Ion Diffusion Coefficients in Ion Exchange Membranes: Significance of

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Ion Diffusion Coefficients in Ion Exchange Membranes: Significance of Counterion Condensation Jovan Kamcev,† Donald R. Paul,† Gerald S. Manning,‡ and Benny D. Freeman*,† †

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McKetta Department of Chemical Engineering, Center for Energy and Environmental Resources, and Texas Materials Institute, The University of Texas at Austin, 10100 Burnet Road Building 133 (CEER), Austin, Texas 78758 United States ‡ Department of Chemistry and Chemical Biology, Rutgers University, 610 Taylor Road, Piscataway, New Jersey 08854-8087, United States S Supporting Information *

ABSTRACT: This study presents a new framework for extracting single ion diffusion coefficients in ion exchange membranes from experimental ion sorption, salt permeability, and ionic conductivity data. The framework was used to calculate cation and anion diffusion coefficients in a series of commercial ion exchange membranes contacted by aqueous NaCl solutions. Counterion diffusion coefficients were greater than co-ion diffusion coefficients for all membranes after accounting for inherent differences due to ion size. A model for ion diffusion coefficients in ion exchange membranes, incorporating ideas from counterion condensation theory, was proposed to interpret the experimental results. The model predicted co-ion diffusion coefficients reasonably well with no adjustable parameters, while a single adjustable parameter was required to accurately describe counterion diffusion coefficients. The results suggest that for cross-linked ion exchange membranes in which counterion condensation occurs condensed counterions migrate along the polymer backbone and contribute to a current in the presence of an externally applied electric field. Moreover, diffusion coefficients for condensed counterions were approximately 2−2.5 times greater than those for uncondensed counterions.

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INTRODUCTION Various membrane-based technologies for water purification and energy generation applications rely on ion exchange membranes (i.e., ionomers, charged membranes, etc.) to selectively and efficiently regulate ion and water transport. Examples of such technologies include electrodialysis,1−4 membrane-assisted capacitive deionization,5,6 photoelectrochemical cells,7 and fuel cells,8−12 among others. A fundamental understanding of ion and water transport in such materials, especially the influence of polymer structure on transport properties, could facilitate rational development of membranes with improved properties, potentially leading to lower process operating costs and increasing the utility of such membranes in technologies that have not previously used them.13−15 However, despite a long-standing academic and industrial interest in ion exchange membranes, the current state of fundamental understanding in the open literature remains incomplete. Ion transport in dense (i.e., nonporous) polymer membranes, including most commercial ion exchange membranes, occurs by diffusion.16−18 Consequently, ion transport in such materials is typically quantified by ion diffusion coefficients. A variety of experimental techniques have been employed to measure ion diffusion coefficients in ion exchange membranes.16,19,20 Early studies have predominantly used measurement techniques involving radioactively labeled ions (i.e., tracer diffusion experiments).16,19,21−26 Ion diffusion coefficients obtained from such experiments are so-called “self”-diffusion coefficients © XXXX American Chemical Society

since these measurements are typically performed at, or near, equilibrium (i.e., in the absence of an external driving force).16,19 From a practical viewpoint, however, ion diffusion coefficients measured in the presence of an external driving force are required for process models since such values are more representative of ion transport occurring during real operation of membrane-based technologies. Some studies report counterion (i.e., ion with opposite charge to that of fixed charges on the backbone of ion containing polymers) diffusion coefficients in ion exchange membranes obtained from ionic conductivity measurements, which employ an externally applied electric field to drive ion transport through a membrane, but this technique is only rigorous for counterion diffusion coefficients in ion exchange membranes equilibrated with pure water, i.e., when counterions are the only mobile ionic species in the membrane. Diffusion cell experiments employing a salt concentration gradient driving force can be used to measure “bulk” (or coupled) salt diffusion coefficients, which contain contributions from both counterions and co-ions (i.e., ions with similar charge to that of the fixed charges).27,28 In this study, we present a new, simple methodology to extract cation and anion diffusion coefficients in ion exchange membranes from experiments in which ion transport is induced Received: March 26, 2018 Revised: May 29, 2018

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solution is approximately 50% higher than that of Na+ ions).31,32 As the faster ion diffuses away from the slower ion, the electroneutrality condition is violated, and an electric field forms between the ions (i.e., the diffusion potential).31,32 The diffusion potential retards the faster ion and accelerates the slower ion so that at steady state cations and anions diffuse at the same rate.31,32 Consequently, experiments in which ion transport is driven by a concentration gradient (e.g., salt permeability measurements) can only yield coupled, average, salt diffusion coefficients. The Nernst−Planck equation, in combination with the electroneutrality condition (∑iziJmi = 0), can be used to derive an expression for the steady-state mobile salt (i.e., the coions and counterions that balance them) diffusive flux across a membrane, Jms , driven only by a concentration gradient.16,20,33 The final result is

by an external driving force. The framework, based on the Nernst−Planck equation, requires equilibrium ion sorption, salt permeability, and ionic conductivity measurements. The framework was used to extract counterion and co-ion diffusion coefficients in commercial ion exchange membranes in contact with aqueous NaCl solutions, with salt concentrations ranging from 0.01 to 1 M. Lastly, a model for ion diffusion coefficients in ion exchange membranes based on ideas from counterion condensation theory is presented and used to rationalize the experimental results.

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BACKGROUND For a system consisting of an ion exchange membrane separating two aqueous salt solutions, the condition for equilibrium requires the electrochemical potential of each ionic species to be equal in every phase.16,29 Altering the electrochemical potential in one of the solutions (e.g., changing the ion concentration or applying an electric potential difference) perturbs this balance and creates an electrochemical potential gradient that drives ion transport across the membrane from the high electrochemical potential side to the low electrochemical potential side. Consequently, the steady-state diffusive molar flux of ion i through a membrane, Jmi , is proportional to the electrochemical potential gradient.16 For one-dimensional ion transport in a membrane16,20 Jim = −

m CimDim dμi̅ RT dx

Jsm

m ij DgmDcm(zg 2Cgm + zc 2Ccm) yz dC m zz s = −Dm dCs = −jjjj 2 m m z s j zg Dg Cg + zc 2DcmCcm zz dx dx k {

(3)

where the subscripts g, c, and s denote counterions, co-ions, and mobile salt, respectively. For 1:1 electrolytes, the mobile salt concentration is equal to the co-ion concentration in the membrane. Application of Fick’s first law to mobile salt transport across a membrane yields an equivalent result. However, an advantage of deriving eq 3 via the Nernst−Planck approach is that it provides a direct relation between the mobile salt diffusion coefficient, Dms , and the individual ion concentrations and diffusion coefficients in a membrane. Equation 3 is valid for the diffusive component of the total mobile salt flux. Frame of reference (i.e., convective) contributions to the total salt flux, largely due to water diffusion in the opposite direction, are not included.33 Moreover, eq 3 ignores nonideal thermodynamic effects, which arise due to ion activity coefficient gradients in the membrane.33 Procedures for accounting for frame of reference and nonideal thermodynamic effects are detailed elsewhere and were fully adopted in the present study.33 From a practical perspective, frame of reference and thermodynamic nonidealities are small effects and act in opposite directions in the membranes of interest for these studies.33 Typical diffusion cell experiments, in which a membrane separates salt solution and pure water compartments, measure the pseudo-steady-state flux of mobile salt across a membrane.28,34 Such experiments can be used to determine mobile salt diffusion coefficients in a membrane, Dms , which, according to eq 3, are defined as

(1)

where Cmi is the concentration of ion i in the membrane, Dmi is the diffusion coefficient of ion i in the membrane, μ̅ mi is the

electrochemical potential of ion i in the membrane, R is the gas constant, T is absolute temperature, and x is the direction of ion transport (i.e., across the membrane thickness). The electrochemical potential of ion i in a membrane is defined as μ̅ mi = μ0m i + RT ln γmi Cmi + ziFψ, where μ0m i is the standard state chemical potential, γmi is the activity coefficient of ion i in the membrane, Cmi is the concentration of ion i in the membrane, F is Faraday’s constant, and ψ is the electrical potential of the phase.16,29 Inserting the expression for ion electrochemical potential into eq 1 and rearranging yields the well-known Nernst−Planck equation:16,20,30 É ÅÄÅ dC m z FC m dψ ÑÑÑÑ Å ÑÑ Jim = −DimÅÅÅ i + i i ÅÅÅ dx RT dx ÑÑÑÖ (2) Ç The Nernst−Planck equation, as presented in eq 2, does not include an ion activity coefficient term, which is only rigorously valid if ion activity coefficients in the membrane are unity or invariant. The consequences of this assumption are discussed below. Another implicit assumption in eq 2 is that ionic mobilities, umi , are related to ion diffusion coefficients via the Einstein equation: umi RT = |zi|FDmi .16,29,31 The Nernst−Planck equation can be used to describe both concentration gradient and electric field driven ion transport in ion exchange membranes, which provides a unifying framework to analyze experimental data obtained with either concentration gradient or electric field driving force, or both. Concentration Gradient Driven Ion Transport. In the absence of an externally applied electric field, cation and anion diffusion in concentration gradient driven ion transport through ion exchange membranes is coupled due to the electroneutrality condition. For common electrolytes, cation and anion mobilities often differ (e.g., the mobility of Cl− ions in an aqueous NaCl

Dsm =

DgmDcm(zg 2Cgm + zc 2Ccm) zg2DgmCgm + zc 2DcmCcm

(4)

Dms

From eq 4, depends on both the counterion and co-ion concentrations and diffusion coefficients in the membrane. Equation 4 clearly demonstrates that experiments in which ion transport is driven by a concentration gradient can only provide information on the coupled transport of counterions and coions. Additional information is required to extract individual ion diffusion coefficients. Such information can be acquired from experiments in which ion transport is driven by an electric field, as demonstrated in the following section. Electric Field Driven Ion Transport. Electric field driven ion transport in ion exchange membranes is often quantified by membrane ionic conductivity or resistivity.16,30,35 In typical electric field driven ion transport experiments, ion concenB

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Figure 1. Chemical structures of the membranes used in this study: (a) CR61-CMP, (b) AR103-QDP, and (c) AR204-SZRA.

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RESULTS AND DISCUSSION Ion Sorption, Mobile Salt Diffusion Coefficients, and Ionic Conductivity. The framework presented in the previous section was used to calculate single ion diffusion coefficients for a series of commercial ion exchange membranes in contact with aqueous NaCl solutions. One cation exchange membrane (CR61-CMP, GE Power and Water) and two anion exchange membranes (AR103-QDP and AR204-SZRA, GE Power and Water) were used in this study. As with other commercial electrodialysis ion exchange membranes, these materials have a composite structure, which incorporates a hydrophobic, highly porous fabric support to enhance membrane mechanical properties. The ion exchange polymer phase is continuous and considered to be reasonably homogeneous. The chemical structures of the ion exchange polymers are presented in Figure 1, and relevant membrane properties are recorded in Table 1.

trations are held constant throughout the membrane (i.e., no concentration gradients are imposed).16,36−41 Ion transport is induced by placing the membrane between electrodes and applying an electric potential difference between the electrodes.16 Cations migrate toward the negatively charged electrode, and anions migrate toward the positively charged electrode. The electric current density, I, is related to the ionic fluxes in the membrane via16 I = F ∑ ziJim

(5)

i

Inserting the Nernst−Planck equation (i.e., eq 2) into eq 5 yields I=−

F2 RT

∑ zi2CimDim i

dψ dx

(6)

The membrane ionic conductivity, κ, is defined as

16

κ = −I /

dψ dx

Table 1. Properties of the Membranes Used in This Study27 (7)

membrane

reported IEC [mequiv/g (dry polymer)]

Cm,p A [mol fixed charge groups/L (swollen polymer)]a

Cm,w A [mol fixed charge groups/L (water sorbed)]a

water volume fraction, ϕw, [L(water)/ L(swollen polymer)]a

CR61 AR103 AR204

2.2 (min) 2.2 (min) 2.4 (min)

1.60 ± 0.04 1.44 ± 0.03 1.44 ± 0.02

3.21 ± 0.08 3.58 ± 0.07 2.82 ± 0.04

0.50 ± 0.014 0.40 ± 0.018 0.52 ± 0.002

Thus κ=

F2 RT

∑ zi 2CimDim = i

F2 2 m m (zg Dg Cg + zc 2DcmCcm) RT

(8)

According to eq 8, membrane ionic conductivity depends on the counterion and co-ion concentrations and diffusion coefficients in the membrane. The second equality in eq 8 follows from restricting the analysis to a single electrolyte. The new procedure set forth in this report for determining individual ion diffusion coefficients in a membrane, Dmi , is as follows. Concentration gradient driven ion transport experiments (i.e., diffusion cell experiments) are combined with equilibrium ion sorption experiments to measure mobile salt diffusion coefficients in the membrane, Dms (cf. eq 4). Electric field driven ion transport experiments are used to measure membrane ionic conductivity, κ (cf. eq 8), at similar external solution salt concentrations. Equilibrium ion concentrations in a membrane contacted by an aqueous salt solution are measured via a desorption technique, as detailed in a separate study.34 Finally, eqs 4 and 8 are solved simultaneously to obtain single ion diffusion coefficients in the membrane for a given external solution salt concentration. A key assumption in this framework is that ion transport in ion exchange membranes occurs via diffusion, regardless of whether the driving force is a concentration gradient or an electric field. Additionally, the Einstein equation relating ionic mobilities to ion diffusion coefficients is assumed to be valid.16,31 The Einstein equation is strictly valid for aqueous salt solutions at infinite dilution,16,31 but it appears to be a reasonable assumption for ion exchange membranes based on limited evidence in the literature.42,43 The validity of this assumption for a broad class of materials, however, remains to be explored.

a

Data reported for membranes equilibrated with deionized water.

Experimental salt permeability, ionic conductivity, and equilibrium ion sorption results were reported previously and are summarized below for convenience.27,33,34 All experimental results are reported on the basis of ion exchange polymer (i.e., contributions from the fabric backing have been subtracted using a procedure described elsewhere34). Membrane water content (i.e., degree of swelling) is a key membrane property that influences ion transport in waterswollen membranes.16 The volume fraction of water in a membrane is a useful parameter for quantifying water content because it approximates the fractional volume available for ion transport.44,45 Typically, membranes having higher water volume fractions exhibit faster ion transport rates than membranes having lower water volume fractions, all other factors being equal.44,45 Water volume fraction values for the membranes in this study are presented in Figure 2 as a function of external NaCl concentration. Water volume fraction values were relatively constant at low external NaCl concentrations (e.g., 0.1 M or less) and slightly decreased at higher external NaCl concentrations (e.g., greater than 0.1 M). The decrease in membrane water content with increasing external NaCl concentration is attributed to osmotic deswelling.46 These membranes are relatively highly swollen, with water occupying C

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importance of using appropriate membrane ion concentration units consistently, as any inconsistencies could lead to sizable errors in subsequent analysis of ion sorption or transport data. Membrane counterion concentrations were relatively constant at low external solution NaCl concentrations (0.1 M), counterion concentrations increased due to a combined effect of increased co-ion sorption, which must be accompanied by an equivalent increase in counterion sorption to maintain electroneutrality, and decreased membrane water content, which lowers the total membrane volume. Both of these factors contribute to the increase in membrane counterion concentration with increasing external NaCl concentration.34 In sharp contrast to the behavior of membrane counterion concentration, co-ion concentrations depended strongly on external solution salt concentration. This behavior is generally observed in highly charged ion exchange membranes, and it is attributed to the electric field discontinuity (i.e., Donnan potential) at the membrane−solution interface that forms due to an inability of the fixed charged groups in the membrane to cross the phase boundary.16,34,48 The strength of the Donnan potential depends on the counterion concentration difference between the membrane and external salt solution.16,48 For highly charged membranes, the fixed charge group concentration (and therefore the counterion concentration) is typically much greater than the ion concentration in the contiguous salt solution, at concentrations of interest for desalination, and it is relatively constant for membranes in contact with dilute salt solutions. Consequently, as salt concentration in the external solution increases, the counterion concentration difference between the membrane and solution decreases, resulting in a weaker Donnan potential and greater co-ion sorption in the membrane. Mobile salt diffusion coefficients, Dms , for the materials in this study were determined previously from salt permeability, ion sorption, and osmotic water permeability experiments.27,33 The results are presented in Figure 4a as a function of external NaCl

Figure 2. Membrane water volume fraction as a function of external NaCl concentration. The data were obtained from a previous study.27 The lines were drawn to guide the eye.

nearly half of the membrane volume, which simplifies modeling ion transport in these materials, as discussed later. Equilibrium ion concentrations in the membranes were measured using a desorption procedure.34 The results are presented in Figure 3 as a function of NaCl concentration in the contiguous aqueous solution. In studies of ion transport in water-swollen membranes, ion concentration units in the membrane are conveniently expressed as moles of ions per liter of swollen membrane since ion fluxes are typically based on the membrane geometric area (i.e., the membrane area measured after salt diffusion experiments, which includes the polymer and water).16,19,33,47 On the other hand, for a thermodynamic analysis of equilibrium ion partitioning in ion exchange membranes, it is more convenient to express ion concentration units as moles of ions per liter of water imbibed in the membrane (i.e., the volume of polymer chains is excluded), as demonstrated previously.34 For the membranes considered in this report, these two concentration scales differ by approximately a factor of 2 (cf. Table 1). This underscores the

Figure 3. (a) Equilibrium counterion and (b) co-ion concentrations in the membranes as a function of external NaCl concentration. The data were obtained from a previous study.34 The lines were drawn to guide the eye. D

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Figure 4. (a) Local NaCl diffusion coefficients as a function of external NaCl concentration. Dms values have been corrected for frame of reference (i.e., convection) and nonideal thermodynamic effects.27,33 (b) Membrane ionic conductivity as a function of external NaCl concentration. The data were obtained from a previous study.49

concentration. Dms values reported in Figure 4a are local (i.e., differential) salt diffusion coefficients that have been corrected for frame of reference (i.e., convection) and nonideal thermodynamic effects.33 Within experimental uncertainties, Dms values did not change significantly over the external NaCl concentration range considered in this study. This behavior is reasonable since membrane water content did not change appreciably over this salt concentration range. In addition, salt diffusion coefficients in aqueous salt solutions typically do not exhibit a strong concentration dependence, which agrees with the behavior observed in Figure 4a.32 Membrane ionic conductivity values were measured via electrochemical impedance spectroscopy using an experimental technique based on direct contact between the membrane and electrodes.49 The results are presented in Figure 4b as a function of external solution NaCl concentration. At low external NaCl concentrations (0.1 M), equilibrium counterion and co-ion concentrations in the membrane increased (cf. Figure 3), while salt diffusion coefficients remained largely unchanged, resulting in increased ionic conductivity values. Ion Diffusion Coefficients. Individual ion diffusion coefficients, Dmg and Dmc , were calculated from the experimental ion sorption, mobile salt diffusion coefficients, and ionic conductivity results presented in Figures 3 and 4 via the framework outlined in the Background section. A representative calculation for membranes CR61 and AR103 contacted by 0.1 M NaCl solution is provided in the Supporting Information. Calculated ion diffusion coefficients are presented in Figure 5 as a function of external NaCl concentration. For all membranes, ion diffusion coefficients were relatively constant over the

external NaCl concentration range explored in this study. However, a slight decrease in ion diffusion coefficients at higher external salt concentrations (>0.3 M) should not be ruled out. The slight decrease in ion diffusion coefficients at higher external NaCl concentration is consistent with the decrease in membrane water content caused by osmotic deswelling (cf. Figure 2). Interestingly, for all materials, counterion diffusion coefficients were systematically greater than co-ion diffusion coefficients, regardless of which ion served as the counterion (Na+ in the CEM and Cl− in the AEMs). For comparison, Cl− ion diffusion coefficients in aqueous NaCl solutions are approximately 1.5 times greater than Na+ ion diffusion coefficients.32 This relative order of diffusion coefficients is preserved in the AEMs, although Cl− diffusion coefficients are 2.3 and 2.7 times greater on average than Na+ diffusion coefficients for AR103 and AR204, respectively. However, the order of diffusion coefficients is reversed in the CEM, where Na+ ion diffusion coefficients are 1.2 times greater on average than Cl− ion diffusion coefficients. Previous reports on ion diffusion coefficients in ion exchange polymers, often measured using radioactive tracers to probe self-diffusion, have suggested that after accounting for differences in ionic mobilities in solution due to differences in ion size, counterion diffusion coefficients are lower than co-ion diffusion coefficients since counterions experience a strong electrostatic attraction to the fixed charge groups, effectively reducing their mobility.16,19,20,25,26 However, the results presented in Figure 5 suggest the opposite behavior; i.e., counterion diffusion is not retarded due to attractive interactions with the fixed charge groups but instead enhanced. This behavior is rationalized below using a model for ion diffusion coefficients in ion exchange polymers based on ideas from counterion condensation theory. First, a simple molecular description of ion diffusion in homogeneous ion exchange membranes is provided. The membranes used in this study are made from relatively highly swollen (water volume fractions of 0.4−0.5) cross-linked polyelectrolytes. These membranes are considered to be reasonably homogeneous.50 Such materials can be viewed as a collection of randomly distributed polymer chains, which are connected by cross-links, surrounded by tortuous and interconnecting regions of water in which ion transport occurs.19 E

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Figure 5. Counterion, Dmg , and co-ion, Dmc , diffusion coefficients as a function of external NaCl concentration for (a) CR61, (b) AR103, and (c) AR204. Uncertainties were calculated via propagation of errors.

transport in highly swollen uncharged membranes where specific solute/polymer interactions are likely to be minimal. However, electrostatic effects are likely to play an important role in influencing counterion diffusion in ion exchange membranes since counterions spend a significantly longer time on average in the vicinity of the fixed charge groups relative to co-ions. Consequently, the Mackie/Meares model has found better success in describing co-ion diffusion coefficients in ion exchange polymers compared to counterion diffusion coefficients.19,47 Additionally, the model predicts that ion diffusion coefficients go to zero when water volume fraction goes to zero, which is also not likely to be realistic. As a result, the model is most useful for rather highly hydrated polymers. Recently, we proposed using a model developed by Manning to capture electrostatic effects on ion diffusion in ion exchange membranes.27,51 The model was originally formulated for ion diffusion in aqueous solutions containing polyelectrolytes and simple ions. Manning assumed that the polyelectrolyte chains create locally inhomogeneous electric fields, which influence the diffusional path of nearby ions. Consistent with the counterion condensation theory for colligative properties,52 Manning used a Debye−Hückel approximation to describe the inhomogeneous electric field created by a polyelectrolyte chain.51 This model provides predictive (i.e., no adjustable parameters) expressions for counterion and co-ion diffusion coefficients in such systems. Manning’s counterion condensation theory is essential for explaining the results presented in Figure 5. The central parameter in Manning’s counterion condensation theory is the dimensionless linear charge density of a polyelectrolyte, which is defined as52

In this molecular description, the fractional volume available for ion transport in a membrane is characterized by the volume fraction of water. The polymer chain segments undergo Brownian motion due to thermal agitation of the system, so the aqueous regions available for ion transport change over time. However, it is reasonable to assume that polymer chain segment motion occurs on a time scale much longer than that of ion diffusion through a membrane.47 Thus, relative to an ion diffusing across a membrane, the polymer chains can be considered stationary and the network rigid. For relatively highly swollen membranes, the average spacing between chain segments is presumed to be much greater than the size of hydrated ions, so that no physical size sieving occurs. To cross a membrane, ions must diffuse in the interconnecting aqueous regions along a tortuous path formed by the randomly distributed, stationary polymer chains. Ions diffusing through such materials are also subject to electrostatic interactions with the fixed charge groups as well as other specific interactions with the polymer chains. Mackie and Meares used a similar molecular viewpoint of ion diffusion in ion exchange polymers to develop a simple model for predicting ion diffusion coefficients.47 Their model assumes that the mechanism for ion diffusion in a membrane is similar to that in solution and that the tortuosity experienced by the mobile ions is the dominant factor influencing ion diffusion. That is, ions diffuse in the membrane with the same mobility as in aqueous salt solutions, but the average distance traveled by an ion to cross a membrane is longer than the distance to cross an aqueous salt solution of equivalent thickness. Mackie and Meares used a simple statistical treatment, which assumes random distribution of polymer chains onto a cubic lattice, to derive an expression for the tortuosity factor, which only depends on the membrane water volume fraction, ϕw. Their final result is47 Dim ijj ϕw yzz zz = jjj z Dis 2 − ϕ w k {

ξ=

λ e2 = B 4πε0εkTb b

(10)

where e is the protonic charge, ε0 is the vacuum permittivity, ε is the dielectric constant, k is Boltzmann’s constant, T is absolute temperature, b is the average distance between fixed charge groups on the polymer chain, and λB is the Bjerrum length. If ξ is greater than a critical value (ξcrit = 1/|ωzg|, where zg and ω are the counterion and fixed charge group valences, respectively), a sufficient number of counterions condense onto the polymer chain to neutralize some of the fixed charge groups and reduce ξ to the critical value. The remaining uncondensed counterions are more or less free to undergo Brownian motion in any direction. Counterion condensation is a general phenomenon

2

(9)

where Dmi is the diffusion coefficient of ion i in the membrane and Dsi is the diffusion coefficient of ion i in aqueous solution. This model neglects electrostatic interactions between the mobile ions and fixed charge groups as well as any other specific interactions between the ions and polymer chains. This assumption is reasonable for co-ion (or neutral solute) transport in relatively highly swollen charged membranes or for solute F

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Macromolecules for polyelectrolyte solutions caused by overlapping electric fields from closely spaced fixed charge groups.52,53 If ξ is less than ξcrit, counterion condensation does not occur. ξ values for the membranes used in this study were calculated previously from the polymer chemical structure, water content, and ion exchange capacity (i.e., millimoles of fixed charges per gram of dry polymer).34 For the membranes in this study, ξ > ξcrit, so subsequent discussion will be restricted to this case. Manning’s counterion condensation theory has been successfully used to predict ion activity coefficients in the membranes used in this study, motivating our use of Manning’s model to capture electrostatic effects on ion diffusion coefficients in ion exchange membranes.34 For 1:1 electrolytes (e.g., NaCl) and when ξ > ξcrit, the expressions for ion diffusion coefficients derived by Manning for polyelectrolyte solutions and applied to ion exchange membranes are given by27,51 Dgm Dgs

ij 1 i X yyz = fu jjj1 − Ajjjj1; zzzzzzz j 3 k ξ {z{ k

Dcm ijj 1 ij X yzyzz jj1 − Ajjj1; zzzzzz s = j Dc 3 k ξ {{ k

Figure 5a. Thus, this framework fails to accurately explain the data presented in Figure 5. Although Manning initially assumed, for simplicity, that condensed counterions were immobile, it was noted that this assumption was unlikely to be true and that condensed counterions were likely to migrate along the polyelectrolyte chain.51 In subsequent studies, Manning classified condensed counterions as being either “territorially” bound or “site” bound.54 Territorially bound counterions are free to move in a region parallel to the polymer chain but cannot diffuse away from the polymer chains due to the strong overlapping electric fields created by the fixed charge groups. In contrast, site bound counterions are in direct contact with the fixed charge groups, leading to the formation of a complex that is close in nature to that formed between ion pairs in aqueous salt solutions. For polyelectrolyte solutions, territorially bound counterions are much more frequently encountered relative to site bound counterions, and this is especially the case for monovalent counterions.54 Condensed (i.e., territorially bound) counterions play an important role in the polarization of polyelectrolyte solutions in the presence of an electric field.55−62 The ability of condensed counterions to migrate along a polymer backbone could have significant implications for electric field driven ion transport in ion exchange membranes. Since the membranes used in this study are made from crosslinked polymers, it is reasonable to suppose that condensed counterions could traverse the membrane thickness by migrating along the polymer backbone, thereby contributing to a current in the presence of an externally applied electric field. Indeed, a model based on counterion condensation for polarization of polyelectrolyte solutions suggests that at high electric fields condensed counterions carry an electrical current along a polyion by dissociating from the ion rich end of a polyelectrolyte and associating on the ion depleted end of a polyelectrolyte.55−57 Thus, counterion diffusion coefficients in highly charged ion exchange membranes extracted from electric field driven experiments (i.e., membrane ionic conductivity) presumably contain contributions from both uncondensed and condensed counterions, provided counterion condensation occurs (i.e., ξ > ξcrit). In light of these ideas and data, the model for counterion diffusion coefficients has been modified by relaxing the assumption that condensed counterions are immobile. The modified expression for counterion diffusion coefficients in ion exchange membranes is given by

(11)

(12)

where Dsg and Dsc are the counterion and co-ion diffusion coefficients in aqueous solutions, respectively, f u is the fraction of uncondensed counterions, and X = CmA /Cms , where CmA is the fixed charge group concentration and Cms is the mobile salt concentration. For 1:1 electrolytes, the fraction of uncondensed counterions can be calculated from f u = (X/ξ + 1)/(X + 1).52 It follows that fc = 1 − f u. The function A is given by27,51 i Xy Ajjjj1; zzzz = k ξ{

∞

∑ m1 =−∞

ÄÅ É−2 ÅÅ 2ξ ÑÑÑÑ 2 2 Å π + + + ( m m ) 1 ∑ ÅÅÅ 1 Ñ 2 X ÑÑÑÖ Ç m2 =−∞ Å ∞

(m1 , m2) ≠ (0, 0) (13)

Equation 9 can be combined with eqs 11−13 to yield expressions for ion diffusion coefficients in ion exchange membranes that account for tortuosity and electrostatic effects. The final expressions require no adjustable parameters and have been successfully used to predict salt permeability coefficients in the membranes considered in this study.27 Originally, Manning assumed that condensed counterions were immobile, while uncondensed counterions were influenced by the inhomogeneous electric fields generated by the polyelectrolyte chains in the same way as co-ions.51 Consequently, Manning’s model predicts that counterion diffusion coefficients are suppressed by the presence of polyelectrolyte chains to a greater extent than co-ion diffusion coefficients, since the condensed counterions do not contribute to the overall counterion diffusion coefficient. However, this result does not agree with the data presented in Figure 5. For example, within the framework of the Manning and Mackie/Meares models, taking the ratio of counterion to co-ion diffusion coefficients yields Dmg /Dmc = f u(Dsg/Dsc) (note that tortuosity effects cancel out since such effects influence counterion and co-ion diffusion coefficients to the same extent). For NaCl transport in a CEM, Dsg/Dsc ≈ 0.66 and f u is less than unity when ξ > ξcrit, so the model predicts counterion diffusion coefficients to be lower than co-ion diffusion coefficients, which contradicts the results presented in

Dgm = fu Dgm, u + fc Dgm, c

(14)

where f u and fc are the fractions of uncondensed and condensed counterions, respectively, and Dmg,u and Dmg,c are the diffusion coefficients of the uncondensed and condensed counterions, respectively, in the membrane. The expression for uncondensed counterion diffusion coefficients, Dmg,u, is obtained from the Manning and Mackie/Meares models, as was done previously.27,47,51 The expression for condensed counterion diffusion coefficients, Dmg,c, proposed in this study is given by 1 Dgm, c = αDgs (15) 3 The factor 1/3 accounts for transport of condensed counterions along a polymer segment oriented in the direction of flux and not in the other two orthogonal directions. The factor α accounts for the molecular architecture of the polymer and the structured water on or near it. It is expected to be less than unity and G

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Figure 6. (a) Counterion and (b) co-ion diffusion coefficients as a function of external NaCl concentration. The filled symbols denote ion diffusion coefficient values calculated from experimental ion sorption, salt diffusion, and ionic conductivity results, and the solid lines denote ion diffusion coefficients calculated via the model presented in this study eqs 15 and 16.

relative mobilities of condensed and uncondensed counterions. The ratio of condensed counterion diffusion coefficients to uncondensed counterion diffusion coefficients is presented in Figure 7 as a function of external NaCl concentration. When

dependent on the type of counterion and polymer. An expression for this factor is currently not available, so it will be treated as an adjustable constant. The final expressions for counterion and co-ion diffusion coefficients in ion exchange membranes are given by 1 ijj X yzzzyzijj ϕw yzz 1 jij zz + fc α j = f 1 − Ajj1; zzzzjjj j s uj z z Dg 3 k ξ {{k 2 − ϕw { 3 k 2

Dgm

(16)

Dcm ijj 1 ij X yzyzzjij ϕw zyz zz j1 − Ajjj1; zzzzzzjjj s = j j Dc 3 k ξ {{k 2 − ϕw z{ (17) k These expressions contain one adjustable parameter, α, which can be calculated by fitting the model to experimental results. Figure 6 presents a comparison between membrane ion diffusion coefficients calculated from experimental ion sorption, salt diffusion, and ionic conductivity data (symbols) and values calculated via eqs 15 and 16 (solid lines). Values for α, which were fitted to the experimental data, and Dmg,c are recorded in Table 2. For all membranes, agreement between counterion 2

Table 2. ξ, b, α, and Dmg,c Values for the Membranes Considered in This Study membrane

ξ

b (nm)

α

CR61 AR103 AR204

1.8 2.2 2.5

0.73 0.73 0.53

0.68 0.38 0.66

Dmg,c

−6

[×10

Figure 7. Ratio of condensed counterion diffusion coefficients, Dmg,c, to uncondensed counterion diffusion coefficients, Dmg,u, as a function of external NaCl concentration.

2

cm /s]

3.05 2.55 4.52

calculating this ratio, condensed counterion diffusion coefficients were extracted at each external NaCl concentration (i.e., the assumption of constant Dmg,c values was relaxed). Interestingly, for all membranes, condensed counterion diffusion coefficients were approximately 2−2.5 times greater than uncondensed counterion diffusion coefficients. That is condensed counterions appear to migrate faster than uncondensed counterions. One way to rationalize this result is by considering the total distance counterions must travel to traverse a membrane. By migrating along the polymer backbone, condensed counterions travel a shorter distance, on average, to cross the membrane than uncondensed counterions since the latter must migrate around the polymer chains and the locally inhomogeneous electric fields created by the fixed charge groups. These results also suggest that there is no significant

diffusion coefficients calculated from experimental ion sorption and transport results and values calculated from the model was excellent over the entire NaCl concentration range using a constant fitted value for α, which was less than unity, as expected. That is, condensed counterion diffusion coefficients appeared to be relatively independent of external NaCl concentration. Agreement between predicted and experimental co-ion diffusion coefficients was reasonably good, considering the model for co-ion diffusion coefficients required no adjustable parameters. To gain deeper insight into counterion diffusion in highly charged ion exchange membranes, it is helpful to consider the H

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distance to traverse the membrane compared to uncondensed counterions, leading to enhanced effective counterion diffusion coefficients. Moreover, it should be noted that the membranes used by other investigators may have significantly different properties than the membranes used in the present study. For example, the fixed charge group concentrations of the membranes used by Ueda et al.25 were in the range 0.04−0.22 mol/L, which are significantly lower than the fixed charge group concentrations of the membranes used in the present study (2.8−3.2 mol/L). It is reasonable to suppose that counterion condensation did not occur in the membranes used by Ueda et al. due to their very low fixed charge group concentrations, so the effects contributing to fast counterion diffusion mentioned above should not be expected to play a role in this case. The results presented in this study have significant implications for correctly interpreting electric field driven ion transport in ion exchange membranes, which is important in many practical applications, as well as designing membranes with enhanced transport properties. One obvious route to enhance membrane ionic conductivity is to introduce more fixed charge groups, and hence more counterions, into the polymer matrix. Membranes in which the majority of counterions are condensed (i.e., territorially bound) could lead to higher ionic conductivity values, since condensed counterions exhibit higher effective diffusion coefficients than uncondensed counterions. Further studies involving different polymers, as well as different types of electrolytes, are necessary to determine the generality of these findings. Of particular interest are studies on polymers that display properties near the limit of counterion condensation (ξ ≈ ξcrit), where enhanced counterion transport should not be observed, and ions that exhibit strong specific interactions with the fixed charge groups or polymer chains (e.g., perchlorate ions and hydrophobic polymers), which could lead to site bound, immobile counterions. Additionally, the applicability of the ideas presented in this study to highly charged ion exchange membranes made from linear polymers (e.g., Nafion) would be of significant interest. It is clear that additional efforts are necessary to bring the level of understanding of electric field driven ion transport in charged membranes to that in polyelectrolyte solutions.63,64

energy barrier for migration of condensed counterions along a polymer backbone. That is condensed counterions are not strongly bound (i.e., site bound) to the fixed charge groups. Simulation studies on electric field driven counterion transport in polyelectrolyte solutions agree with this observation.59 As mentioned previously, territorially bound counterions are likely to lose part of their hydration in undergoing condensation, resulting in a smaller hydrated ionic size, which could also contribute to faster mobility relative to uncondensed counterions. It should be noted that although the diffusion coefficients of condensed counterions are greater than those of the uncondensed counterion, they are still lower than values in aqueous solutions. The ability of condensed counterions to migrate along a polymer backbone could adequately explain the results presented in Figure 5; i.e., counterion diffusion coefficients are greater than co-ion diffusion coefficients after accounting for differences in inherent ion mobilities in aqueous salt solutions. Counterion diffusion coefficients extracted from electric field driven experiments are effective values, containing contributions from both condensed and uncondensed counterions. Condensed counterion diffusion coefficients were approximately 2− 2.5 times greater than uncondensed counterion diffusion coefficients, which resulted in higher than expected overall counterion diffusion coefficients. Both co-ion as well as condensed and uncondensed counterion diffusion coefficients in the membranes were significantly lower than ion diffusion coefficients in aqueous electrolyte solutions. Interestingly, a number of studies on ion diffusion in polyelectrolyte solutions and ion exchange membranes have reported suppressed counterion diffusion coefficients, after accounting for other effects such as tortuosity, presumably due to strong attractive interactions between the counterions and fixed charge groups.16,19,20,25,26 We speculate that these results can be explained by considering the manner in which the experiments were performed. In the literature studies, ion diffusion coefficients were measured using radioactive tracer techniques, which are typically performed at equilibrium (i.e., no concentration gradients or external electric fields are imposed). Essentially, a small quantity of radioactively labeled ions is introduced into an equilibrated system consisting of a membrane surrounded by aqueous salt solutions, and the motion of the labeled ions across the membrane is monitored to determine ion self-diffusion coefficients. Since condensed and uncondensed counterions are in a state of dynamic equilibrium, the radioactively labeled ions will spend a fraction of their time in the uncondensed state and the remaining time in the condensed state. In the uncondensed state, ions can execute diffusion steps relatively freely in any direction, whereas in the condensed state ions can only execute diffusion steps in a direction parallel to the polymer chain. Moreover, condensed counterion diffusion along the polymer chain will likely be restricted due to repulsive interactions with nearby condensed counterions. The measured counterion diffusion coefficient is a time-averaged value of the ion diffusion coefficients in the condensed and uncondensed states. Thus, in the absence of a driving force (e.g., electric field), it is reasonable to assume that the mobility of condensed counterions may be lower than the mobility of uncondensed counterions, since the Brownian motion of condensed counterions is more restricted than that of uncondensed counterions, resulting in lower average counterion diffusion coefficients. However, as mentioned earlier, in the presence of an external electric field, condensed counterions potentially travel a shorter

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CONCLUSIONS A new methodology was presented for obtaining single ion diffusion coefficients in ion exchange membranes in contact with aqueous salt solutions from experimental ion sorption, mobile salt diffusion coefficients, and ionic conductivity results. The framework was used to extract ion diffusion coefficients in a series of commercial ion exchange membranes over an external NaCl concentration range of 0.01−1 M. Ion diffusion coefficients were relatively constant over the NaCl concentration range considered in this study. Counterion diffusion coefficients were greater than co-ion diffusion coefficients for all membranes, even after accounting for differences in Na+ and Cl− mobilities in aqueous salt solutions. The results were interpreted within the framework of a model based on Manning’s counterion condensation theory, which accounts for electrostatic effects, and the Mackie and Meares model, which accounts for geometric (i.e., tortuosity) effects on ion diffusion. Manning’s original model for ion diffusion coefficients was modified by relaxing the assumption of immobile condensed counterions. This modification appeared justified based on strong evidence in the polyelectrolyte literature for the ability of condensed counterions to migrate I

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(4) Xu, T. W.; Huang, C. H. Electrodialysis-Based Separation Technologies: A Critical Review. AIChE J. 2008, 54 (12), 3147−3159. (5) AlMarzooqi, F. A.; Al Ghaferi, A. A.; Saadat, I.; Hilal, N. Application of Capacitive Deionisation in water desalination: A review. Desalination 2014, 342, 3−15. (6) Porada, S.; Zhao, R.; van der Wal, A.; Presser, V.; Biesheuvel, P. M. Review on the science and technology of water desalination by capacitive deionization. Prog. Mater. Sci. 2013, 58 (8), 1388−1442. (7) Hernandez-Pagan, E. A.; Vargas-Barbosa, N. M.; Wang, T. H.; Zhao, Y. X.; Smotkin, E. S.; Mallouk, T. E. Resistance and polarization losses in aqueous buffer-membrane electrolytes for water-splitting photoelectrochemical cells. Energy Environ. Sci. 2012, 5 (6), 7582− 7589. (8) Heinzel, A.; Barragan, V. M. A review of the state-of-the-art of the methanol crossover in direct methanol fuel cells. J. Power Sources 1999, 84 (1), 70−74. (9) Ji, E.; Moon, H.; Piao, J. M.; Ha, P. T.; An, J.; Kim, D.; Woo, J. J.; Lee, Y.; Moon, S. H.; Rittmann, B. E.; Chang, I. S. Interface resistances of anion exchange membranes in microbial fuel cells with low ionic strength. Biosens. Bioelectron. 2011, 26 (7), 3266−3271. (10) Kariduraganavar, M. Y.; Nagarale, R. K.; Kittur, A. A.; Kulkarni, S. S. Ion-exchange membranes: preparative methods for electrodialysis and fuel cell applications. Desalination 2006, 197 (1−3), 225−246. (11) Kreuer, K. D. On the Development of Proton Conducting Polymer Membranes for Hydrogen and Methanol Fuel Cells. J. Membr. Sci. 2001, 185 (1), 29−39. (12) Merle, G.; Wessling, M.; Nijmeijer, K. Anion exchange membranes for alkaline fuel cells: A review. J. Membr. Sci. 2011, 377 (1−2), 1−35. (13) Geise, G. M.; Lee, H. S.; Miller, D. J.; Freeman, B. D.; Mcgrath, J. E.; Paul, D. R. Water Purification by Membranes: The Role of Polymer Science. J. Polym. Sci., Part B: Polym. Phys. 2010, 48 (15), 1685−1718. (14) Kamcev, J.; Freeman, B. D. Charged Polymer Membranes for Environmental/Energy Applications. Annu. Rev. Chem. Biomol. Eng. 2016, 7 (1), 111−133. (15) Park, H. B.; Kamcev, J.; Robeson, L. M.; Elimelech, M.; Freeman, B. D. Maximizing the right stuff: The trade-off between membrane permeability and selectivity. Science 2017, 356 (6343),.eaab0530. (16) Helfferich, F. Ion Exchange; Dover Publications: New York, 1995. (17) Paul, D. R. Reformulation of the solution-diffusion theory of reverse osmosis. J. Membr. Sci. 2004, 241 (2), 371−386. (18) Wijmans, J. G.; Baker, R. W. The solution-diffusion model: a review. J. Membr. Sci. 1995, 107, 1−21. (19) Crank, J.; Park, G. Diffusion in Polymers; Academic Press: London, 1968. (20) Lakshminarayanaiah, N. Transport Phenomena in Membranes; Academic Press: London, 1972. (21) Fernandez-prini, R.; Philipp, M. Tracer Diffusion-Coefficients of Counterions in Homoionic and Heteroionic Poly(Styrenesulfonate) Resins. J. Phys. Chem. 1976, 80 (18), 2041−2046. (22) Goswami, A.; Acharya, A.; Pandey, A. K. Study of self-diffusion of monovalent and divalent cations in Nafion-117 ion-exchange membrane. J. Phys. Chem. B 2001, 105 (38), 9196−9201. (23) Stolwijk, N. A.; Wiencierz, M.; Fogeling, J.; Bastek, J.; Obeidi, S. The Use of Radiotracer Diffusion to Investigate Ionic Transport in Polymer Electrolytes: Examples, Effects, and Their Evaluation. Z. Phys. Chem. 2010, 224 (10−12), 1707−1733. (24) Yeager, H. L.; Steck, A. Cation and Water Diffusion in Nafion Ion-Exchange Membranes - Influence of Polymer Structure. J. Electrochem. Soc. 1981, 128 (9), 1880−1884. (25) Ueda, T.; Ishida, N.; Kamo, N.; Kobatake, Y. Effective Fixed Charge-Density Governing Membrane Phenomena. 4. Further Study of Activity-Coefficients and Mobilities of Small Ions in Charged Membranes. J. Phys. Chem. 1972, 76 (17), 2447−2452. (26) Ueda, T.; Kobatake, Y. Effective Fixed Charge-Density Governing Membrane Phenomena. 6. Activity-Coefficients and Mobilities of Small Ions in Aqueous-Solutions of Poly(Styrenesulfonic Acid). J. Phys. Chem. 1973, 77 (25), 2995−2998.

along a polyelectrolyte backbone. The model predicted co-ion diffusion coefficients reasonably well with no adjustable parameters. However, a single adjustable parameter (the condensed counterion diffusion coefficient) was necessary to obtain excellent agreement between counterion diffusion coefficients calculated from experimental results and values calculated from the model. The results in this study suggest that condensed counterions in highly charged, cross-linked ion exchange membranes migrate along the polymer backbone and contribute to an electrical current in the presence of an externally applied electric field. For the materials considered in this study, condensed counterion diffusion coefficients were approximately 2−2.5 times greater than uncondensed counterion diffusion coefficients, explaining the enhanced counterion diffusion in these membranes. This observation was attributed to the shorter distance traveled by condensed counterions to cross the membrane (i.e., along the polymer chains) compared to the distance traveled by the uncondensed counterions (around the polymer chains and inhomogeneous electric fields) and the likely smaller hydrated size of condensed counterions.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b00645.

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Calculating ion diffusion coefficients; Tables S1 and S2 (PDF)

AUTHOR INFORMATION

Corresponding Author

*(B.D.F.) E-mail [email protected]; Tel +1-512-2322803; Fax +1-512-232-2807. ORCID

Gerald S. Manning: 0000-0002-6210-4353 Benny D. Freeman: 0000-0003-2779-7788 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This material is based upon work supported in part by the National Science Foundation (NSF) Graduate Research Fellowship under Grant No. DGE-1110007, the Welch Foundation Grant No. F-1924-20170325, and by the Australian-American Fulbright Commission for the award to B.D.F. of the U.S. Fulbright Distinguished Chair in Science, Technology and Innovation sponsored by the Commonwealth Scientific and Industrial Research Organization (CSIRO).

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REFERENCES

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K

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