Equilibrium of ion-exchange polymeric membrane with aqueous salt solution and its thermodynamic modeling

Fluid Phase Equilibria 180 (2001) 115–138

Equilibrium of ion-exchange polymeric membrane with aqueous salt solution and its thermodynamic modeling Irina M. Shiryaeva a , Alexey I. Victorov b,∗ a

b

Department of Chemistry, Stanford University, Stanford, CA 94305-5080, USA Department of Chemistry, St. Petersburg State University, Universitetsky prosp. 2, 198904 St. Petersburg, Russia Received 23 May 2000; accepted 4 December 2000

Abstract A thermodynamic model is proposed to describe distribution of the components between a liquid solution and a swollen membrane undergoing structural transformations. Free energy contributions related to formation of solution-filled micro-cavities in the membrane interior are estimated. Formation of the cavities of different shape is accounted for by using the Helfrich expressions for the bending energy of a curved interface. Three adjustable parameters of the model are related to the hydrophobic polymer matrix of the membrane, while the electrostatic contribution is estimated explicitly. Structural changes in the membrane are described as a transition from spherical to cylindrical cavities. Predominance of cavities having definite shape (spheres, cylinders) results in a specific shift of the Donnan equilibrium, which thus, becomes dependent on the structure of the membrane on the mesoscale. The results of model calculations are compared with the experimental data on the distribution of ions (H+ , Li+ , Cs+ , K+ , Na+ , Ca2+ , Mg2+ ) between the aqueous solution and the membrane. Different types of predicted thermodynamic behavior of the membrane in the liquid solution, including the hysteresis of ion-exchange equilibrium curves, are discussed. The model takes into account the effect of micro-inhomogeneties and helps to establish a link between molecular characteristics of the perfluoropolymer membrane and its macroscopic behavior in the liquid solution. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Perfluorosulfonate membrane; Ion-exchange equilibrium; Thermodynamic model; Swelling polyelectrolyte; Non-uniformity on mesoscale; Method of calculation

1. Introduction Ion-exchange membranes play an important role in many technological and biological separation processes. They are involved in living cell metabolism and are widely used in chemical sensors, fuel cells, and production of chemicals by electrochemical synthesis. An important example of such membranes is perflourosulfonate polymeric ionomer membranes, which find an extensive application owing to their ∗ Corresponding author. Tel.: +812-4284066; +812-4286939. E-mail address: [email protected] (A.I. Victorov).

0378-3812/01/$20.00 © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 0 ) 0 0 5 1 6 - 1

116

I.M. Shiryaeva, A.I. Victorov / Fluid Phase Equilibria 180 (2001) 115–138

Fig. 1. Structure of perfluorosulfonate membrane composed of the polymer having chemical formula

where n ∼ 6.5 and Me+ is the counterion [1]. These mobile cations are denoted by + in the clusters.

remarkable thermal and chemical stability, high unipolar electric conductivity with respect to cations, and compatibility with blood. Experimental studies of structural [1–3], transport [4,5], and ion-exchange [6–11] properties of perfluorosulfonate membranes reveal a number of interesting features showing a close link between their structure and thermodynamic behavior. Extensive experimental work in this area is accompanied by attempts to develop theoretical description of such membranes [7,12–14]. Nevertheless, the thermodynamic behavior of ion-exchange membranes is far from being clearly understood and there is a lack of quantitative models adequately describing the membrane structural features and its equilibrium with the liquid solution. The perflourosulfonate ionomer membranes (known after DuPont de Nemours Co., under the commercial name Nafion) are solid electrolytes composed of highly entangled flexible molecules of a linear tetrafluoroethylene and perfluorosulfonate-vinyl ether copolymer. According to spectroscopic data [2,3], the membranes are microscopically inhomogeneous and can be depicted as an amorphous polymer matrix with inclusions of micro-crystallite domains (Fig. 1). In amorphous regions functional groups assemble to form ionic clusters, which are connected with each other through narrow channels. A systematic experimental study of ion-exchange equilibrium has been recently performed for the membranes in aqueous solutions containing two counterions: H+ and one of the following cations: Li+ , Cs+ , K+ , Na+ , Ca2+ , Mg2+ [9,11,15]. The experimental isotherms follow different patterns for different ions. Unsymmetrical curves are observed (as will be illustrated later, see Figs. 5 and 8), which are indicative of a decrease (H+ –Na+ ), an increase (H+ –K+ ), or a reversal (H+ –Li+ ) of the membrane’s selectivity, as well as isotherms demonstrating a plateau (H+ –Ca2+ ) or an inflexion (H+ –Mg2+ ). Such behavior is more typical for the ionites, which, unlike the perfluorosulfonate membranes, have more than one type of functional groups participating in the ion-exchange. A remarkable hysteresis (i.e. a deviation between ion-exchange isotherms for the direct and the reverse ion-exchange processes) is observed for freshly hydrolyzed membranes. Careful reversibility studies have confirmed that the hysteresis is well reproducible and is not due to retarded kinetics or excessive sorption of the electrolyte. This hysteresis is often accompanied by a hysteresis of electric conductivity and optical density [9,10,15]. These experimental observations can be interpreted in terms of the membrane structure. Spherical ion clusters in the amorphous polymer matrix contain most of the water absorbed by the membrane. In the

I.M. Shiryaeva, A.I. Victorov / Fluid Phase Equilibria 180 (2001) 115–138

117

Table 1 Measured characteristics of the membrane [9,15]: selectivity coefficients (KH−Me ), water uptake (Aw , n), electric conductivity (κ) and relative mobility of ions (UMe /UH ) in the membrane Mez + Aw ± 0.1 (wt.%)

n, mol H2 O per mol SO3 −

(κ ± 0.5)·× 103 UMe /UH ( cm)−1

KH−Me Freshly prepared (with hysteresis) H+ → Mez+ Mez+ → H+

H+ Li+ Na+ K+ Cs+ Mg2+ Ca2+

18.5 19 13.8 8.4 6.1 14.5 14.2

12.5 12.9 9.3 5.7 4.1 9.8 9.6

20.3 – 3.5 2.3 – 2.1 1.6

– – 0.17 0.11 – 0.1 0.08

– 1.45 1.94 7.05 – 3.42 5.94

– 0.92 3.50 17.02 – 3.41 14

Stored 1 year in water (without hysteresis) – – 2.40 8.32 16.25 – 9.11

channels, the water content is far lower. Thus, the interaction of counterions with the fixed groups is likely different depending on the location of the ions, and the membrane with sulfonate functional groups can reveal the properties of an ionite having more than one type of ionic functional groups. This explains the occurrence of different types of ion-exchange isotherms. Redistribution of water, fixed groups and counterions results in a change in the membrane structure during the ion-exchange process. According to the experimental data (Table 1), the H+ → Mez+ transition of the membrane is accompanied by a decrease of the water uptake almost in all cases. Water-rich spherical clusters (corresponding to a membrane in the H+ -form) transform into cylindrical channels corresponding to the Mez + -form. Thus, in pure ionic forms the membrane structural states are different, and because the state reached in the ion-exchange process depends on the initial one, the hysteresis can occur. In the present work a thermodynamic model of a polyelectrolyte ion-exchange membrane is proposed to describe an equilibrium distribution of the components between the aqueous solution and the swollen membrane taking into account its structural transformations. The model resembles those proposed for a swelling polyelectrolyte gel by Katchalsky and co-workers [16,17] and reviewed later by Hill [18]. It gives rise to consideration of electromechanical properties, which govern the equilibrium between the swollen polymer and the aqueous salt solution. The approach was further developed via incorporating a more detailed description of deformational swelling [19] and was successfully applied to swelling ionic gels in electrolyte solutions [20]. A similar thermodynamic framework (with a somewhat different definition of the swelling pressure) was presented in a recent paper on swelling gels by Maurer and Prausnitz [21]. In all of the above models the gel phase is treated as if it were spatially uniform. In the present model, however, we take into account the mesoscale inhomogeneity of the polymeric membrane and distinguish between the free energies of swollen membranes having solution-loaded cavities of different shape. The molecular thermodynamic theory used for quantifying the effects of shape was elaborated earlier for aqueous micellar solutions of ionic surfactants [22–24] and describes curvature elasticity of micelles, bilayers, and vesicles [25]. The main difference is that we consider cavities (filled with an aqueous solution) in a hydrophobic polymer medium, rather than micelles in an aqueous salt solution. A similarity between the two classes of the systems is however, obvious, because a transition from cylindrical micelles to an amorphous polymer structure with solution-loaded cavities is observed in an aqueous solution of perfluorinated polymeric surfactant [26,27]. The Helfrich–Safran expressions [25,28] for the work of

118

I.M. Shiryaeva, A.I. Victorov / Fluid Phase Equilibria 180 (2001) 115–138

deformation of a curved interface are used in the present work to estimate free energies of cavities having different shapes. 1.1. The model As supported by the experimental data on the membrane structure and by the hysteresis curves, the ion-exchange properties of the membrane are affected by the membrane structural state. The starting point for the derivation of the model is that the membrane soaked in the aqueous solution can exist in various states depending upon a size, a shape, a number, and a spatial distribution of the solution-loaded cavities inside the hydrophobic polymer matrix. The basic assumptions of the model are summarized below. The membrane is represented as an elastic polymer [29]. Only two structural states, which the membrane can assume depending upon its initial ionic form, are considered: (1) all cavities are spheres, and (2) all cavities are narrow cylinders. The cavity walls carry a uniform charge, which is determined by the amount of fixed charged groups in the polymer matrix. A pseudophase approximation [30] is employed to describe the membrane, i.e. the cavity size is assumed to be large enough, and its physical state is independent upon its location inside the membrane. We neglect the entropy of the distribution of cavities inside the membrane and the interaction between the cavities in the hydrophobic polymer matrix. Thus, the model essentially considers the equilibrium between the external bulk solution and a single spherical or cylindrical cavity inside the membrane (Fig. 2). The condition of material equilibrium between the membrane and the external solution can be written (Appendix A) as ln

a H2 O υH2 O Π = a H2 O Rg T

for the solvent, and
1/zi  1/zq aq a (υi /zq ) − (υi /zi ) = Π; ln i ai aq Rg T

(1)

i = q

(2)

for the mobile ionic species, where ai is the activity, υ i the partial molar volume of component i, prime and double prime refer to the outer and the inner solutions, respectively, zi the ionic charge, and Π = p − p is the swelling pressure [31], i.e. the pressure difference between the external and the internal solutions. This expression is well-known as the Donnan ion-exchange equilibrium [31], and is basically the equation obtained by Maurer and Prausnitz [21] for a swelling polyelectrolyte gel. Eq. (2) shows that the equilibrium distribution of ions between the external solution and the membrane is determined

+

+

Fig. 2. A model of ion-exchange equilibrium between bulk aqueous solution and the membrane. Mez11 , Mez22 are two counterions participating in ion-exchange.

I.M. Shiryaeva, A.I. Victorov / Fluid Phase Equilibria 180 (2001) 115–138

119

Fig. 3. Different types of ion-exchange isotherms described by Eq. (3). N1 , N1 are the equivalent ion fractions inside the membrane and in the external solution, Π is the swelling pressure, and υ i is the partial molar volume.

by the ratio of the ion activities in the inner and the outer solutions, the ionic charge, the difference in the ion partial molar volumes, and by the swelling pressure. The essence of the present work is that we take into account the inhomogeneous structure of the polymeric membrane on the mesoscale in the calculation of the swelling pressure. The specific expression for the latter is determined by the membrane free energy, but before we discuss it in every detail, we consider different types of the ion-exchange isotherms that Eq. (2) can give. The effect of the right-hand-side of the equation can be most clearly seen for the case of two ions of unit charges. In the Debye–Hückel approximation the ratio of their activity coefficients is equal to one both for the inner and for the outer solutions, and, using the so-called equivalent ionic fractions as compositional variables, Ni = ni /(n1 +n2 ) = xi /(x1 +x2 ), one obtains from Eq. (2) ln

N1 N2 υ2 − υ1 = Π N1 N2 Rg T

(3)

If the right-hand-side of this equation does not depend upon concentration (constant values of the swelling pressure and the partial molar volumes), the ion-exchange isotherms are symmetrical (Fig. 3). One can obtain unsymmetrical curves by taking into account a change of the swelling pressure along the isotherm, a compositional dependence of the partial molar volumes also causes a deviation from symmetry, the change in the sign of υ2 − υ1 resulting in the selectivity reversal. A dilute external electrolyte solution (0.1N in the experiments [9,15]) is described by the standard Debye–Hückel theory. The membrane Helmholtz energy is represented as Amb = A nonel (V , {n i }) + A el (V , a, R, {n i }) + AFlory (V ) + AHelf (a, R)

(4)

where AFlory , AHelf , A el are the contributions of the swollen polymer matrix, of the curved hydrophobic wall of the cavity, and of the double electric layer formed in the inner solution, respectively. A nonel describes the free energy of a uniform solution having the same volume and composition as the solution inside the cavity, but with the electrostatic interactions turned off. V , {n i } are the volume and the mole numbers of the components in the internal solution, a, R the surface area and the curvature radius of the cavity surface. Note that non-uniformity of the inner solution is accounted for by the electrostatic and by the Helfrich contributions. In the present simplified version of the model, A nonel is taken to be that of the ideal solution.

120

I.M. Shiryaeva, A.I. Victorov / Fluid Phase Equilibria 180 (2001) 115–138

The swelling pressure is determined by the condition of mechanic equilibrium (Appendix A, Eq. (A.2))
Helf 
el 
Flory 
Helf  ∂A ∂A da dR dR ∂A dA + + + Π= dV ∂a ∂R a dV ∂R a,{n i } dV R dV
el  da ∂A + (5) ∂a R,{n i } dV and consists of three different contributions Π = Π Flory + Π Helf + Π el

(6)

as discussed below. 1.1.1. Bulk elasticity of the polymer matrix The free energy contribution, AFlory (V ), resulting from an elastic deformation of the polymer carcass caused by an uptake of a certain amount, V , of the liquid solution is described by the Flory theory of elastic cross-linked polymers [21,29] AFlory = Rg T ϑk

{3[(V0 + V )/V0 ]2/3 − 3 − ln[(V0 + V )/V0 ]} 2

(7)

Here V0 is the volume of the unswollen polymer, and ϑ k the number of cross-links in the polymer. Although, the Nafion membranes are not cross-linked polymers, the latter quantity has the meaning of some effective number of cross-links, reflecting the mechanical activity of the elastic polymeric matrix. ϑ k is estimated as the number of side chains terminated by sulfonate groups and is set equal to the experimental value of the exchange capacity of the membrane (number of milligram-equivalents of fixed ions per gram of dry membrane, Γ 0 ) [9,15]. It is assumed that the polymeric matrix of the membrane is incompressible (V0 = const), such that the volume changes are determined by the amount of the solution absorbed by the membrane. Bulk elasticity of the polymer matrix does not depend on the shape of the cavities, but is determined by the total volume of the absorbed solution. 
  Rg T ϑk dAFlory V0 + V 2/3 1 Flory Π = = − (8) dV V0 + V V0 2 1.1.2. Curvature elasticity of the cavity wall Another contribution to the membrane free energy is related to the formation of the surface of the cavity wall and its bending. This contribution is expressed using the Helfrich harmonic approximation [28], discussed in detail by Safran [25]. ¯ AHelf = a(f0 − 2kH20 + 2k(H − H0 )2 + kK)

(9)

where f0 is the free energy per unit area of a planar surface, k and k¯ are bending and saddle-splay elastic moduli, respectively, a the surface area. H0 is the spontaneous curvature, H = 1/2(1/R1 + 1/R2 ), and K = 1/R1 R2 , where R1 and R2 are the principal radii of curvature. This contribution to the membrane free energy depends on the shape of the cavity and is different for the spherical and for the cylindrical one. For spheres R1 = R2 = R, while for long narrow cylinders R1 = R, R2 = ∞, and thus, AHelf = AHelf (R)

I.M. Shiryaeva, A.I. Victorov / Fluid Phase Equilibria 180 (2001) 115–138

121

in both cases. It should be mentioned that this contribution reflects only that part of the overall bending energy, which comes from polymeric tails of the hydrophobic matrix surrounding the cavity. One of the main assumptions of the present work is that this contribution can be represented by three adjustable ¯ which characterize a given membrane sample and remain constant in various parameters (f0 , k and k), ion-exchange processes. In the pseudophase approximation V and a can be considered as the volume and the surface area of one (typical) cavity of the radius R: V = V (R), a = a(R). Thus, the contribution of Π Helf to the swelling pressure owing to the formation and bending of the hydrophobic polymer wall of the cavity can be expressed as
Helf 
Helf  da dR ∂A ∂A 2f0 4kH0 Helf Πsph = − + = (spheres) (10) ∂a ∂R a dV R R2 R dV
Helf 
Helf  ∂A ∂A k da dR f0 Helf Πcyl = + − = (cylinders) (11) ∂a ∂R a dV R 2R 3 R dV The contribution to the bending rigidity that comes from the electrolyte solution inside the cavity is taken into account explicitly, as discussed below. 1.1.3. Electrostatic interactions in internal solution An electrostatic contribution to the free energy, Ael = Ael (V , a, R, {n i }), accounts for formation of a double electric layer and its curvature at the interface between the inner aqueous solution and the polymeric matrix. It consists of two parts Ael = ASt + Adif

(12)

where ASt is a contribution of the Stern layer of thickness δ, and Adif a contribution of the diffuse part of the double electric layer [30]. The former can be estimated using the expressions for spherical and cylindrical capacitors [24] and is given by ASt sph a ASt cyl a

=

σ 2 Rδ 2εε0 (R − δ)

(spheres)

(13)

=

σ 2R R ln 2εε0 R − δ

(cylinders)

(14)

where ε is the dielectric constant of the inner solution, ε0 the permittivity of vacuum, σ the surface charge density, and R the radius of a spherical or a cylindrical cavity, respectively. The expressions for Adif are obtained based on the solution of the linearized Poisson–Boltzman equation for the following case: the inner solution contains only counterions, because according to the experimental findings [9,15] Cl− ions, at any studied concentrations, do not penetrate inside the membrane. The solution to this problem, which is essentially given by the Gouy–Chapman theory of colloid [32], is considered in detail in Appendix B. The simplified approximate expressions used in our model can be written as 

 Adif 3δ σ2 1 1 1 1 sph 2 = 1+ 2δ + + 2 3δ + + 2 (spheres) (15) a 2εε0 χ R χ R χ χ

122

I.M. Shiryaeva, A.I. Victorov / Fluid Phase Equilibria 180 (2001) 115–138

Adif cyl a

=



 σ2 1 1 1 3 δ 1+ δ+ + 2 δ2 + + 2εε0 χ R 2χ R χ 8χ 2

(cylinders)

Here χ −1 is the Debye screening length, given by
2 2 1/2 e i zi ci χ= εε0 kT

(16)

(17)

where e is the electron charge, ci the concentration of the counterion, and the summation is performed over the counterions only. The consideration is not restricted to the monovalent ions, but the system may contain ions of arbitrary charge. Comparing the above expressions for the electrostatic free energy with the Hielfrich form, Eq. (9), the elastic constants of the double electric layer can be determined for the Stern part and for the diffuse part, respectively. f0St =

σ 2δ ; 2εε0

k St =

σ 2δ3 ; 3εε0

H0St = −

3 ; 8δ


 σ2 3 σ2 δ ; k dif = δ2 + + ; 2εε0 χ εε0 χ χ 8χ 2 
σ2 δ 1 k¯ dif = − (diffuse part) δ2 + + 2εε0 χ χ 2χ 2

f0dif =

σ δ k¯ St = − 6εε0

2 3

H0dif = −

(Stern layer)

(18)

χ (2χ δ + 1) ; 8 (χ δ)2 + 8χ δ + 3 (19)

Note that f0St and f0dif are the Helmholtz energies (per unit area) for plane capacitors having separations δ and 1/χ. The negative values of the spontaneous curvature reflect the tendency for the double electric layer to bend outwards, owing to the repulsion between the charges of the same sign. In the absence of the Stern layer (δ = 0), the expressions for k¯ dif and kdif reduce to those obtained by Lekkerkerker [33] in the Debye–Hückel limit (see formulae (36–37) of the original paper). If one assumes that the surface charge density is known, one can estimate the electrostatic contribution of the inner solution to the bending rigidity of the polymer–cavity interface, if only δ and 1/χ are known. Several serious (though quite common in the theories of aqueous micellar solutions [23–25,30,34,35]) approximations are involved in the derivation of the electrostatic contribution. We neglect the so-called image charge effects [36], i.e. a contribution to the electrostatic free energy associated with the presence of electric charges located at the boundary of two media having different dielectric constants. Based on spectroscopic arguments [26,37], we do not consider ion-pairing within the concentration range of the inner solution studied in this work [9,15]. An obvious oversimplification is the use of the same value for dielectric constant in both the inner and the outer solutions. Though often applied in the molecular thermodynamic models of micellar systems [22,30], this assumption, of course, cannot be true, and it has been shown [7] that the dielectric constant of the surface layer of the cavities can be approximately half that of the bulk. However, in the present work, aimed at validation of the model and study of the importance of different factors that govern ion-exchange equilibrium, we give merely estimates of different contributions and keep the expressions as simple as possible. The electrostatic part of the swelling pressure Π el also consists of the contributions from the Stern and the diffuse regions of the double electric layer Π el = Π St + Π dif

(20)

I.M. Shiryaeva, A.I. Victorov / Fluid Phase Equilibria 180 (2001) 115–138

123

with St Πsph

St Πcyl

and dif Πsph

dif Πcyl

  2f St 4k St H0St σ 2δ 2 δ = (spheres) + 2 = 0 − 2εε0 R R R R2   f0St k St δ2 σ 2δ 1 − = − = (cylinders) R 2εε0 R 3R 3 2R 3

  4 2δ 7 1 9 1 2 5δ + δ + (spheres) + + + 2 2R R 2 χ 2R 3 χ χ  

 σ2 2 1 1 δ 3 1 = + + 3 + δ+ (cylinders) 2εε0 χ R R 2 χ R χ 4χ 2 σ2 = 2εε0 χ

(21)

(22)



(23)

(24)

Eqs. (13)–(16) have been used for derivation of these expressions. The dependence of the Debye screening length on R is accounted for, but constant surface charge density, σ is assumed, as will be discussed later. At constant σ and δ, Π St depends only on R and falls off with an increase of the cavity size, whereas Π dif is determined by R and the Debye length acting in opposite directions. Eqs. (10) and (11) for the swelling pressure have a form of the Young–Laplace mechanical equilibrium equations for a spherical (#p = 2γ (R)/R) and a cylindrical (#p = γ (R)/R) dividing surfaces [38,39], with the curvature dependence of interfacial tension γ (R) taken into account. Special care should be taken, however, in relating the Helmholtz free energy per unit area, f0 , with the interfacial tension at the boundary between the solution and polymeric matrix. For estimating the surface tension of the micellar hydrocarbon core it is often assumed [40] that f0 does not depend on a and therefore γ = (∂A/∂a)T ,V ,{ni } = (∂af0 /∂a)T ,V ,{ni } = f0

(25)

which leads to interpretation of γ as the free energy per unit area of a flat interface (R → ∞). This approximation is invalid for estimating the contribution of the double electric layer, it even gives the incorrect sign of the interfacial tension. Indeed, taking into account the dependence of the surface charge density and, hence, of f0el (a) on the surface area, Eqs. (18) and (19), we get γ el = f0el + a(∂f0el /∂a) = −f0el

(26)

in agreement with the original work by Evans and Ninham [40]. The negative sign of the interfacial tension reflects the tendency of repelling charges on the surface to move apart. In the order to properly estimate the total value of the surface tension, both the electrostatic and non-electrostatic contributions need to be known. For the hydrophobic polymer matrix this requires a model adequately describing the dependence of f0 on the packing of polymer chains, that is f0 (a), the equation of state for the surface phase. The model proposed in the present work is based on the Helfrich approximation, which does not account for this effect, but pertains to a given packing state of the polymer chains (f0 is a constant). Accordingly, we have also assumed the constancy of the surface charge density, which is supported by the experimental data on the Nafion membranes [1]. It seems that under these conditions attempts to estimate the surface tension would not have given plausible results and we refrain from using this quantity in the present work.

124

I.M. Shiryaeva, A.I. Victorov / Fluid Phase Equilibria 180 (2001) 115–138

1.1.4. Model parameters and calculation procedure From expressions (8), (10), (11), (21)–(24) one can see that the swelling pressure is determined by the volume of the inner solution, by its composition, and by the shape and the size of the cavities formed inside the membrane Π(V , R, {n i }) = Π Flory (V ) + Π Helf (R) + Π St (R) + Π dif (R, {n i })

(27)

Experimental studies [3,9] suggest that membranes in the H+ -form contain predominantly spherical cavities, while narrow cylindrical channels prevail in the Mez + -form. The structure of the membrane in the mixed H+ + Mez+ form depends, most likely, on the initial state, as supported by the hysteresis of the ion-exchange isotherms. It is assumed that the membrane retains the cavities of the same shape as in the initial ionic form during the ion-exchange, i.e. on the H+ → Mez+ isotherm the cavities remain spherical, while they are cylindrical on the reverse curve, Mez+ → H+ . This is an obvious oversimplification, because in reality both the spherical clusters and the channels are always present, only their proportion varies. Because Π is different for spherical and cylindrical cavities, the equilibrium distribution of ions will depend upon the initial ionic form, thus, leading to a hysteresis of the calculated ion-exchange isotherms. According to the Helfrich expression, Eq. (9), the surface free energy is determined by the free energy of a flat interface and by a contribution, related to the bending elasticity of the surface. If the free energy of the flat interface is negligibly small relative to that determined by bending moduli, there is essentially no limitation to form the interface and the equilibrium state of the system is controlled by the elastic properties of the interface, i.e. by its curvature [41]. This situation is actually observed in certain surfactant-containing systems, such as micro-emulsions and some micellar solutions [42] having very low surface tension. Therefore, it is reasonable to neglect the contribution from f0 [41]. A swollen perfluorosulfonic membrane, in which segregation into hydrophobic and aqueous ionic regions occurs, can be viewed [3,27] as a concentrated solution of an ionic surfactant forming an inverted-micelle structure. In this latter case the possibility for the surfactant molecules to adjust their surface is likely to be substantially restricted by the low mobility of long polymeric chains in the membrane. It is then sensible to account for the free energy contribution of the flat interface, and to assume that the hydrophobic matrix-cavity interface can be characterized by a definite way of polymeric chains packing, which remains approximately the same, irrespective of the extent of swelling and ion-exchange. This is supported by the experimental data [1], according to which the surface area per fixed group in perfluorosulfonate membranes (∼70 Å2 ) is practically unaffected by the extent of swelling and upon replacement of mobile ions. Based on this we assumed a constant value of the surface charge density in our calculations, as mentioned above. Parameters f0 , k, H0 are characteristics of the hydrophobic polymer matrix. They are determined by the elastic properties of a given specific sample of the membrane and depend on the conditions of its production and storage (extrusion, hydrolysis, etc.). It is assumed that the parameters do not depend on the ¯ related to topology of the surface, nature of counterions nor on the water uptake. Note that the constant k, does not enter the equilibrium conditions, Eqs. (10) and (11). This is due to the fact that the equilibrium shape of the cavities is not sought for, alternative shapes are postulated as possible metastable states of the membrane and equilibrium with external solution is considered separately for each of them.

I.M. Shiryaeva, A.I. Victorov / Fluid Phase Equilibria 180 (2001) 115–138

At equilibrium the following set of equations should be satisfied
1/z1
1/z2 x2 (γ1 )1/z1 (γ2 )1/z2 x υ2 /z2 − υ1 /z1 Π = ln + ln ln 1 1/z2 1/z1 + x1 x2 (γ2 ) (γ1 ) Rg T     γH 2 O xH 2 O υH O = ln + 2 Π ln xH2 O γH2 O Rg T x1 =

n 1 , n 1 + n 2 + n H2 O

x2 =

n 2 , n 1 + n 2 + n H2 O

x1 + x2 + x− + xH 2 O = 1 z1 n 1 + z2 n 2 = Γ 0 z1 x1 + z2 x2 + z− x− = 0

xH 2 O =

n 1

n H2 O +

n 2

+

125

                       n H2 O

                     

(28)

where (x ) H2 O , (γ ) H2 O are the mole fraction and the activity coefficient in the external solution and in the membrane, x− is the mole fraction of anions in the external solution, and Γ is the exchange capacity. The last two equations express the electroneutrality of the membrane and the external solution, under the assumption that no mobile anions penetrate inside the membrane, which is confirmed experimentally [9,15] for the case of dilute external solutions considered in the present work. Assigning values to any two of the following variables x1 , x2 , x− , xH 2 O , x1 , x2 , xH 2 O , n 1 , n 2 , n H2 O for example, by fixing the concentration of salt outside, x− , and n 1 in the membrane one calculates the ion-exchange equilibrium by solving (28). The cavity radius needed to calculate thermodynamic functions of the inner solution should be obtained from the condition of internal equilibrium [43] of the swollen membrane   ∂(Amb /V ) =0 (29) ∂R V ,{n i } The solution of this equation gives reasonable results for the equilibrium radius of spherical and cylindrical cavities (e.g. R cyl = 4 Å, R sph = 19 Å. for a membrane in the H+ -form). However, as expected, these results should be considered only as an estimate, because the Helfrich approximation does not provide an equation of state for a polymeric interface, but relates to the surface with given characteristics (σ, f0 , k, H0 are constants). Therefore, a simplified procedure based on the experimental data on water uptake for the pure H+ - and Mez + -forms has been used in the present work. In accord with a previous proposal [31,44], we have considered water inside the membrane as consisting of two parts: water bound in the hydration shells of ions and free water. Using the assumption that the hydration number, g, of an ion remains constant, the values of υ i for counterions inside the membrane were estimated as υi = 43 π(ri3∗ + gr3H2 O )

(30)

where ri∗ is the ion crystallographic radius, and rH2 O = 1.85 Å [2]. It is known [6] that hydration numbers correlate with the water uptake in the pure ionic forms. According to the experimental data [15] the water uptake depends linearly (Fig. 4) on the extent of ion-exchange and is nearly the same for “sphere-dominated” and “cylinder-dominated” membranes of the same ionic composition. Hence, the

126

I.M. Shiryaeva, A.I. Victorov / Fluid Phase Equilibria 180 (2001) 115–138

Fig. 4. Experimental (points) [9,15] and correlated (lines) water uptake vs. the ionic fraction in the membrane.

amount of free water inside the membrane has been taken as a constant during the ion-exchange, change of the water content being solely due to the difference in hydration numbers of the exchanging cations. The amount of free water (0.046 wt.%) has been found from the experimental data on the water uptake for Cs+ , for which we assigned g = 1 [45]. The hydration numbers for other ions are calculated from the experimental water uptake data for the pure ionic forms of the membrane. The agreement with the values given by Israelachvili [46] (Table 2) partly validates the approximations made in estimating υ i . Using these approximations one calculates the volume of the inside solution directly thus making one of the Eq. (28) redundant, which allows us to discard the equation for the solvent. The radii of spherical and cylindrical cavities are calculated based on the value of the surface charge density and experimental values of the water uptake. For the pure ionic forms the results are compared with the experiment in Table 2. It is seen that from Li+ and H+ towards Cs+ , as the water uptake decreases, the calculated radius for spherical cavities deviates more and more from the experimental values, whereas for cylinders the calculated and experimental radii become closer. This is in line with the notion that the cavities are predominantly spherical in H+ -form, while the membrane with minimal water uptakes (K+ -, Cs+ -forms) contains basically the narrow channels. It is likely that in Na+ -form, which has an intermediate water uptake, both cylindrical and spherical cavities are essential.

Table 2 Ion hydration numbers and cavity radii for the membrane in the pure ionic forms Ions

H+ Li+ Na+ K+ Cs+

Hydration number g

Cavities R (Å)

Calculated

Spherical

9.4 9.7 6.2 2.6 1

Experimental

10 10 6.6 3–4 1–2

Cylindrical

Calculated

Experimental [3]

Calculated

Experimental [3]

23.7 24.3 18 11.8 9.6

23.7 22.45 21.05 17.25 17.8

15.8 16.2 12 7.9 6.4

5–8 5–8 5–8 5–8 5–8

I.M. Shiryaeva, A.I. Victorov / Fluid Phase Equilibria 180 (2001) 115–138

127

Table 3 Characteristics of the membrane used in calculations Quantity

Value

Dry membrane volume (V0 ) Number of side chains in polymer matrix (ϑ k = Γ 0 ) Stern layer thickness (δ) Surface charge density (σ ) Helmholtz energy of flat interface polymer matrix/inner solution (f0 ) Elastic modulus (k) Spontaneous curvature (H0 )

Experimental [1] Experimental [9,15] Estimated [23,30] Experimental [3,15] Fitted Fitted Fitted

0.5139 m3 /g 0.82 × 10−3 equiv/g 3.85 Å 0.372 C/m2 4.55 mN/m 3.89 × 10−20 J 1.18 nm−1

The surface charge density is estimated based on the spectroscopic data on the cavity radii for a membrane in the H+ -form (R = 23.7 Å) [3] σ =ρ

4/3πR 3 4πR 2

(31)

where ρ = F Γ 0 /V , Γ 0 the experimental value of the exchange capacity (Table 3) [9,15], and the volume of the inner solution, V , is determined from the experimental values of water uptake (Fig. 4), as discussed above. The model parameters f0 , k, H0 (Table 3) were fitted to the experimental data on the H+ –K+ ion-exchange equilibrium. By an order of magnitude k is comparable to the experimental value k = 4.11 × 10−21 J [42] for inverted-micelles in the system AOT–water–isooctane, which is used in the literature for imitating the behavior of swollen perfluorosulfonic membranes [26]. However, the calculated value is somewhat larger, reflecting higher rigidity of the polymeric membrane in comparison with surfactant micelles in a liquid solution. The contribution of electrostatic interactions (k St = 3.79 × 10−21 J, k dif = 9.09 × 10−21 J) to the bending rigidity of the membrane is smaller relative to the contribution from the elastic polymer matrix (Table 3). This agrees with the conclusion of the molecular statistical study by Szleifer et al. [47] that long hydrophobic tails of chain-molecules are responsible for the main contribution to the elastic constants of lipid bilayers. 2. Calculated results The calculated ion-exchange equilibrium for monovalent ions (Fig. 5) is in a satisfactory agreement with the experimental data. The model reflects correctly the selectivity sequence Li+ < Na+ < K+ < Cs+ , it describes the appearance of the hysteresis and the trend of its change for different ions, the predicted hysteresis increases from Li+ to Na+ and K+ in line with the experimental data. Contributions to the swelling pressure, which, according to the model, governs the hysteresis, are illustrated in Fig. 6 for the system H+ –K+ . The values of Πcyl − Πsph calculated for other pure ionic forms are given in Table 4. The overall values of the swelling pressure can be compared with the experimental estimate available in the literature for polystyrene divinyl cationites of lower equivalent weight, which lies in the interval: 5–20 MPa [31]. Thus, the magnitudes of the swelling pressure predicted by the model seem reasonable. According to the calculations, the main contribution to the swelling pressure comes from the Helfrich and double electric layer terms, while the bulk elasticity of the polymer matrix is far less important. There is no

128

I.M. Shiryaeva, A.I. Victorov / Fluid Phase Equilibria 180 (2001) 115–138

Fig. 5. Ion-exchange isotherms (T = 293 K) for monovalent ions plotted as the membrane composition vs. the bulk solution composition. Solid lines: calculated; points: experimental [9,15]. Dark circles: H+ → Me+ exchange; open circles: Me+ → H+ exchange.

experimental data on separate contributions, though a comparison with a computer simulation would be of interest, and work in a very similar field is in progress [47]. It should be noted that the experimental data for Cs+ (Fig. 5d) were obtained for the membrane that did not show hysteresis (the sample was stored for a long time in water). As seen from Fig. 5d these data are described by the curve predicted for cylinders. According to the experimental data, the transition of the membrane into Cs+ -form is accompanied by a substantial decrease of water content (Table 1), which may well lead to transformation of spherical clusters into narrow channels already in the course of the direct process H+ → Cs+ . It is for this system that the model predicts the largest difference between the free energies of the membranes with spherical and cylindrical cavities (Fig. 7). The predicted selectivity of the membrane to different Me2+ ions considered in the present work (Fig. 8) is nearly the same, because the estimated (from the data on the water uptake, Table 1) ionic sizes are

I.M. Shiryaeva, A.I. Victorov / Fluid Phase Equilibria 180 (2001) 115–138

129

Fig. 6. Swelling pressure contributions calculated for spherical and cylindrical cavities as a function of the membrane ionic composition for the system H+ –K+ .

nearly the same. In agreement with the experimental data the selectivity of the membrane toward divalent ions is higher than toward Na+ , for which the water uptake has a similar value (Table 1). The calculated curves for Ca2+ practically coincide with the experimental reverse isotherm, which we associate with the cylinder-dominated structure. The hysteresis almost disappears. This is actually observed experimentally for Mg2+ (Fig. 8a) however, in this case the predicted curves strongly deviate from the experimental ones. An attempt to improve the description has been made using regular solution type expressions for the activity coefficients of ions inside the membrane. The approach is often applied to describing Table 4 Swelling pressure differences calculated for the membranes having spherical and cylindrical cavities Ionic form

H+

Li+

Na+

K+

Cs+

Π cyl −Π sph (Mpa)

6

5

15

42

64

130

I.M. Shiryaeva, A.I. Victorov / Fluid Phase Equilibria 180 (2001) 115–138

Fig. 7. Helmholtz free energy of the membrane with spherical and cylindrical cavities as a function of its ionic composition in the systems H+ –Me+ .

Fig. 8. Ion-exchange isotherms (T = 293 K) for divalent ions. Points: experimental [9,15]; full lines: predicted; dashed lines: correlated using a model with additional term, Eq. (32) with β = −4.

I.M. Shiryaeva, A.I. Victorov / Fluid Phase Equilibria 180 (2001) 115–138

131

micellar pseudophase for mixed ionic surfactants [22,48] and has a molecular interpretation in terms of electrostatic interactions [30]. The chemical potential of an ion inside the membrane is expressed by an equation analogous to Eq. (A.5), with an additional term ln γ˜Me = β x˜ 2H ;

ln γ˜H = β x˜ 2Me

(32)

where γ˜i is the activity coefficient, and x˜i is the mole fraction of the ion on the water free basis: x˜Me + x˜H = 1, γ˜i reduces to 1 for pure ionic forms, x˜i = 1. An empirical parameter β, which effectively reflects various factors influencing the interaction of ions inside the cavity, was adjusted as to improve a correlation between the calculated and the experimental data. This yields a noticeable improvement of calculated results for Mg2+ (Fig. 8a), although there are still substantial deviations in the initial part of the curve. It seems that more careful consideration of factors, which are not taken into account by the model in its present form (e.g. the ion-pairing effects), may be important for multivalent ions. 3. Conclusion A thermodynamic model proposed herein reflects reasonably well the ion-exchange equilibrium properties of perfluorosulfonate membrane/aqueous salt solution system. Thus, the ideas developed originally for describing thermodynamic behavior of surfactant solutions and other systems with self-assembly prove useful in formulating a model of the membrane free energy, while taking into account its structural transformations on the mesoscale. The approach is sufficient for a description of the hysteresis of the ion-exchange isotherms for monovalent ions, for which the model in its present form gives satisfactory results. Using three adjustable parameters related to the polymer matrix and experimental water uptake data for pure ionic forms the model predicts correctly the selectivity sequence for different monovalent ions and the trend of its change. The results for divalent ions are not quantitative. We note that our approach is applied to ion-exchange membranes for the first time, and in order to check its basic validity the most simple version of the model has been formulated. That is why we have chosen rather crude approximations for expressing different contributions to the membrane free energy. Since the model reflects correctly the observed behavior of the membranes, it is sensible to refine the approximations used in its derivation. Thus, more adequate description of the electrostatic contribution, reflecting the ionic size effects should be made. For this purpose the mean spherical approximation, which is most popular now, can be used, its extension for a non-uniform solution inside the cavities being important. Another prospective is a molecular-based model, which incorporates a dependence of the elasticity constants on the polymer chain packing [25,49]. An interesting problem is the transformation of ion-exchange properties of the membrane with time, for instance, the disappearance of the hysteresis upon long storage in water. Modeling of this phenomenon, e.g. through description of the effects of shear in polymer matrix [50] and its relaxation on the elasticity constants, would be of interest. Extending the model to account for a distribution of cavities over their size and shape [23,35,43] would more closely reflect the membrane structure. List of symbols a surface area ai activity of component i A Helmholtz energy

132

Aw ci e f0 F g h H H0 k k¯ K Mez + n N p r∗ R Rg T V V0 xi z

I.M. Shiryaeva, A.I. Victorov / Fluid Phase Equilibria 180 (2001) 115–138

water uptake (wt.%) concentration of the counterion electron charge Helmholtz energy per unit area for flat interface Faraday’s constant ion hydration number length of cylindrical cavity mean curvature spontaneous curvature bending elasticity constant saddle-splay elasticity constant Gaussian curvature metal ion of charge z+ mole numbers equivalent ionic fraction pressure ion crystallographic radius cavity curvature radius gas constant temperature volume volume of unswollen polymer mole fraction of species i ionic valence

Greek symbols γ activity coefficient Γ 0 exchange capacity of the membrane δ the Stern layer thickness ε dielectric constant of the inner solution ε0 permittivity of vacuum ϕ electric potential µi chemical potential of component i Π swelling pressure q electric charge ρ electric charge density σ surface charge density ϑk number of cross-links χ inverse Debye length Superscripts dif the diffuse part of the double electric layer el electrostatic part

I.M. Shiryaeva, A.I. Victorov / Fluid Phase Equilibria 180 (2001) 115–138

Flory g Helf St

–

133

the Flory theory of elastic polymers gas the Helfrich theory of bending elasticity the Stern part of the double electric layer solution inside the cavities external solution the membrane phase

Subscripts cyl cylindrical cavity i component q component sph spherical cavity 1 cations participating in ion-exchange 2 − anion

Acknowledgements We are indebted to Dr. Irina V. Rosenkova for guidance through the experimental techniques and advice. We thank St. Petersburg Sate University Center for Basic Research in Natural Science for financial support (grant RKTZFE 1998–2000).

Appendix A The equilibrium state of the system membrane + external solution is determined by the Gibbs equilibrium principle δ{A +Amb }V ,{ni } = 0

(A.1)

where Amb is the Helmholtz energy of a swollen membrane, A the Helmholtz energy of a uniform external solution, and the variations are performed under the constraint of the constant total system volume and total mole numbers of the components. By a standard thermodynamic manipulation one obtains, under an assumption of incompressibility of the liquid solution, dV = −dV , the conditions of mechanical and material equilibria in the form 
d(Amb − A ) (A.2) Π= dV {n i } µ i (T , p , {n i }) − µ i (T , p , {n i }) = Πυi

(A.3)

In these equations, A is the Helmholtz energy of a uniform solution inside the membrane, so that the difference Amb − A accounts for the deviation of the Helmholtz free energy of a swollen membrane

134

I.M. Shiryaeva, A.I. Victorov / Fluid Phase Equilibria 180 (2001) 115–138

from that of a bulk electrolyte solution, Π = p − p is the swelling pressure, i.e. the pressure difference between the external and the internal solution

 dA ∂A P =− , P =− dV {n i } ∂V {n i } µ i and µ i are the chemical potentials of the component i inside and outside the membrane, respectively. Finally, υ i is the partial molar volume of the component. In an ion-exchange equilibrium the mole numbers of ions are not all independent, because of the electroneutrality condition: zi n i = const, where zi is the ionic charge, which means that the total charge of ions in the inner solution must be counterbalanced by the fixed charge at the cavity walls. Choosing an ionic species, q, as dependent, one gets the equation that governs the equilibrium distribution of ions between the external solution and the membrane [21] µ i − µ i + υi Π −

zi (µ − µ q + υq Π) = 0 zq q

(A.4)

where the chemical potentials refer to the external pressure p . For the electrochemical potential of an ion, one can write µi (T , p, {ni }) = µ0i (T , p) + Rg T ln ai (T , p, {ni }) + zi F Ψ

(A.5)

where F is Faraday’s constant, Ψ the average electric potential of the phase, µ0i (T , p) the standard chemical potential, and ai the activity. For the solvent we have µi (T , p, {ni }) = µ0i (T , p) + Rg T ln ai (T , p, {ni })

(A.6)

Taking infinitely dilute solution as the standard state (for the inside solution we additionally impose the requirement of infinitely large cavity, R → ∞, to approach the limit of bulk uniform solution µ 0i = µ 0i ) we obtain Eqs. (1) and (2) in the text. Note, that the average electrical potentials in the phases cancel out, since electroneutrality of the phases requires that the total charge moved in and out of the phase is zero, and so is the electrical work. The electrical work may however, enter into the equation, if the ion-exchange results in the alteration of the local environment of the ions (e.g. due to the difference in size or charge, or hydration of the ions) as reflected by the corresponding contribution to the activity coefficients. Of all mentioned contributions only the charge difference is accounted for in the framework of the classical Debye–Hückel approximation applied in the present work. Appendix B The Gouy–Chapman theory [32] is extensively used in the literature to describe electrostatic free energy of ionic micelles [24,30,34,35] and curvature elasticity of charged droplets [43] and bilayers [25,33,51]. For 1:1 electrolytes the solution of the linearized Poisson–Boltzmann equation (Debye–Hückel approximation) for particles or cavities of spherical and cylindrical symmetry are obtained by Winterhalter and Helfrich [51]. An approximate solution to the non-linear Poisson–Boltzmann equation [40] is given by Mitchel and Ninham [52], and in a slightly different form (based on the expansion of potential in the vicinity of the charged surface) by Overbeek et al. [44] and Lekkerkerker [33]. The main difference

I.M. Shiryaeva, A.I. Victorov / Fluid Phase Equilibria 180 (2001) 115–138

135

of our system is the absence of added salt (mobile co-ions) inside the membrane, and our goal was to obtain a solution that is not limited to the 1:1 electrolytes. We have chosen the linear approximation in the Poisson–Boltzmann theory, because the resulting proportionality of the electric potential and the surface charge [53] essentially simplifies the expressions, and also because the solution of the non-linear case is rather cumbersome for multivalent electrolytes. With this in mind we write the linearized version of the Poisson–Boltzmann equation for the diffuse part of the double electric layer inside the cavities
 1 d ρ0 2 dϕ r = χ 2ϕ − (spheres) (B.1) 2 r dr dr εε0
 ρ0 dϕ 1 d r = χ 2ϕ − (cylinders) (B.2) r dr dr εε0 where ϕ is the electric potential, r the radial distance from the cavity center, χ the inverse Debye screening length given by Eq. (17), ρ0 = e i zi ci0 is the average charge density in the cavity and ci0 is the average concentration of counterions. The boundary conditions are
 dϕ =0 (B.3) dr r=0
 σ R2 dϕ = (spheres) (B.4) dr r=R−δ εε0 (R − δ)2
 σR dϕ = (cylinders) (B.5) dr r=R−δ εε0 (R − δ) first of which follows from the symmetry requirement at the center of the cavity, and the second expresses the Gauss law. Here R is the cavity radius, δ the Stern layer thickness, and σ = q/a is the surface charge density (a = 4πR 2 for spheres, a = 2πRh for cylinders). We note in passing that the continuity of the electric potential at the boundary between the Stern and the diffuse parts of the double electric layer is also tacitly assumed, as required to have additive contributions of these parts to the total free energy [30]. The potentials that satisfy the above equations can be written as   sinh[χ r] R2 σ ϕsph (r) = εε0 χ(R − δ) cosh[χ(R − δ)] − sinh[χ (R − δ)] r ρ0 + (spheres) (B.6) εε0 χ 2 ϕcyl (r) =

I0 (χr) ρ0 σR + εε0 χ(R − δ) I1 (χ(R − δ)) εε0 χ 2

(cylinders)

(B.7)

where Is is the modified Bessel function of the sth order (see, e.g. [54], page 611), which can be expressed as a power series ∞  χα 2k+s  1 Is (χα) = (B.8) k!(k + s)! 2 k=0 The expressions for the potential are analogous to those given in [43,51] for spherical and cylindrical bilayers in the same approximation but in the limit of high added salt. The difference is the absence of the

136

I.M. Shiryaeva, A.I. Victorov / Fluid Phase Equilibria 180 (2001) 115–138

second term, which emerges in our case because the cavities contain no added salt. This constant term does not alter the expression for the electrostatic contribution to the free energy, which is independent of the choice of zero for the potential. One can obtain the required expression by applying the Debye charging process [55]. A much simpler alternative is to calculate the electrostatic free energy of an aggregate (e.g. micelle, bilayer, cavity) by integration at constant Debye screening length [33,40,43,52]  σ Ael = a ϕa (σ ) dσ (B.9) 0

where ϕ a is the surface potential, though care must be taken, because as discussed by Trizac and Hansen [53] this expression may not always be valid. In our case of the linearized Poisson–Boltzmann equation (by either route) we obtain   Adif σ 2R2 sinh[χ (R − δ)] sph = (spheres) (B.10) a 2εε0 (R − δ) χ(R − δ) cosh[χ (R − δ)] − sinh[χ (R − δ)] Adif cyl a

=

σ 2R I0 (χ(R − δ)) 2εε0 χ(R − δ) I1 (χ(R − δ))

(cylinders)

(B.11)

These expressions can be simplified substantially for χ (R−δ)  1. Applying this limit seems reasonable, because according to experimental data [9,15] χ ≈ (4.9–8.3) nm−1 in the cavities (concentration of counterions in the membrane is about ∼5–10 N, depending upon the water uptake). R = 17–24 Å by spectroscopy [3], thus, leading to an estimate of χ (R − δ) ≈ 10. For large α-values sinh α/cosh α can be expanded in powers of exp(−2α) and hence sinh[χ(R − δ)] = 1 + 0([χ(R − δ)]n ) cosh[χ(R − δ)] i.e. the ratio converges to one faster than any finite power of [χ (R − δ)]. This allows one to simplify the expression for spheres to Adif sph a

=

σ 2R2 2εε0 (R − δ)[χ(R − δ) − 1]

(B.12)

For cylinders in the large χ(R − δ) limit, we get upon applying asymptotic expansions of the Bessel functions (see [54], page 619) and collecting terms up to [χ (R − δ)]−2   Adif σ 2R 1 3 cyl = 1+ + (B.13) a 2εε0 χ(R − δ) 2χ(R − δ) 8[χ (R − δ)]2 It is convenient to write the above expressions as a power series in 1/R 

 Adif σ2 1 1 1 1 3δ sph 2 = 1+ 2δ + + 2 3δ + + 2 (spheres) a 2εε0 χ R χ R χ χ 

 Adif σ2 1 1 1 3 δ cyl 2 = 1+ δ+ + 2 δ + + (cylinders) a 2εε0 χ R 2χ R χ 8χ 2

(B.14) (B.15)

where the first term is the energy of a plane capacitor with the effective plate separation equal to 1/χ , and the other terms represent curvature corrections. Comparable expressions were previously obtained

I.M. Shiryaeva, A.I. Victorov / Fluid Phase Equilibria 180 (2001) 115–138

137

by Winterhalter and Helfrich [51] for a bilayer membrane in a salt solution. Unlike the latter we have considered only the inner side of the membrane and have taken into account the second order corrections for curvature in calculating the surface charge density. Writing the Winterhalter and Helfrich formulae for the inner side of the bilayer and correcting the surface charge density after somewhat lengthy manipulations one can obtain the expressions identical in form to those given by (A.14–A.15), the difference being in the definition of the Debye screening length, in the present work it does not include the salt and corresponds to a weaker screening comparing to the case of added salt, in agreement with a discussion given in [25]. References [1] C. Heinter-Wirguin, J. Membr. Sci. 120 (1996) 1–33. [2] K.A. Mauritz, A.J. Hopfinger, Structural properties of membrane ionomers, in: J.O’M. Bokris, B.E. Conway, R.E. White (Eds.), Modern Aspects of Membrane Electrochemistry, Plenum Press, New York, 14 (1982) 425–508. [3] T.D. Gierke, W.S. Hsu, The cluster-network model of ion clustering in perfluorosulfonated membranes, in: A. Eisenberg, H.L. Yeager (Eds.), Proceedings of the ACS Symposium on Perfluorinated Ionomer Membranes, Series 180, American Chemical Society, Washington, DC, 1982, Chapter 13, pp. 283–307. [4] H.L. Yeager, Transport properties of perfluorosulfonate polymer membranes, in: A. Eisenberg, H.L. Yeager (Eds.), Proceedings of the ACS Symposium on Perfluorinated Ionomer Membranes, Series 180, American Chemical Society, Washington, DC, 1982, Chapter 4, pp. 41–63. [5] N. Berezina, N. Gnusin, O. Dyomina, S. Timofeyev, J. Membr. Sci. 86 (1994) 207–229. [6] H.L. Yeager, Cation exchange selectivity of a perfluorosulfonate polymer, in: A. Eisenberg, H.L. Yeager (Eds.), Proceedings of the ACS Symposium on Perfluorinated Ionomer Membranes, Series 180, American Chemical Society, Washington, DC, 1982, Chapter 3, pp. 25–39. [7] J.R. Bontha, P.N. Pintauro, J. Chem. Eng. Sci. 4 (1994) 3835–3851. [8] F.A. Belinskaya, S.V. Timofeev, L.A. Karmanova, N.Yu. Irshina, Vestnik SPbGU, Series 4, 2 (1998) 67–73 (in Russian). [9] I.M. Shiryaeva, I.V. Rosenkova, Russ. J. Appl. Chem. 71 (1998) 781–785. [10] I.M. Shiryaeva, O.A. Kvyatkovskaya, I.V. Rosenkova, Vestnik SPbGU, Series 4, 3 (1998) 59–67 (in Russian). [11] I.M. Shiryaeva, S.V. Zakharova, I.V. Rosenkova, Vestnik SPbGU, Series 4, 1 (2000) 78–84 (in Russian). [12] J.H.G. Van der Stegen, H. Weerdenburg, A.J. van der Veen, J.A. Hogendoorn, G.F. Versteeg, Fluid Phase Equilibria. 157 (1999) 181–196. [13] S. Mafe, P. Ramirez, A. Tanioka, J. Pellicer, J. Phys. Chem. B 101 (1997) 1851–1856. [14] M.W. Verbrugge, R.F. Hill, J. Phys. Chem. 92 (1988) 6778–6783. [15] I.M. Shiryaeva, Ion-exchange equilibrium between micro-inhomogeneous perfluorosulfonate polymer membrane and aqueous salt solution, Ph.D. Thesis, 2000, St. Petersburg, Russia (in Russian). [16] A. Katchalsky, I. Michaeli, J. Polym. Sci. 15 (1955) 69–86. [17] A. Katchalsky, M. Zwick, J. Polym. Sci. 16 (1955) 221–234. [18] T.L. Hill, An Introduction to Statistical Thermodynamics, Addison-Wesley, Reading, MA, 1960. [19] J. Hasa, M. Ilavsky, K. Dusek, J. Polym. Sci. Polym. Phys. Ed. 13 (1975) 253–262. [20] E. Vasheghani-Farahani, J.H. Vera, D.G. Cooper, M.E. Weber, Ind. Eng. Chem. Res. 29 (1990) 554–560. [21] G. Maurer, J.M. Prausnitz, Fluid Phase Equilibria. 115 (1996) 113–133. [22] C. Sarmoria, S. Puvvada, D. Blankschtein, Langmuir 8 (1992) 2690–2697. [23] R. Nagarajan, E. Ruckenstein, Langmuir 7 (1991) 2934–2969. [24] A. Heindl, H.-H. Kohler, Langmuir 12 (1996) 24–64. [25] S.A. Safran, Statistical Thermodynamics of Surfaces, Interfaces and Membranes, Addison-Wesley, Reading, MA, 1994. [26] S.A. Lossia, S.G. Flore, S. Nimmala, H. Li, S. Schlick, J. Phys. Chem. 96 (1992) 6071–6075. [27] E. Szajdzinska-Pietek, J. Pilar, S. Schlick, J. Phys. Chem. 99 (1995) 313–319. [28] W. Helfrich, Z. Naturforsch. C28 (1973) 693–703. [29] J.P. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, 1953 (Chapter 13).

138 [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]

I.M. Shiryaeva, A.I. Victorov / Fluid Phase Equilibria 180 (2001) 115–138 A. Shiloach, D. Blankschtein, Langmuir 13 (1997) 3968–3981. F. Helfferich, Ion-Exchange, McGraw-Hill, New York, 1962. R.J. Hunter, Foundations of Colloid Science, Vol. 1, Clarendon Press, Oxford, 1987, pp. 330–394. H.N.W. Lekkerkerker, Physica A 159 (1989) 319–328. D. Stigter, J. Colloid Interface Sci. 47 (1974) 473–482. P.J. Missel, N.A. Mazer, G.B. Benedek, C.Y. Young, M.C. Carey, J. Phys. Chem. 84 (1980) 1044–1057. D. Stigter, J. Phys. Chem. 78 (1974) 2480–2485. S. Schlick, M.G. Alonso-Amigo, J. Bednarek, Colloids Surf. A: Physicochem. Eng. Aspects 72 (1993) 1–9. J. Gaydos, Y. Rotenberg, L. Boruvka, P. Chen, A.W. Neumann, The generalized theory of capillarity, in: A.W. Neumann, J.K. Spelt (Eds.), Applied Surface Thermodynamics, Marcel Dekker, New York, 1994, pp. 1–52. S. Ono, S. Kondo, Molecular theory of surface tension in liquids, Handbuch der Physik, Vol. X, Springer, Berlin, 1960. D.F. Evans, B.W. Ninham, J. Phys. Chem. 87 (1983) 5025–5032. S.A. Safran, T. Tlusty, Ber. Bunsenges. Phys. Chem. 100 (1996) 252–263. H. Kellay, J. Meunier, J. Phys. Condens. Matter 8 (1996) A49–A64. J.Th.G. Overbeek, G.J. Verhoeckx, P.L. deBruyn, H.N.W. Lekkerkerker, J. Colloid Interface Sci. 119 (1987) 422–441. S.J. Sondheimer, N.Y. Bunce, C.A. Fyfe, J. Macromol. Sci. Rev. Macromol. Chem. Phys. C26 (1986) 351–411. J.N. Israelachvili, Intermolecular and Surface Forces, Academic Press, San Diego, 1992. I. Szleifer, D. Kramer, A. Ben-Shaul, W.M. Gelbart, S.A. Safran, J. Chem. Phys. 92 (1990) 6800–6817. S. Kutter, J.-P. Hansen, M. Sprik, E. Boek, J. Chem. Phys. 112 (2000) 311–322. J. Rathman, J.F. Scamehorn, Langmuir 2 (1986) 354–361. L.A. Turkevich, S.A. Safran, P.A. Pincus, Theory of shape transitions in micro-emulsions, in: A.W. Neumann, J.K. Spelt (Eds.), Surfactants in Solutions, Vol. 6, Plenum Press, New York, 1986, p. 1177. L.D. Landau, E.M. Lifshitz, Theoretical Physics: Elasticity Theory, Vol. 7, Nauka, Moscow, 1965. M. Winterhalter, W. Helfrich, J. Phys. Chem. 92 (1988) 6865–6867. D.J. Mitchell, B.W. Ninham, Langmuir 5 (1989) 1121–1123. E. Trizac, J.-P. Hansen, J. Phys. Condens. Matter 8 (1996) 9191–9199. G. Arfken, Mathematical Methods for Physicists, Academic Press, New York, 1985.