Competitive absorption of quaternary ammonium and alkali metal cations into a Nafion cation-exchange membrane

Journal of Membrane Science 215 (2003) 103–114

Competitive absorption of quaternary ammonium and alkali metal cations into a Nafion cation-exchange membrane Joaquin Palomo1 , Peter N. Pintauro∗ Department of Chemical Engineering, Tulane University, New Orleans, LA 70118, USA Received 4 September 2001; received in revised form 22 November 2002; accepted 4 December 2002

Abstract The competitive absorption of salt mixtures containing tetramethylammonium (TMA) and/or tetraethylammonium (TEA) cations, respectively, into a NafionTM 117 cation-exchange membrane, under equilibrium conditions, was measured experimentally and modelled theoretically. Uptake experiments were performed, with Nafion equilibrated in an aqueous 0.1 M nitrate salt solution containing either: (a) one of the two quaternary ammonium cations and Li+ , Na+ , or Cs+ or (2) a mixture of TEA and TMA. Equilibrium absorption of the various two-component salt solutions in Nafion was modelled using the molecular-level partition coefficient theory developed by Bontha and Pintauro [Chem. Eng. Sci. 49 (1994) 3835]. Without the use of adjustable parameters, the model predicted accurately cation concentrations for all salt systems except Cs+ /TMA. The observed and computed absorption selectivity ordering in Nafion was (CH3 CH2 )4 N+ > (CH3 )4 N+ > Cs+ > Na+ > Li+ . The failure of the model to predict cation uptake for Cs+ /TMA mixtures was attributed to small errors in the computation of an ion hydration parameter for TMA, coupled with the fact that the Cs+ and TMA cations were nearly the same size. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Quaternary ammonium; Alkali metal cations; Cation-exchange membrane

1. Introduction In order to simulate mathematically a multicomponent ion-exchange membrane separation process, transport equations for each ionic species in the interior of the membrane are combined with equilibrium partition coefficient (ion absorption) models at the two membrane/solution interfaces. Whereas counterion/coion partitioning into an ion-exchange ∗ Corresponding author. Present address: Department of Chemical Engineering, Case Western Reserve University, Cleveland, OH 44106-7217, USA. Tel.: +1-216-368-4150; fax: +1-216-368-3016. E-mail address: [email protected] (P.N. Pintauro). 1 Present address: T3 Inc., Houston, TX 77056, USA.

membrane is reasonably well understood using simple electrostatic attraction/repulsion arguments, the selective (competitive) uptake of counterions from a multicomponent external salt solution is far more complex and only recently has it been examined in some detail. One partition coefficient model, first published by Bontha and Pintauro, has been shown to reproduce accurately the equilibrium uptake of two and three-component mixtures of alkali metal cation salts and alkali metal/divalent cation salt mixtures in a NafionTM 117 cation-exchange membrane [1–3] (Nafion is a registered trademark of E.I. DuPont deNemours and Co. Inc.). This model takes into account: (i) electrostatic interactions between absorbing counterions/coions and the membrane’s fixed charges, (ii)

0376-7388/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0376-7388(02)00606-3

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Nomenclature a Ai Ci e F G IEC Ki

N r R Ri s T V zi

membrane pore radius (m) hydration parameter, defined by Eq. (6) (J/mol) concentration of species i in a membrane pore (mol/m3 ) charge on the electron (1.602 × 10−19 C) Faraday’s constant (96,487 C/eq) ion solvation Gibbs energy (J/mol) membrane ion-exchange capacity (mol/kg) partition coefficient of ion species i (mol/m3 of wet membrane/mol/m3 solution) Avogadro’s number (6.022 × 1023 , mol−1 ) radial position within the membrane pore (m) gas constant (8.314 J/mol K) hard-sphere ion radius (m) membrane area (m2 ) absolute temperature (K) volume (m3 ) charge number of ion species i

Greek letters α dipole moment (C m) χ membrane fixed-ion concentration, mol/m3 of wet membrane ε solvent dielectric constant ε∗ vacuum permittivity (8.8542 × 10−12 F/m) η refractive index (1.33 for water) κ Boltzmann constant (1.381 × 10−23 J/K) θ membrane porosity φ electric potential (V) ρ dry dry membrane density (kg/m3 ) σ membrane pore wall charge density (C/m2 ) Subscripts vac vacuum w water Superscripts b bulk external solution m membrane phase

variations in the membrane-phase water dielectric constant due to solvent dipole alignment by the strong electric field generated by the fixed-charge sites, and (iii) ion solvation Gibbs energy changes that occur when a cationic or anionic species transfers from the bulk electrolytic solution into the low dielectric constant membrane-phase solvent. In the present study, the multicomponent equilibrium absorption of two-component aqueous salt mixtures into a Nafion 117 membrane was modelled, where the solutions outside the membrane contained two quaternary ammonium ions (tetramethylammonium (TMA) and tetraethylammonium (TEA) cations) or a quaternary ammonium ion and an alkali metal cation (either Li+ , Na+ , or Cs+ ). Nitrate anion salts were employed with a total salt mixture concentration (0.1 M) that was well below the membrane’s ion-exchange capacity. The purpose of this study was two-fold. First, quaternary ammonium salts have practical relevance to a variety of membrane-based electrochemical processes/devices and, thus, it is important to understand how these salts interact and absorb into cation-exchange membranes, such as Nafion. Quaternary ammonium salts are used, for example, as the supporting electrolyte in divided organic electrochemical reactors, where a cation-exchange membrane separates the reactor’s anode and cathode compartments [4]. Also, the performance of Nafion-coated electrodes and Nafion-based ion-selective electrodes in the presence of solutions containing aqueous quaternary ammonium and other organic cations has been the subject of numerous studies (see, for example [5–7]). The second goal of this work was to determine whether the Bontha–Pintauro equilibrium ion uptake model could be used to predict accurately the absorption of ionic species that have some organic character and are more complex in their molecular structure that simple “hard-sphere” monovalent and divalent metal cations. Also, it has been found that some quaternary ammonium cations exhibit unusual sorption behaviour in Nafion, which was explained by hydrophobic interactions between the tetra alkyl groups of the quaternary ammonium ions and the fluoropolymer components of Nafion [8]. The presence of such interactions could be deduced indirectly from the present study, depending on whether the model can reproduce experimental uptake data.

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2. Theory The molecular-level equilibrium partition coefficient theory for the prediction of alkali metal and quaternary ammonium cation uptake into a Nafion 117 membrane is the same as that developed by Bontha and Pintauro [1]. The cation-exchange membrane is viewed as an array of parallel cylindrical pores of identical radius with ion-exchange sites distributed uniformly and continuously along the pore wall surface. These fixed-charge groups generate a strong radial-direction electric field that attracts counterions, repels coions, and aligns solvent molecules, thus lowering the dielectric constant of the pore solvent. The model consists of the following equations: 1. A modified Boltzmann equation,
zi F φ(r) Cim (r) = Cib exp − RT   Ai 1 1 − − b RT ε(r) ε Cib

(1)

Cim (r)

where and are the molar concentrations of ionic species “i” in the bulk external solution and membrane phases, respectively, zi is the ion charge number, F the Faraday’s constant (96,487 C/eq), φ the electric potential, ε the dielectric constant in the membrane pore, ε b the dielectric constant of the bulk (external) solution, R and T are the gas constant and absolute temperature, and Ai is an ion hydration (solvation) constant. Both φ and ε are a function of the radial-pore position, r, which is measured from the pore centerline. The first term inside the bracket of the above equation is a measure of the electrostatic energy between absorbed ions and the membrane’s fixed-charge groups (SO3 − for Nafion) and the second exponential term accounts for ion hydration forces that repel all ions from entering into a membrane pore at those pore locations where ε is less than εb . 2. Poisson’s Equation with a variable dielectric constant, n

∇ · [ε(r) ∇ φ(r)] = −

F zi Cim (r) ε∗

(2)

i=1

where ε∗ is the permittivity of vacuum (8.8542 × 10−14 F/cm).

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3. Booth’s Equation [9], which describes the decrease in solvent dielectric constant with electric field strength, 3(ε b − η2 ) ε(r) = η2 + β ∇ φ(r)
× coth[β ∇ φ(r)] −

1 β ∇ φ(r)

 (3)

where β = (5α/2κT )/(η2 + 1), κ is the Boltzmann’s constant, η the solvent’s optical refractive index, and α is the dipole moment of solvent molecules (for water, η = 1.33 and α = 6.17 × 10−30 C m). The boundary conditions for the above equations are symmetry in the electric potential profile at the pore centerline (r = 0) and Gauss’s law at the pore wall (r = a), at r = 0,

dφ =0 dr

at r = a,

σ dφ =− ∗ dr ε ε(a)

and

ε = εb

(4) (5)

where σ (C/m2 ) is the charge density of ion-exchange sites along the pore wall. As was shown previously [1,2], Ai is dependent on the properties of the solvent and the size and charge of the absorbing ion, as quantified by the following equations, Ai =

b m vac Gi |η2 − vac Gi

(1/η2 ) − (1/ε b )

(6)

where m vac Gi |η2 = −

Nz2i e2 Nz2i e2 + 8π ε ∗ Ri 8π ε ∗ Ri η2

(7)

In Eq. (6), bvac Gi is the ion hydration Gibbs energy in the bulk (external) solution and m vac Gi |η2 is the hydration Gibbs energy in a membrane pore where the solvent dielectric constant is at its lowest possible value (η2 , according to Eq. (3)). In Eq. (7), N is the Avogadro’s number, e the charge of an electron, and Ri is the non-hydrated radius of ion species “i”. For alkali metal cations (Li+ , Na+ , and Cs+ ) and nitrate anion, Ri was set equal to the hard-sphere Goldschmidt radius [10]. For tetramethylammonium and tetraethylammonium cations, an equivalent hard-sphere radius

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Table 1 Ion radii, bulk solution free energy of ion hydration, free energy of hydration at the limiting value of the dielectric constant, and Ai for alkali metal, quaternary ammonium, and NO3 − ions in water at 25 ◦ C Ion

Radius (Å)

bvac Gi at ε = 78 (J/mol)

2 m vac Gi at ε = η (J/mol)

Ai (J/mol)

Li+

0.78a

−5.10 −4.11 −2.85 −2.01 −1.72 −3.31

−3.95 −2.96 −1.79 −9.56 −7.99 −1.40

2.12 2.09 1.92 1.94 1.69 3.48

Na+ Cs+ (CH3 )4 N+ (CH3 CH2 )4 N+ NO3 −

1.02a 1.69a 3.22c 3.85c 1.58a

× × × × × ×

105b 105b 105b 105d 105d 105d

× × × × × ×

105 105 105 104 104 105

× × × × × ×

105 105 105 105 105 105

a

Radius based on the Goldschmidt scale [10]. Gibbs energy from Halliwell and Newberg [14] and Rossiensky [15]. c Radius was calculated from bond angle and bond length data in [11–13]. d Gibbs energy from Abraham and Liszi [16] and Friedman and Krishnan [17]. b

(3.22 Å for TMA and 3.85 Å for TEA) was estimated from bond angle and bond length data in [11–13]. Ion radii and bvac Gi (from [14–17]) and computed values of m vac Gi |η2 and Ai for the cation and anion species examined in this study are listed in Table 1. The total concentration of a counterion species in m the membrane (Ci,TOT with units of mol/cm3 of wet membrane) was obtained by integrating an ion concentration profile from the pore centerline to the pore wall, a 2θ 0 Cim (r)r dr m Ci,TOT = (8) a2 where θ is the membrane porosity. Entropic effects associated with the exclusion of counterions from an annular region one ion radius from the pore wall have been considered in some membrane uptake and transport theories (see, for example [18]). Such effects were not incorporated into the present model (i.e. the integration limit in Eq. (8) extends to the pore wall at r = a). Given the use of a simple cylindrical pore structure for the complex micro-morphology of a water-swollen ion-exchange membrane, the omission of entropic effects is not a significant shortcoming of the partition coefficient theory. Also, the uptake model in the present study was tested on a Nafion cation-exchange membrane (manufactured by E.I. DuPont de Nemours and Co. Inc.), which is composed of a poly(tetrafluoroethylene) backbone with side chains that terminate in SO3 − ion-exchange groups. It has been proposed that the solvent and ion-exchange sites phase separate from the fluorocarbon matrix when Nafion is equilibrated in water [19]. For such a situation, one can argue

that the parameter “a” in Eqs. (5) and (8) is the location of the membrane’s ion-exchange sites (which protrude into the pore fluid) and not the pore wall, in which case mobile counterions can coexist with fixed charges in the annular plane at r = a. Eqs. (1)–(3) were solved numerically using a standard finite difference method, with 1500 mesh points in the radial-pore direction and constant mesh-point spacing. Input data to the model were the external salt solution composition, the pore radius, and the pore wall charge density. The model contains no adjustable parameters. Numerical integration of the ion concentration profiles across the membrane pore (cf. Eq. (8)) was performed using Simpson’s rule. Computed valm were then compared to experimentally ues of Ci,TOT measured cation concentrations to test the equilibrium uptake model.

3. Experimental 3.1. Equilibrium uptake experiments Equilibrium ion absorption experiments were performed using a Nafion 117 perfluorosulfonic acid cation-exchange membrane and aqueous mixed-salt solutions containing either LiNO3 + [(CH3 )4 N]NO3 , LiNO3 + [(CH3 CH2 )4 N]NO3 , NaNO3 + [(CH3 )4 N] NO3 , NaNO3 + [(CH3 CH2 )4 N]NO3 , CsNO3 + [(CH3 )4 N]NO3 , or [(CH3 )4 N]NO3 + [(CH3 CH2 )4 N] NO3 . The total external salt concentration was fixed at 0.10 M, the temperature was 25 ◦ C, and the individual cation molar ratio in the external solution

J. Palomo, P.N. Pintauro / Journal of Membrane Science 215 (2003) 103–114

was varied from 1:3 to 3:1. An additional set of 0.10 M single-component salt uptake experiments was performed with LiNO3 , CsNO3 , NaNO3 , [(CH3 )4 N] NO3 , and [(CH3 CH2 )4 N]NO3 . Since the total external concentration of salt was always much less than Nafion’s fixed-ion concentration (0.909 mol/kg of dry membrane), coion (nitrate) absorption was small and was not measured in the experiments. Rectangular strips (5 cm × 2 cm) of Nafion 117 were first pre-treated in the normal manner by boiling for 90 min in 6.0 M HNO3 followed by boiling in deionised and distilled water for 60 min. Conventional methods of soaking, leaching, drying, and analysis [20] were used to determine the membrane-phase cation concentrations and the membrane structure parameters required in the model (the pore radius and pore wall charge density). All experiments were performed in triplicate to insure reproducibility. Initially, the volume of three water-equilibrated Nafion membranes strips was determined by measuring the thickness and area of the films with a micrometer and planimeter. The membrane strips were then boiled under reflux in the desired salt mixture for 90 min, followed by a 24-h equilibration soak in the electrolyte at room temperature. The membranes were removed from the salt solution, excess electrolyte was wiped off their surfaces using filter paper, and the area and thickness were measured. Next, each membrane sample was soaked for 12 h in 100 ml of 2.0 M HNO3 to remove all absorbed cations. The leaching solutions were analysed for alkali metal and/or quaternary ammonium cations using a Dionex DX 500 Ion Chromatography System, with a 4-mm cation-exchange column (IonPac CS12A) and a 4-mm cation self-regenerating suppressor (CSRS-ULTRA). The chromatographic analysis conditions were as follows: eluent flow rate = 0.75 ml/min, eluent composition = 85% acetonitrile and 15% 50 mN sulfuric acid, injection volume = 0.1 ml, and temperature = 25 ◦ C. The variation in membrane-phase cation concentration from duplicate measurements was ±5%. After acid soaking, the membranes were re-equilibrated in deionised and distilled water and then vacuumed dried at 60 ◦ C for 24 h (until the membrane dry weight stabilised). The dry volume (thickness and area) was then measured for the determination of membrane porosity and pore radius (as described in Section 3.2).

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3.2. Determination of membrane pore radius and wall charge density In order to solve the ion uptake model equations, membrane structure parameters (the pore radius and pore wall charge density) must be specified. The average radius (a) of a pore in a salt-equilibrated membrane was determined from the following equation [1,21],
a = aw

s θ sw θw

1/2 (9)

where θ and s are the porosity and area of a membrane sample in a given salt solution and θ w , sw , and aw are the porosity, area, and pore radius of the same sample equilibrated in pure water (membrane in the H+ form). From Gierke et al.’s [22] small angle X-ray diffraction studies with water-soaked Nafion 117, aw was set equal to 27.5 Å. The membrane porosity in Eq. (9) was determined experimentally from the wet and dry volumes of a membrane strip, θ=

Vwet − Vdry Vwet

(10)

The pore wall surface charge density of SO3 − sites in Nafion was calculated from [1,20], σ =

ρdry (1 − θ)(IEC)aF 2θ

(11)

where ρ dry is the density of the dry membrane (1.84 kg/m3 for Nafion 117 [21]) and IEC is the membrane’s ion-exchange capacity (0.909 mol/kg for Nafion 117). The above analysis does not consider explicitly a distribution of different pore sizes in Nafion, because such a distribution is simply not known. The use of a single “average” pore radius simplifies the numerical calculations and eliminates the need to specify pore size distribution parameters. Additionally, the use of a single pore size has been shown to work well for Nafion in prior applications of the multicomponent uptake model [1–3]. Measured Nafion porosities, pore radii, and wall charge densities for the single salt solutions and aqueous salt mixtures in this study are listed in Table 2.

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Table 2 Membrane porosity, pore wall charge density, and pore radius for Nafion 117 in aqueous solutions of a single nitrate salt and two-component nitrate salt mixtures Cation species

External solution cation fraction

Membrane porosity, θ

Pore wall charge density, σ (C/m2 )

Pore radius, a (Å)

A

B

A

B

– – –

Li+

Na+ Cs+

0 0 0

1 1 1

0.242 0.188 0.185

0.611 0.733 0.710

25.1 23.0 20.3

TMA TEA

– –

1 1

0 0

0.173 0.211

0.735 0.679

19.9 24.3

TMA TMA TMA

Li+ Li+ Li+

0.75 0.50 0.25

0.25 0.50 0.75

0.193 0.191 0.186

0.721 0.720 0.736

21.3 21.0 20.8

TEA TEA TEA

Li+ Li+ Li+

0.75 0.50 0.25

0.25 0.50 0.75

0.232 0.207 0.225

0.660 0.708 0.626

25.9 24.9 23.5

TMA TMA TMA

Na+ Na+ Na+

0.75 0.50 0.25

0.25 0.50 0.75

0.162 0.187 0.161

0.773 0.756 0.816

20.6 22.7 21.0

TEA TEA TEA

Na+ Na+ Na+

0.75 0.50 0.25

0.25 0.50 0.75

0.220 0.224 0.238

0.682 0.671 0.636

26.0 26.2 26.2

TMA TMA TMA

Cs+ Cs+ Cs+

0.75 0.50 0.25

0.25 0.50 0.75

0.149 0.177 0.184

0.892 0.767 0.744

20.7 22.0 22.2

TMA TMA TMA

TEA TEA TEA

0.75 0.50 0.25

0.25 0.50 0.75

0.223 0.242 0.281

0.648 0.607 0.522

24.6 26.4 26.9

Total salt concentration = 0.1 M; temperature = 25 ◦ C.

4. Results and discussion An initial set of experiments was performed to determine whether the large tetramethylammonium and tetraethylammonium cations would absorb into a Nafion membrane, associate with the membrane’s SO3 − ion-exchange sites, and then desorb into an acid solution, all in a manner quantitatively similar to that observed in numerous previous studies with alkali metal cations, such as Li+ , Na+ , and Cs+ . To make this determination, the ion-exchange capacity of Nafion 117 was measured experimentally with different cations by means of the usual procedure [19] (i.e. soaking a membrane sample in a given single-component salt solution, followed by washing thoroughly in water, desorbing cations associated with SO3 − sites by soaking in acid, and then analysing

the acid wash for cations by ion chromatography). In Fig. 1, the ion-exchange capacity of Nafion 117, as measured by Na+ , Li+ , Cs+ , TMA, and TEA absorption/desorption, is compared to the published value of 0.909 mol/kg of dry membrane. As can be seen, the same value of the ion-exchange capacity is obtained (within experimental accuracy) for all cation species. Thus, it can be concluded that the process of TMA and TEA ion partitioning is identical to that of simple monovalent cations and that there are no new interactions (e.g. hydrophobic forces) between the membrane and the quaternary ammonium ions that might invalidate the equilibrium uptake model. A comparison of experimental and computed membrane-phase cation concentrations are shown in Fig. 2 for the equilibrium uptake of binary salt mixtures containing one alkali metal cation and either

J. Palomo, P.N. Pintauro / Journal of Membrane Science 215 (2003) 103–114

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absorbed over Li+ and Na+ by Nafion, it was preferentially excluded from the membrane when the external solution was a mixture of TMA and TEA (see Fig. 2f). To further demonstrate the process of selective ion absorption in a Nafion 117 membrane, the change in TMA and TEA partition coefficients as a function of external concentration is shown in Fig. 3. The partition coefficient of species “i” (Ki , with units of mol/m3 of wet membrane/(mol/m3 solution) is defined as the ratio of ion concentrations inside and outside the membrane, m Ci,TOT Ki = (12) Cib

Fig. 1. In-exchange capacity (IEC) for Nafion 117, as measured by Li+ , Na+ , Cs+ , TMA, and TEA absorption/desorption. The reported IEC for Nafion 117 is 0.909 mol/kg.

TMA or TEA (Fig. 2a–e) or a mixture of TMA and TEA (Fig. 2f). Membrane-phase cation concentrations are plotted as a function of the external solution equivalent mole fraction of the smaller cation. Each experimental data point in these figures is the average of three equilibrium uptake experiments. As can be seen, the match of theory and experiments is excellent for all salt systems except Cs+ /TMA (the average error between computed and measured cation concentrations was <6%, except for Cs+ /TMA system, where the average error was 68%). Both the model and experimental data indicate that the monovalent cation with the larger hard-sphere radius (lowest surface charge density) preferentially absorbs into Nafion, regardless of whether the cation is a simple alkali metal species or a more complex organic quaternary ammonium cation. The measured and predicted cation uptake selectivity sequence, as deduced from the results in Fig. 2, is (CH3 CH2 )4 N+ > (CH3 )4 N+ > Cs+ > Na+ > Li+ . For example, equimolar concentrations of Li+ or Na+ and TMA in Nafion were not obtained until the equivalent ion fraction of Li+ or Na+ in the external solution was approximately 0.88. Although TMA was selectively

and is computed theoretically by combining Eqs. (8) and (12). As would be expected from ion-exchange membrane theories, the TEA partition coefficient decreases with increasing TEA concentration outside the membrane. For a dilute salt solution, the counterion concentration in an ion-exchange membrane is equal to that of the membrane’s fixed-charges and the ratio of membrane-phase and external counterion concentration is large. As the external TEA concentration is increased, the membrane counterion concentration approaches that of the bulk solution and the partition coefficient decreases. The dependence of the TMA partition coefficient on the external TMA concentration is quite different from that of TEA and is characteristic of coion uptake by an ion-exchange membrane, i.e. coion exclusion and low membrane-phase concentrations (KTMA → 0) for dilute external salt solutions and coion concentrations that approach the membrane-phase counterion concentration when the external salt solution is more concentrated in TMA. The use of Eq. (9) to quantify the radius of an “average-size” pore in Nafion did not introduce any significant errors into the analysis because the computed membrane-phase cation concentrations are not particularly sensitive to small variations in pore radius. From prior applications of the equilibrium uptake model to dilute salt solutions [1,2], it was found that the majority of the counterions (cations) in a membrane pore were situated near the membrane pore wall (at a dimensionless radial position, r/a, greater than 0.8) where the electrostatic attraction forces are large and the solvent dielectric constant deviates significantly from that in the external solution. A modest

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Fig. 2. Cation concentrations in a Nafion 117 membrane for competitive absorption of two counterions at a total external salt concentration of 0.1 M and a temperature of 25 ◦ C. Symbols are experimental data points and solid lines are model predictions: (a) Li+ /TMA mixtures, (b) Li+ /TEA mixtures, (c) Na+ /TMA mixtures, (d) Na+ /TEA mixtures, (e) Cs+ /TEA mixtures, (f) TMA/TEA mixtures.

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Fig. 3. Cation partition coefficients for competitive TMA/TEA absorption into a Nafion 117 membrane at 25 ◦ C. Symbols are experimental data points and lines are model predictions.

variation in pore size (±10%) altered the value of the pore wall charge density (σ in Eq. (11)) but had essentially no effect on the computed profiles for φ(r), ε(r) and Ci (r) (the change in calculated cation concentration was within the experimental error for measuring salt uptake). The model was unable to reproduce experimental cation uptake data for all mixtures of CsNO3 and [(CH3 )4 N]NO3 , as shown in Fig. 4 (where the solid line model results in this figure were obtained using ATMA = 1.94 × 105 J/mol, from Table 1). The failure of the model for the Cs+ /TMA system was due to an inherent limitation of the theory, namely, that the concentration of one co-absorbing cation from a two-component external mixture is dependent on the difference in the Ai hydration parameters for the two cation species. Since the magnitude of Ai is large and does not differ significantly for ions of similar size (cf. Table 1), relatively small errors in the calculation of the hydration parameters can result in large errors in the A difference. Although it is not intuitively obvious, it can be m shown that Ci,TOT (i.e. the ion concentration measured in an uptake experiment) is a function of Aj − Ai for the case of two co-absorbing cations. To simplify the mathematics, we considered the case of an external solution composed of an equimolar salt mixture with

111

Fig. 4. Measured and computed Cs+ and TMA concentrations in a Nafion 117 membrane during competitive absorption at a total external salt concentration of 0.1 M and a temperature of 25 ◦ C. Symbols are experimental data; (—) model predictions with ATMA = 1.94 × 105 J/mol; (- - -) model predictions with ATMA = 1.84 × 105 J/mol.

a total salt concentration that is much less than the membrane’s ion-exchange capacity (as was the situation in the present study), in which case one can write m m + Cj,TOT =χ Ci,TOT

(13)

and are given by Eq. (8) and χ where is the membrane fixed-ion concentration with units of mol/cm3 of wet membrane. From Eq. (1), the radial-direction concentration profiles for the cation species co-absorbing during a two-component equilibrium uptake experiment are, m Ci,TOT

m Cj,TOT

Cim (r) = Cj (r) exp[ko (r)(Aj − Ai )]

(14)

Cjm (r) = Ci (r) exp[ko (r)(Ai − Aj )]

(15)

where ko =

1 RT



1 1 − ε(r) ε b

 (16)

Using a series expansion for the exponential term in Eq. (14), one can combine Eqs. (8) and (13)–(15) to m (the meaobtain the following relationship for Ci,TOT sured membrane-phase concentration of cation species “i”) in terms of the difference in hydration parameters,  θ  a 1 m Ci,TOT = χ+ Ci (r)[ko A]n 2 n!a 2 0 n=1

×exp[−ko (r) A]r dr

(17)

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J. Palomo, P.N. Pintauro / Journal of Membrane Science 215 (2003) 103–114 m = χ − C bi · Ki Cj,TOT

Fig. 5. Percentage error between the computed and measured TMA partition coefficients as a function of ATMA in the model. (䊐) Cs+ /TMA competitive uptake for an equimolar external solution; (䊉) Li+ /TMA competitive uptake for an equimolar external solution. A∗TMA = 1.94×105 J/mol; ACs+ = 1.92 × 105 J/mol; ALi+ = 2.12 × 105 J/mol.

where A = Aj − Ai . For the alkali metal cation–quaternary ammonium cation systems examined in the present study, the (Aj − Ai ) difference was in the range of 15–43 kJ/mol except for Cs+ /TMA, where (ATMA − ACs+ ) was 2 kJ/mol (see Table 1). Because of this small difference in A, small variations (errors) in the calculation of ATMA and/or ACs+ produced a large change in (ACs+ − ATMA ), which, in turn, impacted significantly on the computed values m m of CCs + ,TOT and CTMA,TOT . Numerical results in support of this argument are shown in Fig. 5, where the % error in the computed TMA partition coefficient is plotted as a function of the % variation in the value of ATMA for two salt systems: TMA + Cs+ and TMA + Li+ (where the external concentrations of each salt was 0.05 M). In Fig. 5, A∗TMA = 1.94 × 105 J/mol (the original value of the TMA hydration parameter from Table 1) is used as a reference. For a dilute two-component external salt mixture (where Eq. (13) applies), the concentrations of both counterion species can be determined from a single counterion partition coefficient measurement and the membrane’s fixed-ion concentration (χ ), according to the following equations, m Ci,TOT = Cib · Ki

(18)

(19)

Thus, any variation in the computed value of Ki inm m fers that both Cj,TOT and Ci,TOT are changing. As can be seen in Fig. 5 for co-absorbing Cs+ and TMA, the exp exp error in (KTMA − KTMA )/KTMA is large (36%) when ((ATMA −1.94×105 )/1.94×105 )×100 is zero, as one would expect from the membrane-phase cation concentration data mismatch in Fig. 4. According to the computed data in Fig. 5, a much better match of the predicted and measured partition coefficients could be obtained if ATM decreased by approximately 5%. Using this new value of ATMA (1.84 × 105 J/mol), the model simulated well all of the Cs+ /TMA equilibrium absorption data, as shown by the dotted lines in Fig. 4 (the average error between the theoretical and experimental concentrations decreased from 68 to 3%). It should be noted that the poor original fit of theory and data for Cs+ /TMA mixtures was attributed to errors in ATMA and not ACs+ . In prior studies, the equilibrium uptake model always predicted accurately multicomponent ion absorption for alkali metal cation mixtures [1], hence there was no reason to suspect that the value of ACs+ was in error. For comparison purposes, the variation in the computed values of KTMA with changes in ATMA for the Li+ /TMA salt system are also plotted in Fig. 5. Here there is a significant difference in the size of the two co-absorbing cations (r = 0.78 Å for Li+ and r = 3.22 Å for TMA+ ) and the difference in magnitude between ALi+ and ATMA (18 KJ/mol) is larger than that for Cs+ and TMA. Not surprisingly, the computed values of the TMA partition coefficient are much less sensitive to small changes in ATMA , which explains why the model fits the Li+ /TMA uptake data equally well (i.e. within experimental error) for ATMA = 1.94×105 and 1.84 × 105 J/mol.

5. Conclusions The competitive absorption of quaternary ammonium cations (tetramethylammonium (TMA) and tetraethylammonium (TEA) cations, respectively) and alkali metal cations into a Nafion 117 cation-exchange membrane, under equilibrium conditions, was measured experimentally and modelled theoretically. Uptake experiments were performed, where Nafion was

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equilibrated in an aqueous 0.1 M nitrate salt solution containing either: (a) one of the two quaternary ammonium cations and Li+ , Na+ , or Cs+ or (2) a mixture of TEA and TMA. Equilibrium absorption of the various two-component salt solutions in Nafion was modelled using the molecular-level partition coefficient theory developed by Bontha and Pintauro [1]. Without the use of adjustable parameters, the model predicted accurately cation concentrations for all salt systems except Cs+ /TMA. The observed and computed absorption selectivity ordering in Nafion was (CH3 CH2 )4 N+ > (CH3 )4 N+ > Cs+ > Na+ > Li+ . The failure of the theory to predict Cs+ /TMA sorption highlighted a limitation of the multicomponent ion uptake model, when the two co-absorbing counterions are nearly the same size. For this situation, the individual ion hydration parameters (Ai in Eq. (1)) are of nearly the same magnitude. Since the calculated membrane-phase concentration of each absorbing species (“i” and “j”) is dependent on the difference between Ai and Aj and since (Ai − Aj ) is always much less than the magnitudes of either Ai or Aj , small errors in the computed values of Ai can generate large errors in the model prediction for ion concentration. Similarly, the model can be made to fit experimental uptake data by making small changes in the calculated value of Ai . For example, when the TMA hydration parameter (ATMA ) was lowered by 5%, the average error in membrane-phase concentration between the model and experimental measurements for the Cs+ /TMA system decreased from 68 to 3%.

[3]

[4] [5] [6]

[7]

[8]

[9] [10] [11] [12]

[13]

[14] [15] [16]

Acknowledgements [17]

This work was funded by the National Science Foundation, Grant No. CTS-0085679. [18]

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