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Approximate evaluation of water transport number in ion-exchange membranes
Electrochimica Acta 49 (2004) 1711–1717
Approximate evaluation of water transport number in ion-exchange membranes C. Larchet a,∗ , B. Auclair a , V. Nikonenko b,1 a
Laboratoire des Matériaux Echangeurs d’Ions, Université Paris 12, av. du Général de Gaulle, Créteil 94010, France b Department of Physical Chemistry, Kuban State University, 149 Stavropolskaya St., 350040 Krasnodar, Russia Received 23 May 2003; received in revised form 26 November 2003; accepted 29 November 2003
Abstract Two approximate methods for the water transport number evaluation in ion-exchange membranes (IEM) are proposed. The first one allows the evaluation from the electrical conductivity, diffusion permeability and emf apparent transport number. It can be considered as a method for complementary IEM characterisation. The second demands only the knowledge of the emf apparent transport number and gives sufficiently approximate values. Experimental data on the water transport numbers for MK-40 in KCl and CM2 in KCl, NaCl and LiCl solutions are presented and the mechanism of the water transport is discussed. © 2004 Elsevier Ltd. All rights reserved. Keywords: Ion-exchange membranes; Characterisation; Transport properties; Water transport number; Water transport mechanism
1. Introduction The water transport number is an important transport characteristic of ion-exchange membranes (IEM). This characteristic has a strong influence on electrodialysis treatment of different solutions [1–3], on membrane operation in fuel cells [4,5] and others [6,7]. The knowledge of the water transport number gives an idea not only on the mechanism of the water transport but on the membrane structure and ion transport mechanism also [4–15]; in particular, this knowledge helps to understand the mechanism of “proton leakage” through anion-exchange membranes [4–7], the selectivity of membranes [13], electrokinetic phenomena [14,15]. The direct measurement of the water transport number is a rather delicate procedure [8–14]. Usually a direct electric current is imposed with the help of Ag/AgCl electrodes. The water passes through the membrane by the electroosmotic mechanism: the electric current is transported mainly by the counter-ions, which carry away water molecules in their hy∗ Corresponding author. Tel.: +33-1-4517-1486; fax: +33-1-4517-1721. E-mail addresses: [email protected] (C. Larchet), v [email protected] (V. Nikonenko). 1 Tel.: +7-8612-699-573; fax: +7-8612-33-9887.
0013-4686/$ – see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.electacta.2003.11.030
dration shell and by “friction” interaction [8,9], interpreted sometimes by Stefan–Maxwell formalism [11]. The volume of the fluid passed through the membrane is measured with two capillaries. Then the electroosmotic permeability coefficient β is determined as the transported volume of the fluid (cm3 ) per unit area (1 cm2 ) per electrical charge passed (A s). The water transport number is defined by the following equation [8]: Jw F (1) tw = i p,π=0 It is calculated from the relation [8]: V¯ w t+ V¯ s + β = tw F z+ ν + F
(2)
where Jw is the water flux density through the membrane; i, current density; F, Faraday constant; t+ , counter-ion transport number (the membrane is supposed to be cation-exchange); z+ , ionic charge of the cation and ν+ , its stoicheiometric coefficient in the salt formula; V¯ w and V¯ S are the molar volumes of water and electrolyte, respectively. When measuring the volume transport, the hydrostatic (p) and osmotic (π) pressure differences across the membrane must be zero. The aim of this paper is to present other indirect methods for the water transport number evaluation issued from
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approximate correlation between different membrane transport properties [16]. The methods are applied for several membrane systems that have given grounds for discussion of the water transport mechanism in ion-exchange membranes.
2. Water transport number approximate evaluation 2.1. Determination of tw from the membrane conductivity, diffusion permeability and apparent transport number (method 1) The following approximate equation connecting transport properties of an IEM equilibrated with a single 1:1 electrolyte was deduced by comparing the Onsager and the Kedem–Katchalsky equation systems [16]: t− =
P ∗ F 2 cs 2RTκt+app g
(3)
where t− is the transport number of the co-ion (t+ +t− = 1); P*, local (differential) diffusion permeability; κ, membrane electrical conductivity; t+app , apparent transport number and g is the activity factor related to the equilibrium solution molar concentration cs : g=1+
d ln a± d ln y± = d ln cs d ln cs
(4)
where y± is the mean molar activity coefficient. The values of g for a number of electrolytes may be determined, for example, from the data presented in ref. [17]. t+app is obtained from membrane potential measurements [4]. A relation equivalent to Eq. (3) was obtained by Paterson and
Gardner [18] and, in the case of zero volume transport (tw = 0), by Kedem and Perry [19] and Gnusin et al., [20]. Comparison of the transport numbers calculated from Eq. (3) and determined by other methods has shown a good agreement [16] (see also Section 3.2 and Table 1). This fact shows that Eq. (3) can be used for an approximate characterisation, especially if the dispersion in properties of samples stemming from different membrane sheets is taken into account. Thus, when knowing P*, κ, t+app and g, one can evaluate the ionic transport numbers t+ and t− in the membrane. Then the water transport number can be calculated from the Scatchard equation [21] resulting from thermodynamic consideration: t+app = t+ − mMw tw
(5)
where m is the molality of the solution, Mw = 0.018 kg/mol is the water molar mass. 2.2. Determination of tw from the apparent transport number (method 2) When the membrane has a sufficiently high exchange capacity and low water content, the counter-ion transport number is very close to 1 (see, for example, the data presented in Table 1 and Fig. 3). Under these conditions the water transport number may be evaluated from Eq. (5) when supposing t+ = 1. We will show that this approximation may be admitted for the concentrations of the equilibrium solution up to 1 and sometimes 2 mol/l. Note that the idea to determine the water transport number from the emf measurements, which are at the origin of the apparent transport number determination, was presented also by Graydon and Stewart in 1955 [22].
Table 1 Properties of studied cation-exchange membranes Membrane
CM2 (Tokuyama Soda Inc., Japan MK-40 (NPO Plastmassy, Russia)
Type
Homogeneous polystyrene divinylbenzene Heterogeneous polystyrene divinylbenzene
Exchange capacity (mmol/g dry membrane)
Water content (at C < 0.2 M) g/g dry membrane
mol H2 O/mol
2.12
0.27–0.30
7.1–7.8
1.64
0.52
11.6
Density (g/cm3 )
Thickness (mm)
Transport number of Na+ in NaCl solutions
1.27
0.135
0.996a
1.12
0.51
0.98a
Nafion 120 (Du Pont de Nemours, USA)
Homogeneous perfluorinated
0.83 [24]
0.30 [24]
20 [24]
–
0.29–0.30
0.985b 0.97c [23] 0.95a
MF-4SK (NPO PlastPolymer, Russia)
Homogeneous perfluorinated
0.77 [12]
0.16 [12]
11.9 [12]
–
0.22 [12]
0.97b 0.97a 0.99b
References are indicated in brackets. a Evaluated in 1 mol/l NaCl solution from κ, t +app and P* with Eq. (3). b Evaluated in 1 mol/l NaCl solution from t +app and tw with Eq. (5). c Measured in 0.52 mol/l NaCl solution by Hittorf’s method [23].
C. Larchet et al. / Electrochimica Acta 49 (2004) 1711–1717
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3. Experimental Two sulphonic polystyrene divinylbenzene cation-exchange membranes were studied: MK-40 (produced by NPO Plastmassy, Russia) in KCl and CM2 (produced by Tokuyama Soda Inc., Japan) in KCl, NaCl and LiCl solutions. The data for two homogeneous perfluorinated sulphocationic membranes Nafion 120 (Du Pont de Nemours, USA) and MF-4SK (NPO PlastPolymer, Russia) found in literature are used for comparison. The main properties of these four membranes are given in Table 1. 3.1. Static properties The membranes were conditioned following the procedure of the French standards [25]. A set of static properties was then determined by conventional methods [25]: the ion-exchange capacity (Q), water content (wc ), the membrane density (ρ) and the membrane thickness (d), all in H+ form. 3.2. Transport properties The following transport properties were measured as functions of the equilibrium solution concentration ranged from 0.01 to 1 mol/l, except the case of the conductivity (up to 4 mol/l): • the specific conductivity was measured under alternating current using a “clamp”-type cell described in ref. [26]; • the salt diffusion flux Js was determined in the system where an electrolyte diffuses from a solution with a given molar concentration cs to the pure water through the membrane under consideration when using a cell described in refs. [27,28]; • the apparent transport number t+app was determined from emf measurements [8] described in ref. [29]. In particular, the problem of relating the value t+app to the equilibrium solution concentration cs was resolved by graphic interpolation known as Prigent’s method [29]. The integral permeability coefficient P is then calculated from the flux value: Js d (6) P= cs p,i=0 the subscripts denote that the pressure gradient across the membrane and the current must be zero. The local diffusion permeability P* enters the differential equation ∗ dcs Js = −P (7) dx p,i=0 where cs denotes the concentration of the “virtual” electroneutral solution being in equilibrium with a thin layer of the membrane localised at normal coordinate x [16]. P ∗ is
Fig. 1. Results of measurements of (a) specific conductivity, (b) apparent transport number and (c) local diffusion permeability as functions of the external solution concentration c for different membrane systems.
linked with the integral coefficient by the following relation [16,30]: dP P ∗ (cs ) = P(cs ) + cs (8) dcs It is convenient to present the function P ∗ − P in the form [16]: d log P ∗ P =P 1+ (9) d log cs where the function log P − log cs may be easily obtained by polynomial regression of the experimental data issued from the diffusion measurements. Usually a second-degree polynomial is suitable. The results of measurements and their treatment with Eqs. (6)–(9) are presented in Fig. 1a–c. As it can be seen from Table 1, the values of “true” counter-ion transport numbers (t+ ) until 1 M are very close
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to 1. Note that in the case of MK-40 three different evaluations of t+ are compared. The t+ calculation from the Scatchard equation (Eq. (5)), whose deduction demands only thermodynamic treatments, can be considered as the most reliable. The t+ calculation from approximate Eq. (3) uses quantities that can be more easily found from experiment. The determination of t+ by Hittorf’s method is based on the direct measurements of concentration changes, but the error of measurements is relatively high. The values of t+ obtained by the three methods are very close that increases the reliability of information.
4. Results of the water transport number determination Fig. 2a–c compares the experimental and calculated water transport numbers determined by two methods described in Sections 2.1 and 2.2 for two homogeneous perfluorinated sulphocationic membranes Nafion 120 and MF-4SK as well as for a MK-40 membrane in NaCl solutions. The experimental data obtained by the conventional method described in Section 1 are taken from [31,32] for the Nafion 120 and from [12,30,33] for the MF-4SK and the MK-40. The activity factor values calculated from the data presented in ref. [17] are given in Table 2. It is seen that the first method gives a better agreement between the experimental and calculated water transport numbers, especially in the cases of the Nafion 120 and the MK-40 membranes. However, in all the cases the values calculated by the first method are lower than experimental ones. This is explained by the fact that the approximation for the counter-ion transport number t+ used in the first method (Eq. (3)) gives the values somewhat lower than the values obtained from the Scatchard equation (Eq. (5)) [16], the latter being considered as more precise. This underestimation Table 2 Activity factor g = 1 + (d ln y± /d ln cs ) for different electrolyte solutions cs (mol/l)
0 0.0001 0.0005 0.001 0.005 0.01 0.05 0.1 0.2 0.5 0.7 1 1.5 2 2.5 3
g = 1 + (d ln y± /d ln cs ) LiCl
NaCl
KCl
1 0.991 0.985 0.980 0.964 0.954 0.936 0.935 0.949 1.022 1.078 1.162 1.312 1.489 1.707 1.933
1 0.991 0.985 0.980 0.963 0.952 0.927 0.918 0.916 0.938 0.960 0.998 1.071 1.152 1.251 1.356
1 0.991 0.985 0.980 0.962 0.951 0.922 0.909 0.900 0.905 0.914 0.932 0.974 1.024 1.080 1.141
Fig. 2. Experimental and calculated (following the methods 1 and 2 described in Section 2) water transport numbers for membranes (a) Nafion 120, (b) MF-4SK with 11.9 mol H2 O per mol fixed groups and (c) MK-40 in NaCl solutions. Experimental data are taken from [31,32] for the Nafion 120 and from [12,30,33] for the MF-4SK and MK-40.
of the calculated transport numbers is due to a relative hydrophobicity of the studied membranes [16,19]. Thus, one can suppose that higher disagreement between calculated and experimental tw in the case of MF-4SK is explained by the fact that this membrane, containing only 0.16 g H2 O/g dry membrane [12], is more hydrophobic than the others presented in this study (Table 1). Conversely, when supposing t+ = 1 in the second method, one overestimates, evidently, the real value of t+ , and the water transport numbers found from Eq. (5) are too high. When cs < 1 mol/l, the two methods give results sufficiently close to the experimental values because the counter-ion transport number is near to 1. This permits to use in this concentration range the very simple second method for approximate
C. Larchet et al. / Electrochimica Acta 49 (2004) 1711–1717
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Fig. 4. The water transport number for a CM2 membrane as function of the concentration of different chloride solutions calculated from Eq. (5) when supposing t+ = 1.
Fig. 3. The (a) K+ transport number calculated from Eq. (3) and the (b) water transport number calculated by method 1 for a CM2 and a MK-40 membranes as functions of the KCl equilibrium solution, solid lines. The measured values of tapp for CM2 and MK-40 are modelled with polynomials t+app = −0.0097c2 − 0.0389c + 0.9998, and t+app = −0.1487c + 1, respectively. The dash lines in (b) show the results of the tw calculations when supposing t+ = 1 (method 2).
evaluation of the water transport numbers. Of cause, when applying this method, one must be sure that the membrane considered is sufficiently permselective, i.e. the t+ value should be greater than 0.95. This is the case of the membranes studied: following the estimation from the apparent transport number and the water transport number with Eq. (5), t+ = 0.97, 0.99 and 0.985, respectively, for Nafion 120, MF-4SK and MK-40 in 1 M NaCl solution. These results were obtained [16] by using the experimental data of Narebska et al. [31,32] for Nafion 120 and Berezina et al. [12,30,33] for MF-4SK and MK-40 membranes. The agreement between experimental tw and that calculated by the second method becomes worse with increasing concentration, evidently because t+ decreases. As shown in ref. [16], the “true” transport number evaluated by Eq. (3) from κ, t+app and P* found in ref. [33] decreases more rapidly with growth of NaCl concentration in the case of MF-4SK in comparison with MK-40. Thus, the error in evaluation of tw by method 2 when supposing t+ = 1 is higher for the MF-4SK. The error in applying method 2 for Nafion 120 in higher concentrations is also relatively large, that is explained as well by low t+ values: its evaluation in 3 M NaCl from Eqs. (3) and (5) by using experimental data [31,32] gives 0.885 and 0.88, respectively.
Fig. 3a shows the transport numbers of K+ calculated from the conductivity, diffusion permeability and apparent transport numbers for CM2 and MK-40 membranes in KCl solution. One can see that the t+ values, especially in the case of the CM2, approach unity. Note also very close results obtained for MK-40 from the data of Berezina et al. for NaCl solutions (t+ = 0.985 in 1 M NaCl) and in this study for KCl solutions (t+ = 0.992 in 1 M KCl). The water transport numbers presented in Fig. 3b were calculated from the Scatchard equation (5) with the counter-ion transport numbers taken from Fig. 3a (solid lines) or supposed to be equal to 1 (broken lines). It can be seen that the difference between two approximations for tw is very small. The diffusion permeability of the membranes studied was measured only in the case of KCl solution, hence, only method 2 can be applied in the cases of LiCl and NaCl solutions. However, it is interesting to compare the water transport numbers for a membrane in contact with different electrolyte solutions. The results of the tw calculation from the apparent transport number (method 2) for a CM2 membrane equilibrated with LiCl, NaCl and KCl solutions are shown in Fig. 4.
5. Discussion It is generally admitted [4,5,8,9,12] that when a current passes through an ion-exchange membrane, the water is transported by two ways. A part of the water (hydration water) is firmly bound with the ions and moves with them with the same velocity. The second part is “free” or “semi-free” water which is dragged (or pushed in narrow channels [4]) in the same direction as the migrating ions but, in general case, with a smaller velocity: tw = nh+ t+ − nh− t− + nvol
(10)
The amount of the water transported by the second mechanism (nvol ) depends rather on the volume of the hydrated ion than on Coulomb or other interactions [4]. It is difficult to evaluate the hydration numbers for cations (nh+ ) and anions
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(nh− ) as well as the number of moles of water dragged (nvol ). For Li+ , Na+ and K+ , Glueckauf and Kitt [34] give the following hydration numbers in sulphonic ion-exchangers: 3.3, 1.5 and 0.6, respectively. Lakshminarayanaiah [8] has obtained similar results interpreted as numbers of moles of water firmly bound (primary hydration) to 1 mol of the ion. Okada et al. [4] report also close values for Nafion membranes: 5.5, 2.8 and 1.4, respectively. To describe nvol , Breslau and Miller [9] have equated the electric force applied to an ion by the external field and the viscous force acting between the hydrated ion and the free water. nvol decreases with increasing the internal solution concentration because the quantity of free water and the degree of swelling decrease. With increasing the external solution concentration, the internal concentration increases also due to the electrolyte penetration in the interstitial solution. Hence, nvol decreases with increasing the external solution concentration that explains the tendency in tw variation. If the concentration of fixed groups in the membrane is high and the water content sufficiently small, the co-ion transport number is small (less than 0.1), and the corresponding term in Eq. (10) may be neglected. Note that Fig. 4 shows that tw varies in the order: LiCl > NaCl > KCl This order is in conformity with the cited hydration numbers of these cations: a more hydrated ion transports not only more water molecules in its hydration shell, but the amount of dragged water is higher also due to a greater size of the hydrated ion. Similar results have been obtained by Berezina et al. [12] (Fig. 5) for a MK-40 membrane. In the two series of data presented, respectively, in Figs. 4 and 5, at high external solution concentrations, the water transport number tends to 2 in the case of KCl, 3 in the case of NaCl and between 6 and 7 in the case of LiCl. Thus, the contribution of nvol in the water transport number becomes small in these conditions. This low limiting quantity of nvol may be evaluated, very approximately, by the following way. We suppose that at relatively high concentrations when the free water amount becomes small and their transport is highly
hindered by the walls of the narrow capillary pore, one water molecule in the first hydration shell can drag only one “free” water molecule and the counter-ion itself can drug (push) also one water molecule. The following relation is then obtained when neglecting the amount of water transported by co-ions: (tw )c→∞ = 2nh+ + 1
(11)
When using for nh+ the values of Glueckauf and Kitt, Eq. (11) gives tw equal to 7.6, 4 and 2.2, respectively, for LiCl, NaCl and KCl, that is not so far from the experimental limiting values. We observe, however, that the limiting values for tw are achieved at concentrations about 1 mol/l in the case of CM2 and 4 mol/l in the case of MK-40. This difference is due to the difference in the structure and the composition of the membranes. The CM2 membrane is more hydrophobic, the water content is almost twice as small in comparison to the MK-40 membrane (Table 1). The concentration of the interstitial solution (found as the exchange capacity divided by the water content) is very high, about 7.5 mol/l, in comparison with 3.2 mol/l in the MK-40 membrane. Hence, the fraction of “free” water which can be dragged by hydrated counter-ions in CM2 is relatively small. And when the external solution concentration increases, hence, the interstitial solution concentration increases also, nh+ achieves sufficiently quickly its limiting values. In the case of MK-40 membrane, the fraction of “free” water at small concentrations is much higher, thus tw is higher. Because of a smaller exchange capacity of MK-40 and higher water content, the rate of increase in internal concentration with increasing the external solution concentration is higher in this membrane than in CM2. However, to achieve the values of the internal solution concentration in MK-40, comparable with those in CM2, the external solution concentration should be more than in the case of CM2. Hence, one can suppose that the conditions of the water transport (the interstitial solution concentration, the free water content, the pore size) in MK-40 become similar to that in CM2 equilibrated with 1 mol/l solution, when the external concentration is close to 4 mol/l.
6. Conclusion
Fig. 5. Water transport numbers for a MK-40-8 membrane (8% of DVB) as function of the concentration of different chloride solutions. Experimental data of Berezina et al. [12].
Two methods of the water transport number evaluation in ion-exchange membranes are proposed. The first consists in the calculation from the counter-ion apparent transport number, membrane electric conductivity and diffusion permeability, with the help of an approximate relation and the Scatchard equation. The second demands only the apparent transport number knowledge, the counter-ion transport number being supposed to be 1. The first method gives values of tw sufficiently close to experimental ones in a wide range of concentrations, whereas the second can be applied only at relatively small concentrations where the counter-ion
C. Larchet et al. / Electrochimica Acta 49 (2004) 1711–1717
transport number does not differ too much from 1 (<1 M for the membranes studied). The comparison of our data for a CM2 membrane in KCl, NaCl and LiCl solutions with the data of Berezina et al. [12] for a MK-40 membrane in the same solutions has shown a similar character of the water transport number behaviour. In two cases tw varies in the order: LiCl > NaCl > KCl, which corresponds to the order of the hydration numbers of the cations. When the external concentration increases, tw tends to its limiting value, which can be evaluated with a new proposed relation. The difference in the behaviour of the membranes is that in the case of CM2, tw achieves its limiting value at external solution concentrations about 1 mol/l, whereas in the case of MK-40 it is about 4 mol/l, that can be related to the difference in the water content and exchange capacity of the membranes. Acknowledgements V.N. thanks the Paris 12 University for giving opportunity to make the research described in this publication. This research was also financially supported by the CNRS, France, grant PICS No. 1811, and by the Russian Foundation of Basic Researches, Grants nos. 02-03-22001 CNRS a and 03-03-96652. References [1] H. Strathmann, in: W.S.W. Ho, K.K. Sirkar (Eds.), Membrane Handbook, Van Nostrand Reinhold, New York, 1992. [2] T.A. Davis, V. Grebenyuk, O. Grebenyuk, in: S.P. Nunes, K.-V. Peinemann (Eds.), Membrane Technology in Chemical Industry, Wiley-VCH, Weinheim, 2001, p. 222. [3] O. Bobreshova, L. Novikova, P. Kulintsov, E. Balavadze, Desalination 149 (2002) 363. [4] T. Okada, G. Xie, O. Gorseth, S. Kjelstrup, N. Nakamura, T. Arimura, Electrochim. Acta 43 (1998) 3741. [5] T. Okada, J. Electroanal. Chem. 465 (1999) 1. [6] G. Pourcelly, I. Tugas, C. Gavach, J. Membr. Sci. 97 (1994) 99.
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[7] B.J. Robbins, R.W. Field, S.T. Kolaczkowski, A.D. Lockett, J. Membr. Sci. 118 (1996) 101. [8] N. Lakshminarayanaiah, Transport Phenomena in Membranes, Academic Press, New York, 1969. [9] B.R. Breslau, I.F. Miller, Ind. Eng. Chem. Fundam. 10 (1971) 554. [10] M. Demarty, A. Maurel, E. Selegny, J. Chem. Phys. Phys. Chem. Biol. 71 (1974) 811. [11] P. Schaetzel, E. Favre, B. Auclair, Q.T. Nguyen, Electrochim. Acta 42 (1997) 2475. [12] N. Berezina, N. Gnusin, O. Demina, S. Timofeyev, J. Membr. Sci. 85 (1994) 207. [13] N.P. Berezina, O.A. Demina, N.P. Gnusin, S.V. Timofeyev, Elektrokhimiya 30 (1994) 1467. [14] G. Schmid, J. Membr. Sci. 150 (1998) 159. [15] A.E. Yaroshchuk, Colloids Surf. A: Physicochem. Eng. Aspects 140 (1998) 169. [16] B. Auclair, V. Nikonenko, C. Larchet, M. Métayer, L. Dammark, J. Membr. Sci. 195 (2002) 89. [17] D.G. Miller, J. Phys. Chem. 70 (1966) 2639. [18] R. Paterson, C.R. Gardner, J. Chem. Soc. (A) 13 (1971) 2254. [19] O. Kedem, M. Perry, J. Membr. Sci. 14 (1983) 249. [20] N.P. Gnusin, N.A. Kononenko, S.B. Parchikov, Russ. J. Electrochemistry 29 (1993) 647 (Translated from Elektrokhimiya 29 (1993) 754). [21] G. Scatchard, J. Am. Chem. Soc. 75 (1953) 2883. [22] W.F. Graydon, R.J. Stewart, J. Phys. Chem. 59 (1955) 86. [23] V.I. Zabolotsky, L.F. El’nikova, N.V. Sheldeshov, A.V. Alekseev, Elektrokhimiya 23 (1987) 1626. [24] E.H. Cwirko, R.G. Carbonnel, J. Membr. Sci. 67 (1992) 211. [25] Norme Française NF X 45-200, AFNOR, December 1995. [26] N.N. Belaid, B. Ngom, L. Dammak, C. Larchet, B. Auclair, Eur. Polym. J. 35 (1999) 879. [27] L. Dammak, C. Larchet, B. Auclair, V.V. Nikonenko, V.I. Zabolotsky, Eur. Polym. J. 32 (1996) 1199. [28] V. Tourreuil, L. Dammak, G. Bulvestre, B. Auclair, New J. Chem. 12 (1998) 1463. [29] R. Lteif, L. Dammak, C. Larchet, B. Auclair, Eur. Polym. J. 37 (2001) 627. [30] N.P. Gnusin, N.P. Berezina, A.A. Shudrenko, O.P. Ivina, Russ. J. Phys. Chem. 68 (1994) 506. [31] A. Narebska, S. Koter, W. Kujawsky, Desalination 51 (1984) 3. [32] A. Narebska, S. Koter, W. Kujawsky, J. Membr. Sci. 25 (1985) 153. [33] N.P. Gnusin, N.P. Berezina, O.A. Demina, N.A. Kononenko, Russ. J. Electrochem. 32 (1996) 154. [34] E. Glueckauf, G.P. Kitt, Proc. R. Soc. (London) A228 (1955) 322 (Cited from F. Helfferich, in: Ion-exchange, McGraw-Hill, New York, 1962, p. 106).
Approximate evaluation of water transport number in ion-exchange membranes C. Larchet a,∗ , B. Auclair a , V. Nikonenko b,1 a
Laboratoire des Matériaux Echangeurs d’Ions, Université Paris 12, av. du Général de Gaulle, Créteil 94010, France b Department of Physical Chemistry, Kuban State University, 149 Stavropolskaya St., 350040 Krasnodar, Russia Received 23 May 2003; received in revised form 26 November 2003; accepted 29 November 2003
Abstract Two approximate methods for the water transport number evaluation in ion-exchange membranes (IEM) are proposed. The first one allows the evaluation from the electrical conductivity, diffusion permeability and emf apparent transport number. It can be considered as a method for complementary IEM characterisation. The second demands only the knowledge of the emf apparent transport number and gives sufficiently approximate values. Experimental data on the water transport numbers for MK-40 in KCl and CM2 in KCl, NaCl and LiCl solutions are presented and the mechanism of the water transport is discussed. © 2004 Elsevier Ltd. All rights reserved. Keywords: Ion-exchange membranes; Characterisation; Transport properties; Water transport number; Water transport mechanism
1. Introduction The water transport number is an important transport characteristic of ion-exchange membranes (IEM). This characteristic has a strong influence on electrodialysis treatment of different solutions [1–3], on membrane operation in fuel cells [4,5] and others [6,7]. The knowledge of the water transport number gives an idea not only on the mechanism of the water transport but on the membrane structure and ion transport mechanism also [4–15]; in particular, this knowledge helps to understand the mechanism of “proton leakage” through anion-exchange membranes [4–7], the selectivity of membranes [13], electrokinetic phenomena [14,15]. The direct measurement of the water transport number is a rather delicate procedure [8–14]. Usually a direct electric current is imposed with the help of Ag/AgCl electrodes. The water passes through the membrane by the electroosmotic mechanism: the electric current is transported mainly by the counter-ions, which carry away water molecules in their hy∗ Corresponding author. Tel.: +33-1-4517-1486; fax: +33-1-4517-1721. E-mail addresses: [email protected] (C. Larchet), v [email protected] (V. Nikonenko). 1 Tel.: +7-8612-699-573; fax: +7-8612-33-9887.
0013-4686/$ – see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.electacta.2003.11.030
dration shell and by “friction” interaction [8,9], interpreted sometimes by Stefan–Maxwell formalism [11]. The volume of the fluid passed through the membrane is measured with two capillaries. Then the electroosmotic permeability coefficient β is determined as the transported volume of the fluid (cm3 ) per unit area (1 cm2 ) per electrical charge passed (A s). The water transport number is defined by the following equation [8]: Jw F (1) tw = i p,π=0 It is calculated from the relation [8]: V¯ w t+ V¯ s + β = tw F z+ ν + F
(2)
where Jw is the water flux density through the membrane; i, current density; F, Faraday constant; t+ , counter-ion transport number (the membrane is supposed to be cation-exchange); z+ , ionic charge of the cation and ν+ , its stoicheiometric coefficient in the salt formula; V¯ w and V¯ S are the molar volumes of water and electrolyte, respectively. When measuring the volume transport, the hydrostatic (p) and osmotic (π) pressure differences across the membrane must be zero. The aim of this paper is to present other indirect methods for the water transport number evaluation issued from
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approximate correlation between different membrane transport properties [16]. The methods are applied for several membrane systems that have given grounds for discussion of the water transport mechanism in ion-exchange membranes.
2. Water transport number approximate evaluation 2.1. Determination of tw from the membrane conductivity, diffusion permeability and apparent transport number (method 1) The following approximate equation connecting transport properties of an IEM equilibrated with a single 1:1 electrolyte was deduced by comparing the Onsager and the Kedem–Katchalsky equation systems [16]: t− =
P ∗ F 2 cs 2RTκt+app g
(3)
where t− is the transport number of the co-ion (t+ +t− = 1); P*, local (differential) diffusion permeability; κ, membrane electrical conductivity; t+app , apparent transport number and g is the activity factor related to the equilibrium solution molar concentration cs : g=1+
d ln a± d ln y± = d ln cs d ln cs
(4)
where y± is the mean molar activity coefficient. The values of g for a number of electrolytes may be determined, for example, from the data presented in ref. [17]. t+app is obtained from membrane potential measurements [4]. A relation equivalent to Eq. (3) was obtained by Paterson and
Gardner [18] and, in the case of zero volume transport (tw = 0), by Kedem and Perry [19] and Gnusin et al., [20]. Comparison of the transport numbers calculated from Eq. (3) and determined by other methods has shown a good agreement [16] (see also Section 3.2 and Table 1). This fact shows that Eq. (3) can be used for an approximate characterisation, especially if the dispersion in properties of samples stemming from different membrane sheets is taken into account. Thus, when knowing P*, κ, t+app and g, one can evaluate the ionic transport numbers t+ and t− in the membrane. Then the water transport number can be calculated from the Scatchard equation [21] resulting from thermodynamic consideration: t+app = t+ − mMw tw
(5)
where m is the molality of the solution, Mw = 0.018 kg/mol is the water molar mass. 2.2. Determination of tw from the apparent transport number (method 2) When the membrane has a sufficiently high exchange capacity and low water content, the counter-ion transport number is very close to 1 (see, for example, the data presented in Table 1 and Fig. 3). Under these conditions the water transport number may be evaluated from Eq. (5) when supposing t+ = 1. We will show that this approximation may be admitted for the concentrations of the equilibrium solution up to 1 and sometimes 2 mol/l. Note that the idea to determine the water transport number from the emf measurements, which are at the origin of the apparent transport number determination, was presented also by Graydon and Stewart in 1955 [22].
Table 1 Properties of studied cation-exchange membranes Membrane
CM2 (Tokuyama Soda Inc., Japan MK-40 (NPO Plastmassy, Russia)
Type
Homogeneous polystyrene divinylbenzene Heterogeneous polystyrene divinylbenzene
Exchange capacity (mmol/g dry membrane)
Water content (at C < 0.2 M) g/g dry membrane
mol H2 O/mol
2.12
0.27–0.30
7.1–7.8
1.64
0.52
11.6
Density (g/cm3 )
Thickness (mm)
Transport number of Na+ in NaCl solutions
1.27
0.135
0.996a
1.12
0.51
0.98a
Nafion 120 (Du Pont de Nemours, USA)
Homogeneous perfluorinated
0.83 [24]
0.30 [24]
20 [24]
–
0.29–0.30
0.985b 0.97c [23] 0.95a
MF-4SK (NPO PlastPolymer, Russia)
Homogeneous perfluorinated
0.77 [12]
0.16 [12]
11.9 [12]
–
0.22 [12]
0.97b 0.97a 0.99b
References are indicated in brackets. a Evaluated in 1 mol/l NaCl solution from κ, t +app and P* with Eq. (3). b Evaluated in 1 mol/l NaCl solution from t +app and tw with Eq. (5). c Measured in 0.52 mol/l NaCl solution by Hittorf’s method [23].
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3. Experimental Two sulphonic polystyrene divinylbenzene cation-exchange membranes were studied: MK-40 (produced by NPO Plastmassy, Russia) in KCl and CM2 (produced by Tokuyama Soda Inc., Japan) in KCl, NaCl and LiCl solutions. The data for two homogeneous perfluorinated sulphocationic membranes Nafion 120 (Du Pont de Nemours, USA) and MF-4SK (NPO PlastPolymer, Russia) found in literature are used for comparison. The main properties of these four membranes are given in Table 1. 3.1. Static properties The membranes were conditioned following the procedure of the French standards [25]. A set of static properties was then determined by conventional methods [25]: the ion-exchange capacity (Q), water content (wc ), the membrane density (ρ) and the membrane thickness (d), all in H+ form. 3.2. Transport properties The following transport properties were measured as functions of the equilibrium solution concentration ranged from 0.01 to 1 mol/l, except the case of the conductivity (up to 4 mol/l): • the specific conductivity was measured under alternating current using a “clamp”-type cell described in ref. [26]; • the salt diffusion flux Js was determined in the system where an electrolyte diffuses from a solution with a given molar concentration cs to the pure water through the membrane under consideration when using a cell described in refs. [27,28]; • the apparent transport number t+app was determined from emf measurements [8] described in ref. [29]. In particular, the problem of relating the value t+app to the equilibrium solution concentration cs was resolved by graphic interpolation known as Prigent’s method [29]. The integral permeability coefficient P is then calculated from the flux value: Js d (6) P= cs p,i=0 the subscripts denote that the pressure gradient across the membrane and the current must be zero. The local diffusion permeability P* enters the differential equation ∗ dcs Js = −P (7) dx p,i=0 where cs denotes the concentration of the “virtual” electroneutral solution being in equilibrium with a thin layer of the membrane localised at normal coordinate x [16]. P ∗ is
Fig. 1. Results of measurements of (a) specific conductivity, (b) apparent transport number and (c) local diffusion permeability as functions of the external solution concentration c for different membrane systems.
linked with the integral coefficient by the following relation [16,30]: dP P ∗ (cs ) = P(cs ) + cs (8) dcs It is convenient to present the function P ∗ − P in the form [16]: d log P ∗ P =P 1+ (9) d log cs where the function log P − log cs may be easily obtained by polynomial regression of the experimental data issued from the diffusion measurements. Usually a second-degree polynomial is suitable. The results of measurements and their treatment with Eqs. (6)–(9) are presented in Fig. 1a–c. As it can be seen from Table 1, the values of “true” counter-ion transport numbers (t+ ) until 1 M are very close
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to 1. Note that in the case of MK-40 three different evaluations of t+ are compared. The t+ calculation from the Scatchard equation (Eq. (5)), whose deduction demands only thermodynamic treatments, can be considered as the most reliable. The t+ calculation from approximate Eq. (3) uses quantities that can be more easily found from experiment. The determination of t+ by Hittorf’s method is based on the direct measurements of concentration changes, but the error of measurements is relatively high. The values of t+ obtained by the three methods are very close that increases the reliability of information.
4. Results of the water transport number determination Fig. 2a–c compares the experimental and calculated water transport numbers determined by two methods described in Sections 2.1 and 2.2 for two homogeneous perfluorinated sulphocationic membranes Nafion 120 and MF-4SK as well as for a MK-40 membrane in NaCl solutions. The experimental data obtained by the conventional method described in Section 1 are taken from [31,32] for the Nafion 120 and from [12,30,33] for the MF-4SK and the MK-40. The activity factor values calculated from the data presented in ref. [17] are given in Table 2. It is seen that the first method gives a better agreement between the experimental and calculated water transport numbers, especially in the cases of the Nafion 120 and the MK-40 membranes. However, in all the cases the values calculated by the first method are lower than experimental ones. This is explained by the fact that the approximation for the counter-ion transport number t+ used in the first method (Eq. (3)) gives the values somewhat lower than the values obtained from the Scatchard equation (Eq. (5)) [16], the latter being considered as more precise. This underestimation Table 2 Activity factor g = 1 + (d ln y± /d ln cs ) for different electrolyte solutions cs (mol/l)
0 0.0001 0.0005 0.001 0.005 0.01 0.05 0.1 0.2 0.5 0.7 1 1.5 2 2.5 3
g = 1 + (d ln y± /d ln cs ) LiCl
NaCl
KCl
1 0.991 0.985 0.980 0.964 0.954 0.936 0.935 0.949 1.022 1.078 1.162 1.312 1.489 1.707 1.933
1 0.991 0.985 0.980 0.963 0.952 0.927 0.918 0.916 0.938 0.960 0.998 1.071 1.152 1.251 1.356
1 0.991 0.985 0.980 0.962 0.951 0.922 0.909 0.900 0.905 0.914 0.932 0.974 1.024 1.080 1.141
Fig. 2. Experimental and calculated (following the methods 1 and 2 described in Section 2) water transport numbers for membranes (a) Nafion 120, (b) MF-4SK with 11.9 mol H2 O per mol fixed groups and (c) MK-40 in NaCl solutions. Experimental data are taken from [31,32] for the Nafion 120 and from [12,30,33] for the MF-4SK and MK-40.
of the calculated transport numbers is due to a relative hydrophobicity of the studied membranes [16,19]. Thus, one can suppose that higher disagreement between calculated and experimental tw in the case of MF-4SK is explained by the fact that this membrane, containing only 0.16 g H2 O/g dry membrane [12], is more hydrophobic than the others presented in this study (Table 1). Conversely, when supposing t+ = 1 in the second method, one overestimates, evidently, the real value of t+ , and the water transport numbers found from Eq. (5) are too high. When cs < 1 mol/l, the two methods give results sufficiently close to the experimental values because the counter-ion transport number is near to 1. This permits to use in this concentration range the very simple second method for approximate
C. Larchet et al. / Electrochimica Acta 49 (2004) 1711–1717
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Fig. 4. The water transport number for a CM2 membrane as function of the concentration of different chloride solutions calculated from Eq. (5) when supposing t+ = 1.
Fig. 3. The (a) K+ transport number calculated from Eq. (3) and the (b) water transport number calculated by method 1 for a CM2 and a MK-40 membranes as functions of the KCl equilibrium solution, solid lines. The measured values of tapp for CM2 and MK-40 are modelled with polynomials t+app = −0.0097c2 − 0.0389c + 0.9998, and t+app = −0.1487c + 1, respectively. The dash lines in (b) show the results of the tw calculations when supposing t+ = 1 (method 2).
evaluation of the water transport numbers. Of cause, when applying this method, one must be sure that the membrane considered is sufficiently permselective, i.e. the t+ value should be greater than 0.95. This is the case of the membranes studied: following the estimation from the apparent transport number and the water transport number with Eq. (5), t+ = 0.97, 0.99 and 0.985, respectively, for Nafion 120, MF-4SK and MK-40 in 1 M NaCl solution. These results were obtained [16] by using the experimental data of Narebska et al. [31,32] for Nafion 120 and Berezina et al. [12,30,33] for MF-4SK and MK-40 membranes. The agreement between experimental tw and that calculated by the second method becomes worse with increasing concentration, evidently because t+ decreases. As shown in ref. [16], the “true” transport number evaluated by Eq. (3) from κ, t+app and P* found in ref. [33] decreases more rapidly with growth of NaCl concentration in the case of MF-4SK in comparison with MK-40. Thus, the error in evaluation of tw by method 2 when supposing t+ = 1 is higher for the MF-4SK. The error in applying method 2 for Nafion 120 in higher concentrations is also relatively large, that is explained as well by low t+ values: its evaluation in 3 M NaCl from Eqs. (3) and (5) by using experimental data [31,32] gives 0.885 and 0.88, respectively.
Fig. 3a shows the transport numbers of K+ calculated from the conductivity, diffusion permeability and apparent transport numbers for CM2 and MK-40 membranes in KCl solution. One can see that the t+ values, especially in the case of the CM2, approach unity. Note also very close results obtained for MK-40 from the data of Berezina et al. for NaCl solutions (t+ = 0.985 in 1 M NaCl) and in this study for KCl solutions (t+ = 0.992 in 1 M KCl). The water transport numbers presented in Fig. 3b were calculated from the Scatchard equation (5) with the counter-ion transport numbers taken from Fig. 3a (solid lines) or supposed to be equal to 1 (broken lines). It can be seen that the difference between two approximations for tw is very small. The diffusion permeability of the membranes studied was measured only in the case of KCl solution, hence, only method 2 can be applied in the cases of LiCl and NaCl solutions. However, it is interesting to compare the water transport numbers for a membrane in contact with different electrolyte solutions. The results of the tw calculation from the apparent transport number (method 2) for a CM2 membrane equilibrated with LiCl, NaCl and KCl solutions are shown in Fig. 4.
5. Discussion It is generally admitted [4,5,8,9,12] that when a current passes through an ion-exchange membrane, the water is transported by two ways. A part of the water (hydration water) is firmly bound with the ions and moves with them with the same velocity. The second part is “free” or “semi-free” water which is dragged (or pushed in narrow channels [4]) in the same direction as the migrating ions but, in general case, with a smaller velocity: tw = nh+ t+ − nh− t− + nvol
(10)
The amount of the water transported by the second mechanism (nvol ) depends rather on the volume of the hydrated ion than on Coulomb or other interactions [4]. It is difficult to evaluate the hydration numbers for cations (nh+ ) and anions
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(nh− ) as well as the number of moles of water dragged (nvol ). For Li+ , Na+ and K+ , Glueckauf and Kitt [34] give the following hydration numbers in sulphonic ion-exchangers: 3.3, 1.5 and 0.6, respectively. Lakshminarayanaiah [8] has obtained similar results interpreted as numbers of moles of water firmly bound (primary hydration) to 1 mol of the ion. Okada et al. [4] report also close values for Nafion membranes: 5.5, 2.8 and 1.4, respectively. To describe nvol , Breslau and Miller [9] have equated the electric force applied to an ion by the external field and the viscous force acting between the hydrated ion and the free water. nvol decreases with increasing the internal solution concentration because the quantity of free water and the degree of swelling decrease. With increasing the external solution concentration, the internal concentration increases also due to the electrolyte penetration in the interstitial solution. Hence, nvol decreases with increasing the external solution concentration that explains the tendency in tw variation. If the concentration of fixed groups in the membrane is high and the water content sufficiently small, the co-ion transport number is small (less than 0.1), and the corresponding term in Eq. (10) may be neglected. Note that Fig. 4 shows that tw varies in the order: LiCl > NaCl > KCl This order is in conformity with the cited hydration numbers of these cations: a more hydrated ion transports not only more water molecules in its hydration shell, but the amount of dragged water is higher also due to a greater size of the hydrated ion. Similar results have been obtained by Berezina et al. [12] (Fig. 5) for a MK-40 membrane. In the two series of data presented, respectively, in Figs. 4 and 5, at high external solution concentrations, the water transport number tends to 2 in the case of KCl, 3 in the case of NaCl and between 6 and 7 in the case of LiCl. Thus, the contribution of nvol in the water transport number becomes small in these conditions. This low limiting quantity of nvol may be evaluated, very approximately, by the following way. We suppose that at relatively high concentrations when the free water amount becomes small and their transport is highly
hindered by the walls of the narrow capillary pore, one water molecule in the first hydration shell can drag only one “free” water molecule and the counter-ion itself can drug (push) also one water molecule. The following relation is then obtained when neglecting the amount of water transported by co-ions: (tw )c→∞ = 2nh+ + 1
(11)
When using for nh+ the values of Glueckauf and Kitt, Eq. (11) gives tw equal to 7.6, 4 and 2.2, respectively, for LiCl, NaCl and KCl, that is not so far from the experimental limiting values. We observe, however, that the limiting values for tw are achieved at concentrations about 1 mol/l in the case of CM2 and 4 mol/l in the case of MK-40. This difference is due to the difference in the structure and the composition of the membranes. The CM2 membrane is more hydrophobic, the water content is almost twice as small in comparison to the MK-40 membrane (Table 1). The concentration of the interstitial solution (found as the exchange capacity divided by the water content) is very high, about 7.5 mol/l, in comparison with 3.2 mol/l in the MK-40 membrane. Hence, the fraction of “free” water which can be dragged by hydrated counter-ions in CM2 is relatively small. And when the external solution concentration increases, hence, the interstitial solution concentration increases also, nh+ achieves sufficiently quickly its limiting values. In the case of MK-40 membrane, the fraction of “free” water at small concentrations is much higher, thus tw is higher. Because of a smaller exchange capacity of MK-40 and higher water content, the rate of increase in internal concentration with increasing the external solution concentration is higher in this membrane than in CM2. However, to achieve the values of the internal solution concentration in MK-40, comparable with those in CM2, the external solution concentration should be more than in the case of CM2. Hence, one can suppose that the conditions of the water transport (the interstitial solution concentration, the free water content, the pore size) in MK-40 become similar to that in CM2 equilibrated with 1 mol/l solution, when the external concentration is close to 4 mol/l.
6. Conclusion
Fig. 5. Water transport numbers for a MK-40-8 membrane (8% of DVB) as function of the concentration of different chloride solutions. Experimental data of Berezina et al. [12].
Two methods of the water transport number evaluation in ion-exchange membranes are proposed. The first consists in the calculation from the counter-ion apparent transport number, membrane electric conductivity and diffusion permeability, with the help of an approximate relation and the Scatchard equation. The second demands only the apparent transport number knowledge, the counter-ion transport number being supposed to be 1. The first method gives values of tw sufficiently close to experimental ones in a wide range of concentrations, whereas the second can be applied only at relatively small concentrations where the counter-ion
C. Larchet et al. / Electrochimica Acta 49 (2004) 1711–1717
transport number does not differ too much from 1 (<1 M for the membranes studied). The comparison of our data for a CM2 membrane in KCl, NaCl and LiCl solutions with the data of Berezina et al. [12] for a MK-40 membrane in the same solutions has shown a similar character of the water transport number behaviour. In two cases tw varies in the order: LiCl > NaCl > KCl, which corresponds to the order of the hydration numbers of the cations. When the external concentration increases, tw tends to its limiting value, which can be evaluated with a new proposed relation. The difference in the behaviour of the membranes is that in the case of CM2, tw achieves its limiting value at external solution concentrations about 1 mol/l, whereas in the case of MK-40 it is about 4 mol/l, that can be related to the difference in the water content and exchange capacity of the membranes. Acknowledgements V.N. thanks the Paris 12 University for giving opportunity to make the research described in this publication. This research was also financially supported by the CNRS, France, grant PICS No. 1811, and by the Russian Foundation of Basic Researches, Grants nos. 02-03-22001 CNRS a and 03-03-96652. References [1] H. Strathmann, in: W.S.W. Ho, K.K. Sirkar (Eds.), Membrane Handbook, Van Nostrand Reinhold, New York, 1992. [2] T.A. Davis, V. Grebenyuk, O. Grebenyuk, in: S.P. Nunes, K.-V. Peinemann (Eds.), Membrane Technology in Chemical Industry, Wiley-VCH, Weinheim, 2001, p. 222. [3] O. Bobreshova, L. Novikova, P. Kulintsov, E. Balavadze, Desalination 149 (2002) 363. [4] T. Okada, G. Xie, O. Gorseth, S. Kjelstrup, N. Nakamura, T. Arimura, Electrochim. Acta 43 (1998) 3741. [5] T. Okada, J. Electroanal. Chem. 465 (1999) 1. [6] G. Pourcelly, I. Tugas, C. Gavach, J. Membr. Sci. 97 (1994) 99.
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[7] B.J. Robbins, R.W. Field, S.T. Kolaczkowski, A.D. Lockett, J. Membr. Sci. 118 (1996) 101. [8] N. Lakshminarayanaiah, Transport Phenomena in Membranes, Academic Press, New York, 1969. [9] B.R. Breslau, I.F. Miller, Ind. Eng. Chem. Fundam. 10 (1971) 554. [10] M. Demarty, A. Maurel, E. Selegny, J. Chem. Phys. Phys. Chem. Biol. 71 (1974) 811. [11] P. Schaetzel, E. Favre, B. Auclair, Q.T. Nguyen, Electrochim. Acta 42 (1997) 2475. [12] N. Berezina, N. Gnusin, O. Demina, S. Timofeyev, J. Membr. Sci. 85 (1994) 207. [13] N.P. Berezina, O.A. Demina, N.P. Gnusin, S.V. Timofeyev, Elektrokhimiya 30 (1994) 1467. [14] G. Schmid, J. Membr. Sci. 150 (1998) 159. [15] A.E. Yaroshchuk, Colloids Surf. A: Physicochem. Eng. Aspects 140 (1998) 169. [16] B. Auclair, V. Nikonenko, C. Larchet, M. Métayer, L. Dammark, J. Membr. Sci. 195 (2002) 89. [17] D.G. Miller, J. Phys. Chem. 70 (1966) 2639. [18] R. Paterson, C.R. Gardner, J. Chem. Soc. (A) 13 (1971) 2254. [19] O. Kedem, M. Perry, J. Membr. Sci. 14 (1983) 249. [20] N.P. Gnusin, N.A. Kononenko, S.B. Parchikov, Russ. J. Electrochemistry 29 (1993) 647 (Translated from Elektrokhimiya 29 (1993) 754). [21] G. Scatchard, J. Am. Chem. Soc. 75 (1953) 2883. [22] W.F. Graydon, R.J. Stewart, J. Phys. Chem. 59 (1955) 86. [23] V.I. Zabolotsky, L.F. El’nikova, N.V. Sheldeshov, A.V. Alekseev, Elektrokhimiya 23 (1987) 1626. [24] E.H. Cwirko, R.G. Carbonnel, J. Membr. Sci. 67 (1992) 211. [25] Norme Française NF X 45-200, AFNOR, December 1995. [26] N.N. Belaid, B. Ngom, L. Dammak, C. Larchet, B. Auclair, Eur. Polym. J. 35 (1999) 879. [27] L. Dammak, C. Larchet, B. Auclair, V.V. Nikonenko, V.I. Zabolotsky, Eur. Polym. J. 32 (1996) 1199. [28] V. Tourreuil, L. Dammak, G. Bulvestre, B. Auclair, New J. Chem. 12 (1998) 1463. [29] R. Lteif, L. Dammak, C. Larchet, B. Auclair, Eur. Polym. J. 37 (2001) 627. [30] N.P. Gnusin, N.P. Berezina, A.A. Shudrenko, O.P. Ivina, Russ. J. Phys. Chem. 68 (1994) 506. [31] A. Narebska, S. Koter, W. Kujawsky, Desalination 51 (1984) 3. [32] A. Narebska, S. Koter, W. Kujawsky, J. Membr. Sci. 25 (1985) 153. [33] N.P. Gnusin, N.P. Berezina, O.A. Demina, N.A. Kononenko, Russ. J. Electrochem. 32 (1996) 154. [34] E. Glueckauf, G.P. Kitt, Proc. R. Soc. (London) A228 (1955) 322 (Cited from F. Helfferich, in: Ion-exchange, McGraw-Hill, New York, 1962, p. 106).