Journal of Membrane Science 360 (2010) 389–396 Contents lists available at ScienceDirect Journal of Membrane Science journal homepage: www.elsevier.com/locate/memsci Phenomenological analysis of transport of mono- and divalent ions in nanofiltration Sarit Bason, Viatcheslav Freger ∗ Zuckerberg Institute for Water Research, Department of Biotechnology and Environmental Engineering, Ben Gurion University of the Negev, Sde-Boqer Campus, Sde-Boqer 84990, Israel a r t i c l e i n f o Article history: Received 26 February 2010 Received in revised form 24 April 2010 Accepted 13 May 2010 Available online 20 May 2010 Keywords: Nanofiltration Modeling Mono- and divalent ions Phenomenological transport coefficients Salt partitioning Ion exclusion mechanism a b s t r a c t A phenomenological approach was used to deduce concentration-dependent permeabilities of salts containing divalent ions (CaCl2 , MgCl2 and Na2 SO4 ) for NF200 membrane and compare them with results for monovalent salts and predictions of the steric–dielectric–Donnan (electric) (SDE) model. As a part of this effort the importance of the correction for concentration polarization was demonstrated and the error associated with the use of single average values of mass transfer coefficients and permeate concentration was analyzed. The dependence of permeability on salt concentration was found incompatible with the standard SDE picture for all salts. The discrepancy was more drastic for divalent cations, which showed a permeability decreasing with concentration in the whole analyzed range, in opposite to the standard model. It may be suggested that, a localized binding or association of counter-ions with fixed charges accompanied by charge reversal for divalent cations may be responsible for the observed behavior, provided the bound ions may retain osmotic activity. Such association may be promoted by the low dielectric constant of the membrane hence a large Bjerrum length commensurate with spacing of fixed charges. © 2010 Elsevier B.V. All rights reserved. 1. Introduction Nanofiltration (NF) membranes show different selectivities towards different ions; thus they typically much better retain divalent ions such as calcium and magnesium than monovalent ions. This may be utilized for water softening [1], water recycling [2], salt crystallization [3], etc. The rejection of different ions strongly depends on the solution composition and filtration conditions and a significant effort is ongoing towards developing predictive models of separation of ionic mixtures by NF. Presently, the accepted framework for physical modeling of NF is the extended Nernst–Planck (ENP) differential equations for each ion coupled with two electroneutrality conditions for ion fluxes and local concentrations [4–12]. In the case of a single salt these reduce to the classic Spiegler–Kedem (SK) equation for the salt transport [13]. However, these fundamental equations require coefficients that in general depend on the concentrations of ions and must be calculated for each ion using appropriate thermodynamic and kinetic models of ion exclusion and transport within the active layer. Virtually all published models of NF utilize, for such calculations, certain relations based on the hindered transport theory ∗ Corresponding author at: Zuckerberg Institute for Water Research, Ben-Gurion University of the Negev, Sde-Boqer Campus, Sde Boqer 84990, Israel. Tel.: +972 8 6563532; fax: +972 8 6563503. E-mail address: vfreger@bgu.ac.il (V. Freger). 0376-7388/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2010.05.037 [14,15] and on three ion exclusion mechanisms, steric, dielectric and Donnan (electric) (SDE) [8,16–19]. The coefficients are then calculated from a few basic physical characteristics, such as the pore size, fixed charge and dielectric properties of the membrane and known size and charge of ions involved and fitting to experimental results allow evaluation of the membrane parameters. Unfortunately, this approach suffers from the difficulty to transparently verify the underlying assumptions and relations used to compute coefficients. Thus it was often observed that the parameters deduced for one salt do not necessarily apply to other salts or their mixture, even though the fitted physical characteristics are not supposed to depend on the salt type and composition [8,17]. Rejection of salts containing divalent ions has been particularly difficult to describe using membrane parameters deduced from data obtained from rejection of monovalent salts [5,8,16,17,20–24]. It is not entirely clear whether the difficulty lies in the poor fitting of different parameters, whose effect on selectivity is not immediately seen, or in some fundamental flaws of the underlying relations for ion exclusion. Some authors observed that the behavior of divalent ions is correlated with a reversal of the surface charge or zeta potential from a negative to a positive value [7,25–30]. However the composition and charge of the surface may not adequately represent the inner selective part of the membrane during filtration [31,32], thereby properly analyzed filtration data should remain the preferred source of information. 390 S. Bason, V. Freger / Journal of Membrane Science 360 (2010) 389–396 Recently, we proposed a phenomenological approach that directly yields the values of coefficients and their variation with concentration and avoids the difficulty of using a specific physical models for calculating the coefficient [33,34]. A similar but slightly simpler approach was proposed by Yaroshchuk a few years earlier [35]. This analysis is based on the SK equation, i.e., essentially, the ENP equations for a single salt. Its utility has been already demonstrated by analyzing NF data for several monovalent salts [36]. Perhaps the most unexpected finding was that the variation of the coefficients with salt concentration showed features incompatible with the regular SDE picture at low salt concentrations pointing to the need for its modification. In this publication we extend this approach to salts containing divalent ions Ca2+ , Mg2+ and SO4 2− . Our focus will be on concentration-dependence of the diffusion permeability ωs and its agreement with the currently accepted picture of ion exclusion. Ultimately, we will see that the results for these salts show even greater incompatibility with the regular SDE relations than monovalent ions and will briefly discuss possible reasons. To see that explicitly we will summarize in the next section the generic relations expected for the regular SDE picture. 2. Theoretical background 2.1. The phenomenological equation The analysis is based on the following form of the SK equation that describes transport of a single salt through a membrane: dc c ′′ , = AJV c − Aωs dx̄   (1) where c is the local corresponding concentration, c′′ the permeate concentration, 0 ≤ x̄ ≤ 1 the distance from the feed side scaled to the total membrane thickness, and the Peclét coefficient A and local diffusion permeability ωs are two phenomenological coefficients. By generalizing arguments for monovalent salts presented earlier [33,34,36] it may be shown that for a single n1 :n2 salt (the subscripts “1” and “2” denote the counter- and co-ions, respectively) the coefficients A and ωs may be related to frictional, thermodynamic and membrane structure characteristics as follows: A= x fsw ,  (n1 + n2 ) RT ωs = (n1 + n2 ) RTK  x f1 Kc/(Kc + X/n1 n2 ) + f2 (2) ≈ (n1 + n2 ) RT K. f2 x (3) Here fsw = f1w + f2w is the salt–water friction coefficient (sum of frictions of all ions with water), f1 = f1w + f1m the total friction coefficient of counter-ion (sum of frictions of counter-ion with water and membrane) and f2 is defined for co-ion similarly to f1 , X the fixed charge content in the membrane, K the partitioning coefficient of the free salt (i.e., co-ion),  the membrane porosity (water volume fraction), and x is the membrane thickness. The last approximate equality in Eq. (3) corresponds to the case where the fixed charge is much larger than the free salt in the membrane, i.e., C = Kc ≪ X. Note that the first part of Eq. (7) in Ref. [36] supposed to be analogous to Eq. (2) here, erroneously contained a tortuosity factor ˛. This factor, however, correctly appeared in the second part of Eq. (7) in Ref. [36], which gives the subsequently used expression for A through the hindered transport theory. In the latter case the tortuosity factor explicitly accounts for the increased velocities and decreased thermodynamic potential gradients in the pores. However, in Eqs. (2) and (3) it is lumped into frictions factors formally defined using superficial velocities and gradients normal to the membrane surface [33]. Since A contains only geometric and frictional parameters, it weakly varies with salt concentration and thereby may be con- sidered constant when Eq. (1) is fitted to experimental data. In contrast, the coefficient ωs is concentration-dependent through K. For fitting to Eq. (1) it is convenient to parameterize the concentration dependence ωs (c) using an appropriate fitting equation, whose particular functional form is not critical, as long as it adequately fits the dependence. For instance, the following one used previously for monovalent salts offers sufficient flexibility [34,36] ωs (c) = a1 c a2 + a3 . (4) Yaroshchuk [35] used the first term (power dependence) of Eq. (4) to parameterize ωs (c) for relatively narrow concentration range on the grounds that it suits the standard SDE mechanism (see Section 2.2). However, we found that (empirical) addition of a finite free term a3 was critical for extending the range to lower concentrations for monovalent salts. From the results on 1:1 salts we could observe that the parameter A in Eq. (1) was small, of the order 103 s/m. As a result, the Peclét number AJV and the parameter Aωs were both much smaller than 1 in all experiments [33,34,36]. This should similarly apply to the divalent salts in this work, since the sizes of divalent ions used here hence their frictions and, ultimately, the values of A (cf. Eq. (2)) are not much different from those of monovalent ions analyzed previously. This turns Eq. (1) to a simpler approximate equation that contains no convection-related parameters [34] (see also Ref. [37]) c ′′ 1 − Aωs dc ≈ −JV ≈ −JV c ′′ . ωs dx̄ ωs (c) (5) The last equation is to be integrated across the membrane to relate c′′ to c′ and will be used in this work for deducing ωs (c) from filtration data. 2.2. SDE salt exclusion relations To analyze the obtained dependence ωs (c) it is useful to compare it with some generic relations that are not expected to qualitatively change within the regular SDE picture. We first note that the concentration dependence is mostly due to the partitioning coefficient K (cf. Eq. (3)), as the frictional parameters vary less significantly with salt concentration. To describe the variation of K, we may follow Yaroshchuk’s suggestion [38] and assume that all non-Donnan contributions for exclusion of an ion may be lumped into a constant non-Donnan partitioning coefficient of an ion k. This simplified approach appears to be useful for identifying expected trends, since the non-Donnan coefficients are not expected to strongly change with concentration. In fact, most recently published simulations relied on relations containing no concentration dependence for non-Donnan contributions. Using the standard assumption that the salt is fully dissociated both in membrane and in solution, the chemical potential of the single n1 :n2 salt must be equal in both phases  n1 ln n1 C+ X n2  +n2 ln (n2 C) = n1 ln (n1 k1 c) + n2 ln (n2 k2 c) , (6) where C is the concentration of free salt (i.e., co-ions and complementary counter-ions) in the membrane and X is the fixed charge content. At high salt concentrations (C ≫ X) this yields a constant K K= 1/(n1 +n2 )  C , ≈ k1n1 k2n2 c (7) and for low concentrations C ≪ X one obtains a power dependence with an exponent depending on salt type  K ≈ k2 k1 n1 n2 c X n1 /n2 . (8) Note that it is the concentration in the membrane C that is compared with X. The total average fixed charge content of polyamide membranes is usually in the range of 0.01–0.1 M [32,39,40]. This is S. Bason, V. Freger / Journal of Membrane Science 360 (2010) 389–396 commensurate with the highest solution concentrations used here. However, since non-Donnan exclusion in NF and RO membranes is strong [41], C must be much smaller that the fixed charge up to solution concentrations c much higher than used here. The system is then most likely in the regime C ≪ X described by Eq. (8). The dependence used by Yaroshchuk [35] was derived from this idealized relation and our Eq. (4) is its modified version. Note that within this model K is never expected to decrease with concentration, which may be also directly seen from Eq. (6). This is also true for any ion in a fully dissociated ionic mixture as well. Indeed, one can generalize Eq. (6) for a mixture using the relations:  C 1/z1i 1i = k1i c1i  −1/z2j C2j =e k2j c2j e/kT for any i, j, (9) where C with a pertinent subscript is the total concentration (i.e., both free and bound to fixed charges) of the specified ion in the membrane and i and j are running indices for counter- and co-ions, respectively. Eq. (9) states that the Donnan potential is the same for all ions. Substituting it to the electroneutrality relations for soluz C z C =X+ z c and membrane z c = tion j 2j 2j i 1i 1i j 2j 2j i 1i 1i (z’s being absolute ionic charges), one sees that the Donnan potential should decrease upon addition of any salt, which means the partitioning coefficient of any co-ion K2j = C2j /c2i or free salt should always increase. For a mixture of monovalent ions (z1i = z2j = 1 for all i, j) at low salt C a compact analytical solution may be obtained. Using X ≈ i 1i and Eq. (9) for C1i one obtains e Eq. (9) for C2j K2j ≈ k2j X e/kT = X/ k c , and then from i 1i 1i (10) k1i c1i . 391 Table 1 Diffusion coefficients D and mass transfer coefficients kd for three salts used in this study. Salt D × 109 (m2 /s) kd (␮m/s) CaCl2 MgCl2 Na2 SO4 1.334a 1.248a 1.229a 22.5 21.5 21.3 a Robinson and Stokes [42]. The pressure was varied between about 0.1 and 2 MPa. The volume flux JV for each pressure was determined by collecting and weighing permeate over a known time and the salt passage—by measuring the conductivity of feed, retentate and permeate. The hydraulic permeability coefficient Lp was determined for each salt and feed concentration as the slope of flux plotted versus net transmembrane pressure (NTP). The latter was calculated by subtracting from the gauge pressure the osmotic pressures difference between feed (corrected for concentration polarization—see next) and permeate which were calculated using the OLI software (OLI System, Inc.). Fig. 1a shows an example of flux versus NTP and Lp calculated for CaCl2 . Similar plots for MgCl2 and Na2 SO4 are presented in Fig. S1 in Supplementary Information. Prior to fitting, all feed concentrations cF were corrected for concentration polarization using the relation c ′ = (cF − c ′′ ) exp J  V kd + c ′′ , (12) where the mass transfer coefficient kd for each salt was calculated for v = 0.47 m/s using [36] kd = v0.5 D2/3 ∗ , (13) i This equation with interchanged indices 1 and 2 and i and j also gives the partitioning coefficient for the free fraction of a counterion (not bound to the fixed charges). This may be verified using Eq. (9) for counter-ions and the above expression for e e/kT and observing that the free fraction of any counter-ion is i k2j c2j C2j C1i − X ≈ C1i j X where D is the salt diffusion coefficient in water and the coefficient * = 28.1 (m/s)0.5 (m2 /s)−2/3 was determined previously for the particular flow cell and flow regime and is supposed to be independent of the salt (see Ref. [36] and its supporting information for details). The salt diffusivities and the calculated values of kd subsequently used for correcting rejection data using Eq. (12) are summarized in Table 1. = j X . (11) i Eq. (8) or (10) was used in our previous study [34] and the predicted linear dependence of permeability on concentrations was not experimentally observed for monovalent salts, particularly for concentration <0.01 M. Our present purpose is to examine Eq. (8) for single divalent salts. 3. Experimental 3.1. Filtration experiments Three single divalent salts of CaCl2 , MgCl2 and Na2 SO4 (all analytical grade) in solutions of four different feed concentrations (0.001, 0.01, 0.05, and 0.1 M) and NF200 membrane (Dow-Filmtec, kindly provided by Dr. Markus Busch) were used in filtration experiments. The setup is fully described elsewhere [34]. Briefly, two flat channel flow cells of membrane area 17.6 cm2 and channel cross-section 22 mm × 2 mm mounted in series were used in each experiment for duplicate tests. The solution from a pressurized feed tank (1.5 L) was circulated at average velocity v = 0.47 m/s (flow rate 1250 mL/min) through a thermostat that maintained a constant temperature of 25 ◦ C and then through the cells back to the tank by means of a gear pump. 4. Results and discussion 4.1. Concentration polarization and correction of flux-rejection data In many previous studies on NF, it was assumed or concluded that the concentration polarization (CP) could be ignored, therefore data treatment was carried out without a CP correction [8,17,43,44]. It appears however that at least for the conditions and the type of NF membrane used in the present work the CP correction was crucial. This view has been advocated by Yaroshchuk and is fully supported by the present results. Indeed, even for the fairly large fluid velocities (up to 60 cm/s) used in our experiments the mass transfer coefficients kd were of the order of 20–30 ␮m/s, while the maximal flux was about 30 ␮m/s. This means the correction was as large as tens or hundreds percent, particularly, for the largest fluxes. An example for CaCl2 shown in Fig. 1b demonstrates the significance of CP correction. (Similar plots for MgCl2 and Na2 SO4 salts are presented in Fig. S2 in Supplementary Information.) Since in practice kd rarely exceeds the above values, the use of fluxes higher than about 25–50 ␮m/s (100–200 L/m2 h) is quite meaningless, as polarization will drastically reduce the membrane selectivity and flux. This means that CP puts a natural limit on the highest usable flux. One important consequence is that it also puts a limit on the Peclét number AJV . In particular, for presently used thin NF membranes the Peclét coefficient A, proportional to membrane 392 S. Bason, V. Freger / Journal of Membrane Science 360 (2010) 389–396 Fig. 1. Filtration results for CaCl2 solutions of different concentrations using NF200 membranes: (a) volume flux vs. NTP for several different feed concentrations also showing the best linear fit; (b) observed (open symbols) and corrected for concentration polarization (solid symbols) passage of CaCl2 vs. volume flux for two different feed concentrations. The legends show feed concentrations in mol/L. thickness, becomes so small that AJV will be much smaller than 1 in any realistic conditions. This was apparently the case in this study therefore the approximate Eq. (5) could be used. The CP correction in this and most other studies was made using a single average value of kd . This value is often determined in separate experiments by varying kd through variation of the fluid velocity. As was pointed out by an anonymous reviewer, this approach may lead to errors, since Eq. (12) essentially addresses local values of concentrations, whereas in a typical cross-flow setup with an open channel kd inherently varies along the channel. Even if cF and Jv stay approximately constant, both c′ and c′′ vary along the channel, since kd and the rejection (depending on c′ ) both vary. However, only the average values of c′′ for all collected permeate is usually measured, therefore it is essential to evaluate the possible error of using average values for kd and c′′ for calculating corrected c′ or passage P = c′′ /c′ . The actual question is how much the value of c′ obtained for measured c′′ using a measured average kd deviates from the value c′ that would have been obtained using Eq. (6), had c′′ and kd been truly uniform over the whole surface. The analysis is simplified, if one requests an upper boundary of the error rather than an accurate error estimate. Such analysis is presented in Appendix A. For the present data Eq. (A9) yields the maximal error of 10% for monovalent salts, 2% for Ca and Mg chlorides and 30% for Na2 SO4 . In the latter case the maximal error was significant, however, it is probably acceptable for the present analysis and is much smaller (<10%) for lower concentrations were the variation of ωs is small. 4.2. Filtration results and model fitting Fig. 1a and analogous plots in Fig. S1 in Supplementary Information show a good linear relation between the flux and NTP, which was insignificantly affected by the salt type and concentration. The observed values of Lp were in the range 18.9–20.3 ␮m/(s MPa) (6.8–7.3 L/(m2 h bar)) for the divalent salts and concentrations involved, which was close to the values 20.2–22.6 (7.2–8.1) Fig. 2. Corrected passages vs. volume fluxes for NF200 with four feed concentrations using CaCl2 MgCl2 , Na2 SO4 and NaCl. The solid lines are the best fits using Eq. (1). The legends show feed concentrations in mol/L. The arrows indicate increasing feed concentration. S. Bason, V. Freger / Journal of Membrane Science 360 (2010) 389–396 393 Table 2 The Stokes, Pauling and Born radii (in nm) of ions involved. Ion 2+ Ca Mg2+ SO4 − Cl− Na+ a Fig. 3. Concentration dependence of divalent salts permeability ωs for CaCl2 , MgCl2 , Na2 SO4 and NaCl deduced from experimental data. obtained for monovalent salts [34] and to the value 21.0 (7.6) for pure water and the manufacturer’s value 21.4 (7.7) [45]. Since Lp is a frictional parameter related to the effective pore geometry [36,46], its insignificant variation indicates that the frictional and structural parameters in Eq. (3) stayed fairly constant for all salts and concentrations and the observed concentration dependence of ωs could be primarily related to the partitioning coefficient K. A similar conclusion could be drawn from AFM-based measurements of swelling of the active layer using the method described in our previous reports [34,47], which did not show a significant variation of swelling in solutions of 0.1 M CaCl2 compared to swelling in pure water. Fig. 2 compares representative fitting results for salts of different types, presently and previously obtained. The striking feature of the salts containing divalent cations, CaCl2 and MgCl2 , is the passage c′′ /c′ decreasing with feed concentration, which is opposite to the salts containing monovalent cations, e.g., NaCl showing an increasing passage [34]. This unexpected behavior of CaCl2 was already noted [37,48]. Interestingly, the “normal” trend shown by Na2 SO4 suggests that it is the charge of the cation that effects the reversal of the trend. The ωs (c) dependence reveals the trends and differences between the salts in more detail. This dependence was obtained by fitting the filtration data for each salt to solution of Eq. (5) using Eq. (4) to parameterize ωs (c). Fig. 3 shows the obtained double-log plots of ωs versus salt concentration for the studied divalent salts and, for comparison, for NaCl. The fitted values of parameters a1 , a2 and a3 used to calculate the plots are presented in Table S1 in Supplementary Information, however it is the plots in Fig. 3, i.e., concentration dependence of ωs (c), rather than the values of parameters that may be regarded as the actual experimental result. Fig. 3 clearly shows that no salt exhibits a dependence that agrees with Eq. (8). Thus at low concentrations both NaCl and Na2 SO4 show slopes (i.e., exponents) that are much slower than the values 1 and 2 predicted by Eq. (8). As in the preceding study on monovalent salts, it seems that the plot for these salts approaches a constant value at very low concentrations. For higher concentrations of NaCl, >0.01 M, the observed slope increases and seems to approach the expected value 1. A similar change could be observed for Na2 SO4 , but it was weaker and the slope certainly remained far below the expected value 2. The weaker change for Na2 SO4 could be related to its much stronger non-Donnan exclusion, in particular, strong dielectric exclusion of divalent anion (see Section 2.2), thereby the internal concentration C was much lower than for NaCl moving this salt further into the “low concentration” regime characterized by a nearly constant permeability. For CaCl2 and MgCl2 not only the slope was different from the expected value 0.5, but it was actually negative in the whole range, Stokesa Pauling [50] Born [49] 0.310 0.347 0.230 0.121 0.184 0.106 0.078 0.23 0.181 0.098 0.173 0.086 0.258 0.202 0.169 Calculated from ion diffusivities in water [51]. whereas Eq. (8) always predicts a positive exponent. The plot of ωs (c) reveals additional features, such as increase of the slope with concentration; thereby it is minimal, about −0.5, at the lowest examined concentration and nearly zero at the highest one. It may also be noted that the permeability of CaCl2 is higher than that of MgCl2 , which is consistent with the dielectric mechanism and the results by Szymczyk et al. [18]. Indeed, based on the experiments with both mono- and divalent ions it was argued that the Pauling (bare) or somewhat larger Born radii rather than the frictional Stokes radii are the ones to be used for calculating the strength of dielectric exclusion (see Table 2). (The Born radius is defined to exactly yield the measured energy of hydration in water using the Born equation [49].) The larger Pauling and Born radii of Ca than of Mg, as seen from Table 2, hence the smaller Born energy (inversely proportional to ion radius [49]) and weaker dielectric exclusion explain larger permeability of CaCl2 . We also attempted to fit Eq. (1) with A as an additional parameter. As for monovalent salts analyzed previously, the values of A were found very small, <3 × 103 s/m. Comparing this value with JV and ωs once again confirms that the terms AJV and Aωs are both small then simplified Eq. (5) was fully adequate for deducing and analyzing ωs (c) in the present case. 4.3. Discussion of observed permeabilities for different salts The results in Fig. 3 clearly show that the variation of permeability with concentration does not obey the power law with exponents predicted by Eq. (8), as expected for the SDE model, the generic framework of most published simulations of NF transport. Revealing the reasons for this failure is then important for finding the correct theoretical basis for modeling. As already noted, one must rule out the variations of the effective pore size and geometry through varying membrane swelling as a possible reason. Despite the fact that the membrane swelling is known to change at extreme pH and salinity [52–54] and pore swelling was hypothesized to be a possible cause of variations in membrane performance in some cases [55], such changes were apparently small for the salt concentrations and conditions used in this study. Neither Lp values nor direct AFM measurements could indicate any such changes in the present case. Another potential interference is the presence of H+ and OH− ions in water. For instance, H+ , even if present at vanishing concentrations may possess a high affinity to carboxylic groups making up most of fixed charge of polyamide membranes [32,39,40]. This means these ions may have large non-Donnan coefficients kH+ or kOH− and compete with the salt ions. However, binding of H+ or OH− to fixed charge groups in polyamide is usually viewed as a reversible chemical reaction, in which the fixed charge totally vanishes, rather than ion exchange, in which the fixed and mobile are not converted to new species. The fixed charge should then be constant at given pH regardless of presence and concentration of other ions. Since pH of the feed and permeate in our experiments was always in the narrow range 6.5–7, the fixed charge should not significantly change. 394 S. Bason, V. Freger / Journal of Membrane Science 360 (2010) 389–396 On the other hand, if binding of H+ or OH− could bear some features of ion exchange, it would have to obey Eq. (11) for ion mixtures and one would still see a linear increase of passage with salt concentration in the dilute range, which was not observed. Such arguments then seem unlikely to satisfactorily explain the unexpected failure of the commonly used SDE relations of Section 2. Correcting this theoretical framework poses a serious challenge that is beyond the scope of this report. However, it may be suggested that variation of the effective fixed charged density X could be a likely reason. For instance, Eq. (8) shows that, if X increases with concentration, the apparent exponents seen as slopes in Fig. 3 could become smaller than the ideal values, as observed in this work. Even though X is presumed to be determined by the chemical structure of the membrane and thus be independent of the salt used, such variation could be possible due to ion association or related effects. To see that this is possible, one can recall that ion association is governed by the Bjerrum length lB signifying the distance, at which the thermal (kT) and electrostatic energies of a counter-ion near a fixed ion become equal. It has a value 0.7 nm for monovalent ions in water and is usually small compared to the inter-ionic distances in aqueous solutions (a few nm for 0.01–0.1 M solutions) thereby ion association in water is negligible. However, for fixed charge densities 0.01–0.1 M (see Section 2.2) the Bjerrum length within the membrane, inversely proportional to the dielectric constant, is many times larger than in water and should become commensurate with inter-ionic spacing. In this situation, much of the nominal fixed charge could be neutralized through association with counter-ions. That also means the concept of a smeared uniform Donnan potential tacitly assumed in the regular SDE relations (Section 2.2) should break down, since many counter-ions will be localized and possibly immobilized in the strong local fields around fixed ions. In a sense, such binding might resemble chemical binding of H+ ions to negatively charged carboxylic groups, upon which both charges vanish (see above). This might suggest that, if an associated fixed charge could be less reactive towards H+ , some kind of competition between H+ and other cations, ruled out above but assumed in some models [48,56], could be reconsidered. In the present context, this also means that binding of a divalent cation by a single fixed charge (a charge reversal) followed by binding of a co-ion might become more favorable than “sharing” of a divalent ion between two fixed charges accompanied by exclusion of co-ions, as the standard Donnan model assumes. It is easy to see that such localized non-orthodox binding might explain the larger permeability of Ca and Mg salts at lower concentrations. Indeed, the concentrations of both co- and counter-ions in the membrane would remain finite as the salt concentration in solution drops to zero, which is equivalent to increasing K and ωs . Since the fixed charge of NF200 is negative at used pH, this is also in line with the fact that it is the type of the cation that determines such behavior. Note that the above arguments bear some resemblance with the recent proposal by Bandini and co-workers who assumed that coions might be selectively bound to some unspecified sites within the membrane [48,56]. The present arguments suggest that such an artificial assumption might not be necessary and binding of co-ions could occur as part of ion association. In order to make the above explanation work the bound divalent counter- and co-ion must retain osmotic activity, i.e., translational freedom, and be able to diffuse in a gradient of chemical potential of the salt. It may seem that ions associated with fixed charges may loose such activity. However, note that immobilization was not part of the classic Bjerrum model, which only assumed that ions forming pairs or triplets loose the “electrostatic activity”, i.e., behave like neutral entities, but do not necessarily loose mobility. As their chemical potential is actually the same as that of dissociated ions, they may diffuse in its gradient. As a last remark, it might be tempting to seek an analogy between the approach to constant K and ωs at low concentrations for monovalent ions and a similar trend of dielectric exclusion in neutral membranes (X = 0) that was demonstrated long ago [57,58]. This might suggest that the fixed charge is inactivated and vanishes at low c. While it appears plausible for monovalent ions, it cannot explain the decreasing permeability of salts with divalent cations, since without a fixed charge there would be no unusual ion binding with charge reversal. Development of the above arguments into a sound physical model is beyond the scope of this paper, however, such effects, intimately related to the membrane phase being a low dielectric medium with a large lB , will have to be accounted for in improved models in the future. 5. Conclusions Using a phenomenological approach we deduced the permeability of the NF200 membrane to several divalent single salts and compared it with the previously obtained results for monovalent salts and available models. It was shown that concentration dependence of all salts contradicts the standard SDE picture. The permeability of Na2 SO4 was much lower than that of monovalent salts obtained previously, but showed a similar trend of permeability approaching a constant value at low concentration. However, for salts of divalent cations (Ca and Mg) the discrepancy was more drastic. Their permeability decreased in entire concentration range, opposite to the standard model. As a possible mechanism we suggest a non-orthodox binding of divalent counter-ions to single fixed charges, which may be caused by the low dielectric constant of the medium and large Bjerrum length commensurate with spacing of fixed charges. The observed behavior may then be explained by fixed charge reversal and binding of a co-ion, provided bound ions may retain osmotic activity. Acknowledgments The authors are grateful to Dr. Markus Busch of Dow-Filmtec for supplying membrane samples and to Mr. Yair Kaufman for the AFM measurements in CaCl2 solutions. This work was partly supported by grants from European Community (Project FP6 MEDINA, EU contract No. 036997) and Water Authority of Israel. Appendix A. Error associated with the use of average mass transfer coefficient ′ It is to be evaluated how  much c approximately calculated for a given cF from average c ′′ under non-uniform transfer conditions deviates from c′ that would be obtained,  if mass transfer was uniform hence c′′ had a single value c ′′ = c ′′ over the entire membrane. Note that the approximate procedure also involves determination of average kd under the same assumption of uniform mass transport. Assume the osmotic pressure and pressure drop along the channel are small then the volume flux JV is uniform along the channel. Also, assume that the total permeate flow is negligible compared to the circulation rate then cF is constant. If the width of an empty channel is large compared to its thickness and the flow is laminar (Re < 1500 for a flat channel), the variation of the mass transfer coefficient kd with the distance from the channel entry y may be written as kd = ˇ−1  y −n L , (A1) where ˇ is a constant, L total channel length and n varies between 1/2 for undeveloped flow in the channel (entrance region) and 1/3 S. Bason, V. Freger / Journal of Membrane Science 360 (2010) 389–396 for developed flow (e.g., [59]). Then c′ varies along the channel as ′ ′′ c (y) = (cF − c ) exp J  V kd  ′′ + c ≈ cF exp JV ˇ  y n  L . (A2) Here the last expression ignores c′′ and thus gives an upper bound, i.e., stronger than actual, variation of upstream concentration c′ (y). One may also upper bound the concentration dependence ω(c) with the function ω(c) = ˛c m , (A3) where ␣ and m are constants; clearly m ≤ a2 (cf. Eq. (4)). It follows from Eq. (5) that 1 JV c ′′ ≈  c′ ˛ m+1 (c ′ ) . JV (m + 1) ω(c) dc ≈ c ′′ (A4) The last relation is obtained by integration from 0 instead of c′′ as the lower limit and thus gives a stronger dependence of c′′ (y) on c′ (y) than the actual one. Since the flux JV is uniform, the average observed permeate concentration is given by an integral  ′′  c = 1 L  L ′′ c (y) dy. (A5) c m+1 = ˛[cF ] JV (m + 1)  1 exp[JV ˇ(m + 1)z n ] dz, ˛[cF ] ¯ exp[JV ˇ(m + 1)]. JV (m + 1) (A7) Using a single averaged value of kd , (A7) implicitly approximates ¯ is experimentally determined by varying ˇ ¯ (actu(A6). Note that ˇ ally, ˇ) and plotting the results in coordinates such that the abscissa ¯ [36]. Expansion of r.h.s. in Eqs. (A6) and is supposed to be linear in ˇ ¯ must be related by ˇ = ˇ ¯ −1 (n + 1) in the (A7) shows that ˇ and ˇ linear region of small JV ˇ, i.e., small polarizations. Therefore the question of interest is how much (A7) deviates from (A6) beyond ¯ + 1). Dropping the identical prefthe linear region keeping ˇ = ˇ(n actor in Eqs. (A6) and (A7), one essentially needs to calculate how ¯ much F = exp [JV ˇ(m + 1)] deviates from the expression 1 ¯ + 1)(m + 1)z n ] dz = exp[JV ˇ(n  0 0 1 exp[(n + 1)z n ln F] dz. (A8) Indeed, F relates to the measured c′′ and cF the hypothetic single c′ that would be obtained for uniform mass transfer including ¯ as part of the procedure, while Eq. (A8) supdetermination of ˇ plies an analogous relation for averages. For n = 1/3 and n = 1/2 the integral (A8) can be evaluated analytically, which yields the upper bound of the relative error for a given F as follows  −1= e2A/3 − 1, (A9) ¯ Note that c ′ /cF ≈ where A = (n + 1) ln F = (n + 1)(m + 1)JV ˇ. ¯ exp [JV ˇ] is the experimentally determined average polarization. For instance, for NaCl in this study c′ /cF did not exceed 2. The hydrodynamic conditions (Re ≤ 1140, see supporting information of Ref. [36]) suggest the flow in the feed channel was an undeveloped laminar one, then n ∼ 1/2 should be used. Thus using m ≤ 0.85 ¯ and F = exp [JV ˇ(m + 1)] ≤21.85 = 3.605, one finds that the maximal error was about 10%. For Na2 SO4 m was smaller (a2 = 0.65, see Supplementary Information of this paper), but the polarization larger, c′ /cF ≤ 4, then second of (A9) yields a significant yet perhaps still acceptable error, about 30%. For Ca and Mg salts the polarization was substantial, c′ /cF ≤ 3, but m < 0, then the maximal error was small, less than 2%. For n = 1/3, i.e., for developed laminar flow, the errors would be smaller by about half and still lower for transient and turbulent regimes that generally show less variation along the channel. All errors calculated above are then overestimates. 1 exp[(n + 1)z n ln F] dz 0 F Appendix B. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.memsci.2010.05.037. (A6) m+1  e2A/3 0 where z = y/L. If, hypothetically, mass transfer were uniform, i.e., ¯ −1 and n = 0, the permeate concentration would be uniform kd = ˇ and upper bounded by c ′′ = ⎧ 1 ⎪   exp[Az 1/3 ] dz ⎪ ⎪ eA (3/A) − (6/A2 ) + (6/A3 ) − (6/A3 ) ⎪ ⎨ 0 −1= − 1, e3A/4 =  e13A/4 ⎪ ⎪   exp[Az 1/2 ] dz ⎪ ⎪ eA (2/A) − (2/A2 ) + (2/A2 ) ⎩ 0 0 To estimate the maximal error of replacing the non-uniform mass transfer with the average, Eqs. (A4) and (A2) are substituted to Eq. (A5) to obtain  ′′  395 −1 Nomenclature A = (1 − )/ω Peclét coefficient (s/m) a1 , a2 , a3 fitting parameters b constant C concentration of free salt in the membrane (M) c local corresponding concentration (M) c′ corrected feed concentration (M) c′′ permeate concentration (M) cF feed concentrations (M) D salt diffusion coefficient (m2 /s) f1 , f2 friction coefficients of counter- and co-ions with membrane and water (J s/mol m2 ) fsw salt–water friction coefficient (J s/mol m2 ) i index for counter-ions j index for co-ions JV volume flux (␮m/s MPa) K partitioning coefficient k non-Donnan partitioning coefficients kd mass transfer coefficient (␮m/s) lB Bjerrum length (m) Lp hydraulic permeability coefficients (m s−1 Pa−1 ) n1 number of counter-ions in the salt n2 number of co-ions in the salt P passage Pobs observed passage R universal gas constant (8.314 J/mol K) rp pore radius (nm) T temperature (K) v linear velocity in the feed channel (m/s) X fixed charge (M) x membrane thickness (m) 396 x̄ z S. 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