Langmuir 2008, 24, 7955-7962
7955
Influence of Steric, Electric, and Dielectric Effects on Membrane Potential Yannick Lanteri,† Anthony Szymczyk,*,‡ and Patrick Fievet† Institut UTINAM, UMR CNRS 6213, UniVersite´ de Franche-Comte´, 16 route de Gray, Besanc¸on Cedex 25030, France, and Chimie et Inge´nierie des Proce´de´s, UMR CNRS 6226, UniVersite´ de Rennes 1/ENSCR, 263 AVenue du Ge´ne´ral Leclerc, Baˆtiment 10 A, CS 74205, 35042 Rennes Cedex, France ReceiVed March 4, 2008. ReVised Manuscript ReceiVed May 5, 2008 The membrane potential arising through nanofiltration membranes separating two aqueous solutions of the same electrolyte at identical hydrostatic pressures but different concentrations is investigated within the scope of the steric, electric, and dielectric exclusion model. The influence of the ion size and the so-called dielectric exclusion on the membrane potential arising through both neutral and electrically charged membranes is investigated. Dielectric phenomena have no influence on the membrane potential through neutral membranes, unlike ion size effects which increase the membrane potential value. For charged membranes, both steric and dielectric effects increase the membrane potential at a given concentration but the diffusion potential (that is the high-concentration limit of the membrane potential) is affected only by steric effects. It is therefore proposed that membrane potential measurements carried out at high salt concentrations could be used to determine the mean pore size of nanofiltration membranes. In practical cases, the membrane volume charge density and the dielectric constant inside pores depend on the physicochemical properties of both the membrane and the surrounding solutions (pH, concentration, and chemical nature of ions). It is shown that the Donnan and dielectric exclusions affect the membrane potential of charged membranes similarly; namely, a higher salt concentration is needed to screen the membrane fixed charge. The membrane volume charge density and the pore dielectric constant cannot then be determined unambiguously by means of membrane potential experiments, and additional independent measurements are in need. It is suggested to carry out rejection rate measurements (together with membrane potential measurements).
1. Introduction Among usual characterization techniques, transversal (or transmembrane) streaming potential measurement has become, thanks to its experimental simplicity, the most commonly used tool for assessing the electrokinetic properties of porous membranes.1 Streaming potential has been successfully applied to investigate fouling of membranes,2 to check the efficiency of cleaning treatments,3 or to investigate the effect of aging on membrane electrokinetic properties.4 However, the interpretation of experimental data is quite cumbersome when measurements are carried out with ionselective membranes such as nanofiltration membranes. This is brought about by pressure-induced concentration gradients arising through pores that lead to the nonlinear variation of the electrical potential difference with the transmembrane pressure difference.5,6 In such cases, the measurement of membrane potential appears as an alternative method. Membrane potential is defined as the electrical potential difference arising through a membrane that separates two solutions of the same electrolyte at the same temperature and hydrostatic pressure but different concentrations. This technique has been applied to a variety of membrane systems, including ion-exchange membranes7–10 and porous membranes * Corresponding author. E-mail:
[email protected]. † Universite´ de Franche-Comte´. ‡ Universite´ de Rennes 1/ENSCR. (1) Szymczyk, A.; Pierre, A.; Reggiani, J. C.; Pagetti, J. J. Membr. Sci. 1997, 134, 59. (2) Nystro¨m, M.; Pihlajama¨ki, A.; Ehsani, N. J. Membr. Sci. 1994, 87, 245. (3) Pontie´, M.; Chasseray, X.; Lemordant, D.; Laine, J. M. J. Membr. Sci. 1997, 129, 125. (4) Pontie´, M. J. Membr. Sci. 1999, 154, 213. (5) Yaroshchuk, A. E.; Boiko, Y. P.; Makovetskiy, A. L. Langmuir 2002, 18, 5154. (6) Szymczyk, A.; Sbaı¨, M.; Fievet, P. Langmuir 2005, 21, 1818. (7) Compan˜, V.; Sørensen, T. S.; Rivera, S. R. J. Phys. Chem. 1995, 99, 12553.
such as microfiltration,11,12 ultrafiltration,13–15 and nanofiltration16–18 membranes. There are basically two approaches to investigate the membrane potential phenomenon. The first one is based on the TeorellMeyer-Sievers (TMS) model,19,20 and the second one on the so-called space charge (SC) model.21 Within the scope of the TMS model, Kobatake et al. derived an equation for the membrane potential arising between two solutions of a symmetric 1-1 electrolyte.22 Westermann-Clark and Anderson compared membrane potentials predicted by these two approaches in the case of 1-1 electrolytes.23 Their findings showed that the TMS model and the much more complex SC model led to similar results for sufficiently low surface charge densities. More recently, Shang et al. completed the pioneer work of Westermann-Clark and Anderson and calculated numerically membrane potentials for (8) Ersoz, M. J. Colloid Interface Sci. 2001, 243, 420. (9) Yamamoto, R.; Matsumoto, H.; Tanioka, A. J. Phys. Chem. B 2003, 107, 10506. (10) Matsumoto, H.; Yamamoto, R.; Tanioka, A. J. Phys. Chem. B 2005, 109, 14130. (11) Takagi, R.; Nakagaki, M. J. Membr. Sci. 1996, 111, 19. (12) Takagi, R.; Nakagaki, M. Sep. Purif. Technol. 2001, 25, 369. (13) Compan˜, V.; Lo´pez, M. L.; Sørensen, T. S.; Garrido, J. J. Phys. Chem. 1994, 98, 9021. (14) Szymczyk, A.; Fievet, P.; Reggiani, J. C.; Pagetti, J. J. Membr. Sci. 1998, 146, 277. (15) Pontie´, M.; Cowache, P.; Klein, L. H.; Maurice, V.; Bedioui, F. J. Membr. Sci. 2001, 184, 165. (16) Can˜as, A.; Benavente, J. J. Colloid Interface Sci. 2002, 246, 328. (17) Yaroshchuk, A. E.; Makovetskiy, A. L.; Boiko, Y. P.; Galinker, E. W. J. Membr. Sci. 2000, 172, 203. (18) Schaep, J.; Vandecasteele, C. J. Membr. Sci. 2001, 188, 129. (19) Theorell, T. Proc. Soc. Exp. Biol. 1935, 33, 282. (20) Meyer, K. H.; Sievers, J.-F. HelV. Chim. Acta 1936, 19, 649; 665; 987. (21) Gross, R. J.; Osterle, J. F. J. Chem. Phys. 1968, 49, 228. (22) Kobatake, Y.; Takeguchi, N.; Toyoshima, Y.; Fujita, H. J. Phys. Chem. 1965, 69, 3981. (23) Westermann-Clark, G. B.; Anderson, J. L. J. Electrochem. Soc. 1983, 130, 839.
10.1021/la800677q CCC: $40.75 2008 American Chemical Society Published on Web 07/11/2008
7956 Langmuir, Vol. 24, No. 15, 2008
1-1, 2-1, 1-2, and 2-2 electrolytes using both TMS and SC models.24 They showed that the two models coincide with each other for slightly charged membranes with pore radii less than 5.0 nm. Otherwise, the TMS model (which neglects radial variations of both electrical potential and ion concentrations) was shown to overestimate the membrane potential. Matsumoto et al. recently extended the TMS theory and proposed a model for membrane potential theory that considers solvent permeation through the membrane.25 Their findings suggest that the effective charge density of reverse osmosis membranes may be pressure dependent. The common feature of the above-mentioned theoretical studies is that ions were treated as point charges. Only a few authors have considered ion size effects in the membrane potential analysis. Within the scope of the space charge model, Cervera et al. included ion size effects in the Poisson-Boltzmann equation and in the transport equations as well.26 They concluded that ion size effects lead to an increase in the membrane potential. Tanioka and co-workers introduced ion size effects in the TMS approach through a Fuoss formalism of ion pairing and analyzed membrane potentials arising through membranes with various water contents.9,10,27,28 Steric hindrance is known to play a role in the separation of solutes by nanofiltration membranes and is taken into account in modern transport models.29,30 With the aim of investigating transport properties of nanofiltration membranes, we recently proposed the steric, electric, and dielectric exclusion (SEDE) model, which is an improved version of the TMS model that describes the exclusion mechanism of charged solutes as the result of ion size effects, the Donnan exclusion, and dielectric effects.31 In the present work, we extend the application of the SEDE model in estimating membrane potential for an asymmetric 2-1 electrolyte. The influence of steric hindrance and dielectric exclusion (which is treated in terms of Born effect and image forces) on membrane potential arising through both neutral and charged membranes is investigated.
2. Theoretical Background When a membrane separates two solutions of the same electrolyte at the same temperature and hydrostatic pressure but different concentrations, the membrane potential (sometimes called exclusion-diffusion potential) is defined as the difference between the electrical potential of the most concentrated solution and that of the most diluted solution. In this work, we focus on the case of asymmetric electrolytes. For all calculations, we set the concentration ratio at 2 and the diffusion coefficients of cations and anions at 0.792 × 10-9 and 2.031× 10-9 m/s2, respectively (which corresponds to ion diffusion coefficients in diluted CaCl2 solutions). It is worth mentioning that the qualitative conclusions drawn below are still valid for other kinds of electrolytes (i.e., 1-1, 2-2, 1-2... electrolytes). For the sake of convenience, all solutions are assumed to be ideal. Within the scope of the TMS model (and related models such as the SEDE model), the membrane potential (∆Ψm) can be split (24) Shang, W. J.; Wang, X. L.; Yu, Y. X. J. Membr. Sci. 2006, 285, 362. (25) Matsumoto, H.; Konosu, Y.; Kimura, N.; Minagawa, M.; Tanioka, A. J. Colloid Interface Sci. 2007, 309, 272. (26) Cervera, J.; Garcia-Morales, V.; Pellicier, J. J. Phys. Chem. B 2003, 107, 8300. (27) Chou, T. J.; Tanioka, A. J. Phys. Chem. B 1998, 102, 129. (28) Tanioka, A.; Matsumoto, H.; Yamamoto, R. Sci. Technol. AdV. Mater. 2004, 5, 461. (29) Bowen, W. R.; Mohammad, A. W.; Hilal, N. J. Membr. Sci. 1997, 126, 91. (30) Wang, X. L.; Tsuru, T.; Nakao, S. I.; Kimura, S. J. Membr. Sci. 1997, 135, 19. (31) Szymczyk, A.; Fievet, P. J. Membr. Sci. 2005, 252, 77.
Lanteri et al.
Figure 1. Schematic representation of the membrane potential (∆Ψm) arising through the membrane. ∆ΨD: Donnan potential. ∆Ψdiff: diffusion potential.
in two components: namely, the difference between the Donnan potentials at each interface (∆ΨD∆xand ∆ΨD0: superscripts ∆x and 0 refer to the interface between the membrane and the most diluted solution and the interface between the membrane and the most concentrated solution, respectively) and the diffusion potential (∆Ψdiff) arising through the membrane pores32
∆Ψm ) ∆ΨD∆x - ∆ΨD0 + ∆Ψdiff
(1)
A schematic representation of the membrane potential arising through the membrane is shown in Figure 1. 2.1. Donnan Potential. The partitioning coefficient of an ion i (Γint i ) is defined as the ratio of its concentration just inside the membrane pores (cjint i ) to its concentration just outside the membrane (cint i ). It is worth mentioning that, throughout this article, the superscript “int” stands for 0 or ∆x, depending on the interface that is considered, and that the bar refers to a magnitude inside the membrane. Within the scope of the SEDE model, the partitioning coefficient accounts for steric hindrance, the Donnan exclusion, and dielectric exclusion31
Γiint )
jciint
(
) κiint exp int
ci
ziF∆ΨDint RT
)
(2)
where zi is the charge number of ion i, F is the Faraday constant, R is the ideal gas constant, T is the absolute temperature and, and κiint is defined as ’int ’int κiint ) φiint exp(-∆Wi,Born ) exp(-∆Wi,image )
(3)
where φint i is the steric partitioning coefficient for ion i (which is defined as the ratio of the available section for ion i to the pore ′int ′int cross section and ∆Wi,Born and ∆Wi,image denote the increase in the interaction energy due to Born dielectric effect and image forces, respectively (the prime symbol indicates that both terms are scaled on kT). For slitlike pores, the steric coefficient can be expressed as follows33
φiint ) 1 - λi
(4)
with λi ) ri/rp, where ri is the Stokes radius of ion and rp is the pore half-width. The work of charge transfer from external solutions characterized by a dielectric constant εb to a medium with the dielectric constant εp is given by (32) Asaka, K. J. Membr. Sci. 1990, 52, 57. (33) Dechadilok, P.; Deen, W. M. Ind. Eng. Chem. Res. 2006, 45, 6953.
Influence of Steric, Electric, and Dielectric Effects ’int ∆Wi,Born )
(
(zie)2 1 1 8πε0kBTri,cav εp εb
Langmuir, Vol. 24, No. 15, 2008 7957
)
NPE reads as follows
where ε0 is the vacuum permittivity, e is the elementary charge, kB is the Boltzmann constant, and ri,cav is the radius of the cavity formed by the ion i in the solvent.34 The dielectric exclusion resulting from image forces was reviewed recently by Yaroshchuk, who derived approximate expressions for the interaction energy due to image forces.35 These equations account for the screening of the interaction between the ion and the polarized interface by the membrane fixed charge and the ionic atmosphere as well. Within the scope of the SEDE model applied to slitlike pores, these equations read as follows31 ’int ∆Wi,image ) -Ri ln[1 -
(
)
εp - εm exp(-2µint)] εp + εm
µint )
κintrp
with
∑
(
( ) ε0εpRT
)
(8)
There is no net electrical current flowing through the membrane pores at steady state. Therefore, we can write
∑ FziJi ) 0
(14)
i
∑ jcizi ) -X
(15)
where X is the membrane volume charge density. According to the TMS theory, the membrane fixed charge is assumed to be homogeneously smeared throughout the pore volume and X is considered to be independent of the ion local concentrations (at given feed concentrations). Thus, from eq 15 we get
dcj
∑ dxi zi ) 0
(9)
2F2Iint
1 2
Ki ) 1 - 1.004λi + 0.418λi3 + 0.210λi4 - 0.169λi5 (13)
-1⁄2
(16)
i
where NA is Avogadro’s number and Iint is the ionic strength defined as
Iint )
(12)
i
2Iint
κint )
)
where Ji is the molar flux of ion i, Di is its diffusion coefficient at infinite dilution, and Ki is the hindrance diffusion coefficient accounting for the effect of finite ion and pore sizes.38 For a slitlike geometry, the hindrance diffusion factor Ki takes the approximate form38
(7)
zie ’int ’int ∆ΨDint - ∆Wi,Born - ∆Wi,image kBT
i
j dcji F dψ + zijci dx RT dx
The electroneutrality condition within the membrane pores reads as follows
(ziF)2 8πε0εpRTNArp
int z2i cint i φi exp -
Ji ) -KiDi
(6)
where εm is the dielectric constant of the membrane, and
Ri )
(
(5)
∑ ciintzi2
(10)
Substituting eq 12 for both cations and anions in eq 14 and considering eq 16, we derived the following expression of the diffusion potential (∆Ψdiff) through the membrane pores after integration between pore ends
i
Rearranging eq 2 yields the following expression for the Donnan potential
( )
j int - ψint ) RT ln ∆ΨDint ) ψ ziF
κiintciint jciint
∆ψdiff ) -
(34) Rashin, A. A.; Honig, B. J. Phys. Chem. 1985, 89, 5588. (35) Yaroshchuk, A. E. AdV. Colloid Interface Sci. 2000, 85, 193. (36) Schlo¨gl, R. Stofftransport durch membranen; D. Steinkopff: Darmstadt, 1964. (37) Dresner, L. J. Phys. Chem. 1972, 76, 2256.
(
ln
(11)
2.2. Diffusion Potential. The extended Nernst-Planck equation (NPE) forms the basis for the description of ion fluxes through nanofiltration membranes.36,37 It describes ion transport in terms of diffusion under the action of concentration gradients, electromigration under the action of spontaneously arising electric field and convection if a volume flux is involved. In the present work, both electrolyte solutions are at the same hydrostatic pressure and a concentration ratio of 2 is set. The volume flux resulting from the osmotic pressure gradient is assumed to be negligible, and therefore, the contribution of convection is neglected in NPE equations throughout this work. It should be stressed that the effect of the osmotic pressure on the volume flux depends on the transmembrane concentration difference, and then this assumption may be not really accurate at high concentrations even if the concentration ratio is only 2. Since pore length in nanofiltration membranes is much larger that their diameter, ion fluxes can be considered one-dimensional and the
(
RT K+D+ - K-DF z+K+D+ - z-K-D-
)
z2+K+D+jc0+ + z2-K-D-jc02 z2+K+D+jc∆x c∆x + + z-K-D-j -
)
(17)
where subscripts + and - denote cations and anions, respectively. 2.3. Membrane Potential. Introducing eqs 11 and 17 into eq 1 yields the expression of the membrane potential in the formalism of the SEDE model. As mentioned previously, we focus on a 2-1 asymmetric electrolyte and we set the concentration ratio at 2 (i.e., c0i ) 2c∆x i ) in our calculations. The expression of the membrane potential reads then as follows
∆Ψm )
( ) (
κi∆xjci0 RT RT K+D+ - K-Dln ∆x 0 ziF F 2K+D+ + K-D2cji κi
(
ln
)
)
4K+D+jc0+ + K-D-jc0(18) 4K D jc∆x + K D jc∆x +
+ +
-
- -
The determination of the membrane potential (as well as the Donnan and the diffusion potentials) therefore requires knowledge of ion concentrations at the pore ends (or equivalently, the partitioning coefficients at both membrane/solution interfaces). This can be done numerically from eqs 2–10.31,39 Nevertheless, (38) Deen, W. M. AIChE J. 1987, 33, 1409. (39) Szymczyk, A.; Sbaı¨, M.; Fievet, P.; Vidonne, A. Langmuir 2006, 22, 3910.
7958 Langmuir, Vol. 24, No. 15, 2008
Lanteri et al.
it is helpful for the analysis of the results presented below to derive analytical expressions of c jint i . Since z+ ) 2 and z- ) -1, we can write from eq 11
∆ΨDint )
( ) ( )
int κint + c+ RT ln 2F jcint +
∆ΨDint ) -
(19a)
int κint - cRT ln F jcint -
(19b)
Identifying the above equations and considering the electroint int neutrality condition outside the membrane (i.e., 2c+ ) c), we can write
jcint + )
int 3 int 2 4κint + (c+ ) (κ- )
(20)
2 (jcint -)
Substituting eq 20 into eq 15 (electroneutrality condition inside pores) leads to the following cubic equation
Figure 2. Membrane potential (s), diffusion potential (-•-), and difference between Donnan potentials (- - -) vs the concentration of 0 the most concentrated compartment (csalt ); X ) 0 mol/m3; rp ) (0.5 nm, 1 nm, 5 nm, no steric effects, i.e., λi ) 0); no dielectric effects. Note: the straight lines and the straight lines with symbols merge; the dashed line and the x-axis merge. int c2,)
{ { 3
int 3 int 2 int 3 2 cint (jcint - ) - X(j - ) - 8(c+ ) (κ- ) κ+ ) 0
(21)
ω
According to Cardano’s formula for normal cubic equation ax3 + bx2 + cx + d ) 0, the three roots (x1, x2, and x3) are given by40
ω2
3
x1 )
3
x2 ) ω
3
x3 ) ω2
-q + √∆ + 2
3
-q - √∆ +h 2
-q + √∆ + ω2 2
3
-q + √∆ +ω 2
3
(22a) (22b)
-q - √∆ +h 2
(22c)
ah3 + bh2 + ch + d b 1 √3 , h ) - , ω ) - + i, and ∆ ) a 3a 2 2 3 4p 3ah2 + 2bh + c with p ) q2 + 27 a
int 3 int 2 int In our case, a ) 1, b ) -X, c ) 0, and d ) -8(c+ ) (κ- ) κ+ . Thus,
q)-
2X3 X X2 int 3 int 2 - 8κint and ∆ ) + (c+ ) (κ- ) , h ) , p ) 27 3 3 2 2X3 4X6 int 3 int 2 κ c + 8κint ( ) ( ) + + 27 729
[
]
int int int The three roots c j1,, c j2,and c j3,of eq 21 read then as follows int ) c1,-
3
3
X3 int 3 int int 2 + 4(c+ ) κ+ (κ- ) + 27 X3 int 3 int int 2 + 4(c+ ) κ+ (κ- ) 27
[ [
] )]
2 X6 X3 int 3 int int 2 κ+ (κ- ) + 4(c+ + ) 27 729
X3 int 3 int int + 4(c+ ) κ+ (κ27
(40) Xing, X. F. J. Cent. UniV. Nation. 2003, 12, 207.
2
2
-
X6 X + 729 3 (23a)
X3 int 3 int int 2 + 4(c+ ) κ+ (κ- ) 27
[ [
] )]
{ { 3
3
X3 int 3 int int 2 + 4(c+ ) κ+ (κ- ) + 27
X3 int 3 int int 2 + 4(c+ ) κ+ (κ- ) 27
} }
2 X3 X6 int 3 int int 2 + 4(c+ + ) κ+ (κ- ) - 729 27
X3 int 3 int int + 4(c+ ) κ+ (κ27
2
2
-
int c3,)
ω
where
q)
3
ω2
-q - √∆ +h 2
X3 int 3 int int 2 + 4(c+ ) κ+ (κ- ) + 27
[ [
] )]
X6 + 729 X (23b) 3
} }
2 X3 X6 int 3 int int 2 + + 4(c+ ) κ+ (κ- ) - 729 27
X3 int 3 int int + 4(c+ ) κ+ (κ27
2
2
-
X6 + 729 X (23c) 3
Three cases can be considered: int int int ∆ > 0: jc1,is a real root and both jc2,and jc3,are imaginary int roots. In this case, only the value of jc1,is considered. int int int int int ∆ ) 0: jc1,, jc2,, jc3,are all real roots, and jc2,) jc3,. int int int int ∆ < 0: jc1,-, jc2,- and jc3,- are all real roots, but only jc1,is positive.
3. Results and Discussion In this section, we investigate the influence of steric hindrance and dielectric exclusion on the membrane potential arising through both neutral and electrically charged membranes. We systematically split the membrane potential in the Donnan contribution (i.e., ∆ΨD∆x - ∆ΨD0) and diffusion potential (eq 17), and results are discussed in terms of impact on these two components. 3.1. Neutral Membranes. Figure 2 shows the variation of the membrane potential and its components (i.e., the difference in Donnan potentials and the diffusion potential) with the concentration of the concentrated compartment for various pore sizes (rp). The only exclusion mechanism that is considered here int is the ion size effect (i.e., κint i ) φi ). The Donnan contribution is null whatever the concentration and the pore size. It means that Donnan potentials are equal at both interfaces although each of them differs from zero. Indeed, since cations and anions have different sizes their steric partitioning coefficients are different. The smallest ions (i.e., the chloride ions on the basis of Stokes radii) are less excluded than the largest ones (i.e., the calcium ions). Consequently, a slight negative Donnan potential arises, pulling calcium ions into pores and expelling chloride ions to
Influence of Steric, Electric, and Dielectric Effects
Langmuir, Vol. 24, No. 15, 2008 7959
Figure 3. Membrane potential (s), diffusion potential (-•-), and difference between Donnan potentials (- - -) vs the concentration of 0 the most concentrated compartment (csalt ); X ) 0 mol/m3; no steric effects (i.e., λi ) 0); no image charges (εm ) εp). The Born effect is considered (i.e., εp * εb), and results are found to be independent of the value of εp. Note: the straight lines and the straight lines with symbols merge; the dashed line and the x-axis merge.
Figure 4. Membrane potential (s), diffusion potential (-•-), and difference between Donnan potentials (- - -) vs the concentration of 0 the most concentrated compartment (csalt ); X ) 0 mol/m3; rp ) 0.5 nm; εm ) (3, 6, 9, 12); no Born effect.
maintain electroneutrality within the membrane. According to eq 11, the difference in Donnan potentials is given by (considering c0i ) 2c∆x i )
∆ΨD∆x - ∆ΨD0 )
( )
κi∆xjci0 RT ln 0 ∆x ziF 2κi jci
(24)
Since only steric effects are considered here, we have κ0i ) Now considering the electroneutrality condition within the membrane (i.e., eq 15), it comes from eq 24 that the Donnan potentials are identical at both interfaces, and thus the Donnan contribution to the membrane potential arising through a neutral membrane is null whatever the concentration and the pore size. Stated otherwise, the membrane potential is equal to the diffusion potential given by eq 17: κ∆x i .
∆Ψm ) -
(
)
RT K+D+ - K-Dln 2 F 2K+D+ + K-D-
(25)
The membrane potential is affected by ion size effects as can be seen in Figure 2. When ions are treated as point charges, the membrane potential is equal to the free diffusion potential (i.e., around 6 mV for CaCl2 solutions with c0i ) 2c∆x i ). On the other hand, the membrane potential is higher than the free diffusion potential as ion size effects are included. These findings are in accordance with the results obtained by Cervera et al. by means of a more sophisticated SC model.26 Since the Donnan contribution is null, the effect of ion size on the membrane potential arises in the Nernst-Planck transport equations through hindered diffusion coefficients K+ and K- (i.e., because of the increased friction between the diffusing solutes and the pore walls). The same is true if Born dielectric effects are included into the overall exclusion mechanism, as can be seen in Figure 3. For the sake of clarity, steric effects were disregarded in Figure 3, that is, κint i ′int ) exp(-∆Wi,Born ). The membrane potential is independent of the dielectric constant inside pores and is equal to the free diffusion potential provided no ion size effects are included (i.e., ions are treated like point charges). Within the scope of the SEDE model, these results are justified because the decrease in the dielectric constant inside pores (i.e., the Born effect) only occurs at the membrane/solution interfaces while the dielectric constant is assumed to be independent of the axial position inside pores. Consequently, κ0i ) κ∆x i and the membrane potential is given by eq 25 just like for neutral membranes with ion size effects only.
Figure 5. Difference between the normalized interaction energies due ′,0 to image forces at both membrane/solution interfaces (∆Wi,image ∆Wi,image′,∆x) vs the concentration of the most concentrated compartment 0 (csalt ); X ) 0 mol/m3; rp ) 0.5 nm; εm ) (3, 6, 9, 12); no Born effect.
The importance of image charges in ion transport through commercial nanofiltration membranes has been recently highlighted by Szymczyk and co-workers.31,39 Now let us consider the effect of image charges on the membrane potential arising through neutral membranes (Figure 4). In this case, the partitioning coefficient at each interface is given by int
’ κiint ) φiint exp(-∆Wi,image )
(26)
The membrane potential appears to be independent of the membrane dielectric constant (εm) although its components are affected by εm. Stated otherwise, the Donnan contribution and the diffusion potential balance each other. Both components exhibit a nonmonotonous behavior with concentration. At low concentrations, the membrane potential tends to the diffusion potential and the Donnan contribution is close to zero. This is because there is almost no screening of the image force interaction at very low concentration.31 As a result, the image force interaction is virtually the same at both interfaces. This is clearly shown in Figure 5, which illustrates the variation of the image charge contribution (i.e., ∆W’0i,image - ∆Wi,image′∆x) versus the concentration of the concentrated compartment. At high salt concentrations, ’0 the image force interaction is totally screened and then ∆Wi,image 31,35 ′∆x ≈ ∆Wi,image ≈ 0. Consequently, the difference in image force interaction is close to zero. According to eq 26, it implies
7960 Langmuir, Vol. 24, No. 15, 2008
κi0 κi∆x
)
φi0 φi∆x
’∆x ’0 ∆Wi,image - ∆Wi,image
exp(
Lanteri et al.
)≈
φi0 φi∆x
(27)
This case is similar to that of a neutral pore with only steric hindrance effects, that is, the Donnan contribution tends to zero and the membrane potential is then equal to the diffusion potential. At intermediate salt concentrations, the image force interaction is strongly dependent upon the concentration and it significantly decreases as the salt concentration increases. Consequently, the difference between the ion concentrations at pore ends increases, which leads to an increase in the diffusion potential (Figure 4) until the screening becomes strong enough to reduce the difference between the concentrations at pore ends (at this time the diffusion potential decreases as shown in Figure 4). 3.2. Charged Membranes. Now let us focus on membranes carrying an electrical fixed charge. Most membranes develop an electrical charge when brought into contact with a polar medium such as an aqueous solution due to the dissociation of ionizable surface groups and/or the adsorption of charged species onto the pore walls. We consider here the case of negatively charged membranes which are frequently encountered in nanofiltration applications. Figure 6 shows the variation of the membrane potential and its components with the concentration of the concentrated compartment for a negatively charged membrane (X ) -1 mol/m3) for various pore half-widths (rp). The membrane potential of charged membranes is strongly dependent on the concentration, unlike neutral membranes. At low salt concentrations (i.e., for high normalized volume charge densities39), the co-ions (i.e., chloride ions in our case) are almost completely excluded from the membrane pores. Indeed, if we consider eq int 21 in the limiting case of c+ ) 0, it appears that ∆ ) 0 and the physically relevant real root given by eqs 23a, 23b, and 23c is int int jc2,) jc3,) 0. Since the membrane volume charge density is assumed to be independent of the axial position inside pores, the concentration of (divalent) counterions inside pores is then constant too. Considering the electroneutrality condition within the membrane pores (i.e., eq 15), we can write
X jc0+ ) jc∆x + )2
∆Ψm ) ∆ΨD∆x - ∆ΨD0 )
(28)
As a result, the diffusion potential tends to zero. The membrane potential is then equal to the Donnan contribution, which is given by eq 24
(29)
As shown in the previous section, κ0i ) κ∆x i when only steric effects are considered (apart from the Donnan exclusion). Equation 29 then reduces to
∆Ψm ) -
RT ln(2) z+F
(30)
which represents the Nernst potential (the value of which is inversely proportional to the charge number of counterions) that corresponds to the asymptotic value of the membrane potential when salt concentration tends to zero. It is worth mentioning that the asymptotic value at low salt concentration is not affected by ion-size effects, unlike the limiting value reached at high concentration. Indeed, the Donnan contribution tends to zero at high concentration due to the screening of the membrane fixed charge density. This can be int shown by considering eq 21 and the limiting case of c+ f +∞. In this case, ∆ > 0 and the co-ion concentration just inside the membrane is given by eq 23a, which becomes int int int 2 jcint - ) 2c+ √κ+ (κ- ) 3
(31)
Substituting eq 31 into eq 24 leads to the following expression of the Donnan contribution:
[( ) ]
0 κ∆x - c+ RT ∆x 0 ∆ΨD - ∆ΨD ) ln z-F 2κ0-c∆x +
3
κ0+(κ0-)2
(32)
∆x 2 κ∆x + (κ- )
Considering that κ0i ) κ∆x i since only steric effects are considered 0 ∆x (apart from the Donnan exclusion) and the condition c+ ) 2c+ , eq 32 shows that the difference in the Donnan potentials is equal int to zero as c+ f +∞. As a result, the asymptotic value of the membrane potential is equal to the diffusion potential. Substituting the electroneutrality condition inside pores (i.e., eq 15) into eq 17 yields
∆Ψm ) ∆Ψdiff ) -
RT K+D+ - K-DF 2K+D+ + K-D-
(
ln
2K+D+(cj0- - X) + K-D-jc02K+D+(cj∆x c∆x - - X) + K-D-j -
)
(33)
int Considering that c+ f +∞ together with eq 31, eq 33 can be rewritten as
∆Ψm ≈
RT K+D+ - K-DF 2K+D+ + K-D-
(
ln
Figure 6. Membrane potential (s), diffusion potential (-•-), and difference between Donnan potentials (- - -) vs the concentration of 0 the most concentrated compartment (csalt ); X ) -1 mol/m3; rp ) (0.5 nm, 1 nm, 5 nm, no steric effects); no dielectric effects.
( )
κ∆x + RT ln 0 z+F 2κ+
4K+D+c0+√κ0+(κ0-)2 + 2K-D-c0+√κ0+(κ0-)2 3
3
d d 2 ∆x d d 2 4K+D+c∆x + √κ+(κ-) + 2K-D-c+ √κ+(κ-) 3
3
)
(34)
0 ∆x Considering once again that c0i ) 2c∆x i and κi ) κi , eq 34 becomes identical to eq 25, and the membrane potential therefore depends on the pore size through the hindered diffusion coefficients K+ and K-. It is worthwhile mentioning that the usual procedure allowing the assessment of the pore size of nanofiltration membranes consists in measuring the rejection rate of neutral solutes at various permeate volume fluxes.29,41,42 The rejection rate of an uncharged solute by a porous membrane depends on two structural features of the membrane, which are the pore size and the thickness to porosity ratio. It can be easily demonstrated that (i) the rejection
Influence of Steric, Electric, and Dielectric Effects
Figure 7. Membrane potential (s), diffusion potential (-•-), and difference between Donnan potentials (- - -) vs the concentration of 0 the most concentrated compartment (csalt ); X ) -3, -2, and -1 mol/m3; no steric effects, no dielectric effects.
Langmuir, Vol. 24, No. 15, 2008 7961
Figure 8. Membrane potential (s), diffusion potential (-•-), and difference between Donnan potentials (- - -) vs concentration of the 0 most concentrated compartment (csalt ); X ) -1 mol/m3; εp ) (48, 58, 68, 78); no steric effects; no image charges.
rate tends to a limiting value at sufficiently high permeate volume fluxes and (ii) this limiting rejection rate depends on a single parameter, namely the pore size. Measurement of the membrane potential at sufficiently high electrolyte concentration could then be an alternative method to assess the pore size of nanofiltration membranes (by means of eq 25). Figure 7 shows the variation of the membrane potential and its components with the concentration of the concentrated compartment for various negative membrane volume charge densities (X). As previously demonstrated, the membrane potential is bound between limiting values corresponding to the Nernst potential and the diffusion potential at low and high concentration, respectively. Ion-size effects have been disregarded in Figure 7. That is why the asymptotic value of the membrane potential at high salt concentrations (i.e., the diffusion potential) tends to the free diffusion potential. As expected, an increase in X (in absolute value) shifts the potential concentration curve toward higher salt concentrations since more counterions are required to screen the fixed charge of a strongly charged membrane. As can be seen in Figure 7, a salt concentration around 102 mol/m3 is high enough to screen charge effects almost completely. Figure 8 shows the membrane potential and its components versus the concentration of the concentrated compartment for various dielectric constants inside pores (i.e., for various magnitudes of the Born effect). For the sake of clarity, the ionsize effect was disregarded in Figure 8, that is, κiint ) ′int exp(-∆Wi,Born ). It can be noted that curves displayed in both Figures 7 and 8 have similar trends. It means that decreasing the pore dielectric constant (εp) or increasing the membrane fixed charge (X) produces the same qualitative effect on the membrane potential and its components. It is worthwhile emphasizing that the results shown in Figure 8 assume that εp is independent of the salt concentration (i.e., the Born effect is considered to be constant whatever the concentration). Indeed, the dependence of the Born effect on concentration does not appear explicitly in the SEDE model (eq 5) since the relationship between εp and the ion concentrations is unknown. According to this assumption, κint i is independent of the membrane/solution interface and the difference in the Donnan potentials (i.e., the Donnan contribution to the membrane potential) tends to zero at sufficiently high electrolyte concentra-
tion, whatever the magnitude of the Born effect. As discussed in the previous section, the Born effect does not affect the diffusion potential, and thus, the high-concentration asymptotic value of the membrane potential corresponds to the free diffusion potential whatever the pore dielectric constant (because no steric effect is taken into account in Figure 8). Nevertheless, it can be noted that a higher salt concentration is needed to screen the membrane fixed charge when the Born effect is taken into account (Figures 7 and 8). Indeed, the decrease in εp leads to a decrease in the ion concentrations inside pores, and thus, a higher external salt concentration is required to screen the membrane fixed charge (otherwise stated, the Born effect strengthens the Donnan exclusion). Membrane potential measurements are sometimes used to determine the membrane volume charge density of nanofiltration membranes.43,44 However, a direct consequence of what was shown above is that there are different couples of values (εp, X) that lead to the same membrane potential value (a set of (εp, X) values leading to identical membrane potential values is plotted in Figure 9 for illustration purpose). The single membrane potential technique therefore does not allow assessing the volume charge density of a membrane, and further independent measurements are needed. For example, rejection rate measurements together with membrane potential experiments should allow
(41) Schaep, J.; Vandecasteele, C.; Mohammad, A. W.; Bowen, W. R. Sep. Sci. Technol. 1999, 34, 3009. (42) Labbez, C.; Fievet, P.; Szymczyk, A.; Vidonne, A.; Foissy, A.; Pagetti, J. J. Membr. Sci. 2002, 208, 315.
(43) Schaep, J. Nanofiltration for the RemoVal of Ionic Components from Water. Thesis, Katholieke Universiteit Leuven, Heverlee, Belgium, 1999. (44) Can˜as, A.; Ariza, M. J.; Benavente, J. J. Colloid Interface Sci. 2002, 246, 150.
Figure 9. Couples of values (εp, X) leading to identical membrane potential values; no steric effects; no image charges.
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Lanteri et al.
exclusion from the membrane pores. As a result, the diffusion potential is negligible on a wider range of concentration (Figure 10; note that the same argument holds as X increases or εp decreases as shown in Figures 7 and 8, respectively). At high electrolyte concentration, both image charge and Donnan effects are totally screened and the membrane potential reaches the diffusion potential (this latter differs from the free diffusion potential since ion size was included in calculations shown in Figure 10).
4. Conclusions
Figure 10. Membrane potential (s), diffusion potential (-•-), and difference between Donnan potentials (- - -) vs concentration of the 0 most concentrated compartment (csalt ); rp ) 0.5 nm; X ) -1 mol/m3; εm ) (3, 6, 9, 12); no Born effect.
determining both X and εp unambiguously since the rejection rate of nanofiltration membranes may be significantly affected by charge and dielectric effects.31,39 Figure 10 shows the variation of the membrane potential (and its components) of a charged membrane (X ) -1 mol/m3) with the concentration of the concentrated compartment when both steric effects and image charges occur. The magnitude of the interaction via image charges increases as the membrane dielectric constant (εm) decreases. As expected for charged membranes, the membrane potential is bound between the Nernst potential (at low concentrations) and the diffusion potential (at high concentrations). As shown in Figure 5, there is almost no screening of the image force interaction at low concentrations, and thus, κ0i ) κ∆x i . According to eq 29, the membrane potential therefore tends to the Nernst limit given by eq 30. Comparing Figures 7, 8, and 10, it clearly appears that the dielectric exclusion phenomenon via image charges has the same qualitative effect as the Born effect and the Donnan-type exclusion (i.e., the membrane potential curves are shifted toward higher concentrations). Otherwise stated, the membrane potential is closer to the Nernst limit (at a fixed concentration) when image charge interaction is taken into account since it results in a higher ion
In the present work, the SEDE transport model that describes the exclusion mechanism of charged solutes by nanofiltration membranes as the result of ion size effects, the Donnan exclusion, and dielectric effects has been extended to investigate the membrane potential arising through nanofiltration membranes separating two solutions of an asymmetric 2-1 electrolyte at different concentrations. Steric effects influence the membrane potential of neutral membranes, unlike dielectric phenomena (i.e., Born and image charge effects). The membrane potential of charged membranes is always bound between the Nernst potential (at low concentrations) and the diffusion potential (at high concentrations). Both ion size and dielectric effects increase the membrane potential value, but the limiting value at high concentration (i.e., the diffusion potential) is affected only by steric effects. Consequently, measuring the membrane potential at high salt concentrations could be an alternative way to determine the mean pore size of nanofiltration membranes. The Donnan exclusion, the Born effect, and the image charges have the same qualitative effect on the membrane potential. Both the membrane volume charge density and the dielectric constant inside pores are known to be dependent upon the physicochemical environment of the membrane, whereas the membrane dielectric constant can be reasonably assumed to be an intrinsic property of the membrane material. As a result, neither the membrane volume charge density nor the dielectric constant inside pores can be deduced from single membrane potential measurements and the unambiguous determination of these magnitudes requires performing additional experiments such as rejection rate measurements. LA800677Q
