Irreversible thermodynamics of transport across charged membranes

Journal of Membrane Science, 25 (1985) 153-1’70 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

IRREVERSIBLE THERMODYNAMICS CHARGEDMEMBRANES

OF TRANSPORT

ACROSS

PART I - MACROSCOPIC RESISTANCE COEFFICIENTS SYSTEM WITH NAFION 120 MEMBRANE ANNA NAREBSKA,

153

FOR A

STANISLAW KOTER and WOJCIECH KUJAWSKI

Institute of Chemistry, Nicolaus Copernicus (Poland)

University, ul. Gagarina 7, 8 7-l 00 Torufi

(Received December 19, 1984; accepted in revised form July 12, 1985)

Summary The transport of aqueous NaCi solutions across the perfluorinated Nafion 120 membrane is studied on the base of irreversible thermodynamics. The straight resistance coefficients rii, partial frictions fiE and diffusion indices RT/cirii are presented and discussed. The results suggest that the main force, which impedes the flow of chloride ions across the membrane is not the friction of these ions with the negatively charged polymer network, but the friction with water. The diffusion indices RT~irii exceed self-diffusion coefficients found by some authors while using tracer technique. Following Meares’ suggestion such results point out to the convective ffow contribution to the transport of ions and water.

Introduction Although the theory of transport across charged membranes in terms of irreversible thermodynamics is known from the fifties, the experimental information about the phenomenological transport coefficients, determined with the necessary number of experimental data and calculated without making any simplifying assumptions, has been published up to date by Meares [l3) and Kumamoto [4] only. Since 1968 Meares et al. [l-3, 5-93 has presented the experimental data for Zeo-Karb 315 membrane and single NaBr, CsBr and SrBr, solutions and discussed the transport behaviour of these systems based on the sets of differential discontinuous, resistance, friction and coupling coefficients. Lately Kumamoto and Kimizuka [4 J published a new solution of the phenomenological transport equations and some selected results for BaCl, and poly(styrenesulphonic acid) cation exchange membrane. In Staverman’s paper [lo] on NaCl transport across sulphonated-styrene/ DVB-copolymer membrane and those by Paterson [ll, 121 for the same electrolyte and AMF C-60 membrane, the transport phenomena have been discussed on the ground of a reduced number of experimental figures, intro0376-7388/85/$03.30

0 1985 Elsevier Science Publishers B.V.

154

ducing simplifying assumptions. Consequently the use of these results in comparative examinations of transport behaviour of different systems is rather restricted. These relatively few papers do not allow for broad interpretation of transport in ionic polymer media. Still more experimental work should be done before all the information supplied by irreversible thermodynamics wil make it possible to construct the comprehensive overview of the transport phenomena in charged membranes. In the present paper we aim at describing quantitatively all the kinds of isothermal transport proceeding in the system Nafion 120 membrane/l:1 electrolyte solutions at room and elevated temperatures. The perfluorosulfonate cation exchange Nafion membranes (DuPont de Nemours and Co., U.S.A.) were devised for the applications in corrosive media and at the temperatures up to 360 K or above. The effects of heating on the membrane morphology and properties may be many. Some of these effects are known from papers by Yeager et al. [13--151, Smith and Jones [ 161, Narebska et al. [17], who studied the properties of Nafion membranes at temperatures up to 363 K. The experiments presented here were planned according to the demands of irreversible thermodynamics and performed at temperatures 298, 313, 333 K. Since to describe the transport of 1:l electrolyte six independent experiments are necessary, the following have been chosen: electrical conductance, electroosmotic volume flow, concentration potential, osmotic volume flow, salt diffusion flow and pressure driven flow. Here we start by presenting the experimental results followed by the interpretation of transport behaviour based on selected phenomenological coefficients, mainly resistance coefficients. In numerous papers on the Nafion 120 membrane some experimental results shown here can be found. However, for calculations of transport coefficients a comprehensive collection of the data determined for identical membrane samples and using equipments ensuring high reproducibility and accuracy are necessary. Therefore we present them here altogether. Mathematical formulation The six independent experiments (see “Results”) enabled computations of all the phenomenological coefficients of transport equations without making any simplifying assumptions. In the computations we followed the procedure described by Meares [ 1, 51 based on phenomenological transport equations adapted to practical fluxes by Kedem and Katchalsky [ 181 and on the definition of differential transport coefficients by Michaeli and Kedem [ 191. The following sequence has been obeyed experimental results + L,,

+ ljk + rik

ir fik * -Dik L 4ik

For the explanation of symbols see list of symbols.

155

The experimental data were taken first to calculate the differential discontinuous coefficients L,, relating the salt, volume and current fluxes J,, J,, I to infinitesimally small thermodynamic forces aIl/ulcs, a(p - Il), 3E. These coefficients characterize the transport across a membrane in equilibrium with a single solution of concentration cS. The procedure itself is rooted in the additivity principle j-l

J”(c~ + Cj)

=

C

Jv(ci

+ ci+l)

(1)

and is particularly- applicable for calculations of these coefficients, which describe transport generated by concentration gradients of the solutions on both sides of the membrane, such as osmotic and salt diffusion, or which become different during transport due to salt rejections, as happens in hydrodynamic volume and salt fluxes. The matrix of molar conductance coefficients [Zik] relating the molar fluxes of permeating components to their forces was calculated from the equation [ 21 [l&I = dr-l [L,fi]P

T

(2)

where I’ is the matrix transforming molar fluxes to the practical fluxes and d is the thickness of the membrane. Next, by inverting the matrix [Z&l the molar resistance coefficients were obtained [rikl

=

(3)

[likl-’

The meaning of these coefficients is relatively simple. The straight coefficient rE defined by equation (4) measures the force per mole of particle i required to generate unit flux of i when all other fluxes are zero [2]. It is related to friction coefficients fg? by equation cirii =

c

(5)

fik

k#i

The friction coefficient fik itself denotes the force per mole of i owing to its interaction with the amount of k normally in its environment at unit relative velocity of i and k [20] . Consequently Zjrii is a measure of the total frictional interactions between one mole of component i and all others in its vicimty. The cross coefficients r& are related to friction coefficients by equation

r ik

= -fik/zk

if

k

(6)

156

rik represents the single interaction between one mole of species i and one mole of k [ZO] . The above equations enabled us to calculate partial frictions fi defined by

f;

=&L Cjrii

(7)

which denote the contribution of the interaction between species i and k to the total interactions of species i with the environment with respect to which it is moving. Both the conductance Eii(at Fk = 0) and the resistance coefficient rii (at Jk = 0) permit calculations of the diffusion indices of the moving particles RT Eti Ej

RT and Zjrii

Meares [2] proved that diffusion indices calculated applying ri coefficients are close to the experimental self-diffusion coefficients of the ions determined using tracer technique. With respect to water, they reflect the convective contribution to the flows. In this paper diffusion indices will be discussed in one of the subsequent sections. In what follows the transport of sodium ions (l), chloride ions (2) and water molecules (w) across Nafion 120 membrane (m) will be discussed considering variations of the straight resistance coefficients rE, some selected friction coefficients and diffusion indices RT/?irii as functions of the molality of the external electrolyte and temperature as the variables. Applying irreversible thermodynamics to the interpretation of transport phenomena, we treat Nafion 120 membrane as macroscopically homogeneous, characterised by concentrations of the components: counterions, coions, water and the fixed charges in mol/dm3 of the swollen membrane. Comments on the morphology of charged membranes and their macroscopic homogeneity in the sense given to the term by irreversible thermodynamics can be found in the papers published by Meares [9] and Paterson [21]. Experimental Systems and variables The system used consisted of a Nafion 120 membrane/NaCl aq., and was studied at T = 298, 313 and 333 K. Other parameters are used as described before [22], namely: m,,t = 0.05-4 mol/kg H*O, p = 0.2-l MPa, I = 2.3 mA/cm*. The methods and equipments used for the determinations of the membrane properties, like swelling, Donnan sorption, membrane resistance and concentration membrane potential, as well as those used for measuring the

157

transport phenomena, i.e. the electroosmotic, diffusion, osmotic and hydrodynamic flows, were described in our previous paper [?2]. In calculations the only difference concerned the computations of the differential practical coefficient LnP. In the view of very high precision needed for measuring the effluent concentration in the hydrodyamic experiment, the term aII/ap for calculating LnP was evaluated by an iteration method under assumption LPn = LnP . Results Swelling and Donnan sorption Since to discuss the membrane properties and transport phenomena the electrolyte and water uptake are fundamental, we measured them despite the number of data of this kind published earlier. The results are presented in Table 1. Detailed investigations on sorption of NaCl into Nafion 120 membrane were presented in our previous paper [17]. TABLE

1

The sorbed electrolyte (C,), 120 membrane equilibrated

mext

0.05 0.1 0.25 0.5 1.0 2.0 3.0 4.0

water (C,) and fixed with NaCl solutions

298 K

charge (mole

313 K

(C,) concentrations per dm3 of swohen

in a Nafion membrane)

333 K

CZ

zw

5,

C,

5w

5,

%

5,

c,

0.0024 0.0058 0.0235 0.0672 0.174 0.379 0.571 0.760

21.3 21.3 21.2 20.9 20.6 19.8 18.9 17.9

1.19 1.20 1.21 1.22 1.25 1.27 1.27 1.28

0.0023 0.0058 0.0237 0.0642 0.169 0.379 0.571 0.762

20.5 20.2 20.1 19.9 19.6 18.8 17.6 16.5

1.20 1.20 1.21 1.23 1.25 1.27 1.27 1.28

0.0028 0.0061 0.0240 0.0659 0.174 0.379 0.572 0.760

20.2 19.9 19.4 19.1 18.9 18.1 16.9 15.7

1.20 1.20 1.21 1.23 1.25 1.27 1.27 1.28

Membrane conductivity The conductivity of Nafion 120 membrane (Fig. 1) increases rather steeply with increasing mext as a result of Donnan sorption, The activation energy of conductance for the membrane equilibrated with NaCl solutions of molality 0.1-l is about 16.3 kJ/mol whereas for the solution itself it is 12.3 kJ/ mol. Apparent transport number of Na’ The apparent transport numbers of sodium counterions tlaPP calculated based on concentration potentials are seen in Fig. 2. The relation of F1aPP to the concentration of the external electrolyte is a well known phenomenon, whereas from Fig. 2 it follows that there is no variation of the transport

158

o 1

298K

0313

K

t

l 333K

0 0

I

1

I

1

2

3

Fig. 1. The conductance

if

L 4

m

of Nafion 120 membrane ( xn, ) in NaCl solutions.

PP ,

Fig. 2. The apparent transport number of Na+ otaPP ) in Nafion 120 membrane vs. the molality of external NaCl solution, at temperatures: 298 K (o), 313 K (0) and 333 K (e).

numbers with temperature within the range of the experimental error or only slightly more than that. Since FIapp is a function of the ratio of cation to anion mobilities, the constancy of fIapp suggests that the mobilities of both ions within the membrane vary with temperature to the same extent. Water transport number A decrease of a flow rate with decreasing water content within the membrane and with increasing sorption of the electrolyte is well established. The

159

transport numbers of water in Nafion 120 membrane calculated from the electroosmotic flows at 298, 313 and 333 K decrease with increasing molality of sodium chloride solution from 10 to 3 mol/F (Fig. 3), however they change only slightly with temperature. Similarly as Z1aPP the transport number of water depend on the mobility ratio of both ions not influenced by temperature. Also it is known from the paper published by Glueckauf [23] that hydration numbers of ions in the internal membrane solution change with temperature only to a slight extent.

O 298K 3f3 K

0

I

I

I

2

3

I

4m

Fig. 3. The transport number of water (7,) external NaCl solution.

in Nafion 120 membrane vs. the modality of

Osmotic volume and salt diffusion flows In Figs. 4 and 5 are presented the composite osmotic and diffusion curves obtained for a system with a solution of constant concentration m’ in one compartment (m’ = 0.01) and m” increasing from 0.01 to 4.0 in another one. These curves were drawn according to Meares’ additivity rule. The applicability of this principle to the system studied had been verified previously. In each experiment the flow was measured at the steady state established after 4 to ‘7 hr with respect to the concentrations m’, m”. In contrast to ion and water transport numbers, the flow under osmotic pressure is very sensitive to temperature. The salt flow in an ideally permselective membrane should be zero. In membranes which are not ideally permselective, transport of salt proceeds, limited by the sorption and mobility of the ions. In the system with a Nafion 120 membrane, the flux of sodium chloride ranges around l-5% of the flux of water (Fig. 5). The activation energy for the diffusional flux of NaCl across the Nafion 120 membrane in contact with the solution of molality 0.05 M was found to be 29.7 kJ/mol and is close to 30.3 kJ/mol published

160

o 298K ‘0 313K ’ 333K

/

P

IO-' Fig. 4. The composite curves of volume flux (Jv) vs. the molality of NaCl solution on the concentrated side of a membrane (m”). The molality of the dilute side m' = 0.01.

Fig. 5. The composite curves of salt flux (J,) vs. the molality of NaCl solution on the concentrated side of a membrane (m"). The molality of the dilute side m' = 0.01.

161 by YWP and Kipling 1241. The activation energy for the diffusion of NaCl in free solutions of the same concentration is 19.6 kJ/mol.

Hydrodynamic flow In experiments with the Nafion 120 membrane and with Ap ranging from 0.2 to 1 MPa, the stationary flows (i.e. Jt , c,ff = constant) were established after 5 to 24 hr. Only the results complying with the requirement of linearity of JF = f(p) were taken for computations, The results drawn in coordinates aJk/ap = f(m) (Fig. 6) prove a strong dependence of the flow on concentration and temperature. We have also measured the pressure driven flow of water which, rather than Jt for solution, can be taken to estimate the resistance imposed by the polymer matrix against the fluid crossing the membrane. The activation energy for water flow for the membrane is 25.5 kJ/ mol, whereas for water in bulk it is 15.2 kJ/mol.

o 298 K h ru

1.2-

0313

K

‘2 -E ”

i.0 -

‘,o LI1

ek$

O8

” Qti-

44-

az-

I

,

lb’ Fig. 6. Hydraulic permeability NaCI solution.

I

100 of NaCl solution

m

(atlap)

vs. the molality of external

Discussion Straight resistance coefficients rc Figs. 7(a)-9(a) present the variation of Fii coefficients with molality of external sodium chloride solution, for three temperatures. These coefficients differ in orders of magnitude and change with molality of NaCl to different

162

lo-

IO0

Fig. 7. The concentration transport I, 1 (a) and partial Fig. 8. The concentration transport rt2 (a) and partial

m

f0”

and temperature dependence of: resistance frictions fls (b);298 K (o), 313 K (a), 333 and temperature dependence of: resistance frictions fz,f: (b); 298 K (o), 313 K (o), 333

too

m

coefficient of Na+ K(e). coefficient of ClK (e).

extents. When discussing the concentration dependence of rii one should stress however that taking into account its reciprocity against &, only the coefficients calculated for finite concentrations have a definite meaning (at Ci + 0 rz --f -). At external molality of sodium chloride of 0.1 m we find rz2:r ll:rww = 2 x lOI

: 1 x 10’” : 3 x 10’

As it has been stated earlier [25], the resistance coefficients of ions are expected to vary inversely in proportion to their concentrations. To some degree the results of Figs. ‘7(a) and 8(a) confirm these expectations, Taking for comparison the coefficients for 298 K, one can see that

163

does not change much, decreasing from 1 X lOlo J-m-set-molP to 0.8 X lOlo J-m-sec-mol-2 or slightly less, when the molarity of sodium ions inside the membrane increases from 1.2 to 2.05 mol/dm3 (at men = 4). (ii) r22 decreases by two orders, i.e. from 2 X 1012 to 2 X 10” J-m-secmolY2, however at the same time the internal molarity of chloride ions rises from 0.006 mol/dm3 at mext = 0.1 to 0.76 mol/dm3 at mext = 4.0. (iii) Instead, the resistance to transport of water increases from 3 X lo7 to 9 X 10’ J-m-set-molV2 when the molarity of water changes from 21.3 to 17.9 mol/dm3. The threefold increase of the r,, coefficient for relatively small variations in water content provoke thoughts on the long known problem of the amount of free water forming the internal membrane solution in a strongly hydrated, charged polymer network. According to eqn. (5) the product of straight resistance coefficient rg and concentration Zi expresses the sum of frictions of species i with all others appearing in the membrane, including the polymer matrix. These frictions in

(i)

rI 1

12-

a)

IO8-

4 f0”

0

1

10”

a 100

L lo*

m

m

Fig. 9. The concentration and temperature transport r,, (a) and partial frictions fEk

dependence (b);

298

K (o),

of: resistance 313

K (a),

coefficient 333

K (0).

of water

164

the transport of cations, anions and water different amounts of the particular species ticle but also due to different strengths comments on rii coefficients can be given to partial frictions f$, defined by eqn. (7).

are different not only due to the in the surroundings of a given parof interactions. More informative by separating the frictional force

Friction in transport of sodium counter-ions The transported sodium ions collide with water (lw), polymer network (lm) and chloride ions (12). At first sight the results seen on Fig. 7(b) substantiate the assumption made by many authors that friction between cations and anions is zero. Here the molarity of chloride ions in the membrane was as high as cZ = 0.76 mol/dm3 at mext = 4.0. Considering the morphology of Nafion membranes, it is not obvious, however, that anions can flow far from cations. When one recalls the hypothesis of channels connecting clusters through which an electrolyte has to pass when crossing the membrane the result that r12 is zero seems to be odd. Nevertheless, with f,? zero, the total friction of cations results from frictions with water molecules (lw) and polymer chains (lm). At a temperature of 298 K and low m,,t the ratio of f,P,/flP, is not far from 1, with predominance to water. With increasing temperature, the friction with water decreases to 0.42 and at 333 K it is not much above one third of a total friction. To some extent the trend in ratios of the partial frictions with water and polymer chains correspond to deswelling and shrinkage of the membrane with heating but, again, it is much more pronounced than changes in swelling.

Friction in transport of chloride coions It has been already stressed in this paper that sorbed salt becomes a sizable part of the total membrane electrolyte concentration above mext = 0.25. Considering the experimental error in determining r& at low mext, and consequently in the calculated f&., we present the partial friction for chloride coions starting from rnext = 0.5. The partial friction of chloride-sodium ions f2T is close to zero at that molality of the external electrolyte. This agrees with expectations. At higher mext f$ is above experimental error. One should remember, however, that due to the high concentration of counterions there are always more cations (1) in the vicinity of anions (2) than vice versa. The partial frictions of chloride ions with the two other components are rather unexpected. Since the Donnan equilibrium and TMS theory were published, it is a well known and documented fact that coions are rejected from a charged polymer by the high potential of the polymer network. It is seen however from Fig. 8(b) that friction of this coion with the charged polymer is not the main force which resists the flow of negative ions in the negatively charged polymer network. Except at 298 K and mext = 0.5 the anion-polymer frictional force (2m) is always below the friction with water (2~) and it de-

165

creases with increasing electrolyte concentration and temperature. As a result, at high temperature and m,,t, the resistance against flowing anions is imposed by water; the lower the amount of water in the membrane, the higher this resistance. These results, although not immediately understandable, seem to show additional forces which influence the selectivity of Nafion membranes. They also suggest that the origins of selectivity in sorption (static) and transport (dynamic) could be different. To some degree the cross effects and couplings between the membrane components, occurring in the real transport, not considered in the analysis based on straight rii coefficients, simplify the explanations of the observed phenomenon. However, since the cross effects are always weaker than the straight effect they can influence this relation but rather not change it. Undoubtedly this effect needs further analysis. Friction in transport of water At low external concentrations the flow resistance of water (rww) is caused only by frictional interactions with counterions Na’ and polymer network (Fig. 9(b)). Since the concentration of water does not change appreciably with mext and T, the partial frictions of water with sodium ions and polymer vary to a limited degree only. As opposed to f&, the friction of p is zero until the molarity of the water molecules with surrounding anions fw2 anions becomes sizable and then it increases. Diffusion indices of ions and water Complementary to the resistance coefficients and partial frictions are the diffusion indices of ions and water within the membrane RT/Z;rii. It is advantageous that they can be compared with diffusion coefficients in Nafion 120 and other membranes, determined by other independent methods. Diffusion index of sodium counterions Using a tracer technique Yeager and Steck [26] determined the selfdiffusion coefficient for sodium ions in Nafion 120 membrane equilibrated with 0.05 M NaCl solution and found &.$ = 0.94 X lo-” m2-set-’ and fiiz = 1.5 X lo-lo m2-sec-‘. At the same concentration the diffusion indices of sodium ions found here are 2 X lo-lo m2-set-’ at 298 K and 3.05 X 10-l’ m2see-l at 313 K (Fig. 10). While comparing the results not only the methods but also the thermal history of the membrane used for experiments should be taken into account. In the procedure adopted in our work, prior to experiments, the Nafion 120 membrane was conditioned by boiling in water for 30 min. It is a well known fact that such a membrane shows a more “opened” structure. Another reason is that with the methods used here, the calculations lead rather to the convective diffusion indices than to pure diffusion coefficients. On the other hand, it is seen from Fig. 10 that variations in membrane composition caused by sorption and dehydration (Table 1) cause the diffusion indices to decrease to almost one half those in pure membrane. This drastic variation proceeds mainly at mext > 1.

166

Fig. 10. The diffusion indices for: counterions Na+ (RZ’fi,r,,), coions Cl-(RT/Z,r,,) and in Nafion 120 membrane in NaCl solutions; 298 K (o), 313 K (o), water (RT~,r,,) 333 K (0).

Diffusion index of chloride coions From the published data [27] it follows that for crosslinked charged gels (ion-exchange resins and membranes) equilibrated with NaCl solutions the self-diffusion coefficient for Cl- exceeds that for Na+. Instead, the diffusion indices for Cl- found here (Fig. 10) are the same as for Na’. On heating, diffusion indices of sodium and chloride ions differ in such a way that RT/ &rza > RT/&rl>. The variation of membrane composition at increasing mext

168

List of symbols

JV 1. & m, mext P

rik R t lam T vi 5s r n Vi

Km

average concentration of salt in the two solutions adjacent to the membrane (mol-m-3) concentration of species i in the membrane (mol-m-3) membrane thickness (m) diffusion coefficient in the membrane ( m2-set-‘) electromotive force (V) friction coefficient (J-set-m*-mol-‘) partial friction driving force for species i (N-mol-‘) electric current density ( A-mm2) flux of species i (mol-m-2-sec-1) volume flux (m-set-’ ) molar conductance coefficient (mol’-J-l -m-l -see-l ) differential discontinuous conductance coefficient external molality of NaCl pressure (Pa) resistance coefficient (J-m-sec-mol-2) universal gas constant apparent transport number of counterion absolute temperature velocity of particle i (m-set-l ) partial molar volume of salt (m3-mol-‘) matrix transforming the molar fluxes to the practical ones osmotic pressure (Pa) number of ions i released per molecule of salt specific conductance of the membrane (Q-l -m-l)

Subscripts 1 - counterion 2 - coion m - membrane s - salt w - water

References 1 2

3

T. Foley, J. Klinowski and P. Meares, Differential conductance coefficients in a cation-exchange membrane, Proc. Roy. Sot., A, 336 (1974) 327. P. Meares, Some uses for membrane transport coefficients, in: E. St%gny (Ed.), Charged Gels and Membranes, Vol. 3, D. Reidel Publishing Company, Dordrecht, Holland, 1976, p. 123. P. Meares, Coupling of ions and water fluxes in synthetic membranes, J. Membrane Sci., 8 (1981) 295.

169

E. Kumamoto, H. Kimizuka, Transport properties of the barium form of a PoIY(stYrenesulfonic acid)cation exchange membrane, J. Phys. Chem., 85 (1981) 635. H. Kramer and P. Meares, Correlation of electrical and permeability properties of ion-selective membranes, Biophys. J., 9 (1969) 1006. P. Meares and A.H. Sutton, Electrical transport phenomena in a cation-exchange membrane. I. The determination of transport numbers and the ratios of tracer fluxes, J. Colloid Interface Sci., 28 (1968) 118. 7 W.J. McHardy, P. Meares, A.H. Sutton and J.F. Thain, Electrical transport phenomena in a cation-exchange membrane, II. Conductance and eiectroosmosis, J. Colloid Interface Sci., 29 (1969) 116. 8 D.G. Dawson and P. Meares, Electrical transport phenomena in a cation-exchange membrane. III. Membrane potentials, J. Colloid Interface Sci., 33 (1970) 117. 9 P. Meares, J.F. Thain and D.G. Dawson, Transport across ion-exchange resin membranes: The frictional model of transport, in: G. Eisenmann (Ed.), Membranes, Vol. 1, Marcel Dekker, New York, 1972, Chap. 2. 10 W. Dorst, A.J. Staverman and R. Caramazza, Transport in charged membranes, Rec. Trav. Chim., 83 (1964) 1329. 11 R. Paterson and C.R. Gardner, Comparison of the transport properties of normal and expanded forms of a cation-exchange membrane by use of an irreversible thermodynamics approach. I. Membranes in the Na+ form in 0.1 m NaCI, J. Chem. Sot. A, (1974) 2254. 12 C.R. Gardner and R. Paterson, Comparison of the transport properties of normal and expanded forms of a cation-exchange membrane. III. Application of irreversible thermodynamics and Nernst-Planck theories to membranes in concentrated NaCl solutions, J. Chem. Sot., Faraday Trans. 1, 68 (1972) 2030. 13 H.L. Yeager, B. Kipling and R.L. Dotson, Sodium ion diffusion in Nafion coionexchange membranes, J. Electrochem. Sot., 127 (1980) 303. 14 H.L. Yeager, B. O’Dell and Z. Twardowski, Transport properties of Nafion membranes in concentrated solution environments, J. Electrochem. Sot., 129 (1982) 85. 15 Z. Twardowski, H.L. Yeager and B. O’Dell, A comparison of perfluorimated carboxylate and sulfonate ion-exchange polymers. II. Sorption and transport properties in concentrated solution environments, J. Electrochem. Sot., 129 (1982) 328. 16 P.J. Smith and T.L. Jones, The determination of ionic transport numbers by use of radiotracers, J. Electrochem. Sot., 130 (1983) 885. 17 A. Narebska, R. Wodzki and K. Erdmann, Properties of perfluorosulfonic acid membranes in concentrated sodium chloride and sodium hydroxide solutions, Angew. Makromol. Chem., 111 (1983) 85, 18 0. Kedem and A. Katchalsky, Permeability of composite membranes, Trans. Faraday Sot., 59 (1963) 1918. 19 L. Michaeli and 0. Kedem, Description of the transport of solvent and ions through membranes in terms of differential coefficients. Part I. Phenomenological characterisation of flows, Trans. Faraday Sot., 57 (1961) 1185. 20 K.S. Spiegler, Transport processes in ionic membranes, Trans. Faraday Sot., 54 (1958) 1408. 21 R. Paterson, R.Q. Cameron and I.S. Burke, Interpretation of membrane phenomena, using irreversible thermodynamics, in: E. SBlOgny (Ed.), Charged Gels and Membranes, Vol. 3, D. Reidel Publishing Company, Dordrecht, Holland, 1976, p. 157. 22 A. Narebska, S. Koter and W. Kujawski, Ions and water transport across charged Nafion membranes. Irreversible thermodynamics approach, Desalination, 51 (1984) 3. 23 E. Glueckauf, The influence of ionic hydration on activity coefficients in concentrated electrolyte solutions, Trans. Faraday Sot., 51 (1955) 1235. 24 H.L. Yeager and B. Kippling, Ionic diffusion and ion clustering in a perfluorinated ion-exchange membrane, J. Phys. Chem., 83 (1979) 1836.

170 25

26 27 28

D.G. Miller, Application of irreversible thermodynamics to electrolyte solutions. I. Determination of ionic transport coefficients 28 for isothermal vector transport processes in binary electrolyte systems, J. Phys. Chem., 70 (1966) 2639. H.L. Yeager and A. S&k, Cation and water diffusion in Nafion ion exchange membranes: Influence of polymer structure, J. Electrochem. Sot., 128 (1981) 1880. N. Lakshminarayanaiah, Transport Phenomena in Membranes, Academic Press, New York, London, 1969. K. Erdmann and A. Narqbska, Self-diffusion coefficients of water in permselective membranes, Bull. Acad. Pol. Sci., Ser. Sci. Chim., 27 (1979) 589.