Theory of Multicomponent Phenomena in Cation-Exchange Membranes: Part I. Thermodynamic Model and Validation

We present and validate a mathematical model for multicomponent thermodynamic activity in phase-separated cation-exchange membranes (e.g., per ﬂ uorinated sulfonic-acid ionomers). The model consists of an expression for the free energy of the membrane and of the surrounding electrolyte solution. A modi ﬁ ed Stokes-Robinson ionic solvation framework treats the solution-like non-idealities resulting from hydration, electrostatics, ion association, and physical interactions in bulk solution and in ionomer hydrophilic domains. Inside the membrane, a mechanics-based composite approach accounts for the swelling of the hydrophobic matrix. Treating the membrane microstructure as a disordered system of domains calculates steric exclusion of ions. Electroneutrality guarantees that the charge of mobile ions in the membrane is equal to the charge on polymer groups. Osmotic coef ﬁ cients for electrolytes from literature parameterize solution-like interactions while mechanical and X-ray scattering characterization gives most membrane-speci ﬁ c parameters. Model predictions compare favorably to measured membrane thermodynamics (i.e., water and ion uptake) in dilute and concentrated binary and ternary salt electrolytes and in water vapor. Interactions between ions in the membrane are similar to those present in bulk electrolytes. Our results reveal that water and ion uptake is dictated by a balance between solution-like energetics and membrane swelling.

Dry membrane Aqueous cation-exchange polymers are an important class of membrane-electrolytes because the imbibed solvent imbues the material with high conductivity while maintaining mechanical stability. [1][2][3] Both of these attributes are essential for membrane electrolytes in numerous energy-storage and conversion devices. [1][2][3][4][5][6][7][8] However, the favorable ion-transport properties of these membranes pose a challenge when multiple ions are present. 4,8,9 Specifically, there is tradeoff between increasing the absorption and transport of certain species that are desirable, such as current-carrying ions, while limiting movement of contaminants, additives, or redox-active species that decrease device performance. 4,10 Electrolyte membranes and hydrogels in energy-storage and conversion devices typically contain fixed ionic groups that are charge balanced by mobile cations to realize high ionic conductivity. 2 The prototypical ion-conducting membranes are perfluorinated sulfonic-acid (PFSA) ionomers, which contain negatively charge sulfonate groups and protons or other cationic counter ions. 2 These membranes absorb water and ions from the surrounding solution. 2 Intense research efforts have yielded compelling approaches for modeling transport and thermodynamics in these membranes, 5,[11][12][13][14][15][16][17][18][19][20][21] yet these systems are still poorly understood due to their chemical complexity and corresponding nanostructural heterogeneity. 22 One source of complexity is the extensive thermodynamic coupling between species. 12,13,16,23 Specifically, the presence, concentration, and properties of one species affect the behavior of all other components. 23,24 The specific and electrostatic interactions of ions with the membrane, solvent, and other ions cause preferential partitioning of some species from the solution over others. [25][26][27][28][29] Furthermore, electroneutrality dictates that charge density everywhere is zero, creating a Donnan potential that induces uptake of ions of opposite charge to the fixed groups on the polymer. 3,[27][28][29][30] Further complicating matters, the presence and concentration of species change speciation via shifting acid and ion-pairing equilibria of active species. 7 Consequently, concentration and speciation of components inside the membrane differ drastically from that in the external solution. 23,25 Because ion and solvent uptake and species identity impact transport properties, permeability and conductivity depend on the concentration of species in the external solution. 5,24,31,32 Accordingly, experimental characterization of the mobility of species inside separators requires both transport and thermodynamic measurements. 32,33 Moreover, ion partitioning, water uptake, conductivity, and other transport properties in the membrane vary drastically when measured in different liquid electrolytes or after different membrane pretreatments. 9,34 Membrane properties that are measured in one electrolyte environment do not set the behavior in another. 34,35 However, mathematical models of molecular thermodynamics can address this challenge by predicting chemical activity across a range of conditions. 36 The influence of species partitioning on transport properties suggests that tuning thermodynamic properties may improve device performance. 4,10,17 Numerous membrane characteristics, including chemical structure, water content, and pretreatments, alter membrane performance. 9,37-39 Microscopic thermodynamic theories provide insights into how characteristics of the membrane influence its properties and inform strategies for improving device performance. This paper is the first of a three-part series that uses mathematical modeling to understand phase-separated cation exchange membranes that contain absorbed solvent and multiple ions. Here in Part I, we develop and validate a semi-empirical microscopic thermodynamic description of the system. Part II 40 develops and validates a microscale concentrated-solution description of multi-ion and solvent transport in these membranes. As a case study, Part III 41 simulates transport in an all-vanadium redox flow-battery separator and elucidates the underlying structure/property/function relationship for membranes in these systems and proposes improved design targets.
The outline of the papers is as follows. The theory section presents a molecular thermodynamic model that calculates ion and water uptake as a function of external solution concentration and separator structure. The model relies on semi-empirical parameters that are mostly available from measurements of bulk solution aqueous electrolytes. We discuss parameter choice and summarize numerical implementation of the model. In the results and discussion section, the model is validated against literature data. The theory reveals how the uptake of species is coupled and quantifies the relative importance of the involved molecular interactions.

Theory
Although numerous water-filled ion-conducting membrane architectures exist, 4,17 here we focus on the perfluorinated sulfonicacid (PFSA) chemistry because of its extensive characterization and widespread use. 2 PFSA ionomers nanophase separate into hydrophilic, water-filled, ion-conducting domains and hydrophobic, structural domains. 2 This section first defines the equilibrium conditions dictating ion and solvent uptake in the membrane. A semi-empirical microscopic model of the system provides a free-energy expression and accompanying electrochemical potentials for each species. We also address how to account for species association. where m d i is the (electro)chemical potential of species i in phase d ( a = or b); water is defined as species 0. The chemical potential of neutral species i is defined as the derivative of the free energy of d G with respect to n i , the moles of species i 43 , where T and p are the temperature and pressure, respectively. We use Eq. 2 to calculate the chemical potentials of charged species. The resulting chemical potentials always appear in neutral pairings to ensure that they are independent of the electrical state of the phase 43 (see Species Chemical Potential). Chemical potential, and subsequent predictions of species partitioning, require a well-defined freeenergy expression.
Free energy.-We define a hypothetical reference state (denoted with superscript q) as the pure liquid solvent, hypothetical ionic species that do not interact with each other (i.e. ideal solution at unit mole fraction) and fully dissociated, and the unswollen polymer membrane (M). The change in free energy from the reference, q G , consists of ideal mixing between ions and solvent, DG id , ion solvation, DG slv , electrostatic interactions between ions, DG els , short-range and non-electrostatic specific physical interaction between ions, DG phy , swelling of the polymer, DG swe , and steric interactions between the ions and polymer, DG stc . With the usual assumption of superposition of free energy processes, 36 id, slv, els, phy, swe, stc, For the external solution, the final two terms are excluded since the polymer is absent where the two superscripts denote contribution and phase. The first change from the reference state free energy comes from ideal mixing of the ions and solvent 36,43 [ ] where R is the gas constant and x i is the mole fraction of species i ( The tethered polymer charged groups are in the hydrophilic domain and partially mix with the absorbed electrolyte solution. 44 We idealize the polymer ionic groups with moles n M as free ions in the hydrophilic domains that contribute to the free energy accordingly. n M is equal to the mass of the polymer divided by its equivalent weight (EW, equal to g-polymer per mole of ionic group). 2 Unlike many models for polymer membranes (i.e. Flory-Huggins theory 45 ), the model neglects mixing between the uncharged volume fraction of the membrane f M and the solution in the hydrophilic domains because the strong segregation of phases means they are not freely interspersed. 2 The terms on the right of Eqs. 3 and 4 except DG id are the "excess" free energy, G ex . 36 For any species i, the excess chemical potential, , is normalized to approach zero (0) as the moles of all species j except the solvent (i.e. ¹ j 0) approaches zero (i.e.  n 0 j ) and the membrane is fully relaxed. The excess free energy in most bulk aqueous electrolyte solutions is known. 46 Literature provides numerous theories for these terms in a G 36 and a few expressions for membrane specific terms, D b G stc, and D b G els, . 2,28,30,47 Despite over 100 years of intense research effort, all accurate and tractable thermodynamic theories for concentrated electrolytes (>1 mol l −1 ) require semi-empirical adjustable parameters. 48 Here, we choose expressions for the excess free energy that are consistent with the physical description of the membrane, are relatively simple to implement numerically, require few membrane-specific adjustable parameters, and are predictive for bulk single and mixed-salt electrolyte solution thermodynamics or mechanical measurements of the membrane. Because the ion molality inside the membrane can exceed 20 mol kg −1 , 2 these expressions must also be valid at extremely high electrolyte concentrations. The parameters, while semi-empirical, are physically grounded. By modeling the thermodynamic nonidealities in the membrane hydrophilic domains with expressions for solution-like excess energies, we assume that molecular interactions in the pores are not drastically altered in the confined environment. This assumption is justified by the relatively short range of solvation, specific, and electrostatic interactions, at the high ionic strengths found in the hydrophilic domains of the membrane (i.e. the range of molecular interactions is less than the size of the hydrophilic domains). 49 Solvation energy accounts for the entropy decrease due to free solvent molecules binding to unsolvated ions and limiting the configurations that can be accessed 50,51 [ ] where a 0 and a i are the ratios of free solvent and unbound ions, respectively, after solvation to before solvation. a ¥ i is the fraction of unsolvated ions i at infinite dilution; the form of the last term in Eq. 6 ensures that excess chemical potential is normalized so that . Supplemental Material (SM) discusses expressions for these variables as derived by Stokes and Robinson, 51 Schonert, 52,53 and Zerres and Prausnitz 50 in terms of the solvent/ion i binding constant k i and the total number of solvent molecules in ion i's solvation shell, N i . Upon charging, the ions interact through long-range electrostatic forces. Debye-Hückel theory characterizes these interactions. 43,53,54 Accordingly, a solution with ionic strength, I ( where the sum is over all mobile ions and excludes the fixed-charged groups of the membrane), has an electrostatic free energy where A is the Debye-Huckel limiting slope (=1.177 m 3/2 mol −1/2 for water at 298 K), B is the Debye-Huckel solvent parameter (= 3.291 m 3/2 mol −1/2 nm −1 for water at 298 K), M 0 is the molecular weight of the solvent (=18 g mol −1 ), r 0 is the density of the solvent (=0.997 ), a is the distance of closest approach between ions (i.e. average diameter), b is the spacing between charged groups in the membrane, and z i is the charge number of species i. The first term accounts for electrostatic interactions between mobile ions and the second term accounts electrostatic interactions between mobile ions and the ions of the membrane. 43,54 The second term is zero in the external solution phase. This expression follows from a statistical-mechanical derivation given in Refs 54,55 that we modify by accounting for the radius of the ions. 43 Equations 6 and 7 are parameterized based only on the properties of the individual ions in the electrolyte (i.e. k i and a) and are independent of which pairs of cations and anions are present. At higher electrolyte concentrations, specific interactions between cation and anion pairs lead to additional short-range interactions (following 50 , we describe this as a physical contribution). Zerres and Prausnitz 50 combined Eqs. 6 and 7 with a semi-empirical term for the specific physical interactions outlined by Scatchard 56 where the interactions are proportional to the concentrations of the ions 43 [ ] is the specific interaction parameter between i and j and is 43 For favorable short-range interactions between species, b < 0 . Hydrophilic domains must swell to make space for the ions and solvent to enter the membrane. 45 Numerous research gives expressions for the excess Helmholtz free energy of an elastically swelling polymer membrane in terms of the integral of the microscopic swelling pressure, p swe , generally given in the form 45,[57][58][59] [ ] where we approximate the excess Helmholtz energy (on the right side) as equal to the excess Gibbs energy (on the left side) due system incompressibility. The free energy DG swe on the left side is normalized by the dry polymer volume,n V M M , whereV M is the partial molar volume of polymer per ionic group (= /r EW M ), f M is the volume fraction of the polymer membrane, and r M is the density of the dry membrane (= 2.1 g cm −3 ). Because there is much less volume of ions than membrane or solvent, we neglect the ion volume fraction for calculations of 0 0 ) and set the partial molar volume to the pure species molar volume. p swe arises from the microscopic deformation of the polymer membrane due to solvent absorption and is unrelated to the thermodynamic pressure p. Specific to phase-separated polymers, we use the expression from Kusoglu et al. 57 for the elastic swelling pressure generated due to deformation of the hydrophobic polymer matrix when solvent enters the hydrophilic domains Young's modulus of the dry polymer, R domain is the size of hydrophilic domains and d is the spacing between hydrophilic domains and superscript 0 denotes the dry membrane property. R domain 0 is the size of the domains in the dry state that includes the volume of charged polymer groups and counter ions. Supplemental Material (SM) discusses the specific expressions for the domain size and spacing dependence on membrane volume fraction, i.e.
The size of the hydrophilic domains also dictates the steric confinement of the ions inside them (e.g. excluded volume that is inaccessible to the ions). The free energy of these steric interactions for an ion i of diameter a i due to confinement induced entropy loss in the membrane is idealized as a system of randomly oriented walls with an average spacing of -  where the chemical potential of component i satisfies Eq. 2 and the superscripts correspond to the respective free-energy terms and phase. Table I gives expressions for each term. We neglect changes in partial molar volume, which are small. 60 Although not explicitly shown, the electrochemical potential of charged species depends on the chemical contributions outlined in Eq. 12 as well as on the electrical state of the phase. To incorporate this latter dependence while ensuring that the chemical contributions to electrochemical potential are thermodynamically accessible, 43 Smyrl and Newman use a reference ion to define a so-called quasielectrostatic potential, F (see, 43 pg. 95). 61 A useful reference for PFSAs with water is the proton because it exists in both the external solution and inside the membrane. Accordingly, F is defined as where F is Faraday's constant, z i is the charge number of i. and the subscript + H denotes protons. We define the electrochemical potential of other species such that the reference and excess chemical potentials (i.e. all the terms on the right sides of Eq. 12 except m i id ) are well-defined and independent of electrical state, At equilibrium, F is referenced to the electrostatic potential in the solution (i.e. F = a 0). We can now use Eq. 2 to calculate the excess chemical potential of charged species because the excess chemical potential terms in Eq. 14 appear only in neutral combinations of species guaranteeing that they are independent of electrical state. 43 Electroneutrality in each phase d is a constraint on (i.e. specifies) the quasi-electrostatic potential, Electroneutrality in cation-exchange membranes requires absorption of fewer anions than cations. Many researchers describe this phenomenon as the result of an electrostatic potential difference Table I. Summary of equations for the chemical potential terms.

Contribution
Term Expression Journal of The Electrochemical Society, 2020 167 013547 between the solution and the membrane (i.e. Donnan potential). 62 As an alternative to using a quasi-electrostatic potential to quantify the electric state of the material, one may instead define the potential as proportional to the electrochemical potential of one of the species (e.g. set it to the potential of a hydrogen reference electrode). 43 The choice between these two definitions does not affect the calculated ion and water uptake. 43 Substituting Eq. 14 for an electroneutral sum of two species (i.e. m m i z z j i j ) into Eq. 1 characterizes the tendency of i and j to sorb into the membrane, which the factor G ij quantifies 18-21 Speciation.-The microscale description of the system's free energy, Eqs. 5 to 11, are based on the physical properties of all distinct species present, which include solvent, free ions, ion pairs, or undissociated acids. Therefore, the model requires specification of component speciation (we will call this the "Molecular Construct")-for example, treating sulfuric acid as protons, sulfates, and bisulfates (see, 43 pg. 119). Conversely, experiments typically report concentrations of fully dissociated species (we will call this the "Experimental Construct")-for example, treating sulfuric acid as only protons and sulfate ions. Both treatments are thermodynamically consistent. 43  . 43 To calculate the contributions to the chemical potential throughout this paper (such as in Table I) Table I are a nonlinear, coupled algebraic system. At a given solution species composition in the Experimental Construct, we calculate the solvent and solute uptake in the membrane as follows: We determine the chemical potential of species in the external solution using Eq. 12 with expressions from Table I. Equations 13 and 14 specify the electrochemical potential of each species in solution. A modified Levenberg-Marquardt 63 algorithm finds f i j , in Eq. 17 for all species in the external solution that satisfies the root of the equilibrium speciation equations (Eq. 18 for each speciation reaction). To determine the excess chemical potential of solvation, a modified Powell hybrid algorithm 64 concurrently solves for a 0 that is the root to Equations in A1 in SM for the solvation expression. The initial guess is a negligible fraction of free solvent.
A modified Levenberg-Marquardt 63 algorithm finds the polymer volume fraction, f b M , solute molality, b ¹ m i M , and quasi-electrostatic potential, F b , inside the membrane that satisfies electroneutrality (Eq. 15) and equality of the electrochemical potential of unassociated species between phases (Eq. 1). Alternatively, Eqs. 15 and 16 also specify partitioning, but without requiring calculation of F. Because Eq. 18 makes electrochemical potential of associating ions and their products dependent, using Eq. 1 on ion-pairs is redundant. Equations 12 and 14 determine the electrochemical potential of species inside the membrane. All root-finding algorithms are implemented in SciPy 0.18.1 package Python 3.6 with a relative tolerance of 1.49 × 10 -8 . The NumPy packaged performed array operations.

Parameters
Operating parameters for this study are ambient temperature (298 K) and pressure (101 kPa). All the electrolytes in this study are aqueous (i.e. water is the solvent). We study the Nafion PFSA chemistry because data are widely available. 2,23,32,33,37,65,66 We use data from Nafion N117, N115, N212, and N211 where the N11x sequence is extruded and the series N21x is cast from a dispersion. Designation x denotes thickness in units of mils. All membranes have an equivalent chemical formula. 2 In addition to different processing methods, the membranes undergo various pretreatments (e.g. boiling) that can alter their properties, such as membrane modulus. 59,67 Variations in water and ion uptake between membranes may be predicted if mechanical properties of the membrane are available. Lacking this information and to ensure that the results are representative of Nafion across treatment history, we use datasets from multiple studies and for membranes from different series at the same environmental conditions. We also note the range of experimental values found in literature whenever possible. Table II lists the physical properties of the individual ions, the ion pairs, and the membrane that parameterize the model. These parameters have varying levels of empiricism. Certain properties are not fit (of the vapor-equilibrated membrane R , domain d, and K j eq ) because they are specified by independent measurements or theories that contain no adjustable properties. Based on molecular dynamics simulations and small-angle X-ray measurements, b is 0.47 nm. 2 M, M , the anion/cation interaction parameter). Because there are many parameters to fit uniquely, physical arguments and previous research suggest simplifications. For instance, the weak solvation of anions supports that their k i 's and N i 's are zero. 50 Further, we follow previous electrolyte models by setting a to a fixed value for all ions. 43 Ample research demonstrates that these simplifications, although somewhat ad hoc, provide accurate predictions of thermodynamic activity of binary and mixed electrolytes. 43,50,53 As SM details, the remaining k i 's and b ¹ ¹ i j M, M 's are fit to the measured osmotic coefficients of 28 salts consisting of pairings of 10 cations and 5 anions across a large range of concentrations (e.g. up to 15 mol kg −1 for LiBr). 46,71 The only parameters adjusted to fit membrane specific data are E b 0 , the modulus of the polymer backbone in liquid water, and b i,M , the cation/membrane interaction parameter. The modulus of the membrane in liquid water is challenging to measure due to a highly nonlinear shear response. 72 The backbone modulus it is estimated to be between 131 to 181 MPa. 72 By fitting water and ion uptake measurements in liquid water, we set E b 0 to136 MPa, which is within the measured range.. Table III gives values for b i,M that were also fit to water and ion membrane uptake measurements, which are presented in the Results and Discussion section. The physical interactions between H + , Li + , Na + , or K + and the membrane have a similar value as these cations have with trifluoromethanesulfonic anions, CF 3 SO 3 − (TFMS − ), which is a simple fluorosulfonate anion structurally similar to the sulfonate group in PFSAs. 71 We did not find thermodynamic activity of other TFMS salt solutions in the literature.
Vanadium ions are relevant in flow-battery applications, which is the focus of Part III, 41 but measurements for thermodynamic activity of vanadium electrolyte solutions are scarce. 73 Consequently, values of k i and b ¹ i j , M for vanadium ions in their various oxidation states are set equal to cations that have the same charge (see SM).

Results and Discussion
Ion-and water-uptake isotherms in a single electrolyte and water vapor.- Figure 1 shows measured (symbols) and calculated membrane water (a) and anion (c) uptake at as a function of external sulfuric acid concentration 37,66 (circles) for various Nafion types (N117, N212, and N211) and hydrobromic acid (diamonds 74 and triangles 75 ). For both water and ion uptake, agreement between the model and experiment is good up to ∼8 molal in the external electrolyte. Note the overlap of HBr and H 2 SO 4 data points at low concentrations. Despite the very different nature of the HBr (a 1-1 salt) and H 2 SO 4 (a 2-1 salt if fully dissociated), water and ion uptake are similar at the same electrolyte concentration This similarity is in part due to the high concentration of H + in the membrane that ensures that the sulfate in the membrane is almost entirely in the bisulfate (HSO 4 − ) form. Consequently, the HBr and H 2 SO 4 both behave as univalent acids.
As the external acid concentration increases, the membrane water content decreases because the difference between solution osmotic decreases. This leads to water content decreasing as the balance between osmotic, which keeps water in the membrane, and swelling pressures (m 0 swe ), which pushes water out of the membrane, shifts in favor of the latter. Using the HBr solution as an example, Fig. 1b shows that the effect of thermodynamic non-idealities on solvent uptake are small but unfavorable. Solution-like interactions (G = G G G 0 sol 0 els 0 slv 0 phy ) of the highly concentrated solution in the membrane are favorable for water uptake, but . membrane swelling (G 0 swe ) is always unfavorable for water uptake. These effects partially balance over a range of concentrations.
Increasing the external acid concentration drives acid into the membrane as the external concentration is greater than that in the membrane. Figure 1d shows the nonidealities associated with hydrobromic acid uptake, G HBr , and the contributions from solution-like interactions, G = G G G HBr sol HBr els HBr slv HBr phy , and steric effects, G HBr stc . There is no swelling contribution to ion chemical potential. Acid partitioning into the membrane is greater than expected for an  Holding all else constant, ions move to a phase with more electrostatic interactions (i.e. to higher ionic strength). Since the membrane has a higher ionic strength than the external solution, the favorable electrostatic interactions ( G ln HBr els = 3.0) compensate for the unfavorable effects of solvation ( G ln HBr slv = −0.5) and specific interactions ( G ln HBr phy = −1.6) in the concentrated electrolyte solution of the membrane.
At higher external acid concentrations, electrostatic interactions in the membrane are similar to those in the external solution, and the energetic benefit of acid partitioning into the membrane is smaller. Moreover, as water dehydrates the membrane, the pores shrink, as Fig. S1 shows (see SM) (available online at stacks.iop.org/JES/167/ 013547/mmedia), and the ions are more sterically hindered. Thus, at high external electrolyte concentrations, anions are excluded more from the membrane than predicted for an ideal solution. However, in the dehydrated membrane, the average hydrophilic domain remains larger than the diameter of the ions and steric hindrance does not fully exclude co-ions. Figure 1 shows that electrolytes with the same cation but different anions partition into PFSA membranes similarly. To explore how the cation impacts electrolyte partitioning, Fig. 2 shows measured (symbols) and calculated (solid lines) membrane water (a) and anion (b) uptake as a function of external concentration of NaCl 76 (squares) and HBr (diamonds 74 and triangles 75 ). The scales in Figs. 1 and 2 are not the same. At low salt concentrations, fractionally changing electrolyte concentrations does not change water content in the membrane. For concentrations above ∼1 molal, the water content decreases with increasing electrolyte concentration in both NaCl and HBr solutions as the membrane absorbs ions and dehydrates due to the high osmotic pressure of the surrounding solution.
At low electrolyte concentrations, membranes in NaCl solutions have a lower water content and higher anion uptake than in HBr at the same electrolyte concentration. In the limit of a membrane in pure water, the H-form of the membrane has higher water uptake than the Na-form in pure water; Fig. 1 shows with the shaded purple and yellow regions, respectively, that provide the range of literaturereported water uptake in these membranes. 2 Table III shows). Consequently, water uptake and dilution of the H + −SO 3 − pairs is more favorable than dilution of Na + −SO 3 − pairs. Similarly, uptake of Na + is slightly more favorable than of H + .
At high concentrations, this effect is eliminated with equal or higher water uptake in NaCl than in HBr solutions. Compared to Na + , the unfavorable interactions between H + and its co-ion   Figure 3 shows that the model effectively calculates (lines) measured (symbols) water uptake of N212 membranes in proton-(diamonds) and sodium-form (squares) in water vapor. Membrane water uptake at 100% relative humidity and salt-free liquid water is different despite a water activity of unity in both systems. This difference is termed Schroder's paradox and has been extensively studied. 2,[80][81][82] Here, we attribute the decreased water uptake in water vapor to structural rearrangement of the hydrophilic domains upon exposure to water vapor leading to different backbone polymer moduli E b 0 , (see SM for details). 72 No additional fitting parameters are used to calculate membrane water uptake in water vapor; agreement between theory and experiment is good. The model's successful thermodynamic predictions of membranes vapor and liquid environments further supports the proposed form of the Gibbs energy in Eqs. 3 and 4.  76 (squares) and HBr measured by Bai 74 (diamonds) and Yeo 75 (triangles). Shaded region denotes the range of values from literature of N117 solvent uptake in liquid water in H-(purple) and Naform (yellow) membranes. 2 Figure 3. Measured (symbols) and calculated (solid lines) N212 membrane water uptake in water vapor as a function of relative humidity for membranes in proton-form (diamonds) and sodium-form (squares) measured by Shi et al. 83 Figure 4c shows that the trend of membrane ion-exchange preference is also exhibited in water uptake. The water content of membranes exchanged with cations follow the order Li + > H + » Na + > K + > Cs + . Using H + -K + exchange as an example, Fig. 4b plots G 0 and the contribution to water uptake from solvation G 0 slv , cation-sulfonate physical interactions G 0 phy , and swelling G 0 swe . The decrease in water content as the membrane exchanges from H-form to K-form is due to the more favorable interactions of K + with the sulfonate compared to H + . The more favorable interactions of K + with the sulfonate groups means that there is a smaller driving force for water to dilute the K + -SO 3 + pairs. G 0 swe and G 0 slv change slightly with ion exchange from H-form to K-form. But these changes are not directly due to ion exchange, but indirectly related to higher cation concentration and decreased swelling of the membrane because of lower water content in K-form membranes. Based on this analysis, stronger favorable interactions between cations and sulfonate groups reduce water uptake. Figure 5 shows calculated (lines) and measured 65 (symbols) proton fraction in the membrane, As the proton content of the external solution decreases, the proton content in the membrane sharply decreases. This is consistent with ideal-solution Donnan theory that shows that multivalent ions i preferentially partition into the membrane over univalent ions j according to Eq. 16 with G = 1 ji . 19 Figure 5a shows that despite very different bulk solution properties, multivalent ions display nearly identical thermodynamic behavior in the membrane.
The model slightly over predicts exchange into the membrane of divalent ions and has larger errors for trivalent ions. If experimental artifacts from incomplete exchange are negligible, then these results demonstrate that upon accounting for solution-like interactions, the membrane does not uptake trivalent ions as much as expected. In addition to ideal-solution Donnan theory, the principle cause of the preferential partitioning of multivalent cations into the membrane is the strong electrostatic interactions between the cations and multiple sulfonate groups. However, the extent of these electrostatic interactions in the membrane is overestimated by the model since sulfonate groups are treated as a line-charge rather than as imbeded at the interface of a phase-seperated structure that cannot freely interact with the trivalent iron cation. This omission may explain the relatively high values of bi,M for multivalent ion i given in Table III. Figure 5b shows that the model very accurately predicts water uptake in these membranes as a function of fractional proton exchange / Consequently, this approach of incorporating solution-like interactions of multivalent electrolyte solutions in the membrane is sufficient to calculate water uptake into the membranes at dilute conditions. As with monovalent cations, the bi,M and k i values determine the water uptake in the membranes.
Ion-and water-uptake isotherms in mixed electrolytes.-We now consider ion uptake from ternary electrolytes that are relevant in vanadium redox-flow-batteries. Figure 6 shows vanadium ion partitioning in a mixed sulfuric-acid electrolyte by plotting measured 23 (symbols) and predicted (lines) uptake of vanadium IV ( + 5 mol L −1 , which we convert to molality using a density of sulfuric acid solution of 1.3 g cm −3 given by 34 . We fit V V ,M to reduce error between the measured and calculated vanadium ion uptake in Fig. 6a (see Table III). As the vanadium concentration in the external solution increases, the membrane absorbs more vanadium. V(V) absorbs less into the membrane because of its lower charge. The water content of the membrane remains constant as the vanadium concentration increases whereas the total sulfate concentration decreases slightly. The model simultaneously calculates these three properties accurately.
In concentrated electrolytes, ion association is an important phenomenon. First, protons and sulfate associate to form bisulfate, as previously discussed. The dotted line in Fig. 6a shows that

Conclusions
This work develops a mathematical model for multicomponent thermodynamic activity of both water and ions in phase-separated cation-exchange membranes. Microscale theory predicts how the free energy of the system changes with electrolyte concentration and membrane water content. The free energy accounts for the thermodynamic behavior of water and ions as they exhibit in bulk electrolyte solutions. We include swelling and steric interactions with the polymer into these expressions to account for the membrane.
The proposed model shows that a balance between solution-like interactions and polymer swelling dictates water uptake of membranes in concentrated electrolytes. The molecular properties of the aqueous cation are particularly important to describe water and ion uptake. More favorable interactions between cations and polymer sulfonate group increase their uptake and reduce water uptake.
The molecular-thermodynamic attributes of the ions in bulk electrolyte solution describe their thermodynamic properties in membranes. The model relies on one membrane-specific adjustable parameter and one membrane-cation specific adjustable parameter. The model quantitatively agrees with experiments over a wide range of electrolyte concentrations and compositions.
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