Nanofiltration Theory: Good Co-Ion Exclusion Approximation for

J. Phys. Chem. B 2005, 109, 5525-5540

5525

Nanofiltration Theory: Good Co-Ion Exclusion Approximation for Single Salts Xavier Lefebvre and John Palmeri* Institut Europe´ en des Membranes (ENSCM CNRS UMR 5635), UniVersite´ Montpellier II, CC047, Place Euge` ne Bataillon, 34095 Montpellier Cedex 5, France ReceiVed: September 13, 2004; In Final Form: January 20, 2005

We applied an approximate analytic method, the good co-ion exclusion (GCE) approximation, to the hindered electrotransport theory describing salt and solution transport across charged nanofiltration membranes. This approximation, which should be valid at sufficiently low feed electrolyte concentration, leads to a considerable simplification of the exact parametrized equations obtained previously for single salt nanofiltration parameters (salt rejection, electric filtration potential, and volume flux density) and therefore provides further insight into ion transfer in nanoporous membranes. We also established the domain of validity of the GCE approximation as a function of the salt type for 1:1, 2:1, 1:2, and 2:2 salts. Our results for the volume flux density, obtained within an extended GCE approximation, confirm that the global osmotic reflection coefficient in the solution flux equation is not equal to the limiting salt rejection.

1. Introduction The present work is devoted to further elucidating the mechanisms at play in the nanofiltration (NF) of electrolyte mixtures. Aqueous nanofiltration is a pressure-driven membrane process that involves using charged nanoporous media to separate multivalent inorganic ions and low molecular weight organic solutes ( +30), while it is valid starting at ξ ≈ -2.5 in the negative one. The results presented in Figure 2 make it possible to assess further the accuracy of the GCE approximation to the HET model (for the ET model see ref 9). Plotting the limiting rejection rate as a function of the normalized membrane charge density clearly shows the domain of validity of GCE as a function of the salt type. We note in Figure 2a that the NaCl curve is nearly symmetric about ξ ) 0. This particularity comes not only from the equality of the absolute values of ion valence

for the chloride and sodium ions but also from the similarity of h Cl- ≈ 1.22). their intramembrane diffusion coefficients (D h Na+/D On the contrary, due to the difference between the diffusion coefficients for CaSO4, Figure 2d shows an enhanced rejection asymmetry despite the equality of ion valences (D h Ca2+/D h SO4222+ ≈ 3.92 . 1 > DCa /DSO4 ) 0.68). In this situation (rp ) 0.5 nm), strong hindered transport effects are clearly responsible for the marked rejection asymmetry, due to the large difference in radius between the calcium and sulfate ions (Table 1). Note that with the ad hoc Stokes-Einstein (SE) choice of ion radius the ratio of intramembrane diffusivities, [D h Ca2+/D h SO42-]SE ≈ 0.27, is much less than the one obtained using the crystal choice (and even less than that obtained using the bulk ion diffusivities); as opposed to the asymmetry observed in Figure 2d, the SE choice for the ion size would therefore lead to an asymmetry skewed toward positive membrane charges (the same would be true if the ions were treated as points). The strong rejection asymmetry for asymmetric salts comes mainly from the difference in valence between the anion and the cation and the inversion of co-ion and counterion as the membrane charge changes sign. When the GCE approximation is valid, Figure 1 shows that the salt rejection curves are monotonically increasing functions of jv. This implies that TGCE(jv) > TGCE lim , and therefore, since > 0, j c (x) is a monotonically decreasing function of jv P h GCE 2 2 (concave downward co-ion concentration profile, see eq 23). 5. Electric Potential The total electric potential across the system is defined as the electric filtration potential Φ ˜ F ) φ˜ f - φ˜ p and is composed

5530 J. Phys. Chem. B, Vol. 109, No. 12, 2005

Lefebvre and Palmeri

Figure 2. Limiting salt rejection, Rlim , as a function of the normalized membrane charge density ξ for four types of |z1|,|z2| salts ((a) 1:1, (b) 1:2, (c) 2:1, and (d) 2:2) (leff ) 50 µm and rp ) 0.5 nm).

of the Donnan interfacial potentials, the diffusion potential, and the streaming potential.1 As explained earlier, during the separation of salt solutions in nanofiltration, counterions are attracted into the pore, and co-ions are repelled. To maintain macroscopic electroneutrality in the permeate in the steady state, an electric force engendered primarily by the streaming potential at high flux is generated that accelerates the transfer of co-ions and retards the transfer of counterions. The exact parametrized equation describing the (dimensionless) electric filtration potential, ∆Φ ˜ F, as a function of volume flux density, jv, has a rather complex form, especially at low and intermediate values.1 Nonetheless, when jv reaches a high enough value, a simplification is possible via an exact analytic determination of the reduced streaming potential

ν˜ j )

d∆φ˜ s d∆Φ ˜F F νj ) lim )Pef∞ djv RT0 djv

where Pe ) jvlm/D h s is the Pećlet number. In the lowest order GCE approximation, the streaming potential φ˜ s depends only on counterion parameters (eq 20), and the diffusion potential vanishes. Thus, in the GCE approximation, the reduced streaming potential

ν˜ GCE ) j

K1,c sgn(Xm)lm |z1|D h1

)

K1,c sgn(Xm)leff |z1|K1,dD1

(30)

is much simpler than the exact expression derived within the

HET model1

(

ν˜ j ) 1 -

)

|z2|ν2kf2cf(K2,c - K1,c) K1,c|Xm|

(

lm

|z1|D h 1kf1

+

|z2|D h 2kf2

)

K1,cXm

ν1|z1|cf

. (31)

(The GCE expression (eq 30) can also be obtained directly from the exact one (eq 31), by neglecting the partition coefficient of the co-ion and then using eq 15 for the partition coefficient of the counterion.) The GCE result for the reduced streaming potential (eq 30) depends only on the sign of the membrane charge, sgn(Xm), not its amplitude, and on counterion parameters (cf. Table 2). In general, the reduced streaming potential is a constant, independent of jv, which means that it should be possible to determine a straight line describing the asymptotic behavior of TABLE 2: Slope, ν˜ GCE , and y-intercept, -∆O˜GCE j lim , Values Used for the Approximate Calculation of the Filtration Potential for the Four Salts Studied (leff ) 50 µm and rp ) 0.5 nm) GCE -∆φ˜ lim

NaCl CaCl2 Na2SO4 CaSO4

ξ ) +2

ξ ) +5

ξ ) +10

ν˜ GCE (L-1 h m2) j

0.884 1.019 -0.305 0.078

1.800 2.852 -0.076 0.536

2.493 4.238 0.097 0.883

0.030 0.030 0.091 0.091

NF Theory: GCE Appoximation for Single Salts

J. Phys. Chem. B, Vol. 109, No. 12, 2005 5531

Figure 3. Dimensionless electric filtration potential, ∆Φ ˜ F, for four types of |z1|,|z2| salts ((a) 1:1, (b) 1:2, (c) 2:1, and (d) 2:2) as a function of the volume flux density, jv, for a positively charged membrane and three different values of ξ ) Xm/cf (leff ) 50 µm and rp ) 0.5 nm).

the exact electric filtration potential at high volume flux densities. Consequently, at high jv we can write

∆Φ ˜ F ≈ -∆φ˜ lim + ν˜ jjv

(32)

where -∆φ˜ lim represents the y-intercept of this straight line.21 Unfortunately, due to the cumulated contributions of the diffusion, streaming, and Donnan potentials to -∆φ˜ lim, its exact analytic expression is complicated. However, in the domain of validity of the GCE approximation, both diffusion and streaming potentials contributions to -∆φ˜ lim become negligible compared to the Donnan ones, ∆φ˜ f(p) D . This simplification allows us to obtain in the GCE approximation

-∆φ˜ GCE lim

)

∆φ˜ pD

-

∆φ˜ fD

( )

kp(GCE) 2 1 ) - ln f(GCE) z2 k 2

(33)

which, after using eq 18 for T ) TGCE lim , can be written as

-∆φ˜ GCE lim )

sgn(Xm) ln[1/TGCE lim ] |z1|

(34)

p f Since the passage TGCE lim is just the concentration ratio c /c , the GCE result (eq 34) for the y-intercept, -∆φ˜ lim, is, in the presence of chloride salts such as KCl, just the Nernst electrode potential for a positively charged membrane and the negative of the Nernst (electrode) potential for a negatively charged membrane. This proves the assertion made in ref 1 that in the

GCE limit the measured electric potential, which includes the electrode potential for anion reversible electrodes

∆Φmes ) ∆ΦF +

[ ]

p RT0 cCl ln f F cCl-

(35)

can be represented at high flux by a straight line with zero y-intercept for a positively charged membrane and a straight line with a y-intercept equal to twice -∆φ˜ lim (eq 34) for a negatively charged membrane (see Figure 10 of ref 1). Equation 30 also clearly shows that the reduced streaming potential increases with increasing effective membrane thickness (by an increase in the real active layer thickness lm or tortuosity τ or a decrease of porosity, φp). Furthermore, since Ki,c/Ki,d increases rapidly with increasing ion to pore radius ratio, λi, increases rapidly with increasing counterion radius or ν˜ GCE j decreasing pore size. We can thus conclude that in the GCE approximation, although the amplitude of the membrane charge density does not influence ν˜ GCE , hindered transport effects can j play an important role in determining this quantity. Since the y-intercept, -∆φ˜ GCE lim , depends on both the sign and the magnitude of the membrane charge density, but not leff, this quantity potentially provides complementary information concerning membrane characteristics. Figure 3, parts a-d, shows the behavior of the electric filtration potential for the four salts studied (1:1, 1:2, 2:1, and 2:2, respectively) and three values of normalized (positive) membrane charge densities of ξf ) |Xm|/cf ) 2, 5, and 10 (leff



TABLE 3: Relative Differences between the Dimensionless Electric Filtration Potential, ∆Φ ˜ F, Calculated Using the Exact Analytical and the Approximate GCE Methods for jv ) 100 L h-1 m-2, leff ) 50 µm, and rp ) 0.5 nm NaCl CaCl2 Na2SO4 CaSO4

ξ ) +2

ξ ) +5

ξ ) +10

6.47% 6.37% 104.34% 47.83%

1.04% 0.41% 29.32% 8.31%

0.30% 0.04% 10.61% 2.12%

) 50 µm and rp ) 0.5 nm). As already observed in Figure 1, we can again conclude that the value ξf ) 2 is not high enough to ensure the validity of the GCE approximation, because the straight lines are not asymptotically superimposed on the curves for any of the cases treated in Figure 3 for this low value of the normalized membrane charge density. However, the calculations concerning the salts containing the chloride ion show that the GCE estimate is accurate starting from a normalized charge density ξf ) 5, while it only becomes valid for CaSO4 starting from ξf ) 10 (consistent with ξˆ f, and not ξf, being the parameter that directly enters into determining the domain of validity of the GCE approximation, see eqs 12 and 17). As already shown in Figures 1 and 2, the GCE limit is not yet reached for Na2SO4 in the positive range of membrane charges studied here. (We consider the GCE approximation valid when the relative difference between the exact and approximated calculations is lower than 2%.) Table 3 summarizes these conclusions by representing the relative differences between exact and asymptotic GCE calculations for the four salts and the three normalized charge densities studied. The small differences observed for NaCl and CaCl2 reveal that the good co-ion exclusion limit is reached at moderate positive values of ξf. The results presented in Figure 3 and Tables 2-3 show that with increasing normalized membrane charge densities, the ˜ F versus jv curve tends to an limiting slope, ν˜ j, of the ∆Φ GCE asymptotic value, ν˜ j (eq 30), that is of the same sign as Xm but independent of its magnitude. We also observe that at high normalized membrane charge densities this slope depends only on the counterion parameters (valence, radius, and bulk diffusion coefficient), also in accordance with eq 30. Figure 3 and Tables 2-3 also show that at sufficiently high ξˆ f the limiting y-intercept, -∆φ˜ lim, of the ∆Φ ˜ F versus jv curve is of the same sign as the membrane charge and increases logarithmically with ξˆ f (via TGCE lim < 1, see eqs 21 and 16), in accordance with the GCE result, -∆φ˜ GCE lim (eq 34). For the normalized membrane charge density range studied, the low rejection of the 2:1 salt (Na2SO4) leads to very low y-intercepts, -∆φ˜ lim (Figure 3c), and the high rejection of the 1:2 salt (CaCl2) leads to very high ones (Figure 3b). 6. Volume Flux Density A. Introduction. The volume flux density for pure water

∆P jwv ) -L0p lm L0p,

First, when combined with the Donnan potentials, the intramembrane electric potentials lead to an osmotic pressure term that subtracts from |∆P|. Second, the streaming potential is at the source of the electroviscous effects that reduce the salt solution permeability from the pure water value and therefore reduce the driving force, |∆P|, by a multiplicative factor. Consequently, the system responds as if the solution viscosity has been increased from the pure water value, η0, to a higher effective value ηeff ) η0(1 + κ) g η0, where κ is the electroviscosity coefficient. The exact analytical determination of the volume flux density as a function of the transmembrane pressure is obtained using the averaged Stokes equation1

1 L0p

∑i cji(x) ∂x (ln γj i)

jv ) -∂x P - F(x) ∂x φ - RT0

(37)

with F(x) ) F∑i zicji(x) the local ion charge density. In the membrane (0+ < x < lm), sufficiently far away from the interfacial regions, eq 37 becomes

1 jv ≈ -∂x P + RT0Xm ∂x φ˜ L0p

(38)

Equation 38 can be integrated across the membrane, leading to

jvlm L0p

) ∆Pm - RT0Xm(∆φ˜ d + ∆φ˜ s)

(39)

which can be written in standard form as1

jv )

Lsp (∆P - Σ∆Π) lm

(40)

where

Lsp )

L0p (1 + κ)

is the salt solution permeability

∆Π ) Πf - Πp ) RT0cf(ν1 + ν2)(1 - T) is the total osmotic pressure difference across the membrane 0 -1 ˜j κ ) -RT0L0pXm(∂x φ˜ s)0+ j-1 v ) RT0LpXmlm ν

(41)

is the electroviscosity coefficient (independent of jv and directly proportional to ν˜ j), and

Σ)1-

kf2 - Tkp2 Xm[∆φ˜ d + ∆φ˜ s - (∂x φ˜ s)0+ lm] 1-T cf(ν + ν )(1 - T) 1

(42)

2

(36)

depends on the pure water permeability of the membrane. During electrolyte transport across a nanofilter, electrical diffusion and streaming potentials are engendered within the membrane to maintain macroscopic electroneutrality in the steady sate. These intramembrane electric potentials give rise to a body force that acts back on the charged fluid to reduce the driving force, |∆P|, and therefore the volume flux density (see, e.g., ref 1). This reduction occurs in two distinct ways.

is the global osmotic reflection coefficient (which, although jv dependent, tends to a constant value at a high volume flux density). (For any quantity Q, ∆Q ≡ Qf - Qp ) -∆Q for the bulk and ∆Qm ≡ Q(0+) - Q(lm) ) -∆Qm for the membrane.) In the limit of high transmembrane pressure, jv asymptotically becomes a linear function of ∆P

jv ≈

Lsp (∆P - Σ lim∆Πlim) lm

(43)



with slope Lsp/lmand x-intercept Σlim∆Πlim. Thus, by measuring the slope and x-intercept of the asymptotic jv versus ∆P curve, we can obtain information concerning membrane properties. Our goal here is to calculate Lsp/lmand Σlim∆Πlim using an extended GCE approximation that we denote GCE+. By employing the GCE approximation for ν˜ j, eq 30, in eq 41 we find an expression for the HET electroviscosity coefficient GCE

κ

RT0L0p|Xm|K1,c

)

|z1|D h1

RT0L0p|Xm|K1,cleff ) |z1|K1,dD1lm

(44)

This GCE result for the HET electroviscosity coefficient is much simpler than the exact HET one (eq 67 of ref 1) and allows one to immediately ascertain how κGCEvaries with system properties. As eq 44 shows, κGCE scales linearly with the absolute value of the effective membrane charge density and reduces to the ENP model one,4 when K1,c and K1,d f 1. Like the reduced streaming potential, the electroviscosity coefficient increases with increasing tortuosity or counterion radius and decreasing porosity or pore size. If, as is usually found for ions in charged NF membranes, leff/lm . 1, then electroviscous effects can be important and much larger than would be predicted by simple capillary pore models that neglect tortuosity by assuming that leff ) lm/φp. Moreover, a glance at the averaged Stokes eq 38 shows that within the HET model nonzero electroviscous effects arise (via the electric body force) only when the membrane is charged. Consequently, provided that the HET model is a reasonable description of salt transport in nanofilters, a difference in slope, Lsp < L0p, is a clear sign of electrostatic interactions. This is in contrast to the streaming potential itself, which, due to hindered transport effects, can be nonzero even in the absence of membrane charge. The GCE approximation turns out, however, to be too restrictive for the calculation of the global osmotic reflection coefficient, Σ. Small terms neglected in this approximation become important once multiplied by the membrane charge density (assumed to be high) appearing in the electric body force term in the averaged Stokes equation (second term on the righthand side of eq 38). Consequently, we have to go beyond the GCE approximation by accounting for higher-order corrections to the co-ion concentration in the membrane. B. Ion Partitioning in the GCE+ Approximation. First of all, we have to calculate the expressions for the Donnan/steric partition coefficients in the GCE+ approximation (cf. ref 4). The GCE approximation to the co-ion partition coefficient, kf(GCE) ) (kf2)I, is given by eq 16. Roman numerals in super2 script will indicate the order of the GCE+ expansion in powers of (kf2)I. Using the normalized electroneutrality eq 6, kf1 ) ξˆ f + kf2, we find the first-order relation for the counterion partition coefficient

(kf1)I ) ξˆ f[1 + (kf2)Iξˆ -1 f ].

(kf2)II ) (kf2)I[1 - τv(kf2)Iξˆ -1 f ].

(46)

Within the HET model, the exact expression for the limiting transmission rate (eq 25) can be written as

Tlim )

[

K2,c + K1,c

( )(

)]

ht 2 1 - K2,cK-1 1,c

ht 1

1+

ht 2ht -1 1

+ TGCE lim )

(47)

[

(kf2)Iω ht 1

1-

ht 2 f I -1 (k2) ξˆ f ht 1

]

(48)

C. Electroviscous and Osmotic Reflection Coefficients in the GCE+ Approximation. Equations 41 and 42 show that although κ depends only on the streaming potential, Σ depends on three different contributions, namely, the interfacial, diffusion, and streaming potentials. Diffusion Potential Contribution. Within the HET model, the exact analytic expression for the diffusion potential is

∆φ˜ d ) -

D h1 - D h2 z2D h 2 - z1D h1

ln

(11 ++ BA)

with A ) (Tkp2)(th1ξˆ f)-1 and B ) (kf2)(th1ξˆ f)-1 (see, e.g., ref 1). Within the GCE+ approximation, the second-order expansion leads to

∆φ˜ IId

(

)-

)(

D h1 - D h2

TGCEkp2

z2D h 2 - z1D h1

and therefore

ht 1ξˆ f

( ))

kf2

f 1 k2 + ht 1ξˆ f 2 ht 1ξˆ f

2

(50)

]( )

[

D h2 kf2 - TGCEkp2 - (kf2)2(2th1ξˆ f)-1 RTXm∆φ˜ IId ) τv 1∆Π (1 - T)(1 + τv) D h 1

(51)

In the GCE approximation, eq 18 can be used to write kp2 in terms of kf2 v ) TτGCE kf(GCE) kp(GCE) 2 2

Streaming Potential Contribution. Within the HET model, the exact expression for the gradient of the streaming potential is1 2

∂x φ˜ s )

ziKi,ccji(x) ∑ i)1 jv

2

∑ i)1

(52)

z2i cji(x)D hi

Using the electroneutrality eq 6 in eq 52, we obtain a relation that depends only on the intramembrane co-ion concentration

∂x φ˜ s )

(45)

Finally, using eq 11, we obtain the second-order GCE+ result for the co-ion partition coefficient

kf2

with ≡ kf2/kf1 ≈ (kf2)Iξˆ -1 f , where the approximation is valid in the GCE limit. In this limit, tends to zero, which allows us keep only the first two terms in an expansion of eq 47 to obtain the GCE+ approximation for the limiting salt transmission

[z2cj2(K1,c - K2,c) - K1,cXm]

(

|z1|D h 1|Xm| 1 +

|z2|cj2 |Xm|th1

)

jv

(53)

Moreover, in the GCE+ limit, we know that cj2 is small (eq 2), which allows us to expand eq 53 and neglect terms of order cj22 or higher. After integrating across the membrane, we find

∆φ˜ s ≈

[

jv |z1|D h1

(

-sgn(Xm)K1,clm +

|z2|sgn(Xm)K1,c |Xm|th1

+

)∫

z2(K1,c - K2,c) |Xm|

lm

0+

]

cj2(x) dx (54)



Furthermore, eq 23, valid in the GCE approximation, can be used for the co-ion concentration to perform the integral in eq 54. We find

∫0

lm +

cj2(x) dx ) I1 + I2

RT0Xm∆φ˜ IId,lim ∆ΠIIlim

)

[

τV(kf2)I

(55) and

with

(2)I RT0Xm∆φ˜ s,lim

I1 ) TGCElmcf2(kf2)I/TGCE lim

∆ΠIIlim and

I2 )

D hs

v+1 - 1)cf2(kf2)I (T τGCE

jv(1 + τv)ω

Thus, it is clear that the streaming potential is composed of two distinct parts, only one of which depends explicitly on the volume flux density, jv (of course the salt transmission, TGCE(jv), which is assumed to be known (eqs 24-29), depends implicitly on jv). Consequently, we can write

RT0Xm∆φ˜ Is ) RT0Xmφ˜ (1)I ˜ (2)I s + RT0Xmφ s

(56)

with

RT0Xm∆φ˜ (1)I s

)

[ (

RT0|Xm|lm |z1|D h1

K1,c -

)

2K1,c - ω ht 1

(kf2)I

TGCE

ξˆ f TGCE lim

]

( )(

)(

1

)

Finally, by letting TGCE f TGCE lim , we see that in the high flux limit ∆φ˜ (1)I contributes only to the electroviscosity coefficient, s κ, and ∆φ˜ (2)I contributes only to the global osmotic reflection s coefficient, Σlim . We can then determine the GCE+ expression for the membrane salt permeability

L0p

+

) Ls(GCE ) p

1 + κGCE

(57)

+

where +

[

(

κGCE ) κGCE 1 - 2 -

)

]

f I ω (k2) K1,c ht ξˆ 1 f

and the limiting (high flux) GCE+ expression for the global osmotic reflection coefficient +

ΣGCE ) lim 1-

where

1+τv ] (kf2)I[1 - (TGCE lim )

(1 + τv)

+

RT0Xm∆φ˜ IId ∆ΠIIlim

+

RT0Xm∆φ˜ (2)I s ∆ΠIIlim (58)

+ τv)

( )[

1-

](

D h2

D h1

)

τv+1 τv 1 - (TGCE 2K1,c lim ) - 1 (kf2)I GCE 1 + τ ω D h1 1 - Tlim v

D h2

+

RGCE lim ) 1 and +

v+1 RT0Xm∆φ˜ (2)I 1 - TτGCE 2K1,c D h 2 τv s ) - 1 (kf2)I ∆Π 1 + τ 1 T ω D h v GCE

(1 -

TGCE lim )(1

Although the above GCE+ approximations for κ and Σ are exact to first order (superscript I terms) or second order (superscript II terms) in the small quantity (kf2)I, higher-order terms are only accounted for in an approximate fashion. Equation 58 reveals explicitly that the interfacial osmotic pressure drops (first two terms), the diffusion (third), and the streaming (fourth) potentials all contribute to the global osmotic reflection coefficient Σlim. Moreover, the dependence of Σlim on the NF system parameters is clearly shown, which demonstrates that, in the case of ionic solutions, it is not equal to the GCE+ result for + + ) 1 - TGCE the limiting rejection, RGCE lim lim , obtained from eq 48. This is easier to see if we keep only the leading order terms + + in RGCE and ΣGCE lim lim

jv

and

)

]( )

1+τv 1 - (TGCE - (kf2)I(2th1ξˆ f)-1 lim )

kf(GCE) 2 ω + ... ht 1

[ ( )(

f(GCE) 1 - 2τv ΣGCE lim ) 1 - k2

D h 2 K1,c -1 ω D h 1

(59)

)]

+ ....

(60)

Figure 4, parts a-d, shows the variation of the volume flux density, jv, as a function of the transmembrane pressure, |∆P|, for four salts and three different values of positive normalized charge density, ξ ) Xm/cf ) +2, +5, and +10. In every case, the salt feed concentration, cf, has been fixed at 10-2 M, L0p/lm ) 10 L h-1 m-2 bar-1, leff ) 50 µm, and rp ) 0.5 nm. At high pressure, there is an excellent asymptotic agreement between the straight line (eq 43) (evaluated in the GCE+ approximation, + and κ f κGCE+) and the exact analytical Σlim f ΣGCE lim calculations for NaCl and CaCl2 for all the normalized membrane charge densities studied here, while the lower order GCE approximation is valid only for ξ g +5. The reasonable agreement observed for CaSO4 for ξ ) +5 and +10 and Na2SO4 for ξ ) +10 allows us to conclude that the GCE+ approximation extends the domain of validity of the GCE approximation to lower normalized charge densities. A study of Table 4 confirms that the GCE+ approximation is accurate for salts that contain the chloride counterion; this is especially true for the electroviscosity coefficient, κ, because the difference between the exact and approximate calculations is less than 1%. Concerning κ, the biggest difference is found for Na2SO4 for low positive membrane charge densities (relative error, ∆κ/κ ≈ 30%, ξ ) +2). However, the calculation of the global osmotic reflection coefficient gives (sometimes, much) larger discrepancies. For Na2SO4 and CaSO4, even the GCE+ approximation is not very accurate for ξ ) +2. Finally, the calculations concerning CaSO4 for ξ ) +10 show a much better agreement in the GCE+ approximation than in the GCE one. The results presented in Figure 4 and Table 4 show that both



Figure 4. Volume flux density, jv, as a function of the transmembrane pressure, |∆P|, for four types of |z1|,|z2| salts ((a) 1:1, (b) 1:2, (c) 2:1, and (d) 2:2) for a positively charged membrane and three different values of ξ ) Xm/cf (cf ) 10-2 M, leff ) 50 µm, and rp ) 0.5 nm). s TABLE 4: Solution Permeability, Lp(ex) /lm, x-intercept, Σlim∆Πlim, and Relative Differences Obtained Using Exact Analytical and GCE+ Methods for the Four Salts Studied for Three Different normalized Charge Densities, ξ (cf ) 10-2 M, L0p/lm ) 10 L h-1 m-2 bar-1, leff ) 50 µm, and rp ) 0.5 nm)

salt

ξ values

(L

Lsp(ex)/lm m-2 bar-1)

h-1

(L

Lsp(app)/lm -1 h m-2 bar-1)

diff (%)

Σ∆Π(ex) (bar)

Σ∆Π(app) (bar)

diff (%)

NaCl

+2 +5 +10

8.831 7.318 5.731

8.864 7.323 5.732

0.37 0.07 0.01

0.2825 0.3964 0.4422

0.3015 0.4049 0.4493

6.73 2.14 1.61

CaCl2

+2 +5 +10

8.893 7.305 5.721

8.942 7.308 5.722

0.55 0.04 0.01

0.5072 0.6922 0.7226

0.5535 0.6982 0.7321

9.13 0.87 1.31

Na2SO4

+2 +5 +10

8.247 5.378 3.305

10.698 5.644 3.343

29.72 5.51 1.15

0.0241 0.0517 0.1609

0.3812 0.1017 0.2149

1481.74 96.71 33.56

CaSO4

+2 +5 +10

7.729 4.923 3.123

8.292 4.952 3.124

7.28 0.59 0.04

0.1958 0.3239 0.4039

0.5000 0.3406 0.4092

155.36 5.16 1.31

electroviscous and osmotic pressure effects can be strong under typical NF conditions, especially at the relatively high normalized membrane charge densities that can be obtained at low salt concentrations. For all the salt studies, the calculated salt solution volume flux density is reduced from the pure water value by about 10-20% for ξ ) +2, about 25% for ξ ) +5, and about 50% for ξ ) +10, mainly due to electroviscous effects. 7. Conclusions The purpose of this work is to extend our study of the influence of the NF system parameters on ion and salt solution transfer. The simplifications to the exact analytical results

obtained using the GCE and GCE+ approximations have allowed us to gain further insight into the role played by the different parameters entering the NF electrotransport model. Furthermore, it has been possible to determine the domain of validity of these approximations as a function of salt type. Clearly, the higher the co-ion valence and the lower the counterion one, the lower is the normalized membrane charge density for which the GCE (or GCE+) becomes valid. Generally speaking, the results presented here allow us to state that the GCE approximation is valid for

ξˆ f ) ξf/(ν1|z1|) g 5

(61)

and the GCE+ is valid for ξˆ f > 1 (the only caveat being for a



positive membrane charge and the 2:1 salt (Na2SO4). Our results (Figure 1c) suggest that in this case the GCE approximation only becomes valid for ξˆ f > 10 and the GCE+ becomes valid for ξˆ f > 3). Since the membrane charge density within a nanofilter is often estimated using electrokinetic measurements (electrophoretic mobility9,27 and tangential streaming potential28,29), it is useful to reformulate the condition for the validity of the GCE approximation directly in terms of the surface ζ-potential. The ζ-potential provides a measure of the electrokinetic surface charge density, σek, of an open surface in equilibrium with a bulk solution. The surface charge density of an open planar surface, σek, can be calculated from the measured ζ-potential via

σek ) (2RT0

∑i ci[exp(-ziFζ/RT0) - 1])1/2

(62)

where ci is the bulk ion concentration of ion i in the salt solution used for the measurements.30 Although, due to the strong overlap of electric double layers within the pores of nanofilters, the ζ-potential is not necessarily equal to the Donnan potential and the electrokinetic surface charge density, σek, is not necessarily equal to the pore wall surface charge density, σw, it is still convenient to estimate the effective volumetric membrane charge density

Xm )

2σw Frp

(63)

by assuming a cylindrical pore geometry and σw ≈ σek.8,9,28 For many salts and typical physical values of ζ, it is also reasonable to use the “weak” ζ-potential approximation relating σek to ζ,30 σek ≈ ζ/λD, where

λD )

( ) RT0

1/2

2F2I

is the Debye length with I ) (1/2)∑iz2i Ci the ionic strength. The “weak” ζ-potential approximation is accurate to less than 20% error when |z1|F|ζ|/(RT0) < 2.5 (or |z1||ζ| < 65 mV at 25 °C). Combining the above relations with eq 61 leads to a critical salt concentration

C/s

≈

4(1 + |z2|/|z1|)ζ2 25ν1RT0r2p

(“weak” ζ-potential)

(64)

such that for Cs < C/s (ζ) the GCE is valid. In this approximation, the critical salt concentration, C/s (ζ), increases as the square of the ζ-potential and varies inversely as the square of the pore radius. If salt concentration is measured in mol/L, ζ-potential in mV, and pore radius, rp, in nm, then for 25 °C we obtain the following convenient formula

[C/s ] ≡

(4.2 × 10-5)(1 + |z2|/|z1|)[ζ]2 ν1[rp]2

(“weak” ζ-potential) (65)

For a 1:1 salt and rp ) 1 nm, we find C/s ) 8.4 × 10-3, 3.4 × 10-2, and 7.6 × 10-2 M for ζ ) 10, 20, and 30 mV, respectively. For salts with multivalent counterions, the “weak” ζ-potential approximation may break down for high values of ζ. In this case, one should use the “exact” formula for σek (eq 62). A

simpler approach, however, which complements the “weak” ζ-potential approximation, is to retain only the counterions in eq 62 (“strong” ζ-potential approximation). This approximation is accurate to less than 20% error when |z1|F|ζ|/(RT0) > 2.5 (or |z1||ζ| > 65 mV at 25 °C) and leads to

σek ≈ (2RT0ν1Cs)1/2 exp[|z1|F|ζ|/(2RT0)] (“strong” ζ-potential) (66) or

C/s ≈

8RT0 25ν1z21F2r2p

exp[|z1|F|ζ|/(RT0)] (“strong” ζ-potential) (67)

If salt concentration is measured in mol/L, ζ-potential in mV, and pore radius, rp, in nm, then for 25 °C we obtain the following convenient formula

[C/s ] )

(5.9 × 10-3) exp(|z1|[ζ]/25.7) V1z21[rp]2 (“strong” ζ-potential)

In this approximation, the critical salt concentration, C/s (ζ), still varies inversely as the square of the pore radius but now increases as the exponential of the ζ-potential. The results presented here have also allowed us to demonstrate that in the case of ionic solutions the global osmotic reflection coefficient (appearing in the volume flux density formula) is not equal to the salt limiting rejection rate (as it is usually assumed in simple transport models for neutral solute solutions31). Indeed, if rejection depends on solute concentration, as is the case for ionic solutions, then Σlim * Rlim. Moreover, since the SE choice of ion radius leads to small anions and large cations (the converse of what is obtained with the crystal choice used here), the salt rejection formulas derived here (Figure 1) reveal that hindered transport effects would lead to different kinds of asymmetries about an isoelectric point (pH at which Xm vanishes) for the two different choices of ion radii. For symmetric salts, the limiting rejection curves would be skewed toward positive membrane charges for the SE choice and skewed toward negative membrane charge for the crystal choice (Figure 2, parts a and d; cf. Figure 6 of ref 9). By examination of salt rejection as a function of pH, this difference in behavior should be experimentally accessible. We would like to underline that the GCE approximation is a limiting form of an homogeneous electrotransport theory, and as such it should not be used in its basic form outside its range of validity. Even if the GCE approximation is not valid under certain NF and UF conditions, it does not mean that the underlying homogeneous electrotransport theory itself is necessarily invalid also (cf. refs 4 and 24-26). By extending the approach developed here to include other interactions, besides the electrostatic and hindered transport ones, one could conceivably shed light on the importance of those interactions not incorporated into the HET model. In a subsequent article, we will extend the GCE approximation to electrolyte mixtures composed of trace ions and a majority salt and also study numerically more general mixtures using our NF transport simulation program, NanoFlux.14,15 Appendix: Virtual Salt Transport and the Spiegler-Kedem Approximation Within the HET model, we examine here the validity of a simplified Spiegler-Kedem-type approach to salt rejection. This



approximation is based on neglecting the concentration dependence of the salt reflection coefficient appearing in the salt molar flux density equation (for the ET model see refs 7, 22, and 23). Through the use of the hindered ENP equations (eq 3), the electroneutrality (eq 6), and the vanishing of the electric current density (eq 9), we obtain a relation between the co-ion concentration and the electric potential

1 - σS(C) ) k2(C) [1 - σ2(cj2(C))]

∂x φ˜ )

[z2cj2(K2,c - K1,c) - K1,cXm]jv - [z2(D h2 - D h 1)] ∂x cj2 [|z2|cj2(|z1|D h 1 + |z2|D h 2) + |z1|D h 1|Xm|]

(

PS(C) ) k2(C)P2(cj2(C)) 1 +

j2 ) -P2(cj2) ∂x cj2 + [1 - σ2(cj2)]cj2jv ) cp2 jv

(69)

with

|z2|2cj2D h 2 [(D h2 - D h 1)] |z2|cj2 (|z1|D h 1 + |z2|D h 2) + |z1|D h 1|Xm|

)

τvξˆ C

(1 + τv)k2(C) + ξˆ C

Although eq 73 can formally be integrated to find

(68)

This result allows us to write the hindered ENP equation for the co-ion (i ) 2) in the following form1

P2(cj2) ) D h2 -

where σS(C) is the local salt reflection coefficient. Through the use of eq 71 to evaluate the derivative, PS(C) can be written as

jvlm )

∫cc f

p

PS(C) dC C[1 - σS(C)] - cp

(74)

it is in general difficult to actually carry out the integration, since PS(C) and σS(C) are complicated functions of C. The low and high flux limiting behavior, however, can be found explicitly. In the limit of low jv, salt rejection is low, and the virtual salt concentration gradient in the membrane is weak; therefore the integrand in eq 74 can be evaluated at C(x) ≈ C(0) ) cf and cp ≈ cf, leading to a linear increase of rejection with jv

the effective co-ion permeability, and

jvlm R(jv) ≈ σS(cf) PS(cf)

[1 - σ2(cj2)] ) |z2| cj2(K1,c - K2,c) + K1,c|Xm| K2,c + |z2|D h2 [|z2| cj2(|z1|D h 1 + |z2|D h 2) + |z1|D h 1|Xm|] where σ2(cj2) is the local co-ion reflection coefficient. We replace the local co-ion concentration cj2(x) with the concentration of a virtual salt, C(x), taken to be in local thermodynamic equilibrium with the membrane at position x7,22,23

cj2(x) ) ν2C(x)k2(C(x))

In the limit of high jv, the virtual salt concentration profile becomes flat (apart from a thin diffusion boundary layer region near the permeate interface), and therefore C can bet taken equal to cf, and the diffusive term in eq 73 can be neglected, leading to R(jv) f σS(cf). In the GCE limit, matters simplify considerably, since |z2|cj2(x)/|Xm| f 0 and therefore f(GCE) v (C) ) Φs2(Φs1)τvξˆ -τ k2 f kGCE 2 C ) k2

(70)

with C(0) ) cf and C(lm) ) cp. The local co-ion partition coefficient, k2(C(x)), obeys eqs 12 and 13 with the salt feed concentration replaced by the virtual one, C(x)

d[ln(k2(C))] d[ln(C)]

() C cf

τv

f τv

P2 f D h2

k1(C) - k2(C) ) (Φs2)1/τvΦs1[k2(C)]-1/τv - k2 ) ξˆ C (71) and where

ξˆ C ≡

(1 - σ2) f (1 - σGCE )) 2

|Xm| |z1|ν1C(x)

ω ht 1

Consequently is a local normalized membrane charge density. Using eq 70, we have

(

∂x cj2(x) ) ν2k2[C(x)] ∂x C(x) 1 +

d[ln k2(C(x))] d[ln C(x)]

)

(1 - σS(C)) f (1 - σGCE S (C)) )

ω GCE k2 (C) ht 1

(72) and

which allows us to obtain the salt molar flux equation in canonical form

jS ) -PS(C) ∂x C(x) + C(x)[1 - σS(C)]jv ) c jv (73)

PS(C) 1 - σS(C)

f

D hs ω

p

with

(

PS(C) ) k2(C) P2(cj2(C)) 1 +

d[ln (k2(C))]

the effective local salt permeability and

d[ln (C)]

)

becomes independent of C. The integral in eq 74 then simplifies to

jvlm ≈ and by using

D hs ω

∫cc f

p

(

dC C -

)

cp [1 - σGCE S (C)]

-1

5538 J. Phys. Chem. B, Vol. 109, No. 12, 2005 GCE [1 - σGCE S (C)] ) Tlim


() C cf

τv

PeSK ≡

the integration can be carried out to recover the nonlinear eq 24 obeyed by the salt transmission in this approximation. In the Spiegler-Kedem (SK) approximation, the concentration dependence of the transport parameters appearing in the salt molar flux eq 73 is neglected by assuming that PS(C(x)) and σS(C(x)) can be approximated by their values at x ) 0 or C(x) ≈ C(0) ) cf. The approximate SK salt transport equation

jS ≈ -PS(cf) ∂x C(x) + C(x)[1 - σS(cf)]jv

(

PS(cf)

) (

c f ∫c 1 - σ (c )

p

f

S

dC C -

cp 1 - σS(cf)

)

GCE RGCE SK (PeSK )

)1-

TGCE lim GCE [1 - RGCE lim G(PeSK )]

-1

(75)

where

PeGCE SK ≡

to find

RSK(PeSK) ) 1 -

Tlim [1 - Rlim G(PeSK)]

with

Tlim ) 1 - Rlim ) 1 - σS(cf) G(Pe) ≡ exp(-Pe) and

PS(cf)

By examining the high and low flux limits of eq 76, we see that the SK approximation yields, as expected, the exact high flux limiting rejection and is also exact to first order in jv at low flux. At intermediate flux, the SK approximation is no longer exact due to the neglect of the concentration dependence of PS(C) and σS(C). This neglect can lead to nonnegligible discrepancies in this flux range (see below). In the GCE approximation, the SK result for salt rejection further simplifies to

can immediately be integrated

jvlm ≈

jvlm[1 - σS(cf)]

(76)

jvlmω D hs

Although the SK-GCE result, PeGCE SK , is also exact for vanishing membrane charge, it is not in general equal to PeSK. Figure A1 shows the calculated rejection as a function of the volume flux density for four salts using a membrane with a normalized membrane charge density of ξ ) Xm/cf ) +10, an effective thickness leff ) 50 µm, and a pore radius rp ) 0.5 nm. The exact analytical calculations were obtained using the results presented in ref 1. For low and high values of volume flux density, Figure A1 shows that when the GCE approximation is valid (Figure A1, parts a, b, and d) there is excellent agreement between the exact,

Figure A1. Salt rejection, R, for four types of |z1|,|z2| salts ((a) 1:1 NaCl, (b) 1:2 CaCl2, (c) 2:1 Na2SO4, and (d) 2:2 CaSO4) as a function of volume flux density, jv, for a positively charged membrane (ξ ) Xm/cf ) +10, leff ) 50 µm, and rp ) 0.5 nm): exact analytical calculations (upper solid lines); GCE results (dotted lines); SK-GCE results (lower bold solid lines).


J. Phys. Chem. B, Vol. 109, No. 12, 2005 5539 T0 ) temperature, K Ti ) passage or transmission (ion i), none or % x ) transverse membrane coordinate, m Xm ) effective membrane charge density (moles per unit pore volume) (mol/l) zi ) ion i valence

Figure A2. Na2SO4 rejection, R, as a function of volume flux density, jv, for a positively charged membrane (ξ ) Xm/cf ) +10, leff ) 50 µm, and rp ) 0.5 nm): exact analytical calculations (circles); enhanced GCE results (upper solid line); enhanced SK-GCE results (lower solid line).

GCE, and SK-GCE results. A nonnegligible discrepancy, however, is observed for intermediate values of jv, where the SK-GCE results leads to an underestimate. For Na2SO4 (Figure A1, part c) under the conditions studied, the GCE itself is not valid. As already stated, we can extend the domain of applicability of the GCE approximation simply by using the exact result for Tlim instead of the GCE value, TGCE lim in TGCE(jv). This approach will be called the enhanced GCE approximation. (Exact analytic results for Tlim have been obtained in ref 1 for symmetric and certain asymmetric (1:2 and 2:1) salts.) We illustrate this point in Figure A2, where we show the exact, enhanced GCE, and enhanced SK-GCE rejection as a function of jv for Na2SO4 and a positive normalized membrane charge density ξ ) Xm/cf ) +10, an effective thickness leff ) 50 µm, and a pore radius rp ) 0.5 nm. (In this case, the exact limiting rejection is Rexact lim ≈ 0.253 and therefore Texact lim ≈ 0.747.) Although the GCE and SK-GCE approximations are about 30% off the exact rejection (Figure A1, part c), the enhanced GCE results are extremely accurate (Figure A2). As expected, the enhanced SK-GCE results still reveal a nonnegligible discrepancy at intermediate volume flux. Nomenclature ci, cji ) bulk and intramembrane concentrations (ion i), M C, Cs ) virtual salt and salt concentrations, M h i ) bulk and intramembrane diffusion coefficients (ion Di, D i), m2 s-1 F ) Faraday constant (F ) 96 500 C mol-1) I ) ionic strength (salt, total), M jv ) volume flux density, m s-1 or L h-1 m-2 ji ) molar flux density (ion i), mol-1 m-2 s-1 kf(p) ) feed (permeate) partition coefficient i Ki,c, Ki,d ) convective and diffusive hindrance factors L0p ) pure water hydraulic membrane permeability, L h-1 -2 m bar-1 Lsp ) salt solution hydraulic membrane permeability, L h-1 -2 m bar-1 lm ) active NF membrane layer thickness, µm leff ) effective active NF membrane layer thickness (leff ) lmτ/φp), µm P ) average fluid pressure, bar ∆P ) transmembrane pressure, bar Pe ) Pećlet number ri ) ion radius, m rp ) effective pore radius, m R ) ideal gas constant (8.314 ) J mol-1 K-1) R,Ri ) rejection (salt, ion i), none or %

Subscripts and Supscripts b ) bulk solution c ) convection d ) diffusion f ) feed i ) ion, i lim ) limiting (high flux) m ) membrane p ) permeate or pore s ) solute or salt Greek Letters j i ) bulk and intramembrane activity coefficient (ion, i) γi, γ ∆ ) finite difference (final - initial) ∆ ) finite difference (initial - final) ∆φf(p) D ) feed (permeate) Donnan potential, mV -∆φlim ) limiting y-intercept of electric filtration potential, mV ∆ΦF ) electric filtration potential (∆ΦF ) Φf - Φp), mV ) water dielectric constant (6.933 × 10-10 C2 J-1 m-1) ζ ) ζ-potential, mV η,ηeff ) dynamic water and effective electrolyte viscosity, Pa s φ ) average (mesoscopic) electric potential, mV φ˜ ) average dimensionless electric potential (φ˜ ) Fφ/RT) φd ) electric diffusion potential, mV φs ) electric streaming potential, mV φf(p) ) bulk feed and permeate electric potential, mV Φs, Φsi ) ion steric partition factor (Φsi ) (1 - λi)2) λi ) relative ion size (λi ) ri/rp) λD ) Debye length based on ionic strength, m λm ) Debye length based on effective membrane charge density, m h i/Ki,c), µˆ i ) intramembrane effective mobility (ion i) (µi ) ziD m2 s-1 κ ) electroviscous coefficient φp ) membrane porosity, none or % νi ) stoichiometric coefficient (ion i) νj ) reduced (flux) streaming potential, mV(L h-1 m-2)-1 Π ) osmotic pressure (bar) ξ ) normalized membrane charge density (ξ ) Xm/cf) ξˆ ) normalized membrane charge density (ξˆ ) Xm/(ν1|z1|cf)) F ) ion charge density, C m-3 σS ) local salt reflection coefficient σek ) electrokinetic surface charge density, C m-2 σw ) pore wall surface charge density, C m-2 Σ ) global osmotic reflection coefficient for an electrolyte mixture τ ) tortuosity τV ) |z2|/|z1| Acronyms ENP ) extended Nernst-Planck ET ) electrotransport GCE ) good co-ion exclusion HT ) hindered transport HET ) hindered electrotransport NF ) nanofiltration

5540 J. Phys. Chem. B, Vol. 109, No. 12, 2005 References and Notes (1) Lefebvre, X.; Palmeri, J.; David, P. J. Phys. Chem. B 2004, 108, 16811. (2) Dresner, L. Desalination 1972, 10, 27. (3) Dresner, L. J. Phys. Chem. 1972, 76, 2256. (4) Sonin, A. A. Osmosis and ion transport in charged porous membranes: macroscopic mechanistic model. In Charged Gels and Membranes; Se´le´gny, E., Ed.; Reidel: Dordrecht, 1976; Vol. 1, p 255. (5) Hijnen, H. J. M.; Van Daalen, J. V.; Smit, J. A. M. J. Colloid Interface Sci. 1985, 107, 525. (6) Smit, J. A. M. J. Colloid Interface Sci. 1989, 132, 413. (7) Yaroshchuk, A. E. AdV. Colloid Interface Sci. 1995, 60, 1. (8) Palmeri, J.; Blanc, P.; Larbot, A.; David, P. J. Membr. Sci. 1999, 160, 141. (9) Palmeri, J.; Blanc, P.; Larbot, A.; David, P. J. Membr. Sci. 2000, 179, 243. (10) Deen, W. M.; Satvat, B.; Jamieson, J. M. Am. J. Physiol. 1980, 238, F126. (11) Wang, X.-L.; Tsuru, T.; Togoh, M.; Nakao, S.; Kimura, S. J. Chem. Eng. Jpn. 1995, 28, 372. (12) Bowen, W. R.; Mukthar, H. J. Membr. Sci. 1996, 112, 263. (13) Schlo¨gl, R. Ber. Bunsen-Ges. Physik. Chem. 1966, 70, 400. (14) Palmeri, J.; Sandeaux, J.; Sandeaux, R.; Lefebvre, X.; David, P.; Guizard, C.; Amblard, P.; Diaz, J.-F.; Lamaze, B. Desalination 2002, 147, 231.

Lefebvre and Palmeri (15) Lefebvre, X.; Palmeri, J.; Sandeaux, J.; Sandeaux, R.; David, P.; Maleyre, B.; Guizard, C.; Amblard, P.; Diaz, J.-F.; Lamaze, B. Sep. Purif. Technol. 2003, 32, 117. (16) Starov, V. M.; Churaev, N. V. AdV. Colloid Interface Sci. 1993, 43, 145 (17) Dresner, L. Desalination 1974, 15, 39. (18) Hagmeyer, G.; Gimbel, R. Desalination 1998, 117, 247. (19) Yaroshchuk, A. E. AdV. Colloid Interface Sci. 2000, 85, 193. (20) Yaroshchuk, A. E. J. Membr. Sci., 2000, 167, 163. (21) Labbez, C. Etude du transport et de la re´tention des solute´s neutres et ioniques par le mode`le DSPM: membranes de Nano et d'Ultrafiltration fine. Ph.D. Thesis, Universite´ de Franche-Comte´, Besanç on, France, 2002. (22) Spiegler, K. S.; Kedem, O. Desalination 1966, 1, 311. (23) Hoffer, E.; Kedem, O. Desalination 1967, 2, 25. (24) Neogi, P.; Ruckenstein, E. J. Colloid Interface Sci. 1981, 79, 159. (25) Cwirko, E. H.; Carbonell, R. J. Colloid Interface Sci.1989, 129, 513. (26) Wang, X.-L.; Tsuru, T.; Nakao, S.; Kimura, S. J. Membr. Sci. 1995, 103, 117. (27) Labbez, C.; Fievet, P.; Thomas, F.; Szymczyk, A.; Foissy, A. J. Colloid Interface Sci, 2003, 262, 200. (28) Hagmeyer, G.; Gimbel, R. Sep. Purif. Technol. 1999, 15, 19. (29) Labbez, C.; Fievet, P.; Szymczyk, A.; Thomas, F.; Simon, C.; Vidone, A.; Foissy, A.; Pagetti, J. Desalination 2002, 147, 223. (30) Hunter, R. J. Foundations of Colloid Science; Clarendon Press: Oxford, 1987/1989; Vol. 1. (31) Deen, W. M. AIChE J. 1987, 33, 1409.

Nanofiltration Theory: Good Co-Ion Exclusion Approximation for

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