Volatile fatty acid selectivity during electrodialysis: effect of a competing counterion on volatile fatty acid transport mechanisms

Abstract

Volatile fatty acids (VFAs) are short-chain carboxylic acids with versatile applications that can be sustainably produced through anaerobic digestion. VFAs must be separated from anaerobic digestate for downstream use. Electrodialysis (ED) is an approach to selectively separate anionic VFAs from other ions, water, and neutral solutes. Towards enhancing selective VFA separation, VFA transport mechanisms during ED need to be understood. Accordingly, our objectives were to (i) determine the relative contributions of the partition and mobility selectivities to the permeability selectivity for VFA-containing counterion pairs, and (ii) determine the effect of a competing counterion on VFA transport mechanisms. To do this, we evaluated the permeability (P_i/P_j), partition (K_i/K_j), and mobility (u_i/u_j) selectivities of six VFA-containing counterion pairs, including VFA/VFA and VFA/Cl⁻ pairs, and a chloride (Cl⁻)/bromide (Br⁻) control under (i) mixed- and (ii) single-counterion conditions. Results showed the partition selectivity drove the permeability selectivity for Cl⁻/VFA and Cl⁻/Br⁻ pairs while the mobility selectivity drove the permeability selectivity for VFA/VFA pairs. Further, for the VFA/Cl⁻ and Cl⁻/Br⁻ pairs, we observed lower mixed-counterion partition selectivities and higher mixed-counterion mobility selectivities than corresponding single-counterion selectivities. Conversely, for the VFA/VFA pairs, we observed generally lower mixed-counterion partition and mobility selectivities than single-counterion selectivities. Overall, results indicate that VFA partitioning decreased by the presence of a counterion with greater affinity for the membrane while the effect of the competing counterion on VFA mobility depended on the competing counterion physicochemical properties. This work provides mechanistic insights into electro-driven VFA transport under conditions with competing counterions which is critical given the complexity of digestates. This study is the first to systematically evaluate VFA transport properties and selectivities in an AEM and provides a robust experimental framework for determining electro-driven counterion permeabilities under mixed-counterion conditions.

Graphical Abstract

1. Introduction

Volatile fatty acids (VFAs) are short-chain carboxylic acids (C2–C5) with diverse applications as both end products and chemical precursors in the pharmaceutical, bioplastics, food, and cosmetics industries.¹ For example, an emerging application of mixed VFAs is as precursor chemicals for production of polyhydroxyalkanoates which are high-value bioplastics used as a sustainable alternative to petroleum-based plastics.^2–4 Additionally, acetic acid (C2 VFA) is used as a preservative in the food industry and as precursor in the pharmaceutical industry.⁵ The versatility of VFAs has driven the market demand for VFAs to grow where the demand for acetic acid (C2 VFA) alone is projected to increase from 17.2 billion USD in 2024 to 29.1 billion USD by 2032.^1,6,7 Production of VFAs is dominated by conventional, petrochemical-based synthesis processes which have negative implications on the environment (e.g., excess greenhouse gas emissions and disposal of harsh chemicals).^8,9 Anaerobic digestion (AD) is a fermentation process that can produce VFAs from organic wastes making AD a sustainable alternative to conventional VFA synthesis processes.^8,10 However, to practically use AD for VFA production, VFAs need to be separated from the uncharged solutes and competing ions in the digestate as well as concentrated for downstream, commercial use.^1,10 Accordingly, separation processes are critical to enable downstream use of VFAs recovered from digestates.^2,11

Electrodialysis (ED) is a separation process that can separate VFAs from complex mixtures (e.g., digestate and other types of fermentation broths) due to the anionic charge of VFAs at circumneutral $\text{pH}\left(\text{p}{K}_a~4.8\right)$.^6,12,13 During ED, an electric potential drives the permeation of anions and cations from the feedwater in a diluate (D) compartment through anion and cation exchange membranes (AEMs/CEMs), respectively, to concentrate the ions in a concentrate (C) compartment.¹⁴ When VFAs are present in fermentation broths,¹⁵ inorganic ions (e.g., chloride, ortho-phosphate, ammonium), organic ions (e.g., lactic¹⁶ or succinic acid), and uncharged soluble products are also present.¹⁷ Therefore, understanding the relative transport (i.e., selectivity) between VFAs of differing carbon-chain lengths and between VFAs and inorganic anions can guide best practices (e.g., AEM selection, AEM design criteria, and ED operating conditions) for recovering, purifying, and concentrating VFAs.¹⁸

Previous work showed inorganic anions selectively permeate over VFAs in AEMs during ED¹⁹, and short carbon-chain VFAs selectively permeate over longer carbon-chain VFAs.^13,20,21 To quantify the relative permeation of VFAs in the presence of other anions, previous studies have evaluated the separation efficiency.^19,20 The separation efficiency is a relative measure of how effectively one counterion (i.e., ions with opposite charge to the IEM they permeate) is removed from the feedwater relative to another counterion; however, the separation efficiency does not provide any fundamental insight on counterion transport mechanisms or intrinsic IEM properties.²² To obtain fundamental insight on counterion transport mechanisms in an IEM, one approach is to determine the transport property selectivities (i.e., permeability selectivity (P_i/P_j), partition selectivity (K_i/K_j), and mobility selectivity (u_i/u_j)) between counterions since transport properties (and accordingly selectivities) control the relative counterion flux via electromigration as described by the Nernst-Planck equation (details in Section 2.1).^22,23 Determination of transport property selectivities has been done previously with inorganic ions^24–32, but remains underexplored for organic ions including VFAs.

Determination of transport property selectivities for VFA-containing counterion pairs reveals the relative contributions of the partition selectivity and the mobility selectivity to the permeability selectivity and reveals the transport property selectivity (partition or mobility) that governs the direction of the permeability selectivity.^25,33 Further, the effect of a competing counterion on VFA permeability selectivity can be elucidated by comparing the transport property selectivities determined under single-counterion conditions with those determined under mixed-counterion conditions. A previous study showed that transport property selectivities for inorganic counterions determined under single-counterion conditions could not be used directly to determine transport property selectivities under mixed-counterion conditions.²⁶ The changes in counterion transport in the IEMs was attributed to counterion-membrane and counterion-counterion interactions that were unique to mixed-counterion conditions.²⁶ While comparing single- with mixed- counterion selectivities provides the necessary information to assess whether the presence of an additional counterion $j$ affects the transport of counterion $i$ in the IEM, this comparison does not reveal whether any observed difference in selectivity stems from (i) an increase/decrease of counterion $i$ transport properties, (ii) an increase/decrease of counterion $j$ transport properties, or (iii) a combination of both. Therefore, to further gain mechanistic insights into the effect of a competing counterion on VFA transport, the counterion transport properties under mixed-counterion conditions need to be evaluated—in addition to their selectivities.

Accordingly, the objectives of this work were to (i) determine the relative contributions of the partition and mobility selectivities to the permeability selectivity during ED for VFA-containing counterion pairs; and to (ii) determine the effect of a competing counterion on VFA transport mechanisms. The counterion pairs studied included (i) Cl⁻ and Br⁻ as a control, (ii) two VFA pairs with different carbon-chain length differentials (i.e., C2/C5 and C3/C4), and (iii) four pairs composed of a VFA and Cl⁻ (i.e., C2/Cl⁻,C3/Cl⁻,C4/Cl⁻, and C5/Cl⁻). Towards achieving our objectives, we measured or calculated from experimental measurements the transport properties and selectivities (i.e., permeability, partitioning, and mobility) for each counterion under both single- and mixed-counterion conditions (details in Section 2 and Section 3). Results showed that for VFA_i/VFA_j counterion pairs, the mobility selectivity governed the direction of the permeability selectivity while for Cl⁻/Br⁻ and VFA/Cl⁻ pairs, the partition selectivity governed the direction of the permeability selectivity. Further, results indicated that for all VFAs, the presence of a competing counterion with a higher affinity for the membrane than the VFA decreased VFA partitioning. Conversely, the presence of a higher affinity competing counterion had variable effects on VFA mobility and depended on the physicochemical properties of the competing counterion. Overall, this work showed that the dominant transport property selectivity (partition or mobility) differed depending on the physicochemical properties of the counterions in each pair, and VFA transport properties were substantially affected by the presence of a competing counterion in the membrane. This study is the first to systematically evaluate VFA transport properties and selectivities in an AEM during ED and provides a robust experimental framework for determining electro-driven counterion permeabilities under mixed-counterion conditions.

2. Theory

2.1. Electric field-driven counterion transport and selectivity

Electric field-driven counterion transport as described by the concentration-normalized counterion flux $(J_{\text{i}}^{\text{m}}/c_{\text{i}}^{\text{s}},\text{m}.{\text{s}}^{-1})$ is governed by transport properties including the permeability $(P_{\text{i}},{\text{m}}^2.{\text{s}}^{-1})$ the partition coefficient $(K_{\text{i}},\text{M}.{\text{M}}^{-1})$, and the diffusion coefficient $(D_{\text{i}},{\text{m}}^2.{\text{s}}^{-1})$ as^23,34–36

$$\raisebox{1ex}{$J_i^m$}\!\left/ \!\raisebox{-1ex}{$c_i^s$}\right.=P_i\left[\frac{z_{i}F}{RT}\right]\frac{d\phi }{dx}$$ (1)

and

$$P_i=K_i{D}_i,$$ (2)

where $z_i$ is the charge of the counterion of interest, $F\left(\text{C}.{\text{mol}}^{-1}\right)$ is the Faraday’s constant, $R(\text{J}.{\text{mol}}^{-1}.{\text{K}}^{-1})$ is the ideal gas constant, $T\left(\text{K}\right)$ is the temperature, $d\phi \left(\text{V}\right)$ is the electric potential across IEM (i.e., transmembrane electric potential — TMEP), and $dx\left(\text{m}\right)$ is the membrane thickness.

The partition coefficient describes the propensity for an ion to partition from the bulk solution to the IEM and is defined by²³

$$K_i=\frac{c_i^m}{c_i^s},$$ (3)

where $c_{\text{i}}^{\text{m}}\left({\text{mol}.\text{L}}^{-1}\right)$ is the counterion concentration in the membrane, and $c_{\text{i}}^{\text{s}}\left({\text{mol}.\text{L}}^{-1}\right)$ is the counterion concentration in bulk solution. In this work, we define the partition coefficient as the total partition coefficient which accounts for counterions that neutralize the fixed charged groups in the membrane ($c_{fix},{\text{mol}.\text{L}}^{-1}$ sorbed water) and the mobile counterions ($c_{\text{i}}^{\text{m},\text{mobile}},{\text{mol}.\text{L}}^{-1}$ sorbed water). Based on equilibrium sorption experiments performed in our previous work, the mobile counterion concentration is negligible compared with $c_{fix}$ under the conditions used in this work $(c_{\text{i}}^{\text{m},\text{mobile}}<0.002\times c_{fix}$).³⁵

The diffusion coefficient is related to the ion mobility in the IEM via³⁷

$$D_i^m=\frac{RT}{z_{i}F}{u}_i^m,$$ (4)

where $u_{\text{i}}^{\text{m}}({\text{m}}^2.{\text{s}}^{-1}.{\text{V}}^{-1})$ is the mobility of counterion $i$. The ion mobility describes the rate of ion transport under an electric field, and therefore, the mobility is more appropriate to describe electric-field driven ion migration than the diffusion coefficient (although both are suitable given their linear relationship).²³ Accordingly, Eqs. 1 – 4 show that electric field-driven counterion transport through IEMs is governed by counterion partitioning, counterion mobility, and the electric potential.^23,25,35

By rearranging Eqs. 1 – 4, the electric field-driven permeability selectivity ($P_j^i$, unitless) between two counterions can be described by^22,25

$$P_j^i=\frac{z_i{P}_i}{z_j{P}_j}=\frac{z_i{K}_i{D}_i}{z_i{K}_i{D}_i}=\frac{K_i{u}_i}{K_j{u}_j}=\frac{\raisebox{1ex}{$J_i^m$}\!\left/ \!\raisebox{-1ex}{$c_i^s$}\right.}{\raisebox{1ex}{$J_j^m$}\!\left/ \!\raisebox{-1ex}{$c_j^s$}\right.},$$ (5)

where subscripts $i$ and $j$ denote counterions $i$ and $j$, respectively. Approaches to determine transport properties and corresponding transport property selectivities of counterions under single-counterion conditions were reported in previous work^25,28,35 are described in Section S1.

Although transport property selectivities can be determined from single-counterion conditions, competing counterions in mixed-counterion conditions are likely to affect each other’s transport properties in an IEM. Counterion-counterion and counterion-membrane interactions are especially prominent for cases of low swelling polymers (e.g., most commercial IEMs) and for cases where the physicochemical properties of the counterions vary substantially (e.g., monovalent vs. multivalent, inorganic vs. organic, point charge vs. resonance structure).^22,25,26,38 Therefore, it is critical to also determine the transport property selectivities under mixed-counterion conditions.^22,25,26

The mixed-counterion permeability selectivity can be determined from ED experiments by measuring over time the counterion concentrations in the D compartment to thereby determine the counterion concentration-normalized fluxes (Eq. 5).²⁵ The partition selectivity between two counterions in mixed-counterion conditions can be measured via equilibrium sorption-desorption experiments.²⁵ Assuming the mobile counterion concentration in the IEM is negligible compared with those that neutralize $c_{\text{fix}}$, the mixed-counterion partition selectivity can be calculated as^22,25

$$K_j^i=\frac{\raisebox{1ex}{$x_i^m$}\!\left/ \!\raisebox{-1ex}{$c_i^s$}\right.}{\raisebox{1ex}{$x_j^m$}\!\left/ \!\raisebox{-1ex}{$c_j^s$}\right.},$$ (6)

where $x_{\text{i}}^{\text{m}}\left(\text{mol}.{\text{mol}}^{-1}\right)$ and $x_{\text{j}}^{\text{m}}\left(\text{mol}.{\text{mol}}^{-1}\right)$ are the fraction of $c_{fix}$ neutralized by counterion $i$ and counterion $j$, respectively. Eq. 6 is a suitable definition of partitioning selectivity when the fixed groups in the membrane possess a monovalent charge (most commercial AEMs), the counterions possess a monovalent charge (this work), and assumes that all fixed groups are neutralized by counterions $i$ or $j$.²² The mobility selectivity between two counterions can be calculated from the mixed-counterion permeability and partition selectivities as $u_j^i=\frac{P_j^i}{K_j^i}$ (Eq. 5).

2.2. Approach to determine transport properties under mixed-counterion conditions

The transport properties (i.e., K, D, P) of counterion $i$ and counterion $j$ under mixed-counterion conditions can be determined from their concentration-normalized fluxes using the following approach. The permeabilities of counterion $i\left(P_{\text{i},\text{mix},}{\text{m}}^2.{\text{s}}^{-1}\right)$ and counterion $j\left(P_{\text{j},\text{mix}},\right.{\text{m}}^2.{\text{s}}^{-1})$ can be calculated from the measured concentration-normalized fluxes (see Eq. 1) as

$$P_{i,mix}=\raisebox{1ex}{$J_i^m$}\!\left/ \!\raisebox{-1ex}{$c_i^s$}\right.{\left[\frac{z_{i}F}{RT}\right]}^{-1}{\left[\frac{d\phi }{dx}\right]}^{-1}$$ (7)

and

$$P_{j,mix}=\raisebox{1ex}{$J_j^m$}\!\left/ \!\raisebox{-1ex}{$c_j^s$}\right.{\left[\frac{z_{j}F}{RT}\right]}^{-1}{\left[\frac{d\phi }{dx}\right]}^{-1},$$ (8)

respectively. Therefore, the only unknown variable to determine the permeabilities is the transmembrane electric potential (TMEP; $d\phi$, V). Although during ED an electric potential can be set to a constant value between reference electrodes closely surrounding the AEM (e.g., 0.4 V used in this work), this “applied” electric potential is only an approximation of the TMEP due to the potential drop contributions from the solutions and boundary layers present between the reference electrodes and the IEM.^39–41 Instead, the TMEP normalized to the membrane thickness can be calculated as^42,43

$$\frac{d\phi }{dx}=\frac{I{R}_m}{dx},$$ (9)

where $I\left(\text{A}\right)$ is the measured current in the ED-cell, $R_{\text{m}}\left(\Omega \right)$ is the membrane resistance, and $dx\left(\text{m}\right)$ is the membrane thickness. Accordingly, the TMEP can be used to calculate the counterion permeabilities via Eqs. 7 and 8.

From the measured fractions of $c_{\text{fix}}$ neutralized by counterions $i$ and $j\left({x_{\text{i}}}^{\text{m}}\text{and}\left.{x_{\text{j}}}^{\text{m}}\right)\right.$, the partition coefficients of counterion $i\left(K_{\text{i},\text{mix}},\text{M}.{\text{M}}^{-1}\right)$ and counterion $j\left(K_{\text{j},\text{mix}},\text{M}.{\text{M}}^{-1}\right)$ can be determined as^22,44

$$K_{i,\mathit{\text{mix}}}=\frac{x_i^m{c}_{fix}}{c_i^s}$$ (10)

and

$$K_{j,mix}=\frac{x_j^m{c}_{fix}}{c_j^s},$$ (11)

respectively. The mobilities of counterion $i\left(u_{\text{i},\text{mix}}\right)$ and counterion $j\left(u_{\text{j},\text{mix}}\right)$ can be calculated from $P_{\text{i},\text{mix}}$ and $K_{\text{i},\text{mix}}$ and from $P_{\text{j},\text{mix}}$ and $K_{\text{j},\text{mix}}$, respectively (Eqs. 2, 4).

3. Materials and methods

3.1. Overview and approach

All mixed-counterion experiments are original to this work, and all single-counterion experiments were originally performed for our previous work.³⁵ For all experiments, the total counterion concentration in the bulk solution (i.e., $i$ only, $j$ only, or $i+j$) was 0.1 M with Na⁺ as the corresponding coion, and the Neosepta AMX (AEM) and Neosepta CMX (CEM) membranes (Tokuyama Soda Corporation, Japan) were used. The AMX membrane was selected due to its moderate ion exchange capacity (1.84 meq.g⁻¹), low to moderate swelling degree under different solution conditions (0.14–0.22 g_H2O.g_polymer⁻¹), moderate thickness $\left(133\mu \text{m}\right)$, and mechanical strength due to the polyvinyl chloride reinforcement backing.^34,35 For all mixed-counterion conditions (i.e., 0.05 M/ 0.05 M counterion $i/\text{counterion}j$), ED experiments were performed to determine the permeability selectivities from the measured concentration-normalized fluxes (Eq. 5), AEM equilibrium sorption-desorption experiments were performed to determine the partition selectivities (Eq. 6), and the mobility selectivities were calculated from the permeability and partition selectivities (Eq. 5). For all single counterion-conditions, the single-counterion transport property selectivities were calculated from the single-counterion transport properties reported in our previous work.³⁵ Further, we calculated the TMEP from the measured ED-cell current and the measured membrane resistance under mixed-counterion conditions to convert the measured concentration-normalized fluxes to permeabilities ($P_{\text{i},\text{mix}}$ or $P_{\text{j},\text{mix}}$, Eqs. 7–8). The counterion partition coefficients under mixed-counterion conditions ($K_{\text{i},\text{mix}}$ or ${\text{K}}_{\text{j},\text{mix}}$) were calculated from the measured fractions of $c_{\text{fix}}$ neutralized by counterions $i$ and $j$ $(x_{\text{i}}^{\text{m}}$ or $x_{\text{j}}^{\text{m}}$; Eqs. 10–11). The corresponding mobilities ($u_{\text{i},\text{mix}}$ or $u_{\text{j},\text{mix}}$) were calculated from $P_{\text{mix}}$ and $K_{\text{mix}}$ (Eqs. 2 and 4).

The Results and Discussion (Section 4) is broken into three sections according to the three sets of counterion pairs including: Cl⁻/Br⁻ – control (Section 4.1), VFA_i/VFA_j pairs (Section 4.2), and VFA/Cl⁻ pairs (Section 4.3). Each set of counterion pairs are written in the notation of counterion $i/\text{counterion}$ $j$ where $i$ denotes the counterion with the lower affinity for the membrane. In each section, we first evaluate the permeability selectivity under mixed-counterion conditions and determine the relative contributions of the partition and mobility selectivities to permeability selectivity. Next, we compare transport property selectivities determined under single- and mixed-counterion conditions and use the transport properties derived from mixed-counterion conditions to elucidate the effect that each competing counterion has on counterion transport mechanisms.

3.2. Chemicals, solutions, and membranes

All solutions were prepared in laboratory grade water ($>17.8\text{m}\Omega .\text{cm}$, Dracor). The VFAs including sodium acetate trihydrate (NaAce: C2), sodium propionate (NaProp: C3), and sodium butyrate (NaBut: C4) were purchased in the sodium salt form and had a pH of ~7.2–7.6 when dissolved in laboratory grade water. Valeric acid was converted to its deprotonated form (valerate: C5) by adding dissolved sodium hydroxide (NaOH) to a valeric acid solution until the pH reached ~7.2. All chemicals in this work including the VFAs, sodium chloride (NaCl), sodium bromide (NaBr), NaOH, and sodium sulfate (Na₂SO₄) were purchased from Fisher Scientific as ACS grade or better, except NaBut which had >98% purity and was purchased from Alfa Aesar. Prior to any experiment, the AMX membrane was equilibrated in the desired solution for at least 24 hours with at least three solution changes. When not in use, CMX membranes were stored in 0.1 M Na₂SO₄ and AMX membranes were stored in 0.5 M NaCl.

3.3. Determination of the permeability selectivities and permeability coefficients

3.3.1. Electrodialysis

ED experiments were performed to evaluate the concentration-normalized flux of each counterion ($i$ and $j$) in the D compartment and thereby calculate the permeability selectivity between counterions (Eq. 5). ED experiments were performed using the same experimental conditions and setup used in our previous work except for the solution conditions in the C and D compartments.³⁵ The C and D compartments surrounding the AEM both contained 0.05 M of each counterion $i$ and counterion $j$ with Na⁺ as the corresponding cation. Fig. 1 shows the custom-made, four-compartment ED-cell with a four-electrode setup used in this work. All ED experiments were performed at a constant voltage of 0.4 V. Additional details regarding the ED-cell setup and operation are described in our previous work³⁵ and in Section S2. The counterion concentrations in the D compartment as a function of time during ED are in Fig. S2.

Figure 1.

ED-cell setup. For all ED experiments, a voltage (V_drop) of 0.4 V was held constant between the reference electrodes for the duration of the ED experiment. ERS stands for electrode rinse solution.

3.3.2. Counterion concentration-normalized flux

The time-dependent counterion flux during ED was determined by first fitting each counterion concentration in the D compartment to a best-fit polynomial function $\left(f\right(t),{\text{mol}.\text{m}}^{-3})$. Using the first-order derivative of the best-fit polynomial function, the volume of the D compartment, and the active membrane area, we calculated the time-dependent counterion flux. Details of the polynomial fitting selection, counterion flux calculation, and the fittings for all counterion pairs are in Section S2. Further, to evaluate whether the counterion flux during ED was predominantly electric field-driven (i.e., transport due to electromigration), we compared the time-dependent flux of counterion $i$ plus that of counterion $j$ determined from the polynomial fitting method to the time-dependent total counterion flux calculated from the measured current in the ED-cell. Calculations and corresponding figures are in Section S2. We generally observed a strong agreement between the flux determined from the polynomial fitting method and the flux calculated from the ED-cell current which confirmed that counterion flux was predominantly driven by electromigration (Fig. S3).

To determine the counterion concentration-normalized flux $(J_{\text{i}}^{\text{m}}/c_{\text{i}}^{\text{s}})$, the counterion flux at each experimental timepoint was divided by the measured counterion concentration in the D compartment at the corresponding timepoint. The concentration-normalized fluxes for each counterion pair are in Fig. S4. The permeability selectivities were calculated at 0, 15, 30, 45, and 60 minutes from the concentration-normalized fluxes (Eq. 5). Permeability selectivities were not calculated for timepoints greater than 60 minutes due to ion depletion in the D compartment (Fig. S2).

3.3.3. Selection of permeability selectivity and electric current for single- vs. mixed-counterion selectivity

Since the permeability selectivity varied slightly to substantially over the duration of ED experiments (results in Section 4), it was necessary to select a single permeability selectivity value to use for comparison between single- and mixed-counterion conditions. Therefore, we used the permeability selectivity determined at the initial ED timepoint (0 minutes) as it ensures consistency with single-counterion permeability experiments in which permeabilities were determined under constant bulk solution concentrations of 0.1 M.³⁵ We also compared the single-counterion permeability selectivities with the mixed-counterion permeability selectivities determined at the 15-minute ED timepoint (Section S3). We observed the same trends between the two sets of comparisons (Section S3 and Section 4). Therefore, the permeability selectivity derived from the initial ED timepoint serves as a representative value for evaluating differences in counterion transport behavior between conditions.

To calculate the mixed-counterion permeability, the TMEP was needed (Eqs. 7–8). The TMEP depends on the electric current, the membrane resistance, and membrane thickness (Eq. 9). For consistency with the permeability selectivity, we used the electric current at the initial ED timepoint $(I_0,A)$ and measured the membrane resistance in 0.05 M/0.05 M counterion i/counterion $j$ (i.e., initial ED conditions; details in Section 3.3.4). The electric current, membrane resistance, and TMEP for each counterion pair is in Section S4. We also calculated the mixed-counterion transport properties using the electric current and concentration-normalized fluxes at the 15-minute ED timepoint and generally observed the same trends between the two sets of comparisons (Section 4 and Section S3).

3.3.4. Resistance and thickness measurements

To determine the membrane resistance and membrane thickness, the AMX membrane was first equilibrated in the mixed-counterion solution of interest. To measure the membrane resistance, we used the difference method where the membrane resistance $({\text{R}}_{\text{m}},\Omega )$ was calculated as the difference between the resistance of the membrane plus solution $({\text{R}}_{\text{m}+\text{s}},\Omega )$ and the resistance of the solution alone $({\text{R}}_{\text{s}},\Omega )$.^45,46 Resistances were determined using a well-established electrochemical impedance spectroscopy method which isolates the membrane resistance from the boundary layer resistances.⁴⁶ Additional experimental details for the resistance measurements are in Section S5. After the resistances were measured, the membrane was quickly and gently dried with a laboratory wipe, and the membrane thickness was measured using a micrometer (model 3732XFL-1, L.S. Starret Co., USA). The membrane thickness was measured at four locations of the membrane to account for variabilities due to the membrane reinforcement.

3.4. Determination of the partition selectivity and partition coefficients

We performed equilibrium sorption-desorption experiments to determine the fraction of $c_{\text{fix}}$ neutralized by each counterion ($x_{\text{i}}^{\text{m}}$ and $x_{\text{j}}^{\text{m}}$) which were needed to calculate the partition selectivity and coefficients for counterions $i$ and $j$ (Eqs. 6, 10, 11). To do this, the AMX membrane was equilibrated in each mixed-counterion solution. After equilibration, the membrane was rinsed thoroughly with N₂-sparged laboratory grade water and placed in a 25 mL solution consisting of 0.25 M Na₂SO₄ for 48 hours to exchange the counterions that neutralized $c_{\text{fix}}$ with SO₄²⁻.²⁴ The Na₂SO₄ solution was changed once during the 48-hour period (around 24 h).²⁴ Both exchange solutions (i.e., exchange solutions A and B) were analyzed to measure the concentrations of counterions $i$ and $j$ using anion chromatography (ICS-3000, ThermoScientific, USA). In the second exchange solution (solution B), all VFA concentrations were non-detectable, and Cl⁻ and Br⁻ were near detection limits. The moles of counterions ($i$ or $j$) that neutralized $c_{\text{fix}}\left(m_{\text{m}}^{\text{i}},\text{mol}\right)$ were calculated as

$$m_i^m=c_{i,A}^{ex}{V}_A^{ex}+c_{i,B}^{ex}{V}_B^{ex},$$ (12)

where ${c_{\text{i},\text{A}}}^{\text{ex}}(\text{mol}.{\text{L}}^{-1})$ and ${c_{\text{i},\text{B}}}^{\text{ex}}(\text{mol}.{\text{L}}^{-1})$ are the concentrations of counterion $i$ in exchange solutions A and B, respectively, and ${V_{\text{A}}}^{\text{ex}}\left(\text{L}\right)$ and ${V_{\text{B}}}^{\text{ex}}\left(\text{L}\right)$ are the volumes of exchange solutions A and B, respectively. The fraction of $c_{\text{fix}}$ neutralized by counterion $i\left(x_{\text{i}}^{\text{m}}\right)$ was calculated as^24,25

$$x_i^m=\frac{m_i^m}{m_i^m+m_j^m}.$$ (13)

The fraction of $C_{\text{fix}}$ neutralized by counterion $j\left(x_{\text{j}}^{\text{m}}\right)$ was calculated as $1-x_{\text{i}}^{\text{m}}$. To determine the partition coefficient of each counterion ($K_{\text{i},\text{mix}}$ or $K_{\text{j},\text{mix}}$; Eqs. 10 and 11), we determined $c_{\text{fix}}$ using the averaged membrane swelling degrees measured in single-counterion solutions of counterion $i$ and counterion $j$ (details in Section S6).

4. Results and Discussion

4.1. Cl⁻/Br⁻ counterion pair (control)

4.1.1. Selectivities under mixed-counterion conditions

The permeability selectivity (P_Cl−/P_Br−) for the Cl⁻/Br⁻ counterion pair was <1 which showed that Br⁻ permeability was greater than Cl⁻ permeability as has been previously observed (Fig. 2A, B).^27,29 From the start of ED to 60 minutes, the permeability selectivity changed by <2% while the concentrations of Cl⁻ and Br⁻ in the D compartment decreased by 75% and 83%, respectively (Fig. 2A, Fig. S2A). Further, the Cl⁻/Br⁻ concentration ratio in the D compartment increased from approximately 1 to 1.48 after 60 minutes of ED operation. Accordingly, the substantial change in Cl⁻ and Br⁻ concentrations and moderate change in Cl⁻/Br⁻ ratio in the D compartment did not substantially affect the Cl⁻/Br⁻ permeability selectivity.

Figure 2.

(A) Permeability selectivity (P_Cl−/P_Br−) between Cl⁻ and Br⁻ during ED for the mixed-counterion condition. (B) Permeability selectivity, (C) partition selectivity (K_Cl−/K_Br−), and (D) mobility selectivity (u_Cl−/u_Br−) between Cl⁻ and Br⁻ for single-counterion and mixed-counterion conditions. The horizontal red line at 1.0 in all panels indicates the direction of each transport property selectivity. For the mixed-counterion condition, the permeability selectivity in Panel B and the calculated mobility selectivity in Panel D are derived from the initial permeability selectivity (vertical dark grey line in Panel A). The total initial counterion concentration for all experiments was 0.1 M with Na⁺ as the corresponding cation. Each point or bar is the average value from at least duplicate experiments, and error bars represent the standard deviation.

Results also showed the Cl⁻/Br⁻ partition selectivity (K_Cl−/K_Br−) governed the direction of the Cl⁻/Br⁻ permeability selectivity given that the permeability selectivity was <1, the partition selectivity was <1, and the mobility selectivity (u_Cl−/u_Br−) was >1 (Fig. 2B–D). The transport property selectivities showed that Br⁻ partitioned into and permeated through the membrane selectively over Cl⁻ while Cl⁻ migrated more easily through the membrane than Br⁻ (i.e., Cl⁻ was more mobile than Br⁻). The finding that Cl⁻/Br⁻ partitioning behavior governed the direction of permeability selectivity is consistent with previous studies.^27,29 Further, results showed the partition and mobility selectivities favored opposite counterions (i.e., K_Cl−<K_Br− while u_Cl−>u_Br−). We hypothesize that this opposing behavior in part stems from the different affinities of the anions for the AEM. Specifically, when Br⁻ and Cl⁻ are both present in the AEM, Br⁻ has a higher affinity for the AEM (as evidenced by K_Cl−/K_Br− <1). The stronger affinity of Br⁻ may restrict Br⁻ mobility within the AEM to thereby allow Cl⁻ to migrate (described by mobility) more freely. Previous studies have similarly shown that in a counterion mixture, the counterion with a comparatively lower affinity for the membrane migrates (or diffuses) more easily through the membrane.^27,31

4.1.2. Mixed-counterion vs. single-counterion selectivities

We next compared the Cl⁻/Br⁻ transport property selectivities between the single- and mixed-counterion conditions. Results showed the directions of the Cl⁻/Br⁻ permeability selectivity (P_Cl−/P_Br−<1), partition selectivity (K_Cl−/K_Br−<1), and mobility selectivity (u_Cl−/u_Br−>1) were consistent between conditions (Fig. 2B–D). However, the permeability selectivity was lower for the mixed-counterion condition which was driven by its lower partition selectivity (Fig. 2B, C). The mobility selectivity did not drive the decrease in permeability selectivity because the mobility selectivity substantially increased under mixed-counterion conditions (Fig. 2C).

4.1.3. Effect of the competing counterion on counterion transport mechanisms

Analysis of the individual counterion transport properties under mixed-counterion conditions indicated that the Cl⁻/Br⁻ permeability selectivity decreased from the single- to mixed-counterion condition because while both Cl⁻ and Br⁻ permeability decreased in mixed-counterion conditions, Cl⁻ permeability decreased to a greater extent (Fig. 2B; Fig. 3A). Further, the partition selectivity decreased because the Cl⁻ partition coefficient decreased while the Br⁻ partition coefficient increased in the presence of the competing counterion (Fig. 2C; Fig. 3B). The mobility selectivity increased because the Cl⁻ mobility increased while the Br⁻ mobility decreased (Fig. 3C). Findings from the mixed-counterion transport property analysis are consistent with our hypothesis that when Cl⁻ and Br⁻ coexist in the AEM, the presence of Br⁻ (higher K) weakened the interactions between Cl⁻ (lower K) and the AEM to thereby allow Cl⁻ to migrate more freely compared to when Cl⁻ is the sole counterion in the AEM. This result was consistent with a molecular dynamics study that evaluated Cl⁻ (higher K) and F⁻ (lower K)⁴⁷ mobility in an AEM and found that the presence of the higher affinity counterion (Cl⁻) enhanced the mobility of the lower affinity counterion (F⁻).⁴⁸ Additionally, we deduced that the presence of Cl⁻ strengthened the Br⁻–AEM interactions, as indicated by the increase in K⁻ for Br⁻, resulting in reduced Br⁻ mobility compared to when Br⁻ was present alone. Overall, results indicate that when two monovalent inorganic anions are in an AEM, the higher affinity counterion may weaken the lower affinity counterion–AEM interactions thereby enhancing its mobility – and vice versa.

Figure 3.

(A) Permeability, P_i, (B) total counterion partition coefficient, K_i, and (C) mobility, u_i, of counterions Cl⁻ and Br⁻ under single-counterion and mixed-counterion conditions. The permeabilities for mixed-counterion conditions were calculated from the measured initial concentration-normalized fluxes, the measured initial current during ED, and the membrane resistance measured in a 0.05 M/0.05 M mixed Cl⁻/Br⁻ solution. All bars are the average values from at least duplicate experiments, and error bars represent the standard deviation.

4.2. VFA_i/VFA_j counterion pairs

4.2.1. Selectivities under mixed-counterion conditions

We evaluated transport property selectivities for two VFA/VFA pairs. For both VFA/VFA pairs, the shorter carbon-chain VFA was assigned VFA_i (numerator), and the longer carbon-chain VFA was assigned VFA_j (denominator). For both VFA_i/VFA_j pairs, the VFA_i/VFA_j permeability selectivity was >1 which showed that the shorter carbon-chain VFA had a higher permeability than the longer carbon-chain VFA (Fig. 4A, B). The VFA_i/VFA_j pair with the greater carbon-chain length differential (i.e., C2/C5) showed higher permeability selectivity for the shorter carbon-chain VFA_i than the VFA_i/VFA_j pair with the lower carbon-chain differential (i.e., C3/C4). This result indicates that the greater the carbon-chain length differential, the greater the permeability selectivity (Fig. 4A,B). In terms of the stability of the permeability selectivity, the VFA_i/VFA_j permeability selectivity changed by less than 15% over 60 min for both VFA_i/VFA_j pairs while 33%–58% of each VFA was removed from the D compartment (Fig. 4A; Fig. S2B–C). Further, over the same time span, the VFA_i/VFA_j concentration ratio in the D compartment decreased from approximately 1 to 0.66. Overall, findings were similar to those from the Cl⁻/Br⁻ case where the moderate changes in counterion concentrations and ratios did not substantially affect the permeability selectivity.

Figure 4.

(A) Permeability selectivity (P_VFAi/P_VFAj) between VFA_i (shorter carbon-chain) and VFA_j (longer carbon-chain) during ED for mixed-counterion conditions. (B) Permeability selectivity, (C) partition selectivity (K_VFAi/K_VFAj), and (D) mobility selectivity (u_VFAi/u_VFAj) between VFA_i and VFA_j under single-counterion and mixed-counterion conditions. The horizontal red line at 1.0 in all panels indicates the direction of each transport property selectivity. For the mixed-counterion condition, the permeability selectivities in Panel B and the calculated mobility selectivities in Panel D are derived from the initial permeability selectivity (vertical dark grey line in Panel A). C2, C3, C4, and C5 represent the number of carbons in acetate, propionate, and valerate, respectively. The total initial counterion concentration for all experiments was 0.1 M with Na⁺ as the corresponding cation. Each point or bar is the average value from at least duplicate experiments, and error bars represent the standard deviation.

The VFA_i/VFA_j mobility selectivity (u_VFAi/u_VFAj>1) governed the direction of the VFA_i/VFA_j permeability selectivity (P_VFAi/P_VFAj>1) for both VFA_i/VFA_j pairs (Fig. 4B, D). Accordingly, the higher mobility of the shorter carbon-chain VFA_i compared with the longer carbon-chain VFA_j controlled the higher permeability of the shorter carbon-chain VFA_i. The transport property selectivity that governed the direction of the VFA_i/VFA_j permeability selectivity differed from that of the Cl⁻/Br⁻ pair where the partition selectivity governed the direction of the permeability selectivity (Fig. 2B–D). We hypothesize that the greater importance of mobility in determining the permeability selectivity in VFA_i/VFA_j pairs compared with the inorganic Cl⁻/Br⁻ pair is the result of the greater steric effects experienced by VFAs due to their larger size and the hydrophobic interactions between VFAs and the AEMs that inorganic ions do not experience.³⁵

Effects of steric hindrance on permeability selectivity can be rationalized by considering the relative sizes of the AEM pores and solutes of interest. Commercial AEMs have estimated pore sizes of 2–11 Å.^49,50 In comparison, VFAs possess a Stokes radii of 2.0–2.4 Å and estimated molecular lengths of 2.6–6.7 Å, while the Stokes radii of Cl⁻ and Br⁻ are ~1.0 Å.^35,51 Thus, VFAs experience significantly greater steric hindrance in the AEM than Cl⁻ and Br⁻ which significantly affects VFA permeability. Further, analysis from our previous study indicated VFA hydration energy, a function of hydration enthalpy and entropy, and temperature, did not drive differences in VFA mobility in part due to the small variations in VFA hydration energy across C2–C5 VFAs (≤3.1%)³⁵ but substantial differences in VFA mobility (up to 104%). Therefore, we do not expect VFA hydration energy to be a determining factor in VFA_i/VFA_j selectivities. Overall, we infer steric hindrance is the determining factor for VFA_i/VFA_j permeability selectivity in the AMX membrane.

The partition and mobility selectivities showed selectivity to opposite counterions within each VFA_i/VFA_j pair (Fig. 4C, D). Specifically, the shorter carbon-chain VFA_i partitioned less selectively into the membrane compared with the longer carbon-chain VFA_j (K_VFAi/K_VFAj<1). By contrast, the shorter carbon-chain VFA_i migrated through the membrane selectively over VFA_j (u_VFAi/u_VFAj>1). Accordingly, these results paralleled those of the Cl⁻/Br⁻ pair where the counterion with the lower affinity for the membrane (i.e., lower K: Cl⁻ or VFA_i) had higher mobility relative to the competing counterion with the higher affinity for the membrane (i.e., higher K: Br⁻ or VFA_j).

4.2.2. Mixed-counterion vs. single-counterion selectivities

We next compared the transport property selectivities of the VFA_i/VFA_j pairs between single-counterion and mixed-counterion conditions. The direction of the transport property selectivities for both VFA_i/VFA_j pairs were consistent between conditions, where the VFA_i/VFA_j permeability selectivities were >1 (selective for the shorter-chain VFA_i; Fig. 4B), the VFA_i/VFA_j partition selectivities were <1 (selective for longer-chain VFA_j; Fig. 4C), and the mobility selectivities were >1 (selective for shorter-chain VFA_i; Fig. 4D). For both VFA_i/VFA_j pairs, the mixed-counterion VFA_i/VFA_j permeability selectivity was lower than the single-counterion VFA_i/VFA_j permeability selectivity; this difference was more substantial for the VFA pair with the greater chain length differential (C2/C5; Fig. 4B). For the C2/C5 pair, combined effects of lower C2/C5 partition and mobility selectivity governed the substantially lower C2/C5 permeability selectivity (Fig. 4B–D). For the C3/C4 pair, the lower C3/C4 permeability selectivity was governed by the lower C3/C4 mobility selectivity (Fig. 4B–D).

4.2.3. Effect of the competing counterion on counterion transport mechanisms

Analysis of the individual counterion transport properties under mixed-counterion conditions indicated that the decrease in VFA_i/VFA_j permeability selectivity from single- to mixed-counterion conditions occurred because the VFA_i permeability decreased in the presence of the competing counterion (i.e., VFA_j) while the permeability of VFA_j either slightly increased or did not change (Fig. 5A). Regarding partitioning behavior, for the C2/C5 case we observed a decrease in the C2 partition coefficient and an increase in the C5 partition coefficient in the presence of the competing counterion (Fig. 5B). The C2/C5 partition results are consistent with those of the Cl⁻/Br⁻ pair where the presence of a higher affinity counterion (i.e., Br⁻ or C5) diminished the affinity of the lower affinity counterion (i.e., Cl⁻ or C2) and vice versa. For the C3/C4 case we did not observe a significant change in partition coefficients from single to mixed-counterion conditions which is likely due to their similar partition coefficients (K ≈ 85) in single-counterion conditions. The similar partitioning means that neither the C3 or C4 VFA had a stronger affinity for the membrane and therefore neither’s affinity was affected by the presence of the other (Fig. 5B).

Figure 5.

(A) Permeability, P_i, (B) partition coefficient, K_i, and (C) mobility, u_i, of each counterion under single-counterion and mixed-counterion conditions. C2, C3, C4, and C5 stand for the number of carbons in acetate, propionate, butyrate, and valerate, respectively. The permeabilities for each VFA under mixed-counterion conditions were determined from the measured initial concentration-normalized fluxes, the measured initial current during ED, and the membrane resistance measured in a 0.05 M/0.05 M mixed VFA_i/VFA_j solution. All bars are the average values from at least duplicate experiments, and error bars represent the standard deviation.

While the observed trends for C2/C5 partition coefficients between single- and mixed-counterion conditions were consistent with those of the Cl⁻/Br⁻ pair (i.e., the partition coefficient of the higher affinity ion, Br⁻ and C5, increased while that of the lower affinity ion, Cl⁻ and C2, decreased), the same was not true for C2/C5 mobility. Specifically, for the C2/C5 pair, the mobility of the ion with the lower affinity for the membrane (C2) substantially decreased in the presence of the ion with the higher affinity for the membrane (C5) (Fig. 5C). By contrast, for the Cl⁻/Br⁻ pair, the mobility of the ion with the lower affinity for the membrane (Cl⁻) increased in the presence of the ion with the higher affinity for the membrane (Br⁻) (Fig. 3C). For the Cl⁻/Br⁻ pair, we rationalized that the mobility of Cl⁻ increased in the mixed-counterion condition because the presence of Br⁻ weakened Cl⁻-AEM interactions. Therefore, we deduce that the C2 mobility decreased (instead of increasing) because the steric hindrance imposed by C5 had a greater negative effect on C2 mobility than the positive effect of weakened C2-AEM interactions caused by C5. The decrease in C2 mobility may also partially stem from the estimated 6.14% lower AEM swelling in the mixed-counterion C2/C5 solution than in the single-counterion C2 VFA solution (Section S6). Previous studies have established that ion size and volume fraction of water are empirically related to ion mobility in membranes where greater ion size and lower volume fraction of water result in lower ion mobility.^52,53

Given that we did not observe a significant change in C3 affinity (denoted by similar K values) in the presence of C4 (Fig. 5B), we attribute the moderate decrease in C3 mobility to steric hindrance imparted by C4 as was the case for the C2/C5 pair (Fig. 5C). The mobility decrease (29.3%) of the C3 VFA in the presence of the C4 VFA was lower than the mobility decrease (68.6%) of the C2 VFA in the presence of the C5 VFA. The greater effects of the C5 VFA on the C2 VFA mobility than that of the C4 VFA on the C3 VFA mobility can be primarily attributed to greater steric hindrance imposed by the C5 VFA and the greater difference in molecular size between VFAs for the C2/C5 pair than the C3/C4 pair. Future studies should evaluate the threshold where effects on VFA mobility by a competing VFA shift from being affinity-driven to sterically-driven.

Further, for the Cl⁻/Br⁻ pair, we rationalized that the mobility of Br⁻ moderately decreased in the mixed-counterion condition because of the strengthened Br⁻-AEM interactions (i.e., increased K in mixed-counterion conditions). Therefore, a similar argument applies to the C5 VFA: the mobility of the C5 VFA moderately decreased (27.6%) in the mixed-counterion condition compared to the single-counterion condition (Fig. 5C) because of the strengthened interaction with the AEM (i.e., increased K; see Fig. 5B). Similarly, because the C4 VFA-AEM interactions were of similar strength in both single and mixed-counterion conditions (i.e., similar K; see Fig. 5B), its mobility did not display significant differences between the two conditions (Fig. 5C). We note that the small increase in the C4 VFA mobility (5.4%) may stem from the 0.05 M decrease in the C4 VFA concentration from single- to mixed-counterion conditions given that organic anion concentration inversely correlates with organic anion diffusion coefficients in AEMs.⁵⁴

4.3. VFA/Cl⁻ counterion pairs

4.3.1. Selectivities under mixed-counterion conditions

The VFA/Cl⁻ permeability selectivity was evaluated between each VFA (C2–C5) and Cl⁻ (Fig. 6). For each VFA/Cl⁻ pair, the VFA was designated as counterion i (numerator), and Cl⁻ was designated as counterion j (denominator). For all VFA/Cl⁻ pairs, the VFA/Cl⁻ permeability selectivity (P_VFA/P_Cl−) was <1 (Fig. 6A–B), and increased substantially (179–530%) from 0 to 60 minutes. Further, from 0 to 60 minutes, the VFA concentration in the D compartment decreased by 25.8–51.3%, and the Cl⁻ concentration decreased by 75.3–90.8% which led to an increase in the VFA/Cl⁻ ratio from 1 to 1.98–7.54 (Fig. 6A, Fig. S2D–G). Overall, results showed that the changes in VFA and Cl⁻ concentrations and VFA/Cl⁻ ratios in the D compartment strongly affected the VFA/Cl⁻ permeability selectivities.

Figure 6.

(A) Permeability selectivity (P_VFA/P_Cl−) between each VFA and Cl⁻ during ED for mixed-counterion conditions. (B) Permeability selectivity, (C) partition selectivity (K_VFA/K_Cl−), and (D) mobility selectivity (u_VFA/u_Cl−) for each VFA/Cl⁻ counterion pair under single-counterion and mixed-counterion conditions. The horizontal red line at 1.0 in all panels indicates the direction of each transport property selectivity. C2, C3, C4, and C5 represent the number of carbons in acetate, propionate, butyrate, and valerate. For the mixed-counterion conditions, the permeability selectivities in Panel B and the calculated mobility selectivities in Panel D are derived from the initial permeability selectivities (vertical dark grey line in Panel A). The total initial counterion concentration for all experiments was 0.1 M with Na⁺ as the corresponding cation. Each point or bar is the average value from at least duplicate experiments, and error bars represent the standard deviation.

The substantial change in VFA/Cl⁻ permeability selectivities over 60 min of ED (Fig. 6A) contrasts with what was observed for the Cl⁻/Br⁻ and VFA_i/VFA_j pairs where the permeability selectivities did not substantially change (<15%; Fig. 2A and Fig. 4A). The discrepancy in the change in permeability selectivities over time for the VFA/Cl⁻ pairs compared with the Cl⁻/Br⁻ and VFA_i/VFA_j pairs is likely due to the substantial differences between the physicochemical properties of VFAs and Cl⁻ compared with the differences between Cl⁻ and Br⁻ or between VFA_i and VFA_j.³⁵ Since VFAs and Cl⁻ have substantially different physicochemical properties (e.g., size, hydrophobicity, hydration energy, structural charge distribution), we hypothesize that as the relative concentrations of VFAs and Cl⁻ in the membrane (i.e., VFA/Cl⁻ ratio) change during ED, the membrane properties (e.g., conductivity and correspondingly permeability) also change to thereby alter counterion transport in the AEM.

To evaluate the role of the differences in counterion physicochemical properties on membrane transport properties, we measured the membrane conductivity, which is linearly related to permeability (Eq. S1), as a function of the fraction of $c_{f\text{ix}}$ neutralized by the C5 VFA. We compared the cases where the C5 VFA was paired with Cl⁻ (C5/Cl⁻ pair) versus the C2 VFA (C5/C2 pair) as competing counterions (Section S7). A previous study applied an empirical model based on Kohlrausch’s law to relate the membrane conductivity to the fraction of charged fixed groups neutralized by a specified counterion under mixed-counterion conditions.²⁶ This model uses an “interaction parameter” as its only fitting parameter. The interaction parameter is a catch-all parameter that captures the effects of interactions between the counterions themselves and/or counterions and the membrane on counterion transport. However, the interaction parameter (see Section S7) does not provide insights into what these interactions are or counterion transport mechanisms (e.g., K and $\mu$).²⁶ Therefore, we focus our discussion on the change in membrane conductivity as a function of the C5 fraction neutralizing $c_{f\text{ix}}$ and not the interaction parameter.

Results showed that when the C5 fraction neutralizing $c_{f\text{ix}}$ increased from 0 to approximately 0.12, the membrane conductivity decreased by 46.9% and 26.5% with Cl⁻ and the C2 VFA, respectively, as the competing counterion (Fig. S14). This indicates that the counterion transport behavior changed more substantially when the C5 fraction neutralizing $c_{f\text{ix}}$ increased relative to Cl⁻ than when relative to the C2 VFA. Overall, this experiment supported our hypothesis that the greater the difference in physicochemical properties between the two counterions in the membrane, the more the counterion transport behavior in the membrane changes with changing counterion concentration ratio in the membrane. Accordingly, the greater changes in VFA/Cl⁻ permeability selectivities (Fig. 6A) compared with those of the Cl⁻/Br⁻ or VFA_i/VFA_j pairs (Figs. 2A and 4A) can in part be attributed to the substantial differences in VFA and Cl⁻ physicochemical properties.

We further observed that the VFA/Cl⁻ partition selectivities (K_VFA/K_Cl−<1) controlled the direction of the VFA/Cl⁻ permeability selectivities (P_VFA/P_Cl−<1; Fig. 6B–C), consistent with the findings for the Cl⁻/Br⁻ pair (Fig. 2B–C). Additionally, results showed that the direction of the VFA/Cl⁻ mobility selectivities under mixed-counterion conditions varied between VFA/Cl⁻ pairs (Fig. 6D). For the case of the C2/Cl⁻ pair, the partition and mobility selectivities showed selectivity for opposite counterions where the partition selectivity was selective for Cl⁻ (K_C2/K_Cl− <1) while the mobility selectivity was selective for C2 (u_C2/u_Cl−>1; Fig. 6C, D). The C2/Cl⁻ results are analogous to what was observed with the Cl⁻/Br⁻ and VFA_i/VFA_j pairs where the counterion with the lower affinity (lower K) for the membrane migrated more easily through the membrane. Conversely, for the C3/Cl⁻, C4/Cl⁻, and C5/Cl⁻ pairs, both partition and mobility selectivities were selective for Cl⁻ over the VFA (C3/Cl⁻ and C4/Cl⁻ pairs) or not selective for either counterion (C5/Cl⁻ pair). We speculate that the reason why in the C3–C5/Cl⁻ pairs the ion with the lesser affinity for the membrane (i.e., the VFA) did not have greater mobility than the ion with the greater affinity for the membrane (i.e., Cl⁻) is because the intrinsically much lower mobility of the C3–C5 VFAs compared to that of Cl⁻ (Fig. 6D, single-counterion condition data). Therefore, even though the presence of Cl⁻ may have weakened the VFA affinity for the membrane, the mobility of the C3–C5 VFAs was not greater than that of the much smaller, more mobile Cl⁻ ion.

4.3.2. Mixed-counterion vs. single-counterion selectivities

We next compared the mixed-counterion selectivities to the single-counterion selectivities for each VFA/Cl⁻ pair. For all VFA/Cl⁻ pairs, the mixed-counterion permeability and partition selectivities followed the same direction as the single-counterion selectivities (P_VFA/P_Cl− and K_VFA/K_Cl− <1; Figs. 6B, C). Conversely, the agreement between the direction of the mobility selectivity in single and mixed-counterion conditions varied across the different VFA/Cl⁻ pairs (Fig. 6D).

The VFA/Cl⁻ permeability selectivities for the mixed-counterion conditions were lower than those for the single-counterion conditions except for the C5/Cl⁻ case (Fig. 6B). The VFA/Cl⁻ partition selectivities were substantially lower for mixed-counterion conditions while the VFA/Cl⁻ mobility selectivities were substantially higher (Fig. 6C, D). Accordingly, the generally lower VFA/Cl⁻ permeability selectivities for the mixed-counterion conditions were primarily driven by the lower VFA/Cl⁻ partition selectivities. Results are consistent with those for the Cl⁻/Br⁻ pair which showed that the lower permeability selectivity in mixed-counterion conditions compared with single-counterion conditions was driven by the decrease in partition selectivity (Fig. 2B, C).

4.3.3. Effect of the competing counterion on counterion transport mechanisms

Analysis of the individual transport properties under mixed-counterion conditions indicated that the VFA/Cl⁻ permeability decreased relative to single-counterion conditions because the VFA permeabilities decreased (with the exception of the C5 case) while the Cl⁻ permeabilities increased (Fig. 7A). These permeability changes were governed by substantial decreases in VFA partition coefficients and increases in Cl⁻ partition coefficients in the presence of the competing counterion (Fig. 7B) which also translated into the substantial decreases observed for the partition selectivity (Fig. 6C). Specifically, the VFA partition coefficients substantially decreased in the presence of Cl⁻ because Cl⁻ outcompeted VFAs for adsorption sites in the membrane due to its smaller size and lower (less negative) hydration energy.³⁵ Overall, consistent with the results for the Cl/Br⁻ and C2/C5 VFA pairs, the partitioning results for VFA/Cl⁻ pairs indicated that interactions between the membrane and the ion with the lower K (VFAs) in single-counterion conditions were weakened by the presence of the ion with the higher K(Cl⁻) and vice versa.

Figure 7.

(A) Permeability, P_i, (B) partition coefficient, K_i, and (C) mobility, u_i, of each counterion under single-counterion and mixed-counterion conditions. C2, C3, C4, and C5 stand for the number of carbons in acetate, propionate, butyrate, and valerate, respectively. The permeabilities for each counterion under mixed-counterion conditions were determined from the measured initial concentration-normalized fluxes, the measured initial current during ED, and the membrane resistance measured in a 0.05 M/0.05 M mixed VFA/Cl⁻ solution. All bars are the average values from at least duplicate experiments, and error bars represent the standard deviation.

The increase in VFA/Cl⁻ mobility selectivity stemmed from an increase in VFA mobility and decrease in Cl⁻ mobility in the presence of the competing counterion (Fig. 7C). The increase in VFA mobility occurred due to weakened VFA-AEM interactions (denoted by lower K) in the presence of Cl⁻. As previously discussed, weaker ion-membrane interactions can facilitate higher ion mobility in membranes.^27,32,55 The increase in VFA mobility was not attributed to changes in AEM swelling or plasticization. This is because the AMX membrane SD is 9.8–25.7% greater in VFA solutions than Cl⁻ solutions (swelling data in Section S6).³⁵ Therefore, considering that IEM deswelling hinders ion mobility⁵⁶, any AEM deswelling caused by Cl⁻ did not contribute to the observed increase in VFA mobility. Further, organic solutes can plasticize membranes to thereby increase the mobility of ions in the membrane.^57,58 However, enhanced plasticization does not explain the observed increase in VFA mobility under mixed-counterion conditions because the VFA concentration in the membrane was 74.0–84.2% lower in VFA-Cl⁻ solutions compared with single-VFA conditions.

Notably, compared to single-counterion conditions, the mobility of the C5 VFA increased to a greater extent than those of the C2–C4 VFAs in the presence of Cl⁻ (617% increase vs. 37.8–106% increase; Fig. 7C). The pronounced increase in C5 mobility may stem from the 0.05 M decrease in VFA concentration under mixed-counterion conditions which reduced steric hindrance in the membrane. This reduction was most prominent for the larger C5 VFA than the other VFAs. The substantial increase in C5 mobility drove the increase in C5 permeability (Fig. 7A). The Cl⁻ mobility decreased for all VFA/Cl⁻ cases albeit only slightly for the C3⁻/Cl⁻ pair (Fig. 7C). The decrease in Cl⁻ mobility was the result of combined effects of strengthened Cl⁻-AEM interactions and increased steric hindrance to migration due to the presence of VFAs.

4.3.4. Effect of VFA carbon-chain length on VFA/CI⁻ selectivities

We next evaluated the effect of VFA carbon-chain length on VFA/Cl⁻ selectivities for single- and mixed-counterion conditions (Fig. 8; statistics in Section S8). For both single- and mixed-counterion conditions, the permeability selectivity decreased, the partition selectivity increased, and the mobility selectivity decreased with increasing VFA carbon-chain length except for the mixed-counterion C5/Cl⁻ permeability and mobility selectivities (Fig. 8). Since the C5/Cl⁻ mixed-counterion pair aligned with the positive relationship between VFA carbon-chain length and partition selectivity (Fig. 8B), the deviation of the C5/Cl⁻ permeability selectivity was governed by its mobility selectivity. We attribute the deviation of the C5/Cl⁻ mixed-counterion pair from the negative relationship between VFA carbon-chain length and mobility selectivity to C5 having the greatest negative effect on Cl⁻ mobility compared with the C2–C4 VFAs (Fig. 7C). We speculate that C5 imparted the greatest steric hindrance on Cl⁻ migration across the VFAs because C5 is the largest of the VFAs. Results are consistent with the C2/C5 pair where the presence of C5 substantially hindered C2 mobility in the membrane (Fig. 5C). Further, for both the single- and mixed- counterion conditions, the mobility selectivity had a stronger effect on the ranking of the permeability selectivity across VFA(Cl⁻) pairs than the partition selectivity (Fig. 8). This shows that steric effects due to the size of the VFAs play a greater role than VFA partitioning on differences in VFA/Cl⁻ permeability selectivity across VFA/Cl⁻ pairs.

Figure 8.

Correlation between VFA carbon-chain length in the VFA/Cl⁻ counterion pair and (A) permeability selectivity (P_VFA/P_Cl−), (B) partition selectivity (K_VFA/K_Cl−), and (C) mobility selectivity (u_VFA/u_Cl−). Dashed lines represent the linear regression analysis between VFA carbon-chain length and transport property selectivities. Selectivity data is reproduced from Fig. 6. The mixed-counterion C5/Cl⁻ permeability and mobility selectivities are indicated by a teal circle.

5. Conclusions

In this work we assessed the transport properties and selectivities of six VFA-containing counterion pairs and a Cl⁻/Br⁻ control under single- and mixed-counterion conditions to elucidate (i) the transport property selectivity (i.e., partition selectivity or mobility selectivity) that governed permeability selectivity, and (ii) the effect of a competing counterion on VFA transport mechanisms. To determine the transport property selectivity that governed the permeability selectivity, we performed ED experiments with each mixed-counterion pair and measured the concentration-normalized fluxes to calculate the permeability selectivities. We also performed competitive sorption-desorption experiments with each mixed-counterion pair to determine the partition selectivities. The mobility selectivities were calculated from the permeability and partition selectivities. Further, to understand the role of the competing counterion on VFA transport mechanisms, the mixed-counterion selectivities and transport properties in mixed-counterion conditions were compared to those from single-counterion conditions. The main findings from this work are summarized as follows:

• For the Cl⁻/Br⁻ pair, the partition selectivity (K_Cl−/K_Br− <1) drove the direction of the permeability selectivity (P_Cl−/P_Br− <1), and the presence of Br⁻ (higher K) led to enhanced Cl⁻ (lower K) mobility in the AEM due to weakened Cl⁻-AEM interactions.

• For the VFA_i/VFA_j pairs, the mobility selectivity (u_VFAi/u_VFAj>1) drove the direction of the permeability selectivity (P_VFAi/P_VFAj>1).

• The presence of VFA_j (higher K) in the AEM reduced the affinity of VFA_i (lower K) for the membrane and hindered VFA_i mobility in the membrane due to steric hindrance imparted by VFA_j.

• For the VFA/Cl⁻ pairs, the partition selectivity (K_VFA/K_Cl−<1) drove the direction of the permeability selectivity (P_VFA/P_Cl−<1) while the mobility selectivity drove the ranking of the VFA/Cl⁻ permeability selectivity between the VFA/Cl⁻ pairs.

• Analogous to the Cl⁻/Br⁻ case, for the VFA/Cl⁻ pairs, the presence of Cl⁻ (higher K) enhanced VFA (lower K) mobility in the AEMs compared to VFAs under single-counterion conditions due to weakened VFA-AEM interactions.

• Across the mixed-counterion VFA/Cl⁻ pairs, there was a negative relationship between permeability selectivity and VFA carbon-chain length except for the C5/Cl⁻ pair. This deviation was likely a result of the steric hindrance imparted by the C5 VFA on Cl⁻ mobility.

Taken together, results indicate that the presence of a competing counterion with a higher affinity for the membrane weakened VFA affinity for the membrane. Conversely, the effects on VFA mobility of a competing counterion with a higher affinity for the membrane depended on the physicochemical properties of the competing counterion. Additionally, this work provided an approach to convert the concentration-normalized fluxes measured during ED to permeabilities using the TMEP which aids in elucidating counterion transport mechanisms under mixed-counterion conditions. The experimental approach developed in this work can be used to investigate counterion transport in IEMs to understand the effects of competing counterion and membrane physicochemical properties on counterion transport mechanisms. Overall, this work provides mechanistic insights into electro-driven VFA transport through AEMs under mixed-counterion conditions. By explicitly evaluating the role of a competing counterion and mixed-counterion effects on VFA transport mechanisms, this work reveals new insights into VFA transport under conditions that begin to emulate digestates—a critical step towards enhancing VFA recovery. Future work should aim to elucidate counterion transport mechanisms under more complex solution conditions (e.g., ternary and quaternary counterion mixtures), investigate VFA transport mechanisms using molecular dynamics simulations, develop frameworks for predictive modeling of VFA transport, and expand evaluation to include AEMs with different physicochemical properties. The latter will enable pinpointing AEM physicochemical properties that give rise to targeted VFA selectivities.

Highlights.

• Mobility selectivity drove the direction of permeability selectivity for VFA/VFA pairs.

• Partition selectivity drove the direction of permeability selectivity for VFA/Cl⁻ and Cl⁻/Br⁻ pairs.

• VFA partitioning decreased in the presence of a higher affinity counterion.

• Transport property selectivities differed between single- and mixed-counterion conditions.

• Presented framework to determine electro-driven permeabilities for mixed-counterions.