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

Chemical equilibrium dictates ion and solvent partitioning between the external solution phase $\alpha$ and the membrane phase $\beta$ follows⁴²

$$\begin{eqnarray}&&{\mu }_{i}^{\alpha }={\mu }_{i}^{\beta }\end{eqnarray} \tag{ 1 }$$

where ${\mu }_{i}^{\delta }$ is the (electro)chemical potential of species $i$ in phase $\delta$ ($=\alpha$ or $\beta$); water is defined as species $0$. The chemical potential of neutral species $i$ is defined as the derivative of the free energy of ${G}^{\delta }\,\,$with respect to ${n}_{i}$, the moles of species $i$⁴³,

$$\begin{eqnarray}&&{\mu }_{i}^{\delta }={\left(\displaystyle \frac{\partial {G}^{\delta }}{\partial {n}_{i}^{\delta }}\right)}_{T,p,{n}_{j\ne i}}\end{eqnarray} \tag{ 2 }$$

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 phase⁴³ (see Species Chemical Potential). Chemical potential, and subsequent predictions of species partitioning, require a well-defined free-energy expression.

We define a hypothetical reference state (denoted with superscript $\theta$) 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 (${\rm{M}}$). The change in free energy from the reference, ${G}^{{\theta }}$, consists of ideal mixing between ions and solvent, ${\rm{\Delta }}{G}^{\mathrm{id}}$, ion solvation, ${\rm{\Delta }}{G}^{\mathrm{slv}}$, electrostatic interactions between ions, ${\rm{\Delta }}{G}^{\mathrm{els}}$, short-range and non-electrostatic specific physical interaction between ions, ${\rm{\Delta }}{G}^{\mathrm{phy}}$, swelling of the polymer, ${\rm{\Delta }}{G}^{\mathrm{swe}}$, and steric interactions between the ions and polymer, ${\rm{\Delta }}{G}^{\mathrm{stc}}$. With the usual assumption of superposition of free energy processes,³⁶ the free energy of the membrane phase is

$$\begin{eqnarray}&&\begin{array}{l}{G}^{\beta }-{G}^{\theta }={\rm{\Delta }}{G}^{{\rm{id}},\beta }+{\rm{\Delta }}{G}^{{\rm{slv}},\beta }\\ \,+\,{\rm{\Delta }}{G}^{{\rm{els}},\beta }+{\rm{\Delta }}{G}^{{\rm{phy}},\beta }+{\rm{\Delta }}{G}^{{\rm{swe}},\beta }+{\rm{\Delta }}{G}^{{\rm{stc}},\beta }.\end{array}\end{eqnarray} \tag{ 3 }$$

For the external solution, the final two terms are excluded since the polymer is absent

$$\begin{eqnarray}&&{G}^{\alpha }-{G}^{\theta }={\rm{\Delta }}{G}^{{\rm{id}},\alpha }+{\rm{\Delta }}{G}^{{\rm{slv}},\alpha }+{\rm{\Delta }}{G}^{{\rm{els}},\alpha }+{\rm{\Delta }}{G}^{{\rm{phy}},\alpha }\end{eqnarray} \tag{ 4 }$$

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

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}{G}^{{\rm{id}}}}{RT}=\displaystyle \displaystyle \sum _{i}{n}_{i}\,\mathrm{ln}\,{x}_{i}\end{eqnarray} \tag{ 5 }$$

where $R$ is the gas constant and ${x}_{i}$ is the mole fraction of species $i$ ($={n}_{i}/\displaystyle \sum _{i}{n}_{i}$). 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 ${n}_{{\rm{M}}}$ as free ions in the hydrophilic domains that contribute to the free energy accordingly. ${n}_{{\rm{M}}}$ 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 theory⁴⁵), the model neglects mixing between the uncharged volume fraction of the membrane ${\phi }_{{\rm{M}}}$ 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 ${\rm{\Delta }}{G}^{{\rm{id}}}$ are the "excess" free energy, ${G}^{\mathrm{ex}}$.³⁶ For any species $i,$ the excess chemical potential, ${\left(\partial {G}^{\mathrm{ex}}/\partial {n}_{i}\right)}_{{n}_{j\ne i},T,p}$, is normalized to approach zero ($\to 0$) as the moles of all species $j$ except the solvent (i.e. $j\ne 0$) approaches zero (i.e. ${n}_{j}\to 0$) 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 ${G}^{\alpha }$³⁶ and a few expressions for membrane specific terms, ${\rm{\Delta }}{G}^{{\rm{stc}},\beta }$ and ${\rm{\Delta }}{G}^{\mathrm{els},\beta }$.^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

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}{G}^{{\rm{slv}}}}{RT}={n}_{0}\,\mathrm{ln}\,{\alpha }_{0}+\displaystyle \displaystyle \sum _{i\ne 0}{n}_{i}\,\mathrm{ln}\,\displaystyle \frac{{\alpha }_{i}}{{\alpha }_{i}^{\infty }}\end{eqnarray} \tag{ 6 }$$

where ${\alpha }_{0}$ and ${\alpha }_{i}$ are the ratios of free solvent and unbound ions, respectively, after solvation to before solvation. ${\alpha }_{i}^{\infty }$ 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 ${\left(\partial {\rm{\Delta }}{G}^{{\rm{slv}}}/\partial {n}_{i}\right)}_{{n}_{j\ne i},T,p}\to 0$ as ${n}_{i\ne 0}/{n}_{0}\to 0$. Supplemental Material (SM) discusses expressions for these variables as derived by Stokes and Robinson,⁵¹ Schonert,^52,53 and Zerres and Prausnitz⁵⁰ in terms of the solvent/ion $i$ binding constant ${k}_{i}$ and the total number of solvent molecules in ion $i$'s solvation shell, ${{\mathscr{N}}}_{i}$.

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, $I$ ($=\frac{{\rho }_{0}}{2{n}_{0}{M}_{0}}\displaystyle {\sum }_{i\ne 0,{\rm{M}}}{z}_{i}^{2}{n}_{i}$), where the sum is over all mobile ions and excludes the fixed-charged groups of the membrane), has an electrostatic free energy

$$\begin{eqnarray}&&\begin{array}{l}\frac{{\rm{\Delta }}{G}^{\mathrm{els}}}{{RT}}\,=\\ \,-\frac{4}{3}A{I}^{3/2}\,{n}_{0}\,{M}_{0}\tau ({aB}{I}^{1/2})/{\rho }_{0}-{n}_{0}{M}_{0}{m}_{{\rm{M}}}{z}_{{\rm{M}}}^{2}\frac{2A}{{bB}}\mathrm{ln}({a}_{M}B{I}^{1/2})\end{array}\end{eqnarray} \tag{ 7 }$$

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), ${\rho }_{0}$ is the density of the solvent (=0.997 g cm⁻³), $\tau \,\,\,$is a function ($\tau \left(x\right)=3/{x}^{3}(\mathrm{ln}\left(1+x\right)-x+{x}^{2}/2)$), $a$ 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 (following⁵⁰, we describe this as a physical contribution). Zerres and Prausnitz⁵⁰ combined Eqs. 6 and 7 with a semi-empirical term for the specific physical interactions outlined by Scatchard⁵⁶ where the interactions are proportional to the concentrations of the ions⁴³

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}{G}^{{\rm{phy}}}}{RT}={n}_{0}{M}_{0}\displaystyle \displaystyle \sum _{i\ne 0}\displaystyle \displaystyle \sum _{j\ne 0}{\beta }_{i,j}{m}_{i}{m}_{j}\end{eqnarray} \tag{ 8 }$$

where ${\beta }_{i,j}$ is the specific interaction parameter between $i$ and $j$ and is symmetric (i.e. ${\beta }_{i,j}={\beta }_{j,i}$). Because of the rarity of interactions between like-charged ions, ${\beta }_{i,j}=0$ for ${z}_{i}{z}_{j}\gt 0$.⁴³ For favorable short-range interactions between species, ${\beta }_{i,j}\lt 0$. whereas unfavorable interactions give ${\beta }_{i,j}\gt 0$.

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, ${p}^{\mathrm{swe}}$, generally given in the form^45,57–59

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}{G}^{{\rm{swe}}}}{{n}_{{\rm{M}}}{\bar{V}}_{{\rm{M}}}}\approx \displaystyle {\int }_{1}^{{\phi }_{{\rm{M}}}}\displaystyle \frac{{p}^{{\rm{swe}}}}{{\phi }_{{\rm{M}}}^{2}}d{\phi }_{{\rm{M}}}\end{eqnarray} \tag{ 9 }$$

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 ${\rm{\Delta }}{G}^{{\rm{swe}}}$ on the left side is normalized by the dry polymer volume, ${n}_{{\rm{M}}}{\bar{V}}_{{\rm{M}}}$, where ${\bar{V}}_{{\rm{M}}}$ is the partial molar volume of polymer per ionic group (= $EW/{\rho }_{{\rm{M}}}$), ${\phi }_{{\rm{M}}}$ is the volume fraction of the polymer membrane, and ${\rho }_{{\rm{M}}}$ 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 ${\phi }_{{\rm{M}}}$ ($\approx {n}_{{\rm{M}}}{\bar{V}}_{{\rm{M}}}/\left({n}_{{\rm{M}}}{\bar{V}}_{{\rm{M}}}+{n}_{0}{\bar{V}}_{0}\right)$) and set the partial molar volume to the pure species molar volume. ${p}^{{\rm{swe}}}$ 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

$$\begin{eqnarray}&&{p}^{{\rm{swe}}}={E}_{{\rm{b}}}^{0}\left(1-\displaystyle \frac{\displaystyle \frac{d}{2}-{R}_{{\rm{domain}}}}{\displaystyle \frac{{d}^{0}}{2}-{R}_{{\rm{domain}}}^{0}}\right)\end{eqnarray} \tag{ 10 }$$

where ${E}_{{\rm{b}}}^{0}$ is Young's modulus of the dry polymer, ${R}_{{\rm{domain}}}$ is the size of hydrophilic domains and $d$ is the spacing between hydrophilic domains and superscript $0$ denotes the dry membrane property. ${R}_{{\rm{domain}}}^{0}$ 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. ${R}_{{\rm{domain}}}({\phi }_{{\rm{M}}})$ and $d\left({\phi }_{{\rm{M}}}\right)$.

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 $d-{d}^{0}$ ⁴⁷

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}{G}^{{\rm{stc}}}}{RT}=\displaystyle \displaystyle \sum _{i\ne 0,{\rm{M}}}{n}_{i}\displaystyle \frac{{a}_{i}}{(d-{d}^{0})}\end{eqnarray} \tag{ 11 }$$

where $d$ the same as in Eq. 10.

Equation 2 defines the chemical potentials of all species. Because the Gibbs energies given in Eqs. 3 and 4 are expressed as the sum of different contributions, chemical potential contributions similarly superimpose

$$\begin{eqnarray}&&\begin{array}{l}{\mu }_{i}^{\beta }-{\mu }_{i}^{\theta }={\mu }_{i}^{{\rm{id}},\beta }+{\mu }_{i}^{{\rm{slv}},\beta }+{\mu }_{i}^{{\rm{els}},\beta }+{\mu }_{i}^{{\rm{phy}},\beta }+{\mu }_{i}^{{\rm{swe}},\beta }+{\mu }_{i}^{{\rm{stc}},\beta }\\ {\mu }_{i}^{\alpha }-{\mu }_{i}^{\theta }={\mu }_{i}^{{\rm{id}},\alpha }+{\mu }_{i}^{{\rm{slv}},\alpha }+{\mu }_{i}^{{\rm{els}},\alpha }+{\mu }_{i}^{{\rm{phy}},\alpha }\end{array}\end{eqnarray} \tag{ 12 }$$

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

Table I.  Summary of equations for the chemical potential terms.

Contribution	Term	Expression
Ideal mixing	${\mu }_{i}^{{\rm{id}}}$	$RT\,\mathrm{ln}\,{x}_{i}$
Solvation	${\mu }_{i\ne 0}^{{\rm{slv}}}$	$RT\,\mathrm{ln}\left(\tfrac{{\alpha }_{i}}{{\alpha }_{i,\infty }}\right)$
	${\mu }_{0}^{{\rm{slv}}}$	$RT\,\mathrm{ln}\left({\alpha }_{0}\right)$
Charging	${\mu }_{i\ne 0}^{{\rm{els}}}$	$-ART{z}_{i}^{2}\left[\frac{{I}^{1/2}}{1+Ba{I}^{1/2}}+\frac{{m}_{{\rm{M}}}{z}_{{\rm{M}}}^{2}}{2bBI}\right]$
	${\mu }_{0}^{{\rm{els}}}$	$ART(\frac{2}{3}{I}^{\frac{3}{2}}\sigma \left({Ba}{I}^{\frac{1}{2}}\right)\frac{{M}_{0}}{{\rho }_{0}}\,+\,\frac{{M}_{0}{m}_{M}{{z}_{M}}^{2}}{{bB}})$
Physical	${\mu }_{i\ne 0}^{{\rm{phy}}}$	$2RT\displaystyle \sum _{i\ne 0}{\beta }_{i,j}{m}_{j}$
	${\mu }_{0}^{{\rm{phy}}}$	$-{M}_{0}RT\displaystyle \sum _{i\ne 0}\displaystyle \sum _{j\ne 0}{\beta }_{i,j}{m}_{i}{m}_{j}$
Swelling	${\mu }_{i\ne 0,{\rm{M}}}^{{\rm{swe}}}$	$0$
	${\mu }_{0}^{{\rm{swe}}}$	${\bar{V}}_{0}{E}_{{\rm{b}}}^{0}\left[1-\tfrac{\tfrac{d\left({\phi }_{{\rm{M}}}\right)}{2}-{R}_{{\rm{domain}}}\left({\phi }_{{\rm{M}}}\right)}{\tfrac{{d}^{0}}{2}-{R}_{{\rm{domain}}}^{0}}\right]$
Steric	${\mu }_{i\ne 0,{\rm{M}}}^{{\rm{stc}}}$	$\tfrac{RT{a}_{i}}{(d-{d}_{0})}$
	${\mu }_{i=0}^{{\rm{stc}}}$	$0$
where
$\begin{array}{c}{\tau }\text{'}\left(\,x\right)={\textstyle \tfrac{d{\tau }}{{dx}}}={\textstyle \tfrac{3}{{x}^{4}}}\left\{-3\,{\rm{ln}}\left(1+x\right)+2x-{\textstyle \tfrac{{x}^{2}}{2}}+{\textstyle \tfrac{x}{1+x}}\right\}\\ {\sigma }\left(x\right)={\textstyle \tfrac{3}{{x}^{3}}}\left\{\left(1+x\right)-2\,{\rm{ln}}\left(1+x\right)-{\textstyle \tfrac{1}{1+x}}\right\}\end{array}$

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 quasi-electrostatic potential, ${\rm{\Phi }}$ (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, ${\rm{\Phi }}$ is defined as

$$\begin{eqnarray}&&{z}_{{{\rm{H}}}^{+}}F{\rm{\Phi }}\equiv {\mu }_{{{\rm{H}}}^{+}}-{\mu }_{{{\rm{H}}}^{+}}^{{\rm{id}}}\end{eqnarray} \tag{ 13 }$$

where $F$ is Faraday's constant, ${z}_{i}$ is the charge number of $i$. and the subscript ${{\rm{H}}}^{+}$ 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 ${\mu }_{i}^{{\rm{id}}}$) are well-defined and independent of electrical state,

$$\begin{eqnarray}&&{\mu }_{i}={\mu }_{i}^{\theta }-\displaystyle \frac{{z}_{i}}{{z}_{{{\rm{H}}}^{+}}}{\mu }_{{{\rm{H}}}^{+}}^{\theta }+{\mu }_{i}^{{\rm{id}}}+{\mu }_{i}^{{\rm{ex}}}-\displaystyle \frac{{z}_{i}}{{z}_{{{\rm{H}}}^{+}}}{\mu }_{{{\rm{H}}}^{+}}^{{\rm{ex}}}+{z}_{i}F{\rm{\Phi }}.\end{eqnarray} \tag{ 14 }$$

At equilibrium, ${\rm{\Phi }}$ is referenced to the electrostatic potential in the solution (i.e. ${{\rm{\Phi }}}^{\alpha }=0$). 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 $\delta$ is a constraint on (i.e. specifies) the quasi-electrostatic potential,

$$\begin{eqnarray}&&\displaystyle \displaystyle \sum _{i}{m}_{i}^{\delta }{z}_{i}=0.\end{eqnarray} \tag{ 15 }$$

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 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. ${\mu }_{i}-\tfrac{{z}_{i}}{{z}_{j}}{\mu }_{j}$) into Eq. 1 characterizes the tendency of $i$ and $j$ to sorb into the membrane, which the factor ${{\rm{\Gamma }}}_{ij}$ quantifies^18–21

$$\begin{eqnarray}&&\displaystyle \frac{{x}_{i}^{\beta }{\left({x}_{j}^{\beta }\right)}^{-\displaystyle \frac{{z}_{i}}{{z}_{j}}}}{{x}_{i}^{\alpha }{\left({x}_{j}^{\alpha }\right)}^{-\displaystyle \frac{{z}_{i}}{{z}_{j}}}}=\exp \left(\displaystyle \frac{\left({\mu }_{i}^{{\rm{ex}},{\rm{\alpha }}}-{\mu }_{i}^{{\rm{ex}},\beta }\right)-\displaystyle \frac{{z}_{i}}{{z}_{j}}\left({\mu }_{j}^{{\rm{ex}},\alpha }-{\mu }_{j}^{{\rm{ex}},\beta }\right)}{RT}\right)={{\rm{\Gamma }}}_{ij}\end{eqnarray} \tag{ 16 }$$

where ${{\rm{\Gamma }}}_{ij}$ is independent of the electrostatic potential for $i,j\ne {\rm{M}}$. For a neutral species $i$, ${z}_{i}$ is zero and we drop the second subscript $j$ for convenience. Because ${\mu }_{i}^{{\rm{ex}}}$ is the sum of different free-energy contributions, ${{\rm{\Gamma }}}_{ij}$ is the product of these contributions, ${{\rm{\Gamma }}}_{ij}\,={{\rm{\Gamma }}}_{ij}^{\mathrm{slv}}{{\rm{\Gamma }}}_{ij}^{\mathrm{els}}{{\rm{\Gamma }}}_{ij}^{\mathrm{phy}}{{\rm{\Gamma }}}_{ij}^{\mathrm{swe}}{{\rm{\Gamma }}}_{ij}^{\mathrm{stc}}$. If $i$ and $j$ are two oppositely charged ions (such as the anion and cation of a salt) and ${{\rm{\Gamma }}}_{ij}=1$, the oppositely charged pairs $i$ and $j$ partition into the membrane according to ideal Donnan equilibrium. ${{\rm{\Gamma }}}_{ij}\gt 1$ indicates that $i$ and $j$ favorably partition into the membrane while if partitioning is unfavorable, ${{\rm{\Gamma }}}_{ij}\lt 1$.

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 The Molecular Construct introduces an additional variable, ${f}_{i,j}$: the moles of species $i$ in the Experimental Construct, ${n}_{i}^{\exp }$, that partially associates into ${n}_{j}^{{\rm{mol}}}$ moles of species $j$ in the Model Construct

$$\begin{eqnarray}&&{f}_{i,j}=-\displaystyle \frac{{n}_{j}^{{\rm{mol}}}{s}_{i}}{{n}_{i}^{\exp }{s}_{j}}\end{eqnarray} \tag{ 17 }$$

where superscript $\exp$ and ${\rm{mol}}$ denote the construct and ${s}_{i}$ and ${s}_{j}$ are the stoichiometric coefficients of the association reaction of species $i$ in species $j,$ respectively. Note that $\displaystyle \sum _{j}{f}_{i,j}=1$. Chemical equilibrium of these reactions specified ${f}_{i,j}$⁴²

$$\begin{eqnarray}&&{s}_{j}{\mu }_{j}=-\displaystyle \displaystyle \sum _{i}{s}_{i}{\mu }_{i}\end{eqnarray} \tag{ 18 }$$

where the sum is over species $i$ that associate to form $j$. The superscript is not applied to the electrochemical potential because it is independent of the chosen construct. For the case $j=i$, ${\mu }_{i}={\mu }_{j}$ and ${f}_{i,i}={n}_{i}^{\mathrm{mol}}/{n}_{i}^{\exp }$. The thermodynamic equilibrium constant for the association reaction to form $j$ is ${K}_{j}^{\mathrm{eq}}=\exp \left(-\tfrac{{\sum }_{i}{s}_{i}{\mu }_{i}^{\theta }+{s}_{j}{\mu }_{j}^{\theta }}{RT}\right)$. This speciation equilibrium constant is defined in terms of the species chemical potential at the hypothetical reference state of unit mole fraction and ideal solution (i.e. ${x}_{i}^{\theta }=1$ and ${\mu }_{i}^{{\rm{ex}}}=0$) and is dimensionless; the more-commonly reported equilibrium constant, ${K}_{j}^{\mathrm{eq}^{\prime} }$, is defined in terms of species chemical potential at the reference state of an ideal solution at unit molarity (i.e. ${c}_{i}^{\theta }=1$ and ${\mu }_{i}^{{\rm{ex}}}=0$). These two constants are related by ${K}_{j}^{{\rm{eq}}^{\prime} }\approx {K}_{j}^{{\rm{eq}}}{\left({c}_{0}^{0}/{c}^{\theta }\right)}^{{s}_{j}+\displaystyle \sum _{i}{s}_{i}}$ where ${c}_{0}^{0}$ is the solvent concentration for an electrolyte with an infinitely dilute salt concentration.^36,43 At infinite salt dilution, ${f}_{i,j}\to 0$ for $i\ne j$ and ${f}_{i,i}\to 1$.⁴³

To calculate the contributions to the chemical potential throughout this paper (such as in Table I), we use composition variables in the Molecular Construct, ${n}_{i}^{\mathrm{mol}}$, ${x}_{i}^{\mathrm{mol}}$, and ${m}_{i}^{\mathrm{mol}}$; the superscript ${\rm{mol}}$ is not explicitly written throughout for simplicity. If one chooses to write the ideal chemical potential in the Experimental Construct (i.e. ${\mu }_{i}^{\mathrm{id}}=RT\,\mathrm{ln}\,{x}_{i}^{\exp }$, as we do in Part III of this series⁴¹) rather than the Molecular Construct (i.e. ${\mu }_{i}^{{\rm{id}}}=RT\,\mathrm{ln}\,{x}_{i}^{{\rm{mol}}}$), the term $RT\,\mathrm{ln}\,{x}_{i}^{{\rm{mol}}}/{x}_{i}^{\exp }$ is added to ${\mu }_{i}^{\mathrm{ex}}$. The calculated chemical potential ${\mu }_{i}$ is independent of the chosen construct. When presenting results, we convert concentrations to the Experimental Construct for consistency with measurements unless otherwise stated.

Equations 1 and 18 with those in 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-Marquardt⁶³ 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 algorithm⁶⁴ concurrently solves for ${\alpha }_{0}$ 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-Marquardt⁶³ algorithm finds the polymer volume fraction, ${\phi }_{{\rm{M}}}^{\beta }$, solute molality, ${m}_{i\ne {\rm{M}}}^{\beta }$, and quasi-electrostatic potential, ${{\rm{\Phi }}}^{\beta }$, 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 ${\rm{\Phi }}$. 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.

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 (diamonds⁷⁴ and triangles⁷⁵). 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 H₂SO₄ data points at low concentrations. Despite the very different nature of the HBr (a 1-1 salt) and H₂SO₄ (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 (HSO₄⁻) form. Consequently, the HBr and H₂SO₄ both behave as univalent acids.

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As the external acid concentration increases, the membrane water content decreases because the difference between solution osmotic pressure ($-({\mu }_{0}^{{\rm{id}},\alpha }+{\mu }_{0}^{{\rm{els}},\alpha }+{\mu }_{0}^{{\rm{slv}},\alpha }+{\mu }_{0}^{{\rm{phy}},\alpha })/{\bar{V}}_{0}$) and the membrane osmotic pressure $\left(-\left({{\mu }}_{0}^{{\rm{id}},{\beta }}+{{\mu }}_{0}^{{\rm{els}},{\beta }}+{{\mu }}_{0}^{{\rm{slv}},{\beta }}+{{\mu }}_{0}^{{\rm{phy}},{\beta }}\right)/{\bar{V}}_{0}\right)$ decreases. This leads to water content decreasing as the balance between osmotic, which keeps water in the membrane, and swelling pressures (${\mu }_{0}^{{\rm{swe}}}$), 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 (${{\rm{\Gamma }}}_{0}^{{\rm{sol}}}={{\rm{\Gamma }}}_{0}^{{\rm{els}}}{{\rm{\Gamma }}}_{0}^{{\rm{slv}}}{{\rm{\Gamma }}}_{0}^{{\rm{phy}}}$) of the highly concentrated solution in the membrane are favorable for water uptake, but . membrane swelling (${{\rm{\Gamma }}}_{0}^{{\rm{swe}}}$) 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, ${{\rm{\Gamma }}}_{\mathrm{HBr}}$, and the contributions from solution-like interactions, ${{\rm{\Gamma }}}_{\mathrm{HBr}}^{\mathrm{sol}}={{\rm{\Gamma }}}_{\mathrm{HBr}}^{\mathrm{els}}{{\rm{\Gamma }}}_{\mathrm{HBr}}^{\mathrm{slv}}{{\rm{\Gamma }}}_{\mathrm{HBr}}^{\mathrm{phy}}$, and steric effects, ${{\rm{\Gamma }}}_{\mathrm{HBr}}^{\mathrm{stc}}$. There is no swelling contribution to ion chemical potential. Acid partitioning into the membrane is greater than expected for an ideal solution at low external acid concentrations. The enhanced uptake is due to favorable electrostatic interactions (i.e. negative excess free energy) that increase with ion concentration (see Eq. 7). 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 ($\mathrm{ln}\,{{\rm{\Gamma }}}_{{\rm{HBr}}}^{{\rm{els}}}$ = 3.0) compensate for the unfavorable effects of solvation ($\mathrm{ln}\,{{\rm{\Gamma }}}_{{\rm{HBr}}}^{{\rm{slv}}}$ = −0.5) and specific interactions ($\mathrm{ln}\,{{\rm{\Gamma }}}_{{\rm{HBr}}}^{{\rm{phy}}}$ = −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 NaCl⁷⁶ (squares) and HBr (diamonds⁷⁴ and triangles⁷⁵). 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.

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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 literature-reported water uptake in these membranes.² Compared to Na⁺, H⁺ has stronger solvation (${k}_{{{\rm{H}}}^{+}}\gt {k}_{{{\rm{Na}}}^{+}},$ provided in SM) and unfavorable physical interaction with the sulfonate group on the membrane (${\beta }_{{\mathrm{Na}}^{+},{{\rm{M}}}^{-}}\lt {\beta }_{{{\rm{H}}}^{+},{{\rm{M}}}^{-}}$, as Table III shows). Consequently, water uptake and dilution of the H⁺−SO₃⁻ pairs is more favorable than dilution of Na⁺−SO₃⁻ 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 (${\beta }_{{\mathrm{Na}}^{+},{\mathrm{Cl}}^{-}}\lt {\beta }_{{{\rm{H}}}^{+},{\mathrm{Br}}^{-}}$, as SM shows) shifts the osmotic pressure balance in favor of the external electrolyte and drives water from the membrane into the solution to dilute H⁺-Br⁻ pairs.

The data shown in Figs. 1 and 2 are used to determine the parameters ${\beta }_{{{\rm{H}}}^{+},{{\rm{M}}}^{-}}$ and ${\beta }_{{\mathrm{Na}}^{+},{{\rm{M}}}^{-}}$, respectively, and ${E}_{{\rm{b}}}^{0}$. These values are the best-eye fit to the data. The fitted values of the specific interactions between cations and the membrane, ${\beta }_{i,{{\rm{M}}}^{-}}$, are very similar to the fitted interactions between alkali cations and CF₃SO₃⁻ in solution, ${\beta }_{i,{\mathrm{TFMS}}^{-}}$. The similarity of ${\beta }_{i,{{\rm{M}}}^{-}}$ and ${\beta }_{i,{{\rm{TFMS}}}^{-}}$ suggests that short-range specific interactions between cations and the membrane sulfonate groups are comparable to the interactions between cations and simple sulfonate anions in solution. Consequently, this approach explains the observed water and ions uptake behavior over a range of concentrations by accounting for solution-like interactions in the membrane. Confined pore geometry or long-range electrostatic ion condensation, as other researchers have proposed, can potentially predict ${{\beta }}_{i,{{\rm{M}}}^{-}}$ without fitting to experiments. ^{18,19,77–79}

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 Schröder's paradox and has been extensively studied.^2,80–82 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 ${E}_{{\rm{b}}}^{0}$, (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.

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We explore ion specificity of the model through partitioning of cations from dilute mixed electrolytes. Focusing on membranes in dilute electrolyte makes anion uptake negligible because of Donnan exclusion. Okada et al. measured water content and cation partitioning in varying compositions of mixed electrolytes of $i{{\rm{Cl}}}_{{{\rm{z}}}_{{\rm{A}}}}$ and HCl (or LiCl) where $i$ is Li⁺, Na⁺, K⁺, Cs⁺, Ni²⁺, Cu²⁺, and Fe³⁺.^32,33,65,84 The external solutions were at 0.06 mol kg⁻¹ total ion (chloride, proton, and cation $i$) concentration.⁶⁵ The fraction of protons (or lithium ions) in the solution is quantified as the fraction of chloride anions in the solution charge balanced by protons ${m}_{{{\rm{H}}}^{+}}^{\alpha }/{m}_{{{\rm{Cl}}}^{-}}^{\alpha }$ and in the membrane as the fraction of sulfonate groups balanced by protons ${m}_{{{\rm{H}}}^{+}}^{\beta }/{m}_{{{\rm{M}}}^{-}}^{\beta }$(where the subscript ${{\rm{H}}}^{+}$ is replaced by ${{\rm{Li}}}^{+}$ for membranes exchanged from lithium-form).

Figure 4 shows that as the measured³² (symbols) and predicted (lines) proton fraction in the membrane, ${m}_{{{\rm{H}}}^{+}}^{\beta }/{m}_{{{\rm{M}}}^{-}}^{\beta }$, as a function of the proton fraction of the surrounding mixed proton-alkali electrolyte, ${m}_{{{\rm{H}}}^{+}}^{\alpha }/{m}_{{{\rm{Cl}}}^{-}}^{\alpha }$ (a) and water content, $\lambda ,$ as a function of ${m}_{{{\rm{H}}}^{+}}^{\beta }/{m}_{{{\rm{M}}}^{-}}^{\beta }$ (c). The values of ${\beta }_{{\mathrm{Li}}^{+},{{\rm{M}}}^{-}}$, ${\beta }_{{{\rm{K}}}^{+},{{\rm{M}}}^{-}}$, ${\beta }_{{{\rm{Cs}}}^{+},{{\rm{M}}}^{-}}$ are best-eye fit to the water uptake and ion partitioning in Fig. 4; we use the values of ${\beta }_{{{\rm{H}}}^{+},{{\rm{M}}}^{-}}$ and ${\beta }_{{{\rm{Na}}}^{+},{{\rm{M}}}^{-}}$ obtained from ion and water uptake isotherms (Figs. 1 and 2) without additional fitting. Agreement is good particularly considering the experimental scatter reported in literature, which Fig. 4 notes for completely exchanged alkali-form membranes with a shaded region.² Figure S2 shows that the model is externally valid by predicting water and ion uptake of membranes in mixed lithium-alkali electrolytes at the same conditions (see SM).

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As the alkali cation fraction in solution increases, the fraction of alkali cations in the membrane increases. If alkali cation $i$ and protons equally partition into the membrane (${{\rm{\Gamma }}}_{{\rm{HCl}}}/{{\rm{\Gamma }}}_{i{\rm{Cl}}}=1$), then ${m}_{{{\rm{H}}}^{+}}^{\beta }/{m}_{{{\rm{M}}}^{-}}^{\beta }$ is a straight line between 0 and 1 for ${m}_{{{\rm{H}}}^{+}}^{\alpha }/{m}_{{{\rm{Cl}}}^{-}}^{\alpha }=0$ and $1$. Ideal partitioning nearly occurs for Na⁺-H⁺ exchange. For ${{\rm{\Gamma }}}_{\mathrm{HCl}}/{{\rm{\Gamma }}}_{i\mathrm{Cl}}\gt 1$, the alkali cations prefer to be in the solution more than do protons; ion exchange shows positive deviation from equal partitioning (i.e. curved upward in Fig. 4a, as is the case for Li⁺-H⁺ exchange) whereas for ${{\rm{\Gamma }}}_{\mathrm{HCl}}/{{\rm{\Gamma }}}_{i\mathrm{Cl}}\lt 1$, the alkali cations prefer to be in the membrane more than protons, and ion-exchange curve shows negative deviation (i.e. curved downwards in Fig. 4a, as is the case for K⁺-H⁺ and Cs⁺-H⁺ exchange). Figure 4a shows that ${{\rm{\Gamma }}}_{\mathrm{LiCl}}\lt {{\rm{\Gamma }}}_{\mathrm{HCl}}\approx {{\rm{\Gamma }}}_{\mathrm{NaCl}}\lt {{\rm{\Gamma }}}_{\mathrm{KCl}}\lt {{\rm{\Gamma }}}_{\mathrm{CsCl}}$. This trend follows a Hofmeister series.⁸⁵

To understand why univalent cations follow this trend, we study K⁺-uptake more closely as an example; Fig. 4b plots ${{\rm{\Gamma }}}_{{\rm{HCl}}}/{{\rm{\Gamma }}}_{{\rm{KCl}}}$ with the contributions of nonideality from solvation, ${{\rm{\Gamma }}}_{\mathrm{HCl}}^{\mathrm{slv}}/{{\rm{\Gamma }}}_{\mathrm{KCl}}^{\mathrm{slv}}$, and from physical interactions between the membrane and alkali cations and protons, ${{\rm{\Gamma }}}_{\mathrm{HCl}}^{\mathrm{phy}}/{{\rm{\Gamma }}}_{\mathrm{KCl}}^{\mathrm{phy}}$, as a function of ${m}_{{{\rm{H}}}^{+}}^{\alpha }/{m}_{{\mathrm{Cl}}^{-}}^{\alpha }$. ${{\rm{\Gamma }}}_{{\rm{HCl}}}/{{\rm{\Gamma }}}_{{\rm{KCl}}}$ is less than unity because K⁺ is less solvated than H⁺ and can therefore exist in the concentrated solution of the membrane without requiring a full solvation shell (i.e. ${{\rm{\Gamma }}}_{{\rm{HCl}}}^{{\rm{slv}}}/{{\rm{\Gamma }}}_{{\rm{KCl}}}^{{\rm{slv}}}\lt 1$). K⁺ has more favorable physical interactions with the sulfonate group (i.e. ${\beta }_{{{\rm{K}}}^{+},{{\rm{M}}}^{-}}\lt {\beta }_{{{\rm{H}}}^{+},{{\rm{M}}}^{-}}$ so that ${{\rm{\Gamma }}}_{{\rm{HCl}}}^{{\rm{phy}}}/{{\rm{\Gamma }}}_{{\rm{KCl}}}^{{\rm{phy}}}\lt 1\,$). Because we use the same cation radii for all univalent ions, steric and electrostatic interactions are essentially identical for H⁺ and K⁺ (i.e. ${{\rm{\Gamma }}}_{{\rm{HCl}}}^{{\rm{stc}}}/{{\rm{\Gamma }}}_{{\rm{KCl}}}^{{\rm{stc}}}\approx {{\rm{\Gamma }}}_{{\rm{HCl}}}^{{\rm{els}}}/{{\rm{\Gamma }}}_{{\rm{KCl}}}^{{\rm{els}}}\approx 1$). Consequently, specific, physical interactions between the cations and sulfate groups primarily dictate uptake preference of ion exchange.

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⁺ $\approx$ Na⁺ > K⁺ > Cs⁺. Using H⁺-K⁺ exchange as an example, Fig. 4b plots ${{\rm{\Gamma }}}_{0}$ and the contribution to water uptake from solvation ${{\rm{\Gamma }}}_{0}^{\mathrm{slv}}$, cation-sulfonate physical interactions ${{\rm{\Gamma }}}_{0}^{\mathrm{phy}}$, and swelling ${{\rm{\Gamma }}}_{0}^{\mathrm{swe}}$. 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⁺-SO₃⁺ pairs. ${{\rm{\Gamma }}}_{0}^{{\rm{swe}}}$ and ${{\rm{\Gamma }}}_{0}^{{\rm{slv}}}$ 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 measured⁶⁵ (symbols) proton fraction in the membrane, ${m}_{{{\rm{H}}}^{+}}^{\beta }/{m}_{{{\rm{M}}}^{-}}^{\beta }$, as a function of the proton fraction of the surrounding mixed electrolyte, ${m}_{{{\rm{H}}}^{+}}^{\alpha }/{m}_{{{\rm{Cl}}}^{-}}^{\alpha }$ (a) and water content, $\lambda ,$ as a function of ${m}_{{{\rm{H}}}^{+}}^{\beta }/{m}_{{{\rm{M}}}^{-}}^{\beta }$ (c), for mixed chloride electrolytes with protons and multivalent cation Ni²⁺, Ca²⁺, Cu²⁺, and Fe³⁺. We fit ${\beta }_{{\mathrm{Ni}}^{2+},{{\rm{M}}}^{-}}$, ${\beta }_{{\mathrm{Cu}}^{2+},{{\rm{M}}}^{-}}$, ${\beta }_{{\mathrm{Cu}}^{2+},{{\rm{M}}}^{-}}$, and ${\beta }_{{{\rm{Fe}}}^{3+},{{\rm{M}}}^{-}}$ to the measured water uptake and ion partitioning in Fig. 5 by eye.

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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 ${{\rm{\Gamma }}}_{ji}=1$.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 ${\beta }_{i,{{\rm{M}}}^{-}}$ 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 ${m}_{{{\rm{H}}}^{+}}^{{\beta }}/{m}_{{{\rm{M}}}^{-}}^{{\beta }}$. 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 ${\beta }_{i,{{\rm{M}}}^{-}}$ and ${k}_{i}$ values determine the water uptake in the membranes.

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 measured²³ (symbols) and predicted (lines) uptake of vanadium IV (${{\rm{VO}}}^{2+}$) and V (${{\rm{VO}}}_{2}^{+}$) (a) and total sulfate (dashed lines, right axis) and water uptake (solid lines, left axis) (b) as a function of total external vanadium concentration. The total sulfate molarity is 5 mol L⁻¹, which we convert to molality using a density of sulfuric acid solution of 1.3 g cm⁻³ given by³⁴. We fit ${\beta }_{{\rm{V}}{\left({\rm{IV}}\right)}^{2+},{{\rm{M}}}^{-}}$ and ${\beta }_{{\rm{V}}{\left({\rm{V}}\right)}^{+},{{\rm{M}}}^{-}}$ 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.

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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 ${{\rm{HSO}}}_{4}^{-}$ further associates with ${{\rm{VO}}}^{2+}$ leading to ${{\rm{HSO}}}_{4}^{-}-{{\rm{VO}}}^{2+}$ ion pairs.⁸⁶ Changing concentrations of various species shifts this equilibrium and couples the thermodynamic behaviors of the different species.