found details calculation of f .
Boundary conditions for Equation (17) are (20) and Φ (ŕ )=Φ (ŕ ) | x=0 − ΔΦ , x=l (21) where n is the unit normal vector to the boundary and −¿ σ ¿ is the charge density on the sulfonate group assuming that the negative charge distributes uniformly on the surface of the representative hemisphere,  Figure S1 displays boundary conditions.

Aqueous Domain Free Energies
The negatively charged sulfonic acid groups impose strong electrostatic fields throughout the unit cell. Figure 4 and S2 in supporting information show 2D intensity maps on the x-y surface of the unit cell to illustrate the impact of the resulting field. Figure S2a shows the electrostatic field leads to a sharp decline in ionic potential near the sulfonate groups. Figure 4a shows that near the sulfonate groups the electrostatic field combined with wall confinement disrupts the bonding structure of water resulting in a lower relative permittivity than that of bulk water (i.e. ~78). The strong electric field orients the water surrounding the sulfonate groups causing dielectric saturation of the water that is solvating the sulfonates. Conversely, water near the center of the channel exhibits bulk-like permittivity. , and (c) the x-component of hydronium mobility, reported as +¿ ∞ , u x /u ¿ colored from white (light) to blue (dark) to represent low to high values. Figure 6 shows that the radial distribution function (RDF) of the cation with respect to the center of the sulfonate group displays three peaks, also consistent with molecular dynamics simulations. 26 Supporting Information gives details of the RDF calculation. The first peak, located at 2.4 Å, is caused by partially desolvated cations that form contact-ion pairs with the sulfonate groups (Inset a). The second peak located near 4-6 Å is caused by solvated cations that separate from the sulfonate groups and reside near the center of the channel (Inset b). The third peak, near 5.5-9 Å, arises from cations that form ion pairs with opposing sulfonate groups (Inset c). Positions of the second and third peaks of the RDF in Figure 6 shift depending on membrane water content. As water content decreases from λ=15 (solid line) to λ=9 (dashed line) and λ=4 22 (dotted line), the unit cell shrinks; the distance between sulfonate groups decreases. The distance between a sulfonate group and cation contact pairs of opposing sulfonate groups decreases; the third RDF peak shifts inward. Similarly, as water content decreases, the distance between a sulfonate group and the center of the channel decreases causing the second RDF peak to shift inward. The radial distribution function specifies cation distribution to the furthest extent of the cubic unit cell (i.e.

Aqueous Domain Transport
The strong electrostatic fields around the negatively charged sulfonate groups increases water viscosity due to dielectric friction, as Figure S2b shows. Viscosified water corresponds to water molecules that solvate the sulfonate groups. Conversely, water near the center of the channel is more bulk-like. This is qualitatively consistent with prior work of Yang and Pintauro 32, 67 , but they attributed increased solvent viscosity directly to ion concentration effects. Figure 4c illustrates that increased water viscosity combined with increased hydrodynamic drag near the walls significantly reduces aqueous cation mobility throughout the channel. Decreased mobility near the walls causes the local conductivity to be maximum near center of the channel, as Figure S2c shows. Cation conductivity is facilitated by solvated cations transporting along the center of the channel. with +¿ ∞ η /η ∞ u=I u ¿ where I is the identity tensor). There is a maximum in the dielectric friction-corrected cation conductivity at λ=4 because below this water content increasing water content decreases the fraction of immobile, bound cations, which increases conductivity. Above this water content, dilution effects dominate and conductivity decreases with increasing water content. Average domain conductivity, ⟨ κ ⟩ , includes the resistance from the hydrodynamic drag on the cations due to confinement (solid line, calculated using Equation (3) with u given by Equation (4)). Nanoscale factors reduce conductivity from the ideal-solution limit the most at low water contents. The resulting domain conductivity changes relatively little with water content (~26% difference between the smallest and largest values of ⟨ κ ⟩ versus ~520% difference for +¿ ∞ κ ¿ ). Figure S4 shows that although nanoscale resistance depends on water content, it is relatively insensitive to how the domains swell (anisotropic vs. isotropic swelling).

Impact of Side-Chain Size
The molecular composition of the PFSA sidechain (e.g. the number of fluorocarbon or fluoroether groups) influences the partial charge on molecular groups neighboring sulfonates 29   The unit-cell model for PFSA membranes captures the essence of known behavior at the nanoscale. 4 We now extend the aqueous-domain results to predict macroscopic transport properties in PFSAs.
Modeling macroscopic properties is challenging because the aqueous-domain model only accounts for phenomena at the nanoscale. It does not account for transport across a network of connected domains.
A bundle-of-tubes model describes transport through the medium. The effective macroscopic conductivity κ eff is 70 where κ eff is the effective macroscopic conductivity of the membrane, τ is the tortuosity of the network, and φ is membrane hydrophilic volume fraction, which is taken as the combined volume fraction of water and sulfonate groups (23) where ρ poly is the mass density of dry polymer (~ 2 g/cm 3 ) 71 and EW is the equivalent weight of the membrane (g polymer/mole of sulfonate groups).
Varying the cation type of the membrane ("cation form") and polymer chemistry changes τ and ⟨ κ ⟩ in Equation (22) independently. For example, for the same membrane chemistry and water content, tortuosity is assumed independent of cation type. Specifying ⟨ κ ⟩ with the nanoscale model and fitting φ /τ 2 to conductivity of one cation-form membrane predicts resistance of other cationforms. Figure 9a shows experimental (symbols) and predicted (lines) membrane conductivity, κ eff , for sodium and proton membranes with φ /τ 2 fit using conductivity of lithium membrane at the same water content. Agreement is good. Proton-form membranes have the highest conductivity because hydronium cations readily dissociate from the sulfonate group and have the largest mobility.
Conductivity of lithium-and sodium-form membranes are similar; sodium ions have a higher bulk mobility than lithium ions but are hindered in the domains because they are more likely to form ion pairs with sulfonate groups. Figure 9. Experimental (symbols) and predicted (lines) conductivity of (a) Nafion membrane (1100 g/mol SO 3 EW) conductivity for lithium-(circles), sodium-(diamonds), and proton-form (squares) membranes and (b) 3M membrane with EWs of 1100 (circles), 1000 (squares), and 825 (diamonds), and 725 g/mol SO 3 (pentagons) in lithium-(blue) and proton-form (red) as a function of water content. Lines are model predictions (Equation (22)).
Additionally, conductivity of different membrane chemistries further validates the model. We assume that network tortuosity is solely a function of the hydrophilic-phase volume fraction and distribution (i.e. τ ( φ) ) and that aqueous domain-scale conductivity is entirely a function of the local water content, cation form, and pore geometry (i.e. ⟨ κ ⟩(λ) ). The effect of network tortuosity and aqueous domainscale conductivity is separated by changing the amount of hydrophobic backbone in the polymer per sulfonate group (i.e. EW) and by measuring the conductivity of the membranes at different water contents. To account for how tortuosity varies with hydrophilic volume fraction we use the empirical where k is a fitting parameter. Substitution of Equation (24) into Equation (22) specifies the effective macroscopic conductivity at a given water content and membrane chemistry. k is 0.93, which was fit so membrane conductivity from Equations (22) and (24) matched measured conductivity of a lithiumform membrane at 90% relative humidity (i.e. at λ=9 ). k was taken constant for all EW membranes in lithium-and proton-form. Figure 9b shows that predicted membrane conductivity from Equation (22) (lines) agrees well with measured conductivity (symbols) for both lithium and proton membranes as a function of water content across a range of equivalent weights. The membrane conductivity increases with decreasing EW at the same water content because the hydrophilic volume fraction increases, thereby lowering network tortuosity.
Discrepancy between theory and experiment shown in Figure 9 results from the breakdown in the assumption that tortuosity is exclusively a function of water content. Hydrophilic domain morphology (i.e. locally flat domains or inverted micelles) and related domain connectivity depend slightly on water content and cation form rather than solely on water volume fraction. 4 (5)). Figure 10 shows the calculated ideal-solution conductivity as a function of water content (dotted line).
+¿ ∞ κ ¿ decreases with hydration because water dilutes the proton charge carriers. The dashed line in Figure 10 is the proton domain-scale conductivity, ⟨ κ ⟩ , which is equivalent to the solid line in Figure 6. The difference between the dotted and dashed lines represents the conduction losses due to cation interactions with the polymer matrix and sulfonate side groups. ⟨ κ ⟩ is relatively constant with water content because the effect of charge carrier concentration is countered by proton/polymer interactions at lower water content that reduce conductivity. Guided by Equation (22) Aqueous microscale conductivity is relatively constant with water content due to the competition of charge-carrier concentration, which increases conductivity with decreasing water content, and cation solvation, which increases conductivity with increasing water content. Macroscale conductivity increases with increased water content because membrane transport is strongly affected by the tortuosity of the network, which decreases with increasing water content. Addressing transport limitations at both the nano-and network-scales offer avenues to improve membrane performance. Conversely, focus on optimizing and exploring transport at a single lengthscale without regard for the other may not be fruitful. The model developed here provides a framework to understand the root causes of ion-transport resistances in ion-conductive polymers.

Notation
is the location of the first peak of the water radial distribution function (RDF) with respect to sulfur. 30 is specified by subtracting the hard sphere radius of water.

Appendix A: Cation Mobility
The Stokes-Einstein equation predicts that ion mobility varies inversely with solvent viscosity, which provides the basis for the η ∞ /η (r ) correction to mobility in Equation (4). Yang and Pintauro corrected the solvent viscosity based on increased ion concentration. 32, 67 We account for increased water viscosity due to dielectric friction of the sulfonate groups, consistent with nonequilibrium statistical-mechanics calculations. 8,32,67 Hubbard determined that the increase in η due to the slower relaxation of dipoles in an electric field to be 48 where τ d is the Debye dielectric relaxation time, and ε ∞ and ε hf are the unperturbed and high-frequency dielectric constants of the solvent, respectively.
Because the fraction of bulk mobility due to hydrodynamic drag parallel to a wall, β ∥ , and perpendicular to wall, β ⊥ , are different β is an anisotropic, diagonal tensor β ⊥ has an exact solution effectively estimated as 45 where d y is the scalar distance from the center of the cation to the nearest wall (i.e. y ' = y or ¿ l− y ). β ∥ is estimated as 47 38

Appendix B: Solvation Energy Calculation
To avoid simultaneously solving Equations (3), (6), (9), and (12), which is a set of coupled integraldifferential equations, we approximate μ solv as the linear superposition of the solvation excess