Identification of dielectric effects in nanofiltration of metallic salts

Journal of Membrane Science 287 (2007) 102–110

Identification of dielectric effects in nanofiltration of metallic salts A. Szymczyk a,∗ , N. Fatin-Rouge a , P. Fievet a , C. Ramseyer b , A. Vidonne a a

b

University of Franche-Comt´e, Laboratoire de Chimie des Mat´eriaux et Interfaces, 25030 Besan¸con Cedex, France University of Franche-Comt´e, Laboratoire de Physique Mol´eculaire, UMR CNRS 6624, 25030 Besan¸con Cedex, France Received 5 July 2006; received in revised form 18 September 2006; accepted 12 October 2006 Available online 17 October 2006

Abstract Transport of four metallic salts (CuCl2 , ZnCl2 , NiCl2 and CaCl2 ) through a polyamide nanofiltration (NF) membrane has been investigated experimentally from rejection rate and tangential streaming potential measurements. Rejection rates have been further analyzed by means of the steric, electric and dielectric exclusion (SEDE) homogeneous model with the effective dielectric constant of the solution inside pores as the single adjustable parameter. The volume charge density inside pores has been assessed from tangential streaming potential experiments by considering the effective dielectric constant inside pores instead of the bulk one. The conventional NF theory (i.e. disregarding dielectric effects) has been found to be unable to describe the experimental sequence of rejection rates (CuCl2 > ZnCl2 ≈ NiCl2 > CaCl2 ). It has been shown that the rejection rate sequence can be explained by the combination of the dielectric exclusion via image charges and the Donnan effect in the case of asymmetric salts with divalent counter-ions. The influence of the characteristic size of ions on the magnitude of the Born effect may explain the almost identical rejection rates measured with ZnCl2 and NiCl2 whereas a significant variation of the membrane volume charge density has been put in evidence by electrokinetic measurements carried out in both media. © 2006 Elsevier B.V. All rights reserved. Keywords: Nanofiltration; Modelling; Dielectric effects; Streaming potential; Metallic salts

1. Introduction Nanofiltration (NF) membranes are made of organic or ceramic materials the most of which develop a surface electric charge when brought into contact with a protic medium. These membranes have a molecular weight cut-off ranging from a few hundreds to a few thousands Dalton, i.e intermediate between reverse osmosis and ultrafiltration membranes. The combination of pore sizes around a few nanometers with electrically charged materials makes the prediction of their filtration performances extremely difficult since a complex mechanism, including steric hindrance, Donnan exclusion, dielectric phenomena and transport effects, is involved in solute separation [1]. Within very narrow pores such as those of NF membranes, the electrical double layers cannot fully develop and then strongly overlap. As a result, both the local electric potential and the

∗

Corresponding author. Tel.: +33 3 81 66 20 32; fax: +33 3 81 66 20 33. E-mail address: [email protected] (A. Szymczyk).

0376-7388/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2006.10.025

ion concentrations are almost radially constant (provided the electric charge carried out by the pore walls is low enough [2]). Neglecting these variations yields the so-called homogeneous approximation that is used in most NF transport models [3–8]. Until recently, fundamental works on electrolyte transport through NF membranes have considered the solute partitioning at the membrane/solution interfaces only on the basis of size and Donnan effects [3,4,6–8]. Some attempts were made to improve the description of ion transport through NF membranes by taking into account the Born (solvation) dielectric effect into partitioning equations at membrane/solution interfaces [9,10] (the use of the Born equation for the description of ion exclusion was first suggested by Yaroshchuk [11]). Recently, Yaroshchuk pointed out the relevance, in NF pores, of the interaction between ions and the polarization charges induced at the dielectric boundary between the pore walls and the pore-filling solution [12]. In a previous work [1], we proposed the steric, electric and dielectric exclusion (SEDE) model that considers the combination of the Donnan effect and the dielectric phenomena. Within the scope of this 1D model, the separation of solutes results from transport

A. Szymczyk et al. / Journal of Membrane Science 287 (2007) 102–110

effects (described by means of extended Nernst–Planck equations) and interfacial phenomena including steric hindrance, Donnan effect and dielectric exclusion (expressed in terms of (i) Born dielectric effect with is connected to the lowering of the dielectric constant of a solution inside nanodimensional pores and (ii) the interaction between ions and the polarization charges induced at the dielectric boundary between the pore walls and the pore-filling solution, i.e. the so-called image charges). It was shown that the conventional theory of nanofiltration (i.e. based only on the steric hindrance and the Donnan exclusion) failed to describe experimental salt rejection rates (in the case of magnesium chloride solutions electrokinetic measurements produced evidence that even the sign of the membrane charge fitted within the scope of the conventional theory was wrong) whereas a good agreement between experiment and theory was obtained when dielectric effects were included into calculations [1]. In the present work the retention properties of a negatively charged NF membrane are investigated with four electrolytes having same co-ions but different divalent counter-ions. With the help of electrokinetic measurements, the SEDE model is used to describe the experimental rejection rates and the theoretical

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predictions are compared with the conventional NF theory (i.e. neglecting the dielectric phenomena). 2. Theoretical background The SEDE model can be used to investigate transport properties of NF membranes by considering either cylindrical pores or slit-like pores [1,13]. The magnitude of image forces is strongly dependent on the pore geometry [12], which makes difficult the description of experimental data because the pore geometry of most commercial membranes is not well defined. The best description of experimental data was obtained by modelling the active layer of the AFC 40 membrane as a bundle of identical slit-like pores of length x and half-width rp (rp  x) separating the feed solution from the permeate one (Fig. 1). The SEDE model has been described in details in [1]. The governing equations of the model are synthetically reported in Tables 1 and 2. The distribution of ions at both 0+ |0− and x− |x+ interfaces is described by Eqs. (1a) and (1b) which are modified Donnan relations including steric hindrance and dielectric exclusion.

Table 1 Partitioning equations used in the SEDE model (all symbols are defined in nomenclature) Partitioning equations at the membrane/solution interfaces ci (0+ ) γi (0− )   = φi exp(−zi Ψ(0+ |0− ) ) exp(−Wi,Born ) exp(−Wi,im(0 + |0− ) ) ci (0− ) γi (0+ ) − + ci (x ) γi (x )   = φi ) exp(−Wi,im(x exp(−zi Ψ(x− |x+ ) ) exp(−Wi,Born − |x+ ) ) ci (x+ ) γi (x− ) with,   ri,Stokes φi = 1 − (slit-like pores) rp e Ψ(0+ |0− ) = ψD(0+ |0− ) kT e ψD(Δx− |Δx+ ) Ψ(x− |x+ ) = kT   1 (zi e)2 1  Wi,Born = − 8πε0 kTri,cav εp εb





 Wi,im(0 + |0− ) = −αi ln 1 −





 Wi,im(x − |x+ ) = −αi ln 1 −

αi =

εp − εm εp + εm



εp − εm εp + εm

μ(0+ |0− ) = κ(0− )rp

(1b)

(2) (3a) (3b) (4)



exp(−2μ(0+ |0− ) )



(slit-like pores)

(5a)

 exp(−2μ(x− |x+ ) )

(slit-like pores)

(zi F )2 8πε0 εp RTNA rp



(1a)

(5b) (6)

   z2i ci (0− )φi (γi (0− )/γi (0+ )) exp(−zi Ψ(0+ |0− ) − Wi,Born − Wi,im(0 + |0− ) )

2I(0− )

    z2i ci (x+ )φi (γi (x+ )/γi (x− )) exp(−zi Ψ(x− |x+ ) − Wi,Born − Wi,im(x − |x+ ) )

(7a)

i

+

μ(x− |x+ ) = κ(x )rp

2I(x+ )

(7b)

i

Electroneutrality conditions 



zi ci (0− ) = 0

(8a)

zi ci (x+ ) = 0

(8b)

i

 i

i

zi ci (x) + X = 0

for 0+ ≤ x ≤ x−

(8c)

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Fig. 1. Schematic representation of a slit-like pore in the SEDE model.

The dielectric exclusion is split into two different contributions. The first one is connected to the lowering of the dielectric constant of a fluid trapped in nanodimensional cavities [14,15] and is referred as the Born effect. The second dielectric phenomenon arises from the difference between the dielectric constant of the membrane material and that of the pore-filling solution. It corresponds to the interaction between the ions and the polarization charges induced (by the ions themselves) at the dielectric boundary (this dielectric effect is often described by the so-called image charges). Within the scope of the SEDE model, the Born effect is described by Eq. (4) which is a modified Born equation that considers the radius of the cavity formed by the ion i in the solvent (ri,cav ). This latter is approximately determined according to the procedure proposed by Rashin and Honig [16], i.e. on the basis of the covalent radius for cations and the ionic radius Table 2 Transport equations used in the SEDE model (all symbols are defined in nomenclature) Zero electric current condition (steady state)



F

zi ji = 0

(9)

i

Transport equations ji = −Ki,d Di,∞

dci JV ci (x+ ) zi ci Ki,d Di,∞ F dψ − + Ki,c ci V = dx RT dx Ak

Concentration gradients inside pores zi Fci dψ dci JV (Ki,c ci − ci (x+ )) − = dx Ki,d Di,∞ Ak RT dx Electrical potential gradient inside pores



− dψ = dx



(F/RT )

(11)



zi Ki,d Di,∞ (dci /dx) + (JV /Ak )

i

(10)

(12)

z2i ci Ki,d Di,∞

i

with,



Ki,d = 1 − 1.004

 + 0.21

Ki,c

(3 − φi2 ) = 2

ri,Stokes rp

ri,Stokes rp



1 1− 3



 + 0.418

4

 − 0.169



ri,Stokes rp

ri,Stokes rp

ri,Stokes rp

(i) Eq. (1a) is introduced into Eq. (8c) that is numerically solved together with Eq. (5a) by considering Eq. (7a). It allows  both Ψ(0+ |0− ) and Wi,im(0 + |0− ) values to be computed. The ion concentration just inside the pore, i.e. ci (0+ ), can then be determined from Eq. (1a). (ii) Next, a first guess is made for ci (x+ ) allowing for Eq. (8b). Introducing Eq. (1b) into Eq. (8c) and solving numerically together with Eq. (5b) by considering Eq. (7b) yields  both Ψ(x− |x+ ) and Wi,im(x − |x+ ) values. Introducing these values into Eq. (1b) gives the initial values of ci (x− ). When these latter are known, Eqs. (11) and (12) are solved simultaneously using the Runge–Kutta method of order four. The values of ci (0+ ) that are obtained from the transport equations are then compared with those deduced from partitioning equations (step i). A new set of ci (x+ ) values is fixed and step ii is repeated until minimization of error function. 3. Experimental

zi Ki,c ci i

for anions (by increasing these radii by 7% for both cations and anions; see [16] for more details). The interaction between the ions and the induced polarization charges was extensively studied by Yaroshchuk who derived approximate expressions for the resulting excess solvation energy. For slit-like pores, these ones are given by Eqs. (5a) and (5b) (considering the undistorted ionic atmosphere approximation [17]). Equivalent relations for cylindrical pores were derived by Yaroshchuk [12,18] and were applied to the SEDE model in previous works [1,13]. The extended Nernst–Planck equation (Eq. (10) in Table 2) forms the basis for the description of solute transport through charged porous membranes [19,20]. It describes ion transport in terms of diffusion under the action of the solute concentration gradient, migration under the action of spontaneously arising electric field and convection due to solvent flow. This equation can be rewritten so as to establish the expression of the concentration gradient inside pores, i.e. for 0+ ≤ x ≤ x− (see Table 2, Eq. (11)). The expression of the electric potential gradient (Eq. (12)) is derived from Eq. (10) and the condition of zero electric current flowing through the membrane at the steady state (Eq. (9)). The numerical scheme followed to compute the solute rejection rate can be summed up as follows:

3

5 (slit-like pores) (13)

2

3.1. Membrane and chemicals The membrane studied here was a AFC 40 PCI nanofiltration membrane. This is a tubular membrane with a polyamide active layer. All electrolytes were of analytical grade and solutions were prepared with milli-Q quality water. The pH of solutions was found to be 4.8 ± 0.4 (no pH adjustment was made). 3.2. Filtration measurements

(slit-like pores)

(14)

All measurements were carried out on a Micro 240 filtration unit. A pump allowed circulation of the solution from a feed

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Table 3 Stokes radius of the various PEGs ri,Stokes (nm) PEG 200 PEG 400 PEG 600 a b

0.43a,b 0.54a to 0.52b 0.58a to 0.61b

Taken from Ref. [21]. Calculated from Ref. [22].

tank of 30 L to the membrane module where it flowed over the active layer of the membrane. The temperature was kept constant at 25 ± 2 ◦ C by means of a heat exchanger. The permeate solution was collected at various transmembrane pressure differences when a constant filtration regime was achieved (i.e. constant permeate flux and conductivity at ±5%). The permeate volume was measured by weighing and conductivity measurements were carried out with an Inolab conductimeter equipped with a WTW TetraCon cell. The solutions of metal ions were prepared by dissolving the salts in pure water. The concentrations of electrolytes in the feed and in the permeate solutions were obtained from their conductivity using a calibration curve. Solution of PEGs (average molecular weight, MW : 200, 400 and 600 Da) at 1 g/L were filtered through the membrane in order to estimate its pore radius. The hydrodynamic radius of PEGs (see Table 3) were taken from Sarrade et al. [21] and Afonso et al. [22]. Their concentration in the permeate was measured by total organic carbon analysis with a Shimatsu 5050 apparatus. 3.3. Tangential streaming potential (TSP) measurements Measurements of tangential streaming potential were carried out with a Zetacad zeta-meter (CAD Inst., France). This apparatus allows the measurement of the electrical potential difference that appears from the convection of an electrolyte solution between two parallel membranes, facing each other and separated by a teflon spacer of height 2 h. The tubular membrane was cut in order to obtain flat sheets that could be inserted within the measuring cell. The cell was equipped with two Ag/AgCl electrodes placed on each side of the channel and linked to a Keithley 2000 multimeter in order to measure the streaming potential (ϕs ). ϕs was measured by increasing the pressure difference (P) from 0 to 500 mbar (the solution was pushed using nitrogen gas). The streaming potential coefficient was obtained from the slope of the plot ϕs versus P as shown in Fig. 2. In addition, the equipment allows the measurement of both the conductivity and the temperature of the solution. A more detailed description of the apparatus can be found elsewhere [23]. The membrane was equilibrated with the electrolyte solution by re-circulation under low P until ϕs versus P plots were reproducible. This operation needed between few hours and 2 days. TSP measurements were made at 25 ± 3 ◦ C and at various channel heights by using teflon spacers of 300, 600 and 900 ␮m thickness.

Fig. 2. Streaming potential ϕs vs. the applied pressure difference P. [CuCl2 ] = 10−3 M; pH 4.4; 2h = 300 ␮m; T = 25 ◦ C.

4. Results and discussion The SEDE model is a four-parameter model (provided the dielectric constant of the active layer material, εm , is known; this one has been fixed at 3 according to available data for polyamide material [24]). Since the pore half-width (rp ), the thickness to porosity ratio (x/Ak ), the membrane volume charge density (X) and the dielectric constant of the solution inside pores (εp ) have to be known so as to compute the intrinsic rejection rate (Ri ) which is defined for a solute i as Ri = 1 −

ci (x+ ) ci (0− )

(15)

The effective pore size has been determined from the limiting rejection rates measured with polyethyleneglycols of various molecular masses (since it can be easily shown that lim Ri = JV →∞

1 − φi Ki,c for neutral solutes) and a mean value of 0.54 nm has been found for rp . The second structural parameter, i.e. x/Ak , has been assessed from water flux measurements, JW , carried out at various transmembrane pressure differences, P (see Fig. 3) by

Fig. 3. Pure water flux vs. transmembrane pressure difference.

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active layer and inside pores. For slit-like pores, the volume charge density inferred from tangential streaming potential measurements (XTSP ) reads as follows: XTSP = −

Fig. 4. Plot of the reciprocal streaming potential coefficient (P/ϕs ) vs. the reciprocal channel height (1/2h) for the calculation of zeta potential according to the extrapolation method [27,28]. [CuCl2 ] = 5 × 10−4 M; pH 4.4; T = 25 ◦ C.

considering the Hagen–Poiseuille equation for slit geometry: JW =

rp2 3η(x/Ak )

(16)

P

where η is the dynamic viscosity of water (0.89 × 10−3 Pa s at 298 K). It is highly likely that physical properties such as viscosity are affected by structural changes of water in confined systems. However, because of the lack of quantitative studies dealing with water viscosity in nanopores, the correct value of η to be used in Eq. (16) is hardly predictable. That is why we used the viscosity of bulk water in our calculations. By using Eq. (16), we make the implicit assumption that the pressure drop occurring through the macroporous sublayer(s) is negligible so that P can be almost entirely attributed to the active layer. Unfortunately, it is pretty difficult to justify such a rough approximation (although it is almost always made in the literature) since support layers are (most often) not available separated from their active layer (see Refs. [25,26] for detailed investigations on this topic). An approximate value of 7.81 ␮m has been found for x/Ak . As shown in Fig. 4, the zeta potential (ζ) can be obtained from the plot P/ϕs versus the reciprocal channel height (1/2h) according to the extrapolation method that was originally suggested by Yaroshchuk and Ribitsch [27] and applied by Fievet et al. [28]. The zeta potential has been determined for various chloride salts with divalent metallic cations, Ca2+ , Ni2+ , Cu2+ , Zn2+ (see Table 4). The electrokinetic charge density has been further computed according to:

       Fζ −z i −1 ci (0− ) exp σek = −sign(ζ) 2ε0 εb RT RT i

(17) Eq. (17) can be used to compute the effective volume charge density of the active layer under the assumption that the electrokinetic charge density is the same on the top surface of the

σek rp F

(18)

This corresponds to a limiting case called “the constant charge density assumption”. However, it is known that the surface charge density decreases as two surfaces come closer to each other (the so-called charge regulation process) [29] and thus the volume charge density inside pores, X, should be less than XTSP . Accounting for this phenomenon would require the use of a charge regulation model but this would increase the number of fitting parameters and make the use of the SEDE model less appropriate for predictive purposes. As shown by Eq. (17) the electrokinetic charge density is proportional to the square root of the medium dielectric constant. We therefore propose to correct the volume charge density determined from tangential streaming potential measurements by considering the dielectric constant of the solution inside pores (εp ) instead of the bulk value (εb ). Within this scope, the volume charge density used in the SEDE model (X) is computed from XTSP according to:  εp (19) X = XTSP εb It should be stressed that the derivation of Eq. (19) is based on the (arbitrary) assumption that the zeta potential inside the pores is the same as that of the external membrane surface despite the change in the dielectric constant inside the pores. Both X and XTSP values are collected in Table 4 for the different electrolytes. The general mechanism of membrane charge formation results from (i) the acid/base dissociation of functional groups at the pore walls and (ii) the site-binding of counter-ions on the charged surface sites (which depends on the affinity of the metal ion for the membrane). From the tangential streaming potential measurements it can be inferred that the AFC 40 membrane is negatively charged at natural pH (i.e. without any pH adjustment) whatever the electrolyte. This finding is in agreement with the available literature on polyamide membranes and is usually explained by the presence of residual carboxylic groups on the polymer chains. The site-binding of divalent cations on charged sites plays a nonnegligible role in the charge formation of the AFC 40 membrane as indicated by the different values of XTSP obtained with the various electrolytes, the sequence of counter-ions site-binding being Zn2+ > Cu2+ > Ni2+ > Ca2+ (counter-ions with a stronger Table 4 Zeta potential (ζ) and membrane volume charge density (XTSP : volume charge density obtained from tangential streaming potential measurements; X: volume charge density inside pores) for the various electrolytes Electrolyte

pH

ζ (mV)

CaCl2 CuCl2 NiCl2 ZnCl2

5.1 4.4 5.2 4.8

−21.4 −15.5 −20.7 −13.1

± ± ± ±

9.3 2.4 7.4 3.9

XTSP (equiv. m−3 )

X (equiv. m−3 )

−44.1 −31.3 −42.4 −24.0

−41.3 −25.0 −36.7 −21.5

A. Szymczyk et al. / Journal of Membrane Science 287 (2007) 102–110

affinity to the surface lead to a smaller amount of free surface sites, otherwise stated, to a smaller effective charge density X). As a result of the lowering of the dielectric constant inside pores (εp ), the volume charge density inside pores is less than XTSP (which can be viewed as the volume charge density in infinitely wide pores). This decrease in the pore volume charge density can be interpreted in terms of ion pairing inside pores [30]. Because of the lowering of the dielectric constant inside pores, the generation of charged surface sites is decreased and the binding of the counter-ions with the charged sites is more favorable since the attractive electrostatic interaction between the surface sites and the counter-ions is stronger. The volume charge density inside pores decreases by 6–20% (with respect to the volume charge density inferred from streaming potential measurements) depending on the electrolyte. It is worth mentioning that Eq. (19) does not account for the charge regulation process and then the values of X used in our calculations are still overestimated values of the fixed charge density inside the pores. Fig. 5a–d shows the comparison between the experimental rejection rates and the theoretical ones for the four electrolytes at various permeate volume fluxes. Experimental rejection rates have been corrected for the concentration polarization phenomenon. The concentration profiles across the unstirred layer have been calculated numerically from the extended

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Nernst–Planck equation by modelling the unstirred layer as an uncharged membrane of thickness δ with neither steric hindrance nor dielectric effects. The thickness of the unstirred layer, δ, has been assessed from Dittus and Boelter correlation [31]: Sh =

Kdh = 0.04Re0.75 Sc0.33 D

(20)

where Sh is the Sherwood number, dh the hydraulic diameter of the membrane, Re the Reynolds number, Sc the Schmidt number, D represents the solute diffusion coefficient for neutral solutes and the so-called global diffusion coefficient for binary electrolytes (the global diffusion coefficient of a binary electrolyte is defined as D+ D− (z+ − z− )/(z+ D+ − z− D− )) and K is the mass transfer coefficient defined as D/δ. The theoretical rejection rates have been fitted to experimental data with the dielectric constant inside pores as the single adjustable parameter. The SEDE model provides a relatively good description of the experimental data with εp ranging from 50 to 69 depending on the electrolyte (see Table 5). The decrease in the dielectric constant inside pores originates from both confinement and fixed charged effects [13]. The effective dielectric constants are in good agreement with recent results obtained with another polyamide NF membrane [1].

Fig. 5. (a) Comparison between theoretical and experimental salt rejection rates for a 1 equiv. m−3 CuCl2 solution. (b) Comparison between theoretical and experimental salt rejection rates for a 1 equiv. m−3 ZnCl2 solution. (c) Comparison between theoretical and experimental salt rejection rates for a 1 equiv. m−3 CaCl2 solution. (d) Comparison between theoretical and experimental salt rejection rates for a 1 equiv. m−3 NiCl2 solution.

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Table 5 Effective dielectric constant inside pores (εp ) computed from the SEDE model for the various electrolytes Electrolyte

εp (best-fit value using X inferred from Eq. (19))

εp (best-fit value for the upper limiting case, i.e. using XTSP )

εp (best-fit value for the lower limiting case, i.e. using X = 0)

CaCl2 CuCl2 NiCl2 ZnCl2

69 50 59 63

69 49 59 62

>εb 55 69 71

The analysis of the rejection rate sequence is of great interest. As can be seen from Fig. 5, RCu2+ > RNi2+ > RCa2+ whereas XCu2+ < XNi2+ < XCa2+ (see Table 4), i.e. the rejection rate increases as the membrane charge decreases. Such a result cannot be explained by the standard NF theory, i.e. based on a steric/Donnan exclusion mechanism, since the fixed charge theory predicts an increase in the rejection rate with higher volume charge densities. On the other hand, the decrease in the rejection rate of asymmetric electrolytes having divalent counter-ions with increasing volume charge densities can be predicted provided the dielectric exclusion via image charges be taken into account (see Fig. 6). Indeed, unlike Donnan exclusion, image charges do not (qualitatively) differentiate between co-ions and counter-ions since the interaction between ions and induced polarization charges varies with the square of the ion charge as shown by Eqs. (5) and (6). As a result, both co-ions and counter-ions are excluded from the membrane pores by this dielectric effect. By decreasing the ion concentrations inside pores, the image charges therefore weaken the screening of the membrane fixed charge. In other words, the image charges strengthen the Donnan exclusion. Meanwhile, the fixed charge at the pore walls screens the interaction between ions and the polarization charges. Indeed, the counter-ions interact with their own images as well as with the images (carrying out a charge of opposite sign) induced by the membrane fixed charge [1,12]. For electrolytes having divalent counter-ions, the weak increase in the Donnan exclusion as the membrane fixed charge density increases is overcompensated by the strong decrease in the interaction with image charges. As a result, the rejection rate decreases with increasing volume charge densities as illustrated in Fig. 6.

Fig. 6. Theoretical rejection rate (Rsalt ) for a solution of a fictitious 2:1 electrolyte (D+ = D− = 10−9 m2 s−1 ) at 1 equiv. m−3 vs. effective volume charge density (X); rp = 0.5 nm; x/Ak = 5 ␮m; εm = 3; no Born effect (i.e. εp = εb ); JV = 10−5 m s−1 .

It can be noted from Fig. 5b and d that the AFC 40 membrane exhibits similar retention performances for ZnCl2 and NiCl2 although the volume charge density is much higher (by 70%) when the membrane is filled with NiCl2 (due to the preferential site-binding of Zn2+ cations on the charged surface sites of the AFC 40 membrane). According to the results shown in Fig. 6, the combination of the Donnan exclusion and the image charges should lead to a higher rejection rate for ZnCl2 than for NiCl2 . Our experimental results can be explained by the second dielectric effect, i.e. the Born effect, which is greater for NiCl2 than for ZnCl2 since Ni2+ is smaller than Zn2+ (their cavity radii, calculated according to the Rashin and Honig’s procedure [16] are equal to 0.123 and 0.134 nm, respectively). As can be seen from Eq. (4), the Born energy is inversely proportional to the ion cavity radius. Moreover, the smaller the ion, the stronger the electric field at the surface of the ion. As a result, the solvent molecules are more strongly polarized, which should lead, within a confined space like membrane pores, to a stronger lowering of the effective pore dielectric constant. This general trend appears in Fig. 7 where we have plotted the effective dielectric constant inside pores (inferred from the SEDE model) versus the cavity radius of the various cations. As expected, the effective dielectric constant decreases with the ion cavity radius. It appears from our calculations that the Born effect for Ni2+ is about 3.85 kT whereas it is only 2.63 kT for Zn2+ . Consequently, the effect of the combination of the Donnan exclusion and the image charges is balanced by the Born dielectric effect, thus leading to similar rejection rates for ZnCl2 than for NiCl2 . As mentioned previously, the fixed charge density X corrected by means of Eq. (19) remains an overestimated value of the fixed charge density inside pores since Eq. (19) does not account for the charge regulation process. We have carried out a set of additional calculations by setting the effective volume charge density

Fig. 7. Effective dielectric constant inside pores (εp ) vs. ion cavity radius (rcav ); cavity radii: 0.123 nm (Ni2+ ); 0.134 nm (Zn2+ ); 0.186 nm (Ca2+ ); 0.125 nm (Cu2+ ).

A. Szymczyk et al. / Journal of Membrane Science 287 (2007) 102–110

to zero (this would represent the lower limiting case for which the fixed charge density would totally vanish because of the charge regulation process) or equal to XTSP (this would represent the upper limiting case, i.e. with neither charge regulation nor influence of the decreased dielectric constant inside the pores). The various effective dielectric constants inside pores are collected in Table 5. As can be seen, the use of either X or XTSP does not produce any significant change in the best-fit εp values. That appears to be a consequence of the fact that for asymmetric electrolytes with divalent counter-ions, the Donnan exclusion plays only a secondary role. Calculations performed for zero fixed charge density (lower limiting case) systematically lead to a higher dielectric constant inside pores (for CaCl2 , the best-fit value would be even higher than the bulk dielectric constant, which means that the fixed charge density of the AFC 40 membrane cannot be reduced to zero even if the charge regulation process does occur). Otherwise stated, a weaker Born effect is needed to describe the rejection rate of asymmetric electrolytes with double-charge counter-ions when neutral pores are considered. This is due to the great increase in the image charge contribution for neutral pores since, in this case, the interaction with the induced polarization charges is no more screened by the membrane fixed charge. 5. Conclusion Transport properties of a nanofiltration polyamide membrane have been investigated by means of the steric, electric and dielectric exclusion (SEDE) homogeneous model. Within the scope of this one-dimensional model, the separation of solutes results from transport effects (described by means of extended Nernst–Planck equations) and interfacial phenomena including steric hindrance, Donnan effect and dielectric exclusion (expressed in terms of (i) Born dielectric effect resulting from the lowering of the effective dielectric constant inside pores and (ii) the interaction between ions and the polarization charges induced at the pore walls by ions themselves (the so-called image charges). Four metallic salts with same anions but different doublecharged cations (Cu2+ , Zn2+ , Ni2+ and Ca2+ ) have been studied. With the help of tangential streaming potential measurements, it has been shown that the conventional nanofiltration theory fails to describe the experimental data. Indeed, our findings show that the membrane efficiency decreases as its volume charge density increases. Such a result is unpredictable by the standard theory of nanofiltration that predicts that the rejection rate monotonously increases with the membrane volume charge density. On the other hand, the SEDE model has been found to be able to provide a relatively good description of the experimental data. The sequence of rejection rates, CuCl2 > NiCl2 > CaCl2 , has been explained by the combination of the Donnan effect and the interaction with image charges. Similar rejection rates have been observed for NiCl2 and ZnCl2 although the volume charge density is much higher (by 70%) when the membrane is filled with NiCl2 due to the strongest affinity of Zn2+ for the membrane surface. The rather high rejection rate measured for ZnCl2 can be explained by the stronger Born dielectric effect for Zn2+

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with respect to Ni2+ because of the smaller characteristic size of the Zn2+ cations.

Nomenclature Ak ci dh Di,∞

porosity of the membrane active layer concentration of ion i hydraulic diameter of the membrane bulk diffusion coefficient of ion i at infinite dilution e elementary charge F Faraday constant h channel half-height in tangential streaming potential experiments I ionic strength ji molar flux density of ion i permeate volume flux JV JW pure water volume flux k Boltzmann constant K mass transfer coefficient Ki,c hydrodynamic coefficient accounting for the effect of pore walls on convective transport Ki,d hydrodynamic coefficient for hindered diffusion inside pores NA Avogadro number P hydrostatic pressure P hydrostatic pressure difference ri,cav cavity radius of ion i ri,Stokes Stokes radius of ion i rp pore half-width R ideal gas constant Re Reynolds number Ri rejection rate of ion i Sc Schmidt number Sh Sherwood number T temperature V fluid velocity inside pores  Wi,Born dimensionless excess solvation energy due to Born effect for ion i  Wi,im dimensionless excess solvation energy due to “image charges” for ion i x coordinate x effective thickness of the active layer X effective volume charge density inside the membrane pores (calculated from Eq. (19)) XTSP volume charge density inferred from tangential streaming potential measurements zi charge number of ion i Greek symbols γi activity coefficient for ion i δ thickness of the unstirred layer εb dielectric constant of the bulk solution outside pores εm dielectric constant of the membrane active layer

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εp ε0 ζ η κ μ

σ ek φi ϕs ψ Ψ

A. Szymczyk et al. / Journal of Membrane Science 287 (2007) 102–110

effective dielectric constant inside pores vacuum permittivity zeta potential dynamic viscosity of the fluid Debye parameter effective dimensionless reciprocal screening length for interaction between ions and induced polarization charges electrokinetic charge density steric partitioning coefficient for ion i streaming potential local electrical potential inside pore dimensionless Donnan potential

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