Electrochemical Measurement of Water Transport Numbers in Anion-Exchange Membranes

The microelectrode working electrode (WE) was Pt, 50 μm in diameter (Metrohm, Florida, USA), and the counter (CE) and reference electrodes (RE) were gas-diffusion electrodes (GDE), again Pt. For acid-based experiments, commercial GDEs were used (0.5 mg cm⁻² Pt, Ion Power, Delaware, USA), fabricated from commercial gas-diffusion layers (Sigracet 25BC, SGL Carbon, Wiesbaden, Germany); for alkaline-based experiments, custom GDEs were fabricated using a Sono-Tek spray coater (Sono-Tek Corporation, New York, USA), gas-diffusion media (AVCarb 370, microporous layer loading 0.5 mg cm⁻², Pt loading 0.5 mg cm⁻², AVCarb Material Solutions, MA, USA) and Versogen ionomer (PiperION-A (PAP-TP-85), 5 wt% in ethanol, Versogen, Delaware, USA). ⁵⁷ Additional fabrication details are available elsewhere. ⁵⁷

The CEMs used were Nafion 211 (25 μm, 1100 EW, used pre-treated by nitric acid and boiling ³ and as-received, Ion Power, Delaware, USA) and sulfonated polystyrene cast from solution (poly(4-styrenesulfonic acid) solution, 18 wt% in water, 100 g, Sigma-Aldrich, Massachusetts, USA). Nafion 211 was also exchanged to sodium form by submersion in 0.5 M Na₂SO₄ aqueous solution (Sodium sulfate, 1 kg, ACS reagent, $\geqslant$99.0%, Sigma-Aldrich, Massachusetts, USA). AEMs studied were Versogen (20 and 80 μm, ∼425 EW, Versogen, Delaware, USA) and Sustainion (X37–50, 50 μm, ∼900 EW, Fuel Cell Store, Colorado, USA). ^13,21 1 M anion solutions determined the membrane ion-exchange form. Potassium carbonate (500 g, ACS reagent, $\geqslant$99.0%, Sigma-Aldrich, Massachusetts, USA), potassium bicarbonate (500 g, ACS reagent, 99.7%, Sigma-Aldrich, Massachusetts, USA), and potassium hydroxide (3 kg, Certified ACS pellets, Thermo Fisher Scientific, Massachusetts, USA) were used for carbonate, bicarbonate, and hydroxide AEM ion-exchanged forms, respectively. Before use, membranes were placed in three successive water baths using Millipore-grade water (18.2 MΩ-cm) to rinse excess ions, for at least 4 h, overnight, and 30 min, respectively. In the case of hydroxide-form membranes, the water was degassed with nitrogen to prevent conversion to carbonate form due to dissolved carbon dioxide.

Membrane water uptake as a function of RH were taken from previous studies at 25 °C for Nafion in the proton form, for Sustainion in the hydroxide form, and for Sustainion and Versogen in the carbonate and bicarbonate forms. ^3,13,21 Water content was assumed to be invariant with temperature, as the impact of temperature on water content for Nafion in the studied temperature range is inconclusive. ³ The water uptake of hydroxide-form Versogen was measured in this study using a dynamic-vapor sorption (DVS) system (Surface Measurement Systems, UK) with the chamber flushed with dry nitrogen gas for 1 h to remove residual CO₂ and set a dry weight prior to the humidification step. The sample was placed directly in the sample holder from a degassed water reservoir to minimize carbonate formation and held at 25 °C and 70% RH for 3 h to equilibrate the membrane. Next, the humidity was increased to 98% RH to begin equilibration for the water-uptake measurements. The 98% RH step was followed by 95% RH and 90% RH before the humidity set points were reduced in intervals of 10% RH down to 0% RH and then ramped back to 98% RH in a mirrored manner. all humidity set points were held for 60 min. The second, increasing-humidity sorption curve was used to analyze the water uptake. Additional details are found in prior studies. ^13,58

The microelectrode cell used was as described previously; a description can also be found in SI. ^55,56 Hydrogen gas (2 or 4 vol% in Ar, Linde Gas, or 100%, H2PEM-510 H₂ generator, Parker Hannifin, Ohio, USA) was humidified via external humidifiers (Humidification System, Fuel Cell Technologies, New Mexico, USA) or a custom wet/dry gas mixing bubbler before entering the cell. Cell temperature was controlled by a heating pad below the cell controlled by a proportional-integral-derivative (PID) controller (Red Lion, Pennsylvania, USA). Gas lines entering the cell were held at 80 or 90 °C for the 40/50 °C and 70 °C trials, respectively, to minimize water-droplet formation. A syringe tube heater (New Era Pump Systems, New York, USA) maintained temperature between the humidifier and the microelectrode cell. Electrochemical measurements were performed with a VSP-300 potentiostat using an ultra-low current cable (Bio-Logic, Seyssinet-Pariset, France).

OCV was measured between the working and reference electrodes. OCV was held for 30 min or until a stable reading was achieved at each tested humidity. An OCV was considered stable if there was a less than 2 mV continuous increase or decrease in at least a 5–10 min period as a baseline, or if the OCV was fluctuating in a ±5-mV range in the same time period. For the hydroxide-form AEMs, a 15 h potential hold at 0.5 V was performed to remove trace carbon dioxide before the OCV measurements. A hydrogen reference electrode was used in all cases.

All membranes and counterion-exchange forms were tested by fixing the reference electrode (RE) at a constant RH and varying the WE/CE side in 5% RH intervals ±10% from each of these setpoints, e.g., when the RE was held constant at 50% RH, the WE/CE side was varied in increments at 40, 45, 50, 55, and 60% RH. A table of all test conditions may be found in Table S1 of SI. The standard test for AEMs held the RE at 50, 70, and 90% RH while the WE/CE side was varied accordingly. 100% RH was not used when the RE was held at 90% RH due to water-droplet formation. Temperatures tested include 30, 40, 50, and 70 °C. At the non-50 °C temperatures, only 50% RH RE (30, 70 °C) or 90% RH RE (30, 40 °C) were tested. Experiments involving Nafion 211 used the standard procedure at 50 °C, but at 20 °C, the RE was held at 39, 57, 79, and 89% RH, and the WE/CE varied. At 20 °C, experimental error sometimes resulted in RH differences between the WE and RE of up to 14% instead of 10%. Raw OCV versus RH data can be found in the SI as an attached spreadsheet.

OCV, the open circuit voltage, U, corresponds to zero net current. The counter-ion water transport number, ${{\rm{{\rm T}}}}_{i},$ describes the flow of ions to the flow of water in the membrane. To relate these two properties, both cell thermodynamics and species fluxes must be considered. We follow closely the development for hydronium ions in Nafion by Fuller and Newman. ³⁸

The electrochemical cell consists of two Pt electrodes (RE and WE/CE) separated by a PEM (or AEM), with the membrane enclosed in two regions each with a different water activity (as set by the RH). The Pt electrode and membrane enclosed on the "left" side are arbitrarily denoted as phases α and δ, respectively, and the electrode and membrane enclosed on the "right" side as β and γ. A cell schematic is given in Fig. 1a, and a corresponding thermodynamic phase diagram is depicted in Fig. 1b. The cell analysis of Fuller and Newman ³⁸ for the water transport number of hydrogen ions in a PEM reduces to

$$\begin{eqnarray}&&{{\rm{{\rm T}}}}_{{{\rm{H}}}^{+}}=\frac{F}{{RT}}\frac{{dU}}{{dln}\left(\frac{{a}_{0}^{\delta }}{{a}_{0}^{\gamma }}\right)}\end{eqnarray} \tag{ 6 }$$

where F is Faraday's constant, U is the OCV, R is the ideal-gas constant, T is the absolute temperature, and phases δ and γ are as defined in Fig. 1. ³⁸ ${a}_{0}^{i}$ is the water activity in phase i controlled by exposure to water vapor of known RH. Equality of water activity, signifying equilibrium, is imposed between the membrane phase and water vapor on each side. For an isothermal membrane, the ratio of upstream and downstream water activities is given by the ratio of the corresponding RHs. Thus, the hydronium-counterion water transport number, and those to follow, are ascertained by measuring the OCV as a function of the water-vapor RH ratio between the working and reference electrodes. We note that these correspond to in-plane transport numbers. ^3,59

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Equation 6 demands isothermal cell operation with condensation of water on the upstream membrane side and vaporization on the downstream membrane side. Gas convection and small counter-electro-osmotic water flows justify this assumption. The membrane is also assumed homogeneous with membrane charge and neutralizing counterions uniformly distributed spatially. In addition, the membrane must be a single-neutralizing ion-exchange form. Hydrogen partial pressures on each side of the membrane do not appear in Eq. 6 because equal composition of hydrogen gas is supplied to both the working and reference electrode chambers. Partial pressure differences caused by different humidities are not significant, as calculated from the Nernst equation.

Transport numbers of single cations other than hydronium can likewise be quantified by Eq. 3 even in the absence of electrode reactions (e.g., sodium ions). In this situation, the completely polarized electrodes sense the electric potentials enforced at the boundary between the electrode surface and the membrane. We report results below for the water transport number of sodium ions in Nafion 211. Implicit is the assumption that the sodium does not react at the electrode surface.

When an AEM is in the single hydroxide-exchanged form, i.e., HEM, the hydrogen-oxidation reaction ⁸ is

$$\begin{eqnarray}&&{{\rm{H}}}_{2}+2{\rm{O}}{{\rm{H}}}^{-}\leftrightarrow 2{{\rm{H}}}_{2}{\rm{O}}+2{{\rm{e}}}^{-}\end{eqnarray} \tag{ 7 }$$

and thus different than that for a PEM. This reaction necessitates a different expression for the OCV than that used in the hydrogen-ion PEM case. We extend the framework of Fuller and Newman ³⁸ to arrive at

$$\begin{eqnarray}&&{FU}={\int }_{{x}_{\gamma }}^{{x}_{\delta }}{\rm{\nabla }}{\mu }_{{{\rm{OH}}}^{-}}{dx}-{\int }_{{x}_{\gamma }}^{{x}_{\delta }}{\rm{\nabla }}{\mu }_{0}{dx}\,\end{eqnarray} \tag{ 8 }$$

where x is the coordinate parallel to the membrane between the two electrodes with subscript phases defined in Fig. 1, and ${\mu }_{i}$ is the electrochemical potential of species i. Water is given the subscript 0. The last term on the right involving the water gradient arises from the alkaline reaction equilibrium in Eq. 7.

Zero current ³⁸ of hydroxide ion demands that

$$\begin{eqnarray}&&{\rm{\nabla }}{\mu }_{{\rm{O}}{{\rm{H}}}^{-}}=-|{z}_{{\rm{O}}{{\rm{H}}}^{-}}|{{\rm{{\rm T}}}}_{{\rm{O}}{{\rm{H}}}^{-}}{\rm{\nabla }}{\mu }_{0}\end{eqnarray} \tag{ 9 }$$

where ${{\rm{{\rm T}}}}_{{\rm{O}}{{\rm{H}}}^{-}}$ is the transport number of water in a HEM and ${z}_{{\rm{O}}{{\rm{H}}}^{-}}$ is the valence of hydroxide ion. Substitution of Eq. 9 into Eq. 8 and differentiation results in the desired expression for the hydroxide-ion water transport number

$$\begin{eqnarray}&&{{\rm{{\rm T}}}}_{{\rm{O}}{{\rm{H}}}^{-}}=-\frac{F}{{RT}}\frac{{dU}}{{dln}\left(\frac{{a}_{0}^{\delta }}{{a}_{0}^{\gamma }}\right)}-1\end{eqnarray} \tag{ 10 }$$

This result has previously been derived by Wang et al. ⁴⁹ and Roy. ⁵⁰

In the presence of atmospheric carbon dioxide, alkaline AEMs can transform into multi ion-exchange forms of hydroxide, bicarbonate, and carbonate counter ions. Accordingly, we desire the water transport numbers of bicarbonate and carbonate ions in addition to that of hydroxide ion. However, the general multi-ion case is highly involved and requires additional unavailable transport parameters. ⁶⁰ To simplify the analysis, we ion exchange the AEMs into predominately bicarbonate or carbonate forms where each counter ion is present singly. By analogy to Eq. 9, we write for bicarbonate and carbonate ions, respectively, that

$$\begin{eqnarray}&&{\rm{\nabla }}{\mu }_{{HC}{O}_{3}^{-}}=-{\rm{|}}{z}_{{HC}{O}_{3}^{-}}{\rm{|}}{{\rm{{\rm T}}}}_{{HC}{O}_{3}^{-}}{\rm{\nabla }}{\mu }_{0}\end{eqnarray} \tag{ 11 }$$

and

$$\begin{eqnarray}&&{\rm{\nabla }}{\mu }_{C{O}_{3}^{2-}}=-{\left|{z}_{C{O}_{3}^{2-}}\right|{\rm{{\rm T}}}}_{C{O}_{3}^{2-}}{\rm{\nabla }}{\mu }_{0}\end{eqnarray} \tag{ 12 }$$

When AEMs are either in the carbonate or bicarbonate forms, the reaction at the electrode remains the same as Eq. 7, as the system is still alkaline, and thus Eq. 8 still holds. However, the primary mobile ion is no longer hydroxide but is either bicarbonate or carbonate. From Eqs. 1 and 2, respectively, the hydroxide electrochemical potential in Eq. 7 is replaced by either that of bicarbonate or carbonate species. Upon neglect of the small quantity of dissolved carbon dioxide, one finds that

$$\begin{eqnarray}&&{\rm{\nabla }}{\mu }_{O{H}^{-}}={\rm{\nabla }}{\mu }_{{HC}{O}_{3}^{-}}\,\end{eqnarray} \tag{ 13 }$$

or

$$\begin{eqnarray}&&{\rm{\nabla }}{\mu }_{O{H}^{-}}=\frac{1}{2}{\rm{\nabla }}{\mu }_{0}+\frac{1}{2}{\rm{\nabla }}{\mu }_{C{O}_{3}^{2-}}\end{eqnarray} \tag{ 14 }$$

for bicarbonate and carbonate ions, respectively. These expressions follow from the criteria for carbonate chemical-reaction equilibria, and the derivation can be found in SI. Following the development above for HEM, Eqs. 13 and 14 are sequentially substituted into Eq. 8 and the result differentiated to yield

$$\begin{eqnarray}&&{{\rm{{\rm T}}}}_{{HC}{O}_{3}^{-}}=-\frac{F}{{RT}}\frac{{dU}}{{dln}\left(\frac{{a}_{0}^{\delta }}{{a}_{0}^{\gamma }}\right)}-1\end{eqnarray} \tag{ 15 }$$

and

$$\begin{eqnarray}&&{{\rm{{\rm T}}}}_{C{O}_{3}^{2-}}=-\frac{F}{{RT}}\frac{{dU}}{{dln}\left(\frac{{a}_{0}^{\delta }}{{a}_{0}^{\gamma }}\right)}-\frac{1}{2}\,\end{eqnarray} \tag{ 16 }$$

for bicarbonate and carbonate water-transport numbers, respectively.

To evaluate the various water transport numbers as a function of water content, a differential approach is used with Eqs. 6, 10, 15, or 16. In previous studies, an integral method was used to evaluate the water transport number, i.e., the reference side was held at one RH and the working side varied over the entire tested range of RHs. ³⁸ The water-transport number was then evaluated from the slope of the data over the entire range of water contents. Herein, differential measurements were used as described above, where the reference-side RH was held at discrete points and the working-side RH was varied in a small range around this value. To quantify the water-transport number as a function of water activity, measured OCV data were plotted versus the natural logarithm of RH (i.e., water activity) and the local slopes calculated. Figure S2 in SI illustrates the procedure. Differential data analysis better accounts for the variation of the water transport number with water content as it minimizes the magnitude of the water diffusion gradient and error caused by overly large gradients, especially with properties dependent on membrane water content, and also better represents a quantitative water transport number. However, this approach also results in inherently more data scatter and error, as the fewer data around each reference RH datum results in more variability in slope (due to intrinsic variability in RH measurement), as documented in Fig. S2. Figure S3 reports typical error bars in water transport numbers for counterions in AEMs; similar data scatter was witnessed for Nafion.

To validate the microelectrode experimental apparatus and interpretation scheme, measurements were performed with the hydronium ion in Nafion 211 at 20 °C. Results are shown as filled circles in Fig. 2a compared with existing literature (remaining symbols). ^38,42,43 All reported water contents in Fig. 2 and in subsequent figures are the average between the working and reference electrodes. Measured water transport numbers increase from approximately 1.3 at low membrane water contents to nearly 2 at high water contents, or 1.3 to 2 water molecules moving per proton migrated. An increase is expected as higher water contents raise the availability of free water in the membrane, ³ which then migrates with the counterion via electro-osmosis. With less free water, the solvation shell of the protons may not be fully filled, resulting in the lower measured water transport numbers. Our measured values are somewhat higher than previously reported, with a previous Nafion-211 maximum in water vapor reported as ∼1.4. ^3,38 Membranes used in these studies were pretreated using different techniques—nitric acid and boiling in this work, whereas Fuller and Newman pretreated with sulfuric acid, which could contribute to the observed diference. ³⁸ Cell design differences between studies are another possible source of the difference in reported values.

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The proton water transport number of Nafion 211 at 50 °C is compared to that at 20 °C in Fig. 2b. There is a modest increase in the water transport number from a high of 2 at 20 °C to approximately 3 at 50 °C. This result is somewhat higher than expected based on prior studies of the water transport numbers for Nafion. ^44–46 In those studies over the same temperature range, the water transport numbers increased ∼20% for liquid-equilibrated Nafion. It has been shown that the primary solvation shell of aqueous ions does not change significantly in this temperature regime. ^61,62 As a result, it is likely that only the secondary solvation shell (i.e., long-range interactions) is impacted by the increasing temperature. We suggest that at higher temperature, local water viscosity in the secondary shell around each ion is relatively smaller than that at lower temperature leading to lessened viscous forces on the primary shell and more water transporting with the migrating ion. Additional analysis of temperature effects in the AEMs is addressed below.

Understanding water uptake of the different ion-exchange forms of Versogen is requisite to describe fully how the water transport number changes with RH. The water uptake of Nafion as a function of RH is well-studied. ^3,48,63,64 Likewise, water uptake in hydroxide and carbonate forms of Sustainion is available. ²¹ Figure 3 shows our measurement of water uptake of hydroxide-form Versogen (solid line) as a function of RH, compared to available literature data for carbonate and bicarbonate forms (dashed and dotted lines). ¹³ Literature results were performed in the same research group in a prior study with procedures consistent with those used in this work. The water uptake for the hydroxide counterion is slightly lower than that of carbonate and bicarbonate counterions, which, in turn, are almost identical. ¹³ It should be noted that water uptake is defined here as per fixed functional group—if defined per transport ion, the water uptake values for carbonate will be half that displayed in Fig. 3.

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Figure 4a demonstrates the water transport number for hydroxide and carbonate ion-exchange forms of Versogen as a function of membrane water content at 50 °C (Fig. 4a can be found as a function of RH in Fig. S4 in SI). Carbonate-form Versogen has the highest water transport number at all water contents, ranging from ∼3 to ∼11, with the difference between ion-types more pronounced at higher water contents. Hydroxide form consistently has the lowest water transport number, ranging from ∼0.5 to ∼6, with bicarbonate-ion form falling between carbonate- and hydroxide-form values. The values for all ions in Versogen are similar to that for protons in Nafion at low water content and exceed that of protons at high water content. A similar trend is seen for Sustainion in Fig. 4b, with the water transport number for the carbonate species typically higher than the hydroxide counterion, particularly at high water content. The carbonate and hydroxide forms of Versogen are given with error bars in Fig. S3. These error bars are somewhat large, particularly at high water content, and are likely due to two primary factors. The first is the nature of small-interval slope measurements, as mentioned previously, which gives rise to larger error bars. The second is water-droplet formation at high water content (RH). Water droplets result in significant variation in the OCV as a function of time, making it difficult to discern the OCV at a given RH. Even very small quantities of liquid water disturb the measurement.

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The origin of the differences in water transport number between specific ion forms is likely related to the solvation shells of the different ions. One interpretation of water transport number is that the amount of water dragged by an ion as it migrates is at least partially related to the water complexed around the ion in its solvation shells. ³ In this case, the transported solvation shell of protons is 2 or 4 waters, hydroxide ion is 4 or 5 waters, bicarbonate is 6-7 waters, and carbonate is 8-9 waters, as depicted in Fig. 5. ^65–68 With this framework, the origin of the trend with ion type becomes clear: the larger is the number of water molecules solvated around an ion, the larger is the amount of water transported via electro-osmosis. The results indicate that the ion water transport numbers exceed the hydration number of the first solvation shell at high water content. Likely, this is due to the impact of secondary solvation shells and the presence of more free water with higher water uptake. ³ The solvation shells listed above are primary solvation shells only, and do not consider long-range interactions, such as water outside the primary solvation shell transporting with the moving shell due to viscous effects, which would increase the measured water transport number. The carbonate ion, in particular, appears to transport water in amounts far exceeding that of its first solvation shell (multiplying the values in Figs. 4a and 4b by the valence yields a value of 20–24 water molecules transported per carbonate ion in Versogen, for example). Although this result is somewhat surprising, it is known that carbonate ions impact surrounding water molecules beyond the primary solvation shell. ⁶⁷ In addition, divalent ions (e.g., Ni²⁺, Cu²⁺) have similarly high electro-osmotic coefficients in Nafion, and thus transport similar large quantities of water. ^69,70 Our results for the divalent carbonate ion in AEMs are roughly in line with those for divalent ions in Nafion. ^69,70

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To verify that the origin of differences in water transport number between Nafion and the AEMs is primarily related to solvation shells, and not membrane microstructure, the water transport number of proton-form Nafion is compared in Fig. 6a with sodium-form Nafion and proton-form sulfonated polystyrene (PSS) (a non-phase-separated PEM) as a function of RH at 50 °C. ²⁰ Proton PSS exhibits a water transport number equal to or slightly less than that of proton Nafion 211 at all RHs; transport numbers for PSS at high humidity are not shown due to membrane dissolution under those conditions. In contrast, sodium-form Nafion exhibits a water transport number of up to 8 at high humidity, greatly exceeding that of proton-form Nafion. Sodium ions display a solvation shell of 6 waters, again larger than that of a proton. ⁷¹ The only ion/membrane combination that transports less water than its ion solvation shell at high water content is protons in PEMs. This is likely due to Grotthüss hopping that protons undergo in a hydrogen-bonded water network at high membrane water contents. ⁷² Any proton that moves via hopping does not transport water with it, reducing the measured transport number compared to that of an ion that only moves vehicularly. Taken together, ion solvation shells are the primary driver of differences in water transport numbers between membranes (see Fig. 6b) as the impact of membrane microstructure appears smaller, with the lack of phase separation appearing to dampen slightly the transport number, in agreement with prior studies examining the water transport numbers in hydrocarbon-based membranes that similarly lack phase separation. ^73,74

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Most electrochemical devices do not run solely at 50 °C. Accordingly, it is of interest to determine how the water transport number varies with temperature. Figures 7a and 7b report the influence of temperature on water transport numbers for both the carbonate and hydroxide forms of Versogen as a function of RH. There appears to be a weak but inconsistent increase in water transport number with temperature, suggesting that the impact of temperature is limited. This assertion is supported by examining the activation energies of the transport number: 5 and –166 J/mol for carbonate form at low and high water contents, respectively, and 153 and 705 J/mol for hydroxide form at low and high water content, respectively. Three of the four activation energies indicate weak increases with temperature, whereas the activation energy of electro-osmosis in high water content carbonate form actually demonstrates a weak decrease with temperature; the large amount of scatter in the data prevents drawing definitive physical conclusions.

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To correlate different transport properties, especially for cases where there is a lack of experimental data, a model is developed for the water transport number. The Stefan-Maxwell-Onsager framework of frictional interactions between chemical species in a membrane is pertinent for electro-osmotic phenomena. ^75,76 A detailed description of the model can be found in section 1.4 of SI, where we expand upon the specific adaptation of Crothers et al. ^76,77 In the Stefan-Maxwell framework of multicomponent diffusion, the water transport number is defined as ^38,77

$$\begin{eqnarray}&&{{\rm{{\rm T}}}}_{i}=\frac{{L}_{i0}}{{{\rm{|}}z}_{i}{\rm{|}}{L}_{{ii}}}\end{eqnarray} \tag{ 17 }$$

where L_ij is the binary interaction parameter between species i and species j. As outlined in section 1.4 of SI, we estimate ${L}_{i0}$ and ${L}_{{ii}}$ by calculation of binary-species friction coefficients. ⁷⁷ In this approach, the ion-water friction coefficient is determined by scaling the microscale friction coefficient obtained from a Stokes-Einstein estimate of the ion-water binary diffusion coefficient to account for temperature and viscosity. ^76,77 The water-membrane and ion-membrane friction coefficients are determined from Darcy's law for porous-media flow following the work of Crothers et al. ^76,77 This procedure yields

$$\begin{eqnarray}&&\displaystyle {{\rm{{\rm T}}}}_{i}=\frac{{c}_{0}}{\left|{z}_{i}\right|{c}_{i}}\left(\frac{1}{1+\frac{\eta {w}_{i}{c}_{T}{D}_{i0}\phi }{{RT}{c}_{i}{c}_{0}\tau k}}\right)\,\end{eqnarray} \tag{ 18 }$$

for the transport number, where c_i is the interstitial concentration of species i, $\eta$ is the effective viscosity of water in the membrane, w_i is the mass fraction of species i, c_T is the total concentration, D_i0 is the binary diffusion coefficient between species i and water, ϕ is the water volume fraction, τ is the tortuosity, and k is the Darcy permeability. Equation 18 is used here to correlate experimental water transport number results with other useful membrane parameters. All experimental results used in this section correspond to 50 °C. In all cases, curves are calculated at a few discrete datum points.

Two unknown parameters assess the water transport number using the Stefan-Maxwell formulation in Eq. 18: the Darcy absolute permeability of water in the membrane (k) as a function of water content and the power-law parameter $\chi$ in Archie's law for membrane tortuosity (see Eq. S9, $\tau ={\phi }^{-\chi },$ where τ is the membrane tortuosity and $\phi$ is the water volume fraction). ⁷⁸ Because there is one permeability for each experimental water-content datum, plus an unknown tortuosity power-law index, additional information is needed to quantify the transport numbers. For Nafion membranes, literature water transport coefficients, representing water flow in response to a chemical potential gradient, are available and are converted to a Darcy permeability and inserted into the model. ^3,24,79 The $\chi$ parameter was varied over a physically acceptable range to arrive at a value of approximately 2. The solid line in Fig. 8a compares the water transport number calculated via this approach with the experimental data from Nafion 211. Over the range of water contents examined, the fit of the proton transport number is good, with a maximum deviation of ∼33%, although it is typically less (there is also a point with much higher deviation at low water content). Figure 8b shows the resulting tortuosity as a function of water content, which decreases with increasing hydration as expected, although it is somewhat high compared to models in previous literature. ^76,77,80 Nevertheless, the Archie law power index of 2 is well within the physical range determined for other porous media. ⁸¹ Further, the tortuosity of liquid-equilibrated Nafion (λ = 22) is approximately 5.5 when using $\chi =2,$ which is quite reasonable based on correlation of membrane transport properties to bulk values (e.g., comparing water diffusion in the membrane to bulk water diffusion gives a tortuosity around one order of magnitude), as well as more complex multi-scale transport models. ^3,77,81 The water transport coefficients and Darcy permeabilities used from literature are presented as a function of water volume fraction in Fig. S5.

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The acceptable fit of the water transport number and the magnitude of the tortuosity for Nafion PEM suggest that the proposed model may be accurately applied for Versogen AEMs. Literature water transport coefficients are not available for Versogen. We thus adopt a different approach. To determine Versogen tortuosity, a reasonable range of $\chi$ parameters is considered, with that used for Nafion ($\chi =2$) as the maximum and the value calculated from the tortuosity of a liquid-equilibrated hydrogel with similar water uptake as Versogen as the minimum ($\chi =1.3$). ⁸² The latter value is reasonable because Versogen has a domain-network architecture similar to a hydrogel, as opposed to a phase-separated Nafion, as demonstrated by Fig. S7 in SI. ^13,83,84 A total of 5 different $\chi$ values are chosen in the 1.3 to 2 range. The water transport number is then fit by nonlinear least-squares to extract the Darcy permeability as a function of water content.

Figures 9a–9c give the water transport number, Darcy permeability (k), and tortuosity for hydroxide-form Versogen, respectively, as a function of water content. Each curve represents one $\chi$ parameter. The fit in Fig. 9a is good and independent of the $\chi$-value chosen. Corresponding Darcy permeability and tortuosity increase and decrease, respectively, with water content in all cases. Figures 9b and 9c also reveal that permeability and tortuosity are inversely proportional: more tortuous transport pathways result in less water transport. The permeability of Versogen appears to be higher than that of Nafion, with the vapor-equilibrated Versogen permeability at high water content approximately equivalent to that of liquid-equilibrated Nafion. ⁸⁵ This finding is likely caused by the fact that Versogen has a higher water volume fraction (calculated assuming additive molar volumes) than does Nafion at the same water content due to the significantly lower equivalent weight and lower dry density of Versogen. ^3,13 A higher water volume fraction lowers tortuosity and increases permeability. It is also likely that the more hydrogel-like structure enables faster water movement than in the phase-separated Nafion where such transport occurs only within the hydrophilic nano-domains. The calculated water transport coefficients are also shown as a function of water volume fraction in Fig. S6 in SI.

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Actual Versogen tortuosity and Darcy permeability are expected to lie between the two extremes in Fig. 9, highlighting the importance of independent verification of either Darcy permeability or tortuosity. Nevertheless, the values of the permeability across the investigated range of tortuosities are within an order of magnitude of each other at high water content (the most likely condition for device operation), and thus useful as an initial estimate of the water transport parameters in the absence of data. The tortuosity can be used to study other, related physical properties of the membrane by applying other membrane models that rely on tortuosity, such as those used for hydrogels. ^86–88 The water permeability can be used to determine more accurate water balances in AEM fuel-cell and electrolyzer models, ^89,90 thereby improving predictions of device performance.