Predicting Salt Permeability Coefficients in Highly Swollen, Highly

Subscriber access provided by UNIV OF CALIFORNIA SAN DIEGO LIBRARIES

Article

Predicting Salt Permeability Coefficients in Highly Swollen, Highly Charged Ion Exchange Membranes Jovan Kamcev, Donald R. Paul, Gerald S. Manning, and Benny D. Freeman ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.6b14902 • Publication Date (Web): 10 Jan 2017 Downloaded from http://pubs.acs.org on January 13, 2017

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

ACS Applied Materials & Interfaces is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Applied Materials & Interfaces

Predicting Salt Permeability Coefficients in Highly Swollen, Highly Charged Ion Exchange Membranes

Jovan Kamcev1, Donald R. Paul1, Gerald S. Manning2, Benny D. Freeman1*

1. 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, TX 78758 USA 2. Department of Chemistry and Chemical Biology, Rutgers University, 610 Taylor Road, Piscataway, NJ 08854-8087

To whom correspondence should be addressed: [email protected]

(Tel: +1-512-232-2803, Fax: +1-512-232-2807)

Manuscript prepared for submission to: ACS Applied Materials & Interfaces

ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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 counter-ion 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 ( , is related to the salt partition (i.e., sorption)

coefficient, K s , and apparent salt diffusion coefficient, < Dsm* > , via < Ps >= K s < Dsm* > . Salt permeability coefficients, defined as the steady state salt flux normalized by membrane thickness and concentration difference between the upstream and downstream contiguous solutions, are often used to quantify concentration gradient driven salt transport in dense membranes.1-3, 31, 32, 34, 37, 38

The solution-diffusion model assumes that the membrane/solution interfaces at the upstream and downstream faces are at equilibrium, so salt partitioning into the membrane is a thermodynamic process that can be modeled using standard thermodynamic relations.31, 35, 36 Salt diffusion through the membrane is a kinetic phenomenon driven by random Brownian motion. 2 ACS Paragon Plus Environment

Page 4 of 65

Page 5 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Quantitatively capturing these phenomena (i.e., salt sorption and diffusion in a membrane) using fundamental models is necessary for ultimately predicting salt permeability coefficients in such membranes. As discussed elsewhere, ion partitioning between a charged membrane and an aqueous salt solution is strongly influenced by the fixed charge groups on a polymer backbone.1-3, 32, 39-41 A key consequence of the fixed charge groups is to create an unequal distribution of counter-ions (i.e., ions with opposite charge to that of the fixed charge groups on the polymer backbone) and co-ions (i.e., ions with similar charge to that of the fixed charge groups on the polymer backbone) between the membrane and contiguous solution.32, 41 Typically, the concentration of counter-ions in many commercial IEMs is significantly greater than the concentration of co-ions, since counter-ions are required to electrically balance the fixed charge groups and any sorbed coions.32, 41 In the absence of an externally applied electric field, counter-ions and co-ions must partition into the membrane and diffuse through it together to maintain electroneutrality. That is, co-ions are free to enter and leave the membrane, as long as they are accompanied by an equivalent number of counter-ions. However, many counter-ions must remain in the membrane to balance the fixed charge groups. So, for 1:1 electrolytes, the so-called “mobile” salt concentration is equivalent to the co-ion concentration. Consequently, salt transport in highly charged membranes, such as commercial IEMs, is largely governed by co-ion transport.32 In this study, ion partitioning between a charged membrane and an aqueous salt solution was modeled using Donnan theory combined with Manning’s counter-ion condensation theory and the Pitzer model to account for non-ideal ion behavior in the membrane and contiguous solution, respectively.40 Ion diffusion coefficients in the membrane were predicted using the Mackie and Meares model, which presumes ion diffusion in a water-swollen polymer is affected

3 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 65

by a tortuosity factor due to the presence of the polymer chains, and Manning’s model, which accounts for the effect of fixed charge groups on ion diffusion coefficients (i.e., electrostatic effects).42-44 Like Manning’s counter-ion condensation theory for colligative properties, Manning’s model of ion diffusion was originally formulated for polyelectrolyte solutions.44 This model for ion diffusion was extended to ion exchange membranes in the present study. Finally, salt permeability coefficients were calculated using the solution-diffusion model, and the predicted values were compared to those measured experimentally. Model Development The model for predicting salt permeability coefficients for ion exchange membranes developed in this section was tested against previously reported experimental data for a series of commercial ion exchange membranes provided by GE Power and Water.40,

41

Membrane

chemical structures and relevant properties are presented in Figure 1 and Table 1. The properties reported in Table 1 are taken from the literature.41


Page 7 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 1: Chemical structures of the commercial membranes used in this study: (a) CR61-CMP, (b) AR103-QDP, and (c) AR204-SZRA.41



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Sample

Reported IEC [meq/g (dry polymer)]a

C m,p [mol A fixed charge groups/L (swollen polymer)]b

C m,w [mol fixed A

Pure water uptake, wu ,

Page 8 of 65

Pure water volume fraction, φw ,

charge groups /L (water sorbed)]b

[g (water)/g (dry polymer)]

[L(water)/L(s wollen polymer)]

ξ

CR61

2.2 (min.)

1.60 ± 0.04

3.21 ± 0.08

0.84 ± 0.01

0.50 ± 0.014

1.83

AR103

2.2 (min.)

1.44 ± 0.03

3.58 ± 0.07

0.65 ± 0.01

0.40 ± 0.018

2.21

AR204

2.4 (min.)

1.44 ± 0.02

2.82 ± 0.04

0.96 ± 0.01

0.52 ± 0.002

2.52

Table 1: Properties of the membranes used for this study.40, 41

a

These values are reported by the manufacturer.

b

Estimated as the counter-ion concentration in the ion exchange membrane at an external NaCl

concentration of 0.01 M. Here, “swollen polymer” refers to the ion-containing polymer (i.e., contribution of the membrane backing material is neglected). Measured values reported in Table 1 are based on the ion-containing polymer (i.e., contributions of the membrane backing material (i.e., membrane support) are excluded41).


Page 9 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Equilibrium ion partitioning between an ion exchange membrane in contact with an aqueous salt solution is modeled using Donnan theory.32 This approach is detailed elsewhere, and the main steps are summarized below.40 At equilibrium, the electrochemical potential of each type of ion in the membrane phase is equal to that of its analog in the solution phase (e.g., the Na+ electrochemical potential in a membrane is equal to the Na+ electrochemical potential in the contiguous solution). Then, the electrochemical potentials of all ions in the membrane phase are summed and equated to the sum of the electrochemical potentials of all ions in the solution phase. The resulting expression is combined with a charge balance in the membrane to obtain an expression for the mobile salt concentration in the membrane. A charge balance in the membrane yields

∑z C i

i

m,w i

+ ω C Am,w = 0 , where zi and C im,w are the valence and concentration of ion i ,

respectively, ω is the fixed charge group valance, and C Am,w is the fixed charge group concentration. For a 1:1 electrolyte (e.g., NaCl), the final expression is:40

(C

m,w A

) ( )(

) ( ) (C )

+ C sm,w C sm,w γ mg γ cm = γ ±s

2

s s

2

(1)

where C sm,w is the mobile salt (i.e., co-ion) concentration in the membrane (mol of ions per L of water sorbed in the membrane), γ mg and γ cm are the counter-ion and co-ion activity coefficients in the membrane, respectively, γ ±s is the salt mean activity coefficient in the external solution, and C ss is the salt concentration in the external solution. The membrane fixed charge group concentration, C Am,w , can be measured experimentally or calculated from the polymer ion exchange capacity (IEC, mmols of fixed charge groups per gram of dry polymer) and water uptake (grams of water per gram of dry polymer) via

(

)

C m,w = IEC ⋅ ρ w / wu , where ρw is the density of water and wu is the water uptake.40 Salt mean A 7



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 65

activity coefficients in the external solution, γ ±s , can be calculated using, for example, the Pitzer model.45 Ion activity coefficients in the membrane phase, γ mg γ cm , can be predicted using

Manning’s counter-ion condensation theory, as reported elsewhere.40,

41, 46

Application of

Manning’s model to calculate ion activity coefficients in ion exchange membranes requires the dimensionless linear charge density of the polymer, ξ , which is given by:46

ξ=

e2 4πε 0ε kTb

=

λB

(2)

b

where e is the protonic charge, ε 0 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 λ B is the Bjerrum length. For a homogeneous membrane, ξ can be calculated from the polymer chemical structure, IEC, and water content, as demonstrated elsewhere.40 ξ values for the membranes considered in this study were reported previously and are recorded in Table 1 for convenience.40 As a first approximation, ξ is taken to be a membrane property that does not depend significantly on ion type or salt concentration in the external solution.40, 41 When ξ is greater than a critical value ξcrit ( ξcrit = 1/ ω z g , where z g is the counter-ion valence), counter-ions condense (or localize) on the polymer backbone to reduce ξ to the critical value.40 When ξ is less than the critical value, counter-ion condensation does not occur. For the materials considered in this study, ξ > ξcrit (cf. Table 1).40,

41

For ion exchange membranes

equilibrated with 1:1 electrolytes and when ξ > ξcrit , the counter-ion and co-ion activity coefficients are calculated as follows:41

8


Page 11 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


  1 m,w m,w m,w CA     C / ξ + C γ mg =  A m,w m,ws  exp  − m,w2  m,w  C A + Cs   C A + 2ξC s   

(3)

  1 m,w CA   γ cm = exp  − m,w2  m,w  C A + 2ξC s   

(4)

Combining Eqns. 1, 3, and 4 yields:

(

 C m,w / ξ + C m,w    C Am,w s s C Am,w + C sm,w C sm,w  A m,w exp −  = γ±  m,w m,w m,w   C A + Cs  C A + 2ξC s 

)( )

( )( ) 2

C ss

2

(5)

Eqn. 5 can be solved numerically to obtain C sm,w for a given C ss , provided C Am,w , ξ , and γ ±s are known. Expressions equivalent to Eqn. 5 for electrolytes containing divalent ions are reported elsewhere.40 As mentioned previously, the units of C sm,w are mols of mobile salt (i.e., co-ions) per L of water sorbed in the membrane. When considering ion transport in water-swollen membranes, salt concentrations in the membrane are conveniently expressed as mols of salt per L of swollen membrane (i.e., volume includes polymer, water, and ions), C sm,p .42, 43 The two concentration scales are related by:41

C

m,w s

=

C sm,p

(6)

φw

where φw is the volume fraction of water in the membrane. Then, the salt (i.e., co-ion) partition coefficient, K s , can be calculated from:39

Ks =

C sm,p

9

C ss


(7)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 65

Membrane salt diffusion coefficients can be obtained from experimental values of salt permeability coefficients and salt partition coefficients via the solution-diffusion model (i.e.,

< Dsm* >=< Ps > /K s ).37 Salt diffusion coefficients obtained in this manner are apparent values, since they inherently contain contributions from convection (i.e., frame of reference) and nonideal thermodynamic effects.47 Procedures for correcting salt diffusion coefficients for these effects are reported elsewhere.47 The convection effects arise from properly accounting for the frame of reference effects in Fick’s law. For the materials considered here, convection effects were negligible at low upstream salt concentrations (0.1 M), convection corrected salt diffusion coefficients were greater than non-corrected values by approximately 20-30 % for these materials. Non-ideal thermodynamic effects stem from ion activity coefficient gradients in the membrane. Non-ideal effects were negligible at low upstream salt concentrations. At higher salt concentrations, salt diffusion coefficients corrected for nonideal effects were lower than non-corrected values by approximately 20 % for these materials. For NaCl transport in CR61 and AR103, frame of reference and non-ideal thermodynamic effects nearly cancel one another, leading to similar values for apparent salt diffusion coefficients (i.e., calculated from the solution-diffusion model from salt permeability and sorption data),

< Dm* > , and local salt diffusion coefficients corrected for frame of reference and non-ideal s thermodynamic effects, Dsm .47 Thus, for these materials, fundamental models developed for predicting local salt diffusion in ion exchange membranes due to Brownian motion may be used to approximate apparent salt diffusion coefficients necessary for predicting salt permeability coefficients (i.e., < Dsm* >≈ Dsm ).

10


Page 13 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Using the Nernst-Planck equation, the local salt diffusion coefficient in the absence of frame of reference and non-ideal thermodynamic effects, Dsm , is given by:47

(

DgmDcm z 2gC gm,p + zc2Ccm,p

D = 2 m m,p 2 m m,p z g Dg C g + zc Dc C c m s

)

(8)

where z g and zc are the counter-ion and co-ion valences, respectively, Dgm and Dcm are the counter-ion and co-ion diffusion coefficients in the membrane, respectively, and C gm,p and Ccm,p are the counter-ion and co-ion concentrations in the membrane (mols of ions per L of swollen membrane), respectively. Co-ion and counter-ion 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 counter-ion 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 water-swollen 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

11



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 65

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 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 travelled by ions across a membrane of a given thickness is greater than the distance travelled 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.

12


Page 15 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Cross-linked IEM = Fixed charge group = Counter-ion = Co-ion

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

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, Dim . The final result is:

 φ  = w  s Di  2 − φw 

Dim

13

2


(9)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 65

where Dis is the diffusion coefficient of ion i in an aqueous solution, which can be found in the literature50, 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 counter-ion 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 counter-ion condensation theory, he described the inhomogeneous electric field using a Debye-Hückel approximation. In this treatment, it was assumed that condensed counter-ions have no mobility. Manning did note, however, that condensed counter-ions 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 counter-ions are mobile.54-56 In our approach for modeling salt permeability in ion exchange membranes, we preserve the assumption of immobile condensed counter-ions. 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 counter-ion transport32, the assumption of immobile condensed counter-ions will be relaxed. For the case where ξ > ξcrit , the general equations for counter-ion and co-ion diffusion coefficients, Dgm and Dcm , respectively, are:44

14


Page 17 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


 X  +1    2   D   1− 1 z 2 A  1 ; X    z gν gξ =  3 g  z z ξ  Dgs  X  g g  +1    z g ν g 

(10)

   1 2 1 X   = 1− z A  ; Dcs  3 c  z g z g ξ    

(11)

m g

Dcm

where Dgs and Dcs are the counter-ion and co-ion diffusion coefficients in an aqueous solution,

respectively, z g and zc are the counter-ion and co-ion valences, respectively, ν g and ν c are the number of counter-ions and co-ions in a salt molecule, respectively, and X = C Am,w /C sm,w . The function A is given by:44

(

)

  ∞ ∞  ν g + ν c z g zc z g ξ  1 X   = ∑ ∑ π z g m12 + m22 + z g + A ; X   z g z g ξ  m1 =−∞ m2=−∞    (m1 ,m2 )≠(0,0)

(

)

−2

(12)

Manning’s equations for ion diffusion coefficients require no adjustable parameters since ξ and

X are known for the materials considered in this study. Eqns. 10-12 can be combined with Eqn. 9 to yield final expressions for counter-ion and co-ion diffusion coefficients in charged membranes that account for tortuosity and electrostatic effects. For a 1:1 electrolyte, the final expressions are:

 X / ξ +1   1  X    φw  =   1 − A 1;   Dgs  X +1   3  ξ    2 − φw 

Dmg

 1  X  φ  =  1− A  1;    w  s Dc  3  ξ    2− φw 

Dcm

15

2

(13)

2


(14)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 65

where:

 X A  1;  =  ξ

 2ξ  ∑ ∑ π m12 + m22 +1+ X  m1 =−∞ m2 =−∞   (m1 ,m2 )≠(0,0) ∞

∞

(

)

−2

(15)

The local salt diffusion coefficient in a membrane, Dsm , (cf. Eqn. 8) can be predicted by using Eqns. 9-12 to calculate counter-ion and co-ion diffusion coefficients in the membrane and the Donnan/Manning model to calculate counter-ion 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 >= K s ⋅Dsm . 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 Eqn. 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 one 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

16


Page 19 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


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. The filled symbols denote experimental data, and the dashed lines denote values calculated using the Donnan/Manning approach. The experimental data were obtained at ambient conditions.40, 41

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 electrical potential (i.e., the Donnan potential) at the membrane/solution interface that inhibits counter-ions 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

17



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 65

The strength of the Donnan potential dictates the extent of co-ion exclusion from a charged membrane and depends on the counter-ion activity (i.e., concentration) difference between the membrane and contiguous solution.32 The counter-ion concentration in the membrane depends largely on the membrane fixed charge group concentration, C Am,w , which changes little over the salt concentration range considered here41, since most counter-ions 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, C ss . Consequently, the extent of Donnan exclusion, and, in turn, salt partition coefficients, depend on C ss . As external salt concentration increases, the counter-ion concentration difference between the membrane and solution decreases, and the Donnan potential decreases. This decrease in Donnan potential enhances coion sorption in the membrane.32 The strength of the Donnan potential also depends on the counter-ion and co-ion valences.32 All other factors being equal, the Donnan potential is stronger for co-ions having a higher valance, resulting in stronger co-ion exclusion.32 For example, for a given counter-ion, 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 counter-ions having a higher valance, 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 counter-ions with a higher

18


Page 21 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


valence. Thus, the concentration 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 (=< Ps > /K s ).37 Apparent salt diffusion coefficients obtained in this manner are presented in Figure 6 as a function of upstream salt concentration. For the materials considered in this study, apparent salt diffusion coefficients were relatively constant over the salt concentration range considered, within the experimental uncertainties. This behavior is reasonable within the framework set forth by Mackie and Meares regarding ion diffusion in highly swollen ion exchange polymers. In this framework, ion diffusion coefficients in a membrane are lower than ion diffusion coefficients in an aqueous solution due to a tortuosity factor arising from the presence of the polymer chains.42, 43

That is, the polymer chains are considered stationary and impenetrable, so ions must diffuse

around them to traverse the membrane. The distance traveled by the ions to cross the membrane is longer than the distance they would have to travel to cross an aqueous salt solution of equivalent thickness. Mackie and Meares related this tortuosity factor to the water volume fraction.42, 43 Since the water volume fraction for the membranes considered in this study changes little with salt concentration (cf. Figure 5), apparent salt diffusion coefficients also change little with salt concentration, provided this is the main factor influencing ion diffusion in the

24


Page 27 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


membrane. The salt concentration dependence of the apparent salt diffusion coefficients in Figure 6 is also consistent with that observed in aqueous solutions, where salt diffusion coefficients typically do not depend strongly on salt concentration.50

Figure 6: (a) NaCl and MgCl2 diffusion coefficients as a function of upstream salt concentration for cation exchange membrane CR61. (b) NaCl diffusion coefficients as a function of upstream salt concentration for anion exchange membranes AR103 and AR204. The filled symbols denote apparent salt diffusion coefficients, < Dsm* > , calculated from the solution-diffusion model using experimental salt permeability and ion sorption data, and the dashed lines denote local salt diffusion coefficients predicted by the model described in the Model Development section.

Local salt diffusion coefficients calculated using the framework in the Model Development section are compared to apparent salt diffusion coefficients computed from experimental salt permeability and sorption data in Figure 6. Apparent salt diffusion coefficients calculated from experimental permeability and sorption data are effective average values,

25



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 65

whereas those predicted by the model are local values. As a first approximation, comparison between the two is plausible for these materials since it was demonstrated elsewhere that apparent and local salt diffusion coefficients were similar, within the experimental uncertainties.47 Considering the simplicity of the model, and the lack of adjustable parameters, remarkably good agreement was observed between predicted salt diffusion coefficients and values calculated from experimental salt permeability and sorption data with no adjustable parameters. The order of diffusion coefficients for NaCl and MgCl2 diffusion in the CEM was correctly predicted. The model somewhat overestimated apparent salt diffusion coefficients calculated from experimental data, in particular at low upstream salt concentrations. For example, at an upstream salt concentration of 0.01 M, predicted salt diffusion coefficients were 19 % and 23 % higher than those calculated from experimental salt permeability and salt sorption data for NaCl and MgCl2 transport in CR61, respectively. The model prediction for salt diffusion coefficients was better at higher upstream salt concentrations for all materials. The molecular basis for the discrepancy between predicted salt diffusion coefficients and apparent values calculated from experimental salt permeability and salt sorption data at low upstream salt concentration is currently not well understood. As mentioned earlier, Manning’s model considers only electrostatic interactions on a single polymer segment to describe the inhomogeneous electric field created by the fixed charge groups on a polyelectrolyte chain, and it ignores electrostatic interactions between different polymer chains.44 Such electrostatic interactions may play a more important role in influencing ion diffusion in such materials at lower upstream salt concentrations since the concentration of sorbed salt is considerably less than at high upstream salt concentration, so screening of electrostatic interactions between

26


Page 29 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


different polymer chains may be weak at low upstream salt concentrations. Additionally, the Mackie and Meares view of such materials as consisting of rigid, immobile polymer segments on a regular lattice where the polymer segments do not interact with the ions, is clearly a highly simplified view of such materials. Further investigation is necessary to elucidate this issue. Salt transport in highly charged ion exchange membranes driven by a concentration gradient is mainly governed by co-ion transport (i.e., the minority species), since the membrane co-ion concentration is typically significantly lower than the membrane counter-ion concentration over the external salt concentration range considered in this study.32, 41 This fact can be seen by introducing the assumption C gm,p >> C cm,p into Eqn. 8, which leads to the approximation Dsm ≈ Dcm . Thus, the model for co-ion diffusion coefficients presented in the Model Development section can be used to examine the relative influence of the tortuosity and electrostatic effects on salt diffusion for the materials considered here. The co-ion diffusion coefficient ratio, Dcm / Dcs , predicted by the Mackie/Meares and Manning models is presented as a function of upstream salt concentration in Figure 7. For all materials, the tortuosity effect decreases membrane co-ion diffusion coefficients by approximately an order of magnitude, while the electrostatic effect lowers membrane co-ion diffusion coefficients by approximately 15 % for NaCl transport in the IEMs and approximately 5 % for MgCl2 transport in the CEM. Thus, within this framework, the tortuosity effect is the dominant factor influencing salt diffusion in these highly swollen, highly charged membranes.

27



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 65

Figure 7: Ratio of membrane to solution co-ion diffusion coefficients predicted by the Mackie and Meares model (i.e., tortuosity effect, Eqn. 9) and Manning’s model (i.e., electrostatic effect, Eqn. 11) as a function of upstream salt concentration for: (a) NaCl and MgCl2 transport in CEM CR61 and (b) NaCl transport in AEMs AR103 and AR204.

Salt Permeability Coefficients Salt permeability coefficients were measured as a function of upstream salt concentration using a standard diffusion cell at 25 °C, and the results for cation exchange membrane CR61 and anion exchange membrane AR103 were reported previously.63 The experimental protocol for measuring salt permeability coefficients reported by Kamcev et al. was used to measure NaCl permeability coefficients for anion exchange membrane AR204 and MgCl2 permeability coefficients for cation exchange membrane CR61.63 Experimental salt permeability coefficients as a function of upstream salt concentration for the materials considered in this study are presented in Figure 8. As indicated earlier, these salt permeability coefficients, like the salt partition coefficients, have been corrected for the effect of the membrane backing, which is 28


Page 31 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


impermeable to ions and water, so the < Ps > values in Figure 8 represent salt permeability through the ion exchange material in the membrane.

Figure 8: (a) NaCl and MgCl2 permeability coefficients as a function of external salt concentration for cation exchange membrane CR61. (b) NaCl permeability coefficients as a function of external NaCl concentration for anion exchange membranes AR103 and AR204. The filled symbols denote experimental data, and the dashed line denotes values predicted using the model. The experimental data were obtained at 25 °C.63

For all membranes, NaCl permeability coefficients exhibited a strong dependence on upstream solution salt concentration, increasing nearly an order of magnitude as upstream NaCl concentration increased from 0.01 to 1 M. The solution-diffusion model dictates that both salt diffusivity and salt partition coefficients contribute to the behavior of salt permeability coefficients.36 However, for the materials considered in this study, the increase in NaCl permeability coefficients with increasing salt concentration is due almost entirely to the increase 29



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 32 of 65

of salt partition coefficients with increasing external solution salt concentration (cf. Figure 3), since salt diffusion coefficients were relatively constant over the salt concentration range considered (cf. Figure 6). MgCl2 permeability coefficients in cation exchange membrane CR61 exhibited a weaker dependence on salt concentration in the upstream solution relative to that observed for NaCl permeability coefficients. This result is consistent with the salt partition coefficient trend observed for this material (cf. Figure 3). The concentration dependence of the Donnan potential strongly influences salt partition coefficients and, in turn, salt permeability coefficients. In sharp contrast, salt permeability coefficients for uncharged membranes are almost invariant with upstream salt concentration, which is reasonable since there is no Donnan potential at the membrane/solution interface for such materials.2, 37, 48 Salt permeability coefficients calculated from the solution-diffusion model, where salt partition coefficients were predicted using the Donnan/Manning model and ion diffusion coefficients were predicted using the Mackie/Meares and Manning models, are compared to experimental values in Figure 8. The results are also summarized in a parity plot in Figure 9. Quantitative agreement between experimental and predicted salt permeability coefficients was remarkably good for all materials, considering that no adjustable parameters were used. Moreover, the same ξ value was used to predict MgCl2 and NaCl permeability coefficients for CR61, the cation exchange membrane. The model predicted NaCl permeability coefficients with greater accuracy at higher upstream salt concentrations (e.g., >0.1 M). However, some discrepancy between predicted and experimental values was evident at low upstream salt concentrations (e.g., > Ccm ). Moreover, the membrane counter-ion concentration is essentially equal to the membrane fixed charge group concentration. Introducing the assumption C Am,w >> C sm,w into Eqn. 5 yields:

C sm,w ≈

( ) (C ) ξ

2.718 γ ±s

2

s s

2

(16)

C Am,w

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: 1

( )( )

 8.468 γ s 3 C s 3 ξ  2 ± s  C sm,w ≈  m,w   CA  

(17)

Introducing the assumption C gm,p >> C cm,p in Eqn. 8 and using Eqn. 9 for the membrane co-ion diffusion coefficient yields:

 φ  D ≈D =D  w   2− φw  m s

m c

2

s c

(18)

In deriving Eqn. 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 Eqns. 16 and 18 yields:

( )

2 2   2.718 γ ±s C ssξ s  φ  w D < Ps >= s D = φw    c  2 − φ  Cs C Am,w w  

C sm,p

m s

For 1:2 electrolyte (e.g., MgCl2) transport in CEMs, combining Eqns. 17 and 18 yields:

32


(19)

Page 35 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


( )

1

3 2  2 8.468 γ ±s C ssξ  φw  s   < Ps >= φw D   c  2− φw  C Am,w  

(20)

Salt permeability coefficients calculated using Eqns. 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., C ss 0.1 M). This finding is reasonable since the membrane co-ion concentration becomes significant relative to the membrane counter-ion concentration as C ss increases, making the assumption C gm,p >> C cm,p less valid.

33



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 36 of 65

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. The dashed lines denote salt permeability coefficients calculated by the rigorous form of the model, and the solid lines denote salt permeability coefficients calculated by the approximate form of the model (i.e., Eqns. 19 and 20).

In addition to being useful approximations, Eqns. 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 Eqns. 19 and 20, decreasing ξ should decrease salt permeability coefficients, since lower ξ values

34


Page 37 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


lead to fewer condensed counter-ions, 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 experimental data is rather remarkable considering the simplicity of the assumptions inherent in the framework presented here and the lack of adjustable parameters. The Donnan/Manning model assumes that membrane phase non-idealities (i.e., the deviation of ion activity coefficients from unity) are entirely due to local electrostatic interactions between the fixed charge groups and 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. Based on 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-

35



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 38 of 65

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 counter-ions spend a significant time in close proximity to the fixed charge groups, so counter-ion diffusion is expected to be more strongly influenced by the fixed charge groups than co-ion diffusion. Indeed, activation energies for counter-ion 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. 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 counter-ion condensation theory, and solution ion activity coefficients were calculated using the Pitzer model. Predicted values were compared to experimental NaCl and MgCl2 partition coefficients in a CEM and NaCl partition coefficients in

36


Page 39 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


two AEMs. Agreement between predicted and experimental salt partition coefficients was exceptionally good at higher external salt concentrations (e.g., C ss >0.1 M), but some deviation between the two was observed at lower external salt concentrations (e.g., C ss s Dm s Dm g Dm c Ds g Ds c e

ε ε0

Distance between fixed charge groups on a polymer chain Membrane fixed charge group concentration (mols per L of sorbed water) Membrane mobile salt concentration (mols per L of sorbed water) Membrane mobile salt concentration (mols per L of swollen membrane) Membrane counter-ion concentration (mols per L of swollen membrane) Membrane co-ion concentration (mols per L of swollen membrane) External solution salt concentration Membrane apparent salt diffusion coefficient Membrane local salt diffusion coefficient Membrane counter-ion diffusion coefficient Membrane co-ion diffusion coefficient External solution counter-ion diffusion coefficient External solution co-ion diffusion coefficient Protonic charge Dielectric constant Vacuum permittivity

φw

Water volume fraction

γ gm

Membrane counter-ion activity coefficient

38


Page 40 of 65

Page 41 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


γ cm

Membrane co-ion activity coefficient

γ ±s

External solution salt mean activity coefficient

K s k

Salt sorption coefficient

λB

Bjerrum length

ω

Fixed charge group valence

s

Boltzmann constant

Average salt permeability coefficient

ρw

Density of water

T

Temperature

w u

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

z g z c

Counter-ion valence Co-ion valence

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 Pol 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. 39



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 42 of 65

6.

Glater, J.; Hong, S. K.; Elimelech, M. The Search for a Chlorine-Resistant ReverseOsmosis 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 Research 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 Membrane 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 Membrane Sci 2012, 396, 1-21.

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.

40


Page 43 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


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 Membrane 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 Membrane 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 Membrane 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 Membrane 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 Membrane Sci 2001, 185 (1), 29-39.

29.

Merle, G.; Wessling, M.; Nijmeijer, K. Anion Exchange Membranes for Alkaline Fuel Cells: A Review. J Membrane Sci 2011, 377 (1-2), 1-35.

41



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 44 of 65

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 Membrane Sci 2004, 241 (2), 371-386.

36.

Wijmans, J. G.; Baker, R. W. The Solution-Diffusion Model - a Review. J Membrane Sci 1995, 107 (1-2), 1-21.

37.

Geise, G. M.; Freeman, B. D.; Paul, D. R. Sodium Chloride Diffusion in Sulfonated Polymers for Membrane Applications. J Membrane 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 Membrane 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 Membrane Sci 2012, 423, 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 Lon Ser-A 1955, 232 (1191), 498-509. 42


Page 45 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


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 Non-Idealities When Calculating Salt Diffusion Coefficients in Ion Exchange Membranes. submitted 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 .1. Theoretical. Proc R Soc Lon Ser-A 1955, 232 (1191), 498-509.

50.

Robinson, R. A.; Stokes, R. H., Electrolyte Solutions. Second 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 Mat 2004, 16 (49), R1423-R1463.

55.

Netz, R. R. Polyelectrolytes in Electric Fields. J Phys Chem B 2003, 107 (32), 82088217.

56.

Penafiel, L. M.; Litovitz, T. A. High-Frequency Dielectric-Dispersion of Polyelectrolyte Solutions and Its Relation to Counterion Condensation. J Chem Phys 1992, 97 (1), 559567.

43



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 46 of 65

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 Membrane 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 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 Membrane 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 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.

44


Page 47 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table of Contents Graphic 35x19mm (600 x 600 DPI)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment

Page 48 of 65

Page 49 of 65





Page 50 of 65

Page 51 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48


Cross-linked IEM


= Fixed charge group = Counter-ion = Co-ion


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 3a 159x158mm (300 x 300 DPI)


Page 52 of 65

Page 53 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 3b 159x157mm (300 x 300 DPI)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



Page 54 of 65

Page 55 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60





1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 5 156x148mm (300 x 300 DPI)


Page 56 of 65

Page 57 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60





1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



Page 58 of 65

Page 59 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60





1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



Page 60 of 65

Page 61 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60





1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



Page 62 of 65

Page 63 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 9 160x150mm (300 x 300 DPI)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 10a 160x157mm (300 x 300 DPI)


Page 64 of 65

Page 65 of 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 10b 162x160mm (300 x 300 DPI)


Predicting Salt Permeability Coefficients in Highly Swollen, Highly

Recommend Documents