Predicting Salt Permeability Coefficients in Highly ... - ACS Publications

Research Article www.acsami.org

Predicting Salt Permeability Coefficients in Highly Swollen, Highly Charged Ion Exchange Membranes Jovan Kamcev,† Donald R. Paul,† Gerald S. Manning,‡ and Benny D. Freeman*,† †

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 ABSTRACT: This study presents a framework for predicting salt permeability coefficients in ion exchange membranes in contact with an aqueous salt solution. The model, based on the solution−diffusion mechanism, was tested using experimental salt permeability data for a series of commercial ion exchange membranes. Equilibrium salt partition coefficients were calculated using a thermodynamic framework (i.e., Donnan theory), incorporating Manning’s counterion condensation theory to calculate ion activity coefficients in the membrane phase and the Pitzer model to calculate ion activity coefficients in the solution phase. The model predicted NaCl partition coefficients in a cation exchange membrane and two anion exchange membranes, as well as MgCl2 partition coefficients in a cation exchange membrane, remarkably well at higher external salt concentrations (>0.1 M) and reasonably well at lower external salt concentrations ( ξcrit (cf. Table 1).40,41 For ion exchange membranes equilibrated with 1:1 electrolytes and when ξ > ξcrit, the counterion and co-ion activity coefficients are calculated as follows41 1 m,w ⎤ ⎛ C m , w/ξ + Csm , w ⎞ ⎡ C 2 A ⎢ ⎥ γgm = ⎜ A m , w m , w ⎟exp⎢ − m , w m,w ⎥ ⎝ CA + Cs ⎠ ⎣ CA + 2ξCs ⎦

(5) Cm,w s

(3)

(4)

Dsm =

Combining eqs 1, 3, and 4 yields 4046

DgmDcm(zg2Cgm , p + zc2Ccm , p) zg2DgmCgm , p + zc2DcmCcm , p

(8) DOI: 10.1021/acsami.6b14902 ACS Appl. Mater. Interfaces 2017, 9, 4044−4056

Research Article

ACS Applied Materials & Interfaces 2 Dim ⎛ ϕw ⎞ ⎟ ⎜ = ⎟ ⎜ Dis ⎝ 2 − ϕw ⎠

where zg and zc are the counterion and co-ion valences, respectively, Dmg and Dmc are the counterion and co-ion diffusion coefficients in the m,p membrane, respectively, and Cm,p g and Cc are the counterion and coion concentrations in the membrane (moles of ions per liter of swollen membrane), respectively. Co-ion and counterion concentrations in the membrane can be predicted using the Donnan/Manning approach outlined above.40 In this study, we presume ion diffusion coefficients in highly swollen, highly charged membranes are affected by tortuosity effects and by electrostatic effects due to the membrane’s fixed charge groups. A model proposed by Mackie and Meares43 was used to account for the tortuosity effects, and Manning’s counterion condensation theory extended to ion diffusion44 was used to account for the electrostatic effects. Another commonly used physical model for describing salt transport in water-swollen polymers is Yasuda’s free volume theory, although this model requires the use of an adjustable constant.2,48 In this study, we elected to use the Mackie and Meares model due to its simplicity and predictive nature. It is believed that free volume theory could also accurately describe salt diffusion coefficients for the materials in this study, but the framework for predicting salt permeability coefficients would no longer be predictive. Mackie and Meares assumed that ion diffusion coefficients in waterswollen ion exchange membranes are closely related to ion diffusion coefficients in aqueous solutions.43 In their picture, the swollen membrane consists of a rigid polymer network structure surrounded by interconnecting, tortuous capillaries containing sorbed water and ions. The fixed charge groups bonded to the polymer chains protrude into the solvent phase. A simple illustration of this concept is presented in Figure 2. Molecular motions of the polymer chains are

(9)

Dsi

where is the diffusion coefficient of ion i in an aqueous solution, which can be found in the literature,50 and ϕw is the volume fraction of water in the membrane. This model requires no adjustable parameters, provided that ϕw is known or can be measured. Manning applied the principles from his counterion condensation theory to treat ion diffusion in dilute solutions containing polyelectrolytes.44 Manning presumed that the polyelectrolyte chains form a locally inhomogeneous electric field, which affects ion diffusion in such systems. In accordance with his counterion condensation theory, he described the inhomogeneous electric field using a Debye− Hückel approximation. In this treatment, it was assumed that condensed counterions have no mobility. Manning did note, however, that condensed counterions might be able to move along a polyelectrolyte chain, but they cannot easily move away from the chain.44,51−53 Indeed, in the polyelectrolyte literature, there is good evidence that condensed counterions are mobile.54−56 In our approach for modeling salt permeability in ion exchange membranes, we preserve the assumption of immobile condensed counterions. This approximation is unlikely to have a significant effect on the final results since concentration gradient-driven transport in highly charged ion exchange membranes is largely governed by co-ion transport.32 In a future study of electric field-driven transport in ion exchange membranes (i.e., ionic conductivity), which is largely governed by counterion transport,32 the assumption of immobile condensed counterions will be relaxed. For the case where ξ > ξcrit, the general equations for counterion and co-ion diffusion coefficients, Dmg and Dmc , respectively, are44 ⎛ X + 1 ⎞⎛ ⎞ ⎟⎜ ⎜ zg2νgξ 1 2 ⎛⎜ 1 X ⎞⎟⎟ 1 z A ; = − g ⎜ ⎟ ⎜ X ⎟⎟ Dgs 3 ⎜ + 1 ⎟⎜⎝ ⎝ |zg | |zg |ξ ⎠⎠ ⎠ ⎝ |zg | νg

(10)

⎛ ⎞ Dcm ⎜ 1 2 ⎛⎜ 1 X ⎞⎟⎟ = 1 − z A ; c ⎜ ⎟⎟ ⎜ Dcs 3 ⎝ |zg | |zg |ξ ⎠⎠ ⎝

(11)

Dgm

Dsg

Dsc

where and are the counterion and co-ion diffusion coefficients in an aqueous solution, respectively, zg and zc are the counterion and coion valences, respectively, νg and νc are the number of counterions and m,w co-ions in a salt molecule, respectively, and X = Cm,w A /Cs . The function A is given by44

Figure 2. Illustration of salt diffusion in an ion exchange membrane.

much slower than molecular motions of the mobile ions, so the polymer chains are essentially fixed entities relative to the mobile ions,49 that is, ion diffusion is envisioned to occur within an aqueous phase inside the membrane, and the polymer chains act as impenetrable obstructions. According to Mackie and Meares, the ion mobility at any point in the membrane is equivalent to the ion mobility in an aqueous solution. However, the distance traveled by ions across a membrane of a given thickness is greater than the distance traveled by ions in an aqueous solution of equivalent thickness, since the ions must diffuse around the polymer chains, thereby leading to a measured salt diffusion coefficient lower than that observed in aqueous solution. Mackie and Meares used a lattice approach, in which the polymer chain segments were randomly distributed on a cubic lattice, to estimate the increase in diffusional path length due to the presence of stationary polymer chains, resulting in a simple expression for the diffusion coefficient of ion i in the membrane, Dmi . The final result is

Manning’s equations for ion diffusion coefficients require no adjustable parameters since ξ and X are known for the materials considered in this study. Equations 10−12 can be combined with eq 9 to yield final expressions for counterion and co-ion diffusion coefficients in charged membranes that account for tortuosity and electrostatic effects. For a 1:1 electrolyte, the final expressions are 2 ⎛ X /ξ + 1 ⎞⎛ 1 ⎛ X ⎞⎞⎛ ϕw ⎞ ⎜ ⎟ ⎟⎟ ⎜1 − A⎜1; ⎟⎟⎜⎜ = ⎝ X + 1 ⎠⎝ Dgs 3 ⎝ ξ ⎠⎠⎝ 2 − ϕw ⎠

Dgm

4047

(13)

DOI: 10.1021/acsami.6b14902 ACS Appl. Mater. Interfaces 2017, 9, 4044−4056

Research Article

ACS Applied Materials & Interfaces 2 Dcm ⎛ 1 ⎛ X ⎞⎞⎛ ϕw ⎞ ⎟ ⎜ ⎜ ⎟ 1 A 1; = − ⎜ ⎟ ⎟ ⎜ Dcs 3 ⎝ ξ ⎠⎠⎝ 2 − ϕw ⎠ ⎝

electrical potential (i.e., the Donnan potential) at the membrane/solution interface that inhibits counterions from desorbing from the membrane into the external solution and co-ions from sorbing from the external solution into the membrane.32,41 This phenomenon is often referred to as Donnan exclusion.32 The strength of the Donnan potential dictates the extent of co-ion exclusion from a charged membrane and depends on the counterion activity (i.e., concentration) difference between the membrane and the contiguous solution.32 The counterion concentration in the membrane depends largely on the membrane fixed charge group concentration, Cm,w A , which changes little over the salt concentration range considered here,41 since most counterions are present to electrically balance the fixed charge groups, and osmotic deswelling is not significant (see later discussion on Membrane Water Content). Thus, for a given membrane, the Donnan potential depends strongly on the salt activity in the external solution, which is readily expressed in terms of the external solution salt concentration, Css. Consequently, the extent of Donnan exclusion and, in turn, salt partition coefficients depend on Css. As external salt concentration increases, the counterion concentration difference between the membrane and the solution decreases and the Donnan potential decreases. This decrease in Donnan potential enhances co-ion sorption in the membrane.32 The strength of the Donnan potential also depends on the counterion and co-ion valences.32 All other factors being equal, the Donnan potential is stronger for co-ions having a higher valence, resulting in stronger co-ion exclusion.32 For example, for a given counterion, negatively charged membranes exclude electrolytes having multivalent co-ions (e.g., Na2SO4) much more efficiently than electrolytes having monovalent co-ions (e.g., NaCl).31 However, the Donnan potential is weaker for counterions having a higher valence, leading to weaker co-ion exclusion.32 Consequently, MgCl2 partition coefficients in the cation exchange membrane were greater than NaCl partition coefficients at a given external salt concentration. Moreover, the concentration dependence of the Donnan potential is weaker for counterions with a higher valence. Thus, the concentration

(14)

where

The local salt diffusion coefficient in a membrane, Dms (cf. eq 8), can be predicted by using eqs 9−12 to calculate counterion and co-ion diffusion coefficients in the membrane and the Donnan/Manning model to calculate counterion and co-ion concentrations in the membrane. The salt partition coefficient can be predicted using the Donnan/Manning model.40 Thus, the salt permeability coefficient, ⟨Ps⟩, can be predicted from ⟨Ps⟩ = Ks·Dms . For the membranes considered in this study, this approach required no adjustable parameters.

■

RESULTS AND DISCUSSION Salt Partition Coefficients. For the materials considered in this study, salt partition coefficients as a function of external salt concentration were calculated from previously reported ion sorption data via eq 7.40,41 The results are presented in Figure 3. For all membranes, salt partition coefficients increased with increasing external salt concentration. For membranes equilibrated with an aqueous NaCl solution, NaCl partition coefficients increased by approximately 1 order of magnitude as the NaCl concentration in the external solution increased from 0.01 to 1 M. However, for MgCl2 sorption in the cation exchange membrane, the increase in salt partition coefficients with external MgCl2 concentration was much less pronounced. This behavior is in contrast to that observed for uncharged membranes, where salt partition coefficients are almost independent of external salt concentration.38,39 The strong concentration dependence of salt partition coefficients in charged polymers results from the presence of fixed charge groups on their backbone. As mentioned previously, the distribution of mobile ions between a membrane and a contiguous aqueous salt solution is unequal for charged membranes in contact with such solutions.32,41 This unequal distribution of mobile ions between the two phases results in an

Figure 3. (a) NaCl and MgCl2 partition coefficients as a function of external salt concentration for cation exchange membrane CR61. (b) NaCl partition coefficients as a function of external NaCl concentration for anion exchange membranes AR103 and AR204. Filled symbols denote experimental data, and dashed lines denote values calculated using the Donnan/Manning approach. Experimental data were obtained at ambient conditions.40,41 4048

DOI: 10.1021/acsami.6b14902 ACS Appl. Mater. Interfaces 2017, 9, 4044−4056

Research Article

ACS Applied Materials & Interfaces

Figure 4. Membrane counterion concentration as a function of external salt concentration for (a) NaCl and MgCl2 sorption in cation exchange membrane CR61 and (b) NaCl sorption in anion exchange membranes AR103 and AR204. Filled symbols denote experimental data, and dashed line denotes values calculated using the Donnan/Manning approach. Experimental data were obtained at ambient conditions.40

concentration, CmA is the fixed charge group concentration, and Cms is the mobile salt concentration.32,40,41 The analogous expression for 1:2 electrolyte (e.g., MgCl2) sorption in a cation exchange membrane is Cmg = CmA /2 + Cms .32,40,41 Membrane counterion concentrations predicted using the Donnan/ Manning approach, as well as those measured experimentally, for the materials considered in this study are presented in Figure 4 as a function of external salt concentration.41 Membrane counterion concentrations increased with increasing external salt concentration. This increase is due to increased mobile salt sorption and decreased membrane water content (i.e., osmotic deswelling).41 Agreement between model and experimental data was remarkably good for all membranes over the entire salt concentration range studied, particularly considering that the model contains no adjustable parameters. In contrast to the results presented for co-ion sorption (i.e., salt partition coefficients), agreement between calculated and experimental counterion concentrations at low external salt concentrations was excellent. At these conditions, the membrane co-ion concentration is negligible compared to the membrane fixed charge group concentration, which is accurately known from IEC and water content, so the disagreement between model and experimental results for coion sorption does not influence the calculated membrane counterion concentrations. Membrane Water Content. The volume fraction of water in a polymer, ϕw, is often used to describe salt diffusion in water-swollen polymers, since it approximates the free volume (i.e., the volume in which diffusion occurs) of a polymer.1,38,48,58−61 Typically, ion diffusion occurs faster in membranes having higher water volume fractions (i.e., higher free volume) than in membranes having lower water volume fraction, all other factors being equal.1,38,48,58−61 The water volume fraction, ϕw, and ion diffusion coefficient in pure solution, Dsi , are the only parameters required to predict ion diffusion coefficients in a water-swollen membrane using the Mackie and Meares model.42,43 Experimentally measured water volume fractions for the materials considered in this study are presented in Figure 5 as a function of external salt concentration. For all materials, water volume fractions decreased somewhat with increasing external salt concentration due to osmotic deswelling. As the salt concentration in the external solution

dependence of MgCl2 partition coefficients is less pronounced than that for NaCl partition coefficients. Salt partition coefficients calculated using the Donnan/ Manning approach presented in the Model Development section are compared to experimental values in Figure 3. For the materials in this study, the model predicted salt partition coefficients reasonably well over the external salt concentration range considered, particularly given that the model uses no adjustable parameters and is, therefore, entirely predictive. Agreement between model and experimental data was remarkably good at higher salt concentration (>0.03 M) (cf. Figure 3). Some discrepancy between model and experimental data was evident at low salt concentrations (0.1 M). However, some discrepancy between predicted and experimental values was evident at low upstream salt concentrations (e.g., < 0.1 M). This result is reasonable since both the sorption and the diffusion models yielded better predictions for salt sorption and salt diffusion coefficients, respectively, at higher salt concentrations com4052

DOI: 10.1021/acsami.6b14902 ACS Appl. Mater. Interfaces 2017, 9, 4044−4056

Research Article

ACS Applied Materials & Interfaces

Figure 10. (a) Predicted NaCl and MgCl2 permeability coefficients for cation exchange CR61. (b) Predicted NaCl permeability coefficients for anion exchange membranes AR103 and AR204. Dashed lines denote salt permeability coefficients calculated by the rigorous form of the model, and solid lines denote salt permeability coefficients calculated by the approximate form of the model (i.e., eqs 19 and 20).

⎛ 2.718(γ s )2 C sξ ⎞ ⎛ ϕ ⎞2 Csm , p m s ± ⎜ ⎟⎟Dcs⎜⎜ w ⎟⎟ Ps = s Ds = ϕw ⎜ m,w CA Cs ⎝ ⎠ ⎝ 2 − ϕw ⎠

pared to lower salt concentrations. At low salt concentrations, the Donnan/Manning model underestimated salt partition coefficients while the Mackie/Meares and Manning models overestimated salt diffusion coefficients, so these two effects partially cancel each other. This result is perhaps a fortuitous coincidence for these materials. Nevertheless, further investigation is required to improve both models at lower salt concentrations. For highly charged membranes in contact with relatively dilute aqueous salt solutions, a simple analytical expression for salt permeability coefficients can be derived using the framework presented in this study. For this case, the membrane counterion concentration is significantly greater than the membrane co-ion concentration (i.e., Cmg ≫ Cmc ). Moreover, the membrane counterion concentration is essentially equal to the membrane fixed charge group concentration. Introducing m,w the assumption Cm,w into eq 5 yields A ≫ Cs Csm , w ≈

For 1:2 electrolyte (e.g., MgCl2) transport in CEMs, combining eqs 17 and 18 yields ⎛ 8.468(γ s )3 C sξ ⎞1/2 ⎛ ϕ ⎞2 s ± ⎟⎟ Dcs⎜⎜ w ⎟⎟ Ps = ϕw⎜⎜ m,w C ⎝ 2 − ϕw ⎠ A ⎝ ⎠

(16)

This expression is valid for highly charged membranes equilibrated with relatively dilute 1:1 salt solutions (e.g., NaCl). The equivalent expression for CEMs equilibrated with 1:2 electrolytes (e.g., MgCl2) is given by Csm , w

⎛ 8.468(γ s )3 (C s)3 ξ ⎞1/2 s ± ⎟⎟ ≈ ⎜⎜ m,w C A ⎝ ⎠

(17)

m,p Introducing the assumption Cm,p g ≫ Cc in eq 8 and using eq 9 for the membrane co-ion diffusion coefficient yields

Dsm

≈

Dcm

⎛

=

⎞2 ⎟⎟ ⎝ 2 − ϕw ⎠

Dcs⎜⎜

(20)

Salt permeability coefficients calculated using eqs 19 and 20 were compared to those calculated using the more rigorous form of the model. The results are presented in Figure 10. At lower upstream salt concentrations (e.g., Css < 0.3 M), agreement between the approximate and rigorous forms of the model was reasonably good, as expected. However, deviation between the two models is observed at higher upstream salt concentration (e.g., Css > 0.1 M). This finding is reasonable since the membrane co-ion concentration becomes significant relative to the membrane counterion concentration m,p as Css increases, making the assumption Cm,p less valid. g ≫ Cc In addition to being useful approximations, eqs 19 and 20 reveal a connection between membrane properties and salt permeability coefficients for ion exchange membranes. For example, increasing the membrane fixed charge group concentration should decrease salt permeability coefficients, all other factors being equal. Moreover, for a given increase in membrane fixed charge group concentration, the salt permeability decrease for NaCl would be greater than that for MgCl2 in a cation exchange membrane. Within the framework of this model, salt permeability coefficients also depend on the Manning parameter, ξ, which, in turn, depends on membrane morphology (i.e., average distance between charged groups). According to eqs 19 and 20, decreasing ξ should decrease salt permeability coefficients, since lower ξ values lead to fewer condensed counterions, which results in enhanced co-ion exclusion (i.e., lower salt partition coefficients) due to a stronger Donnan potential at the membrane/solution interface.64 The reasonably good agreement between the model and the experimental data is rather remarkable considering the simplicity of the assumptions inherent in the framework presented here and the lack of adjustable parameters. The

2.718(γ±s )2 (Css)2 ξ CAm , w

(19)

ϕw

(18)

In deriving eq 18, electrostatic effects on co-ion diffusion coefficients were neglected. This approach is a reasonable first approximation, since the tortuosity effect is the main factor influencing co-ion diffusion in these materials (cf. Figure 7). However, this assumption could be easily relaxed. Finally, for 1:1 electrolytes, combining eqs 16 and 18 yields 4053

DOI: 10.1021/acsami.6b14902 ACS Appl. Mater. Interfaces 2017, 9, 4044−4056

Research Article

ACS Applied Materials & Interfaces

Predicted values were compared to experimental NaCl and MgCl2 partition coefficients in a CEM and NaCl partition coefficients in two AEMs. Agreement between predicted and experimental salt partition coefficients was exceptionally good at higher external salt concentrations (e.g., Css > 0.1 M), but some deviation between the two was observed at lower external salt concentrations (e.g., Css < 0.1 M). Ion diffusion coefficients in the membrane were predicted using the Mackie and Meares model to account for tortuosity effects and the Manning model to account for electrostatic effects. The model did a remarkably good job of predicting salt diffusion coefficients in the charged membranes. On the basis of the model, the tortuosity effect is the dominant factor influencing salt diffusion in the materials considered in this study, accounting for about an order of magnitude decrease in membrane ion diffusion coefficients relative to values in aqueous solution. Electrostatic effects accounted for about a 15% decrease in membrane ion diffusion coefficients relative to values in aqueous solution for NaCl transport in the IEMs and about a 5% decrease for MgCl2 transport in the CEM. Salt permeability coefficients for the charged membranes were predicted using the solution− diffusion model. Agreement between predicted and experimental values was remarkably good with no adjustable parameters. As with salt diffusion and partition coefficients, agreement between model and experimental values was better at higher upstream salt concentration compared to that at lower upstream salt concentration. To the best of our knowledge, this study reports the first example where salt permeability coefficients in charged membranes were predicted, to a reasonable first approximation, with no adjustable parameters.

Donnan/Manning model assumes that membrane phase nonidealities (i.e., the deviation of ion activity coefficients from unity) are entirely due to local electrostatic interactions between the fixed charge groups and the mobile ions.40,41,46 Electrostatic interactions between different polymer chains or between distant segments on the same chain as well as specific polymer/ion interactions are neglected.46 A breakdown in these assumptions may occur at low salt concentrations, as discussed previously (cf. Figure 2). The Mackie and Meares model assumes that ion diffusion coefficients in charged membranes are affected by a tortuosity factor, while the Manning model assumes ion diffusion coefficients are affected by a locally inhomogeneous electric field, which is described by a Debye−Hückel approximation.42,43 Specific interactions between polymer chains and mobile ions are neglected. On the basis of the reasonably good agreement between model and experimental data, the assumptions in the Manning and Mackie/Meares models appear to be plausible for the materials considered in this study and, perhaps, for other commercial ion exchange membranes having similar properties. Indeed, previous studies reported activation energies for co-ion diffusion in ion exchange polymers to be similar to those in aqueous solutions.42 Moreover, from the Manning model for ion diffusion, electrostatic effects account for approximately 15% of the reduction in co-ion diffusion coefficients for NaCl transport in the materials considered in this study, which supports the notion that fixed charge groups do not significantly influence co-ion diffusion in such materials. This result is reasonable for highly swollen membranes, since co-ions are likely to diffuse in aqueous regions as far away as possible from the fixed charge groups to minimize electrostatic repulsion between co-ions and fixed charge groups. In contrast, a large portion of counterions spend a significant time in close proximity to the fixed charge groups, so counterion diffusion is expected to be more strongly influenced by the fixed charge groups than co-ion diffusion. Indeed, activation energies for counterion diffusion in ion exchange polymers are typically lower than those observed in aqueous solutions.42 Further investigation is necessary to test the applicability of the proposed model for predicting salt permeability coefficients in membranes having different chemical and physical structures as well as to identify conditions where the model breaks down. Charged membranes having lower water content are of particular interest, since electrostatic interactions between different polymer chains and specific polymer/ion interactions may play a bigger role in affecting salt sorption and diffusion coefficients in such systems. Moreover, charged membranes having ordered microstructures (e.g., phase separated membranes such as Nafion) are of great scientific and practical interest.65,66 An improved fundamental understanding of ion sorption and transport in charged membranes having a broad range of membrane properties could catalyze development of high-performance next-generation materials.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: 512-232-2803. Fax: 512-232-2807. ORCID

Benny D. Freeman: 0000-0003-2779-7788 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This material is based upon work supported in part by the National Science Foundation (NSF) Graduate Research Fellowship under Grant No. DGE-1110007 and by the NSF CBET program (CBET-1160128). The authors thank Dr. Neil Moe and Dr. John Barber from GE Power and Water for kindly providing the membranes used in this study.

■

■

CONCLUSIONS A theoretical framework for predicting salt permeability coefficients in ion exchange membranes was developed and tested using experimental salt permeability data. Salt partition coefficients were predicted using Donnan theory, where membrane ion activity coefficients were calculated using Manning’s counterion condensation theory, and solution ion activity coefficients were calculated using the Pitzer model. 4054

NOMENCLATURE b, distance between fixed charge groups on a polymer chain Cm,w A , membrane fixed charge group concentration (mols per L of sorbed water) Cm,w s , membrane mobile salt concentration (mols per L of sorbed water) Cm,p s , membrane mobile salt concentration (mols per L of swollen membrane) Cm,p g , membrane counterion concentration (mols per L of swollen membrane) Cm,p c , membrane co-ion concentration (mols per L of swollen membrane) Css, external solution salt concentration ⟨D̅ ms *⟩, membrane apparent salt diffusion coefficient DOI: 10.1021/acsami.6b14902 ACS Appl. Mater. Interfaces 2017, 9, 4044−4056

Research Article

ACS Applied Materials & Interfaces Dms , membrane local salt diffusion coefficient Dmg , membrane counterion diffusion coefficient Dmc , membrane co-ion diffusion coefficient Dsg, external solution counterion diffusion coefficient Dsc, external solution co-ion diffusion coefficient e, protonic charge ε, dielectric constant ε0, vacuum permittivity ϕw, water volume fraction γmg , membrane counterion activity coefficient γmc , membrane co-ion activity coefficient γs±, external solution salt mean activity coefficient Ks, salt sorption coefficient k, Boltzmann constant λB, Bjerrum length ω, fixed charge group valence ⟨Ps⟩, average salt permeability coefficient ρw, density of water T, temperature wu, water uptake (g of water per g of dry polymer) ξ, dimensionless linear charge density of a polymer (Manning parameter) ξcrit, critical dimensionless linear charge density of a polymer zg, counterion valence zc, co-ion valence

■

(14) 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. (15) Roquebert, V.; Booth, S.; Cushing, R. S.; Crozes, G.; Hansen, E. Electrodialysis Reversal (Edr) and Ion Exchange as Polishing Treatment for Perchlorate Treatment. Desalination 2000, 131 (1−3), 285−291. (16) Xu, T. W.; Huang, C. H. Electrodialysis-Based Separation Technologies: A Critical Review. AIChE J. 2008, 54 (12), 3147−3159. (17) Anderson, M. A.; Cudero, A. L.; Palma, J. Capacitive Deionization as an Electrochemical Means of Saving Energy and Delivering Clean Water. Comparison to Present Desalination Practices: Will It Compete? Electrochim. Acta 2010, 55 (12), 3845− 3856. (18) 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. (19) Kim, Y. C.; Elimelech, M. Potential of Osmotic Power Generation by Pressure Retarded Osmosis Using Seawater as Feed Solution: Analysis and Experiments. J. Membr. Sci. 2013, 429, 330− 337. (20) Post, J. W.; Veerman, J.; Hamelers, H. V. M; Euverink, G. J. W.; Metz, S. J.; Nymeijer, K.; Buisman, C. J. N. Salinity-Gradient Power: Evaluation of Pressure-Retarded Osmosis and Reverse Electrodialysis. J. Membr. Sci. 2007, 288 (1−2), 218−230. (21) Thorsen, T.; Holt, T. The Potential for Power Production from Salinity Gradients by Pressure Retarded Osmosis. J. Membr. Sci. 2009, 335 (1−2), 103−110. (22) Dlugolecki, P.; Gambier, A.; Nijmeijer, K.; Wessling, M. Practical Potential of Reverse Electrodialysis as Process for Sustainable Energy Generation. Environ. Sci. Technol. 2009, 43 (17), 6888−6894. (23) Turek, M.; Bandura, B. Renewable Energy by Reverse Electrodialysis. Desalination 2007, 205 (1−3), 67−74. (24) Veerman, J.; de Jong, R. M.; Saakes, M.; Metz, S. J.; Harmsen, G. J. Reverse Electrodialysis: Comparison of Six Commercial Membrane Pairs on the Thermodynamic Efficiency and Power Density. J. Membr. Sci. 2009, 343 (1−2), 7−15. (25) Yip, N. Y.; Vermaas, D. A.; Nijmeijer, K.; Elimelech, M. Thermodynamic, Energy Efficiency, and Power Density Analysis of Reverse Electrodialysis Power Generation with Natural Salinity Gradients. Environ. Sci. Technol. 2014, 48 (9), 4925−4936. (26) Hickner, M. A. Ion-Containing Polymers: New Energy & Clean Water. Mater. Today 2010, 13 (5), 34−41. (27) Hickner, M. A.; Ghassemi, H.; Kim, Y. S.; Einsla, B. R.; McGrath, J. E. Alternative Polymer Systems for Proton Exchange Membranes (Pems). Chem. Rev. 2004, 104 (10), 4587−4611. (28) 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. (29) Merle, G.; Wessling, M.; Nijmeijer, K. Anion Exchange Membranes for Alkaline Fuel Cells: A Review. J. Membr. Sci. 2011, 377 (1−2), 1−35. (30) Ghaffour, N.; Missimer, T. M.; Amy, G. L. Technical Review and Evaluation of the Economics of Water Desalination: Current and Future Challenges for Better Water Supply Sustainability. Desalination 2013, 309, 197−207. (31) Baker, R. W. Membrane Technology and Applications, 2nd ed.; J. Wiley: Chichester, New York, 2004. (32) Helfferich, F. Ion Exchange; Dover Publications: New York, 1995. (33) Sata, T. Ion Exchange Membranes; The Royal Society of Chemistry: Cambridge, 2004. (34) Strathmann, H. Ion-Exchange Membrane Separation Processes; Elsevier: Amsterdam, 2004. (35) Paul, D. R. Reformulation of the Solution-Diffusion Theory of Reverse Osmosis. J. Membr. Sci. 2004, 241 (2), 371−386. (36) Wijmans, J. G.; Baker, R. W. The Solution-Diffusion Model - a Review. J. Membr. Sci. 1995, 107 (1−2), 1−21.

REFERENCES

(1) 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. (2) Geise, G. M.; Paul, D. R.; Freeman, B. D. Fundamental Water and Salt Transport Properties of Polymeric Materials. Prog. Polym. Sci. 2014, 39 (1), 1−42. (3) Kamcev, J.; Freeman, B. D. Charged Polymer Membranes for Environmental/Energy Applications. Annu. Rev. Chem. Biomol. Eng. 2016, 7 (1), 111−133. (4) Allegrezza, A. E.; Parekh, B. S.; Parise, P. L.; Swiniarski, E. J.; White, J. L. Chlorine Resistant Polysulfone Reverse-Osmosis Modules. Desalination 1987, 64, 285−304. (5) Brousse, C.; Chapurlat, R.; Quentin, J. P. New Membranes for Reverse-Osmosis. 1. Characteristics of Base Polymer - Sulfonated Polysulfones. Desalination 1976, 18 (2), 137−153. (6) Glater, J.; Hong, S. K.; Elimelech, M. The Search for a ChlorineResistant Reverse-Osmosis Membrane. Desalination 1994, 95 (3), 325−345. (7) Greenlee, L. F.; Lawler, D. F.; Freeman, B. D.; Marrot, B.; Moulin, P. Reverse Osmosis Desalination: Water Sources, Technology, and Today’s Challenges. Water Res. 2009, 43 (9), 2317−2348. (8) Kimura, S. G. Reverse Osmosis Performance of Sulfonated Poly(2,6-Dimethylphenylene Ether) Ion Exchange Membranes. Ind. Eng. Chem. Prod. Res. Dev. 1971, 10 (3), 335−339. (9) Cath, T. Y.; Childress, A. E.; Elimelech, M. Forward Osmosis: Principles, Applications, and Recent Developments. J. Membr. Sci. 2006, 281 (1−2), 70−87. (10) Chung, T. S.; Zhang, S.; Wang, K. Y.; Su, J. C.; Ling, M. M. Forward Osmosis Processes: Yesterday, Today and Tomorrow. Desalination 2012, 287, 78−81. (11) McCutcheon, J. R.; McGinnis, R. L.; Elimelech, M. A Novel Ammonia-Carbon Dioxide Forward (Direct) Osmosis Desalination Process. Desalination 2005, 174 (1), 1−11. (12) Shaffer, D. L.; Werber, J. R.; Jaramillo, H.; Lin, S. H.; Elimelech, M. Forward Osmosis: Where Are We Now? Desalination 2015, 356, 271−284. (13) Zhao, S. F.; Zou, L.; Tang, C. Y. Y.; Mulcahy, D. Recent Developments in Forward Osmosis: Opportunities and Challenges. J. Membr. Sci. 2012, 396, 1−21. 4055

DOI: 10.1021/acsami.6b14902 ACS Appl. Mater. Interfaces 2017, 9, 4044−4056

Research Article

ACS Applied Materials & Interfaces

Transport Properties in Directly Copolymerized Disulfonated Poly(Arylene Ether Sulfone) Random Copolymers. Macromolecules 2011, 44 (11), 4428−4438. (61) Yasuda, H.; Lamaze, C. E.; Peterlin, A. Diffusive and Hydraulic Permeabilities of Water in Water-Swollen Polymer Membranes. J. Polym. Sci. A2 1971, 9 (6), 1117−1131. (62) Khare, A. R.; Peppas, N. A. Swelling Deswelling of Anionic Copolymer Gels. Biomaterials 1995, 16 (7), 559−567. (63) Kamcev, J.; Jang, E. S.; Yan, N.; Paul, D. R.; Freeman, B. D. Effect of Ambient Carbon Dioxide on Salt Permeability and Sorption Measurements in Ion-Exchange Membranes. J. Membr. Sci. 2015, 479, 55−66. (64) Kamcev, J.; Paul, D. R.; Freeman, B. D. Effect of Fixed Charge Group Concentration on Equilibrium Ion Sorption in Ion Exchange Polymers. Submitted for publication, 2016. (65) Beers, K. M.; Hallinan, D. T.; Wang, X.; Pople, J. A.; Balsara, N. P. Counterion Condensation in Nafion. Macromolecules 2011, 44 (22), 8866−8870. (66) Mauritz, K. A.; Moore, R. B. State of Understanding of Nafion. Chem. Rev. 2004, 104 (10), 4535−4585.

(37) Geise, G. M.; Freeman, B. D.; Paul, D. R. Sodium Chloride Diffusion in Sulfonated Polymers for Membrane Applications. J. Membr. Sci. 2013, 427, 186−196. (38) Geise, G. M.; Park, H. B.; Sagle, A. C.; Freeman, B. D.; McGrath, J. E. Water Permeability and Water/Salt Selectivity Tradeoff in Polymers for Desalination. J. Membr. Sci. 2011, 369 (1−2), 130− 138. (39) Geise, G. M.; Falcon, L. P.; Freeman, B. D.; Paul, D. R. Sodium Chloride Sorption in Sulfonated Polymers for Membrane Applications. J. Membr. Sci. 2012, 423-424, 195−208. (40) Kamcev, J.; Galizia, M.; Benedetti, F. M.; Jang, E. S.; Paul, D. R.; Freeman, B. D.; Manning, G. S. Partitioning of Mobile Ions between Ion Exchange Polymers and Aqueous Salt Solutions: Importance of Counter-Ion Condensation. Phys. Chem. Chem. Phys. 2016, 18, 6021− 6031. (41) Kamcev, J.; Paul, D. R.; Freeman, B. D. Ion Activity Coefficients in Ion Exchange Polymers: Applicability of Manning’s Counterion Condensation Theory. Macromolecules 2015, 48 (21), 8011−8024. (42) Crank, J.; Park, G. Diffusion in Polymers; Academic Press: London, 1968. (43) Mackie, J. S.; Meares, P. The Diffusion of Electrolytes in a Cation-Exchange Resin Membrane. 1. Theoretical. Proc. R. Soc. London, Ser. A 1955, 232 (1191), 498−509. (44) Manning, G. S. Limiting Laws and Counterion Condensation in Polyelectrolyte Solutions Ii. Self-Diffusion of the Small Ions. J. Chem. Phys. 1969, 51, 934−938. (45) Pitzer, K. S. A Thermodynamic Model for Aqueous-Solutions of Liquid-Like Density. Rev. Mineral. 1987, 17, 97−142. (46) Manning, G. S. Limiting Laws and Counterion Condensation in Polyelectrolyte Solutions. I. Colligative Properties. J. Chem. Phys. 1969, 51 (3), 924−933. (47) Kamcev, J.; Paul, D. R.; Manning, G. S.; Freeman, B. D. Accounting for Frame of Reference and Thermodynamic NonIdealities When Calculating Salt Diffusion Coefficients in Ion Exchange Membranes. Submitted for publicaton, 2016. (48) Yasuda, H.; Lamaze, C. E.; Ikenberry, L. D. Permeability of Solutes through Hydrated Polymer Membranes I. Diffusion of Sodium Chloride. Makromol. Chem. 1968, 118 (Nov), 19−35. (49) Mackie, J. S.; Meares, P. The Diffusion of Electrolytes in a Cation-Exchange Resin Membrane 0.1. Theoretical. Proc. R. Soc. London, Ser. A 1955, 232 (1191), 498−509. (50) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed.; Butterworths: London, 1959. (51) Manning, G. S. Polyelectrolytes. Annu. Rev. Phys. Chem. 1972, 23, 117−140. (52) Manning, G. S. The Molecular Theory of Polyelectrolyte Solutions with Applications to the Electrostatic Properties of Polynucleotides. Q. Rev. Biophys. 1978, 11 (2), 179−246. (53) Manning, G. S. Counterion Binding in Polyelectrolyte Theory. Acc. Chem. Res. 1979, 12 (12), 443−449. (54) Bordi, F.; Cametti, C.; Colby, R. H. Dielectric Spectroscopy and Conductivity of Polyelectrolyte Solutions. J. Phys.: Condens. Matter 2004, 16 (49), R1423−R1463. (55) Netz, R. R. Polyelectrolytes in Electric Fields. J. Phys. Chem. B 2003, 107 (32), 8208−8217. (56) Penafiel, L. M.; Litovitz, T. A. High-Frequency DielectricDispersion of Polyelectrolyte Solutions and Its Relation to Counterion Condensation. J. Chem. Phys. 1992, 97 (1), 559−567. (57) Lakshminarayanaiah, N. Transport Phenomena in Membranes; Academic Press: London, 1972. (58) Geise, G. M.; Doherty, C. M.; Hill, A. J.; Freeman, B. D.; Paul, D. R. Free Volume Characterization of Sulfonated Styrenic Pentablock Copolymers Using Positron Annihilation Lifetime Spectroscopy. J. Membr. Sci. 2014, 453, 425−434. (59) Geise, G. M.; Freeman, B. D.; Paul, D. R. Characterization of a Novel Sulfonated Pentablock Copolymer for Desalination Applications. Polymer 2010, 51, 5815−5822. (60) Xie, W.; Ju, H.; Geise, G. M.; Freeman, B. D.; Mardel, J. I.; Hill, A. J.; McGrath, J. E. Effect of Free Volume on Water and Salt 4056

DOI: 10.1021/acsami.6b14902 ACS Appl. Mater. Interfaces 2017, 9, 4044−4056