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Hafnia ceramic nanofiltration membranes Part II. Modeling of pressure-driven transport of neutral solutes and ions J. Palmeri∗ , P. Blanc, A. Larbot, P. David Laboratoire des Matériaux et Procédés Membranaires (LMPM) ENSCM, CNRS UMR 5635, Université de Montpellier II, 8, Rue de l’Ecole Normale, 34296 Montpellier Cedex 5, France Received 6 November 1997; received in revised form 20 June 2000; accepted 21 June 2000

Abstract We use the hindered transport theory for neutral solutes and the macroscopic homogeneous electro-transport theory for ions to predict without any adjustable parameters the limiting rejection of, respectively, four neutral molecules and five binary electrolytes by granular porous hafnia nanofilters. Membrane parameters entering into the theory, namely the membrane effective pore size and effective charge density, are estimated from independent measurements performed on the membrane material. The theory is in reasonably good overall agreement with the rejection measurements performed on hafnia nanofilters using uncharged sucrose and PEG (400, 600, 1000) molecules and 10−3 M electrolytes (NaCl, NaNO3 , CaSO4 , CaCl2 , and Na2 SO4 ) at acidic, neutral, and basic pH values (3, 6, and 9). © 2000 Elsevier Science B.V. All rights reserved. Keywords: Nanofiltration; Liquid permeability and separations; Ceramic membranes; Microporous and porous membranes; Theory

1. Introduction Ceramic membranes, with comparatively high thermal, mechanical, and chemical resistances, represent an important class of inorganic membranes with many current and potential applications in crossflow microfiltration, ultrafiltration, and recently also nanofiltration (NF, micropores, pore diameter dp < 2 nm) [1–4]. Many, but not all [1,5], ceramic membranes synthesized by a sol–gel process from metal oxides like alumina [6,7], titania [8], zirconia [9], or hafnia [10] can be modeled at the pore level as charged granular porous media with the passages between ∗ Corresponding author. Tel.: +33-4-67-14-72-32; fax: +33-4-67-14-43-47. E-mail address: [email protected] (J. Palmeri).

the roughly spherical grains forming interconnected funnel-shaped pores [11–15]. Compared with the large number of studies devoted to the older, more numerous, and more readily available organic nanofilters (see, e.g. [16–23]), there are relatively few detailed and comprehensive combined experimental and theoretical studies of the transport behavior of ceramic NF membranes (see, however, [7,14,15,24–28]). This situation is related in part to the relatively recent availability of these membranes. The delay was due in part to the inherent practical difficulties that need to be overcome in order to prepare lightly sintered granular porous membranes with grains sizes in the 5–10 nm range and a highly connected porosity. In this work we model ionic and neutral solute transport in hafnia ceramic nanofiltration membranes. Our purpose here is to apply the transport theory

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Nomenclature ai and bi (i = 1, 2) Ci Cis cf cp cif p ci c¯i (x) D¯ i Di Ds D¯ s dp F I I ji jV kif f k2(z:z)

f k2(1:2)

f k2(2:1)

`

constants appearing in the Langmuir specific ion adsorption formula bulk ionic concentrations used in electrophoretic mobility measurements (M) ionic concentrations at the agglomerate shear plane (M) feed concentration (M) permeate concentration (M) ionic feed concentration (M) ionic permeate concentration (M) ionic concentration in the membrane (M) ionic diffusion coefficients in the membrane (cm2 /s) ionic diffusion coefficients in bulk solution (cm2 /s) neutral solute or salt diffusion coefficient (cm2 /s) neutral solute or salt diffusion coefficient in the membrane (cm2 /s) effective membrane pore diameter (cm) Faraday constant (96 487 C/mol) macroscopic electric current in the membrane (C/(cm2 s)) ionic strength (M) macroscopic ionic flux densities (mol/(cm2 s)) macroscopic volume flux density (cm/s) Donnan partition coefficients at the feed–membrane interface Donnan co-ion partition coefficient at the feed–membrane interface for a symmetric (z:z) electrolyte Donnan co-ion partition coefficient at the feed–membrane interface for an asymmetric (1:2) electrolyte Donnan co-ion partition coefficient at the feed–membrane interface for an asymmetric (2:1) electrolyte membrane thickness (cm)

Pecp Pem (pH)mf (pH)s R R(GCE) Rweak Rz:z rH rp rs T ti Xm x zi

P´eclet number (concentration polarization layer) P´eclet number (membrane) pH at x = 0+ in the membrane pH at the agglomerate shear plane limiting electrolyte rejection limiting electrolyte rejection at high membrane charge (good co-ion exclusion) limiting electrolyte rejection at weak membrane charge limiting electrolyte rejection for a symmetric (z:z) electrolyte membrane pore hydraulic radius (cm) effective membrane pore radius (cm) solute Stokes radius (cm) temperature (K) ionic transport number effective membrane charge density (M) transverse position coordinate in the membrane (cm) ionic electrochemical valence, i = 1 (counter-ion) and i = 2 (co-ion)

Greek letters αi and βi (i = 1, 2) constants appearing in the Langmuir specific ion adsorption formula α same as α1 δ thickness of the concentration polarization layer (cm) dielectric constant of the solution (4.327 × 104 e2 /(mV cm)) Φ(x) macroscopic electric potential in the membrane (mV) Donnan potential at the 1ΦDf membrane–feed interface (mV) p Donnan potential at the 1ΦD membrane–permeate interface (mV) dimensionless Donnan potential at 1Φ˜ Df the membrane–feed interface

J. Palmeri et al. / Journal of Membrane Science 179 (2000) 243–266 p 1Φ˜ D

Γ

λ λD λm µ¯ 0 µE νi σeff

σek

σf σw σ∗ ξeff

ξm ζ ζ˜

dimensionless Donnan potential at the membrane–feed interface auxiliary function used in the formula for the Donnan co-ion partition f coefficient, k2(2:1) rs /rp Debye length based on the bulk ionic strength (cm) Debye length based on the effective membrane charge density, Xm (cm) RT/(ηF ), reference electrophoretic mobility (1.8(m/s)/(V/cm)) electrophoretic mobility ((m/s)/(V/cm)) electrolyte stoichiometric coefficients effective surface charge density appearing in the specific counter-ion adsorption model (C/cm2 ) electrokinetic surface charge density (derived from electrophoretic mobility) (C/cm2 ) neutral solute reflection coefficient in the capillary pore hindered transport model membrane pore wall effective surface charge density (C/cm2 ) dimensionless pore wall surface charge density effective normalized membrane charge density appearing in the specific counter-ion adsorption model normalized membrane charge density zeta potential (mV) reduced zeta potential

Superscripts f feed

mf p

245

membrane–feed interface permeate

Subscripts B Boltzmann D Donnan or Debye–Hückel eff effective ek electrokinetic (GCE) good co-ion exclusion H hydraulic radius i ith ion (i = 1, counter-ion; i = 2, co-ion) m membrane p pore s shear plane or solute V volume flux w pore wall weak weak membrane charge (z:z) symmetric (z:z) electrolyte (1:2) asymmetric (1:2) electrolyte (2:1) asymmetric (2:1) electrolyte developed in [15] to the experimental results presented in [10]. In order to do so, we treat the hafnia nanofiltration membranes prepared and studied in our laboratory as consolidated granular porous media and use the previously developed continuum hydrodynamic pore-level transport models (without any adjustable parameters) to account for the observed limiting rejection of neutral and charged species. For neutral solutes we employ a hindered transport model and for ions we use a macroscopic electro-transport theory that is the homogeneous limit of the pore-level electrokinetic space–charge model [15]. The hafnia nanofilters studied in [10] are lightly sintered nanophase materials possessing pores (the intergranular passages) with diameters in the nanometer range. Since in this case the pore size is close to the size of small neutral organic molecules, the strong steric and hydrodynamic interactions between these small organic molecules and the granular membrane structure lead to high solute rejections and relatively low molecular weight cut-offs (MWCO) in the order of 500 Da. Furthermore, the pore walls of these ceramic membranes become charged due to the amphoteric behavior of metal oxide surfaces. In microporous NF membranes this surface charge can, at sufficiently low electrolyte concentration, give rise to a strong electri-

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cal interaction between the membrane and the ions in solution, leading to high limiting electrolyte rejections despite ionic ‘sizes’ much smaller than the pore size. The work presented here is part of an ongoing project whose aim is to better understand, using fundamental transport theory, the relationships between a ceramic membrane’s microstructure on the one hand and its macroscopic transport (separation and transfer) properties on the other. Although the usefulness of similar projects in the MF and UF ranges has often been seriously undermined by the almost uncontrollable complications arising from fouling phenomena, we believe that fundamental transport modeling on the contrary might be of great use in practical nanofiltration applications. The reason for this optimism is that NF membranes are often called upon to perform more delicate tasks than their larger pore counterparts (such as metal recovery from dilute multi-electrolyte mixtures), and therefore the fouling and related phenomena that complicate matters in realistic MF and UF separation processes can in certain cases be unimportant in the NF range. This paper is organized as follows: in Section 2 we first present a short discussion of the grain consolidation model developed in [15] to model the porous granular structure of certain ceramic membranes. This model allows us to relate the directly measurable pore hydraulic radius to the effective pore size. We then use the hindered transport theory for neutral solutes to predict their limiting rejection (obtained at high volume flux) as a function of the ratio of solute radius to effective pore radius. In Section 3 we show how the electrophoretic mobility, measured on the same ceramic powders used to prepare the membranes, can be used together with the measured pore hydraulic radius to estimate the effective membrane charge density, a crucial parameter entering into the electro-transport theory. In Section 4 we use an electro-transport theory derived as the homogeneous limit of the 3D space–charge model to model the limiting rejection of symmetric and asymmetric binary electrolytes. Using the effective membrane charge densities estimated from electrophoretic mobility and pore hydraulic radius measurements as input, we compare our theoretical results with the measured limiting rejection of five binary electrolytes at a feed concentration of 10−3 M and feed pH values of 3, 6, and 9. It is shown that the electrolyte rejection patterns obtained with the hafnia nanofilter match

reasonably well the signatures expected [15] for amphoteric membranes that reject ions by electrostatic interactions alone. Finally, in Section 5, we present a short discussion and summarize our main results. The results presented here suggest that ceramic NF membranes can be usefully modeled as consolidated granular porous media in which pore level continuum transport theories can be used to estimate the rejection of neutral solutes and ions. A brief account of the results presented here appears in [14]. 2. Neutral solute rejection In this section we treat the hafnia ceramic membranes as granular porous media and use the cylindrical capillary pore hindered transport theory to predict the limiting rejection of four neutral solutes. The sintered granular structure and narrow grain size distribution of the hafnia nanofilters under study are visible in the scanning electron micrographs presented in Figs. 9 and 10 of [10]. These figures, which should be compared with Fig. 1 of [11] and Fig. 1 provide justification for our use of the spherical grain consolidation model to model the morphology of the NF layer.

Fig. 1. A portion of a random, monodisperse, packing of grains used to model a porous granular membrane.

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In [15] we have argued that if the ceramic membrane can be modeled as a sintered packings of nearly monodisperse, roughly spherical grains, then the effective membrane pore radius rp , can be estimated from the hydraulic radius, rH = pore volume/pore surface area, using rp ≈ 1.4rH ,

(1)

over the porosity range of 20–50% commonly encountered in ceramic membranes. The validity of Eq. (1) has been verified theoretically using regular and random model packings of spherical grains. As discussed in [15], a well-defined rp can be found even for a random packing of grains by mapping the 3D connected pore structure onto a discrete effective ‘resistor’ network [15]. Since the hydraulic radius of the hafnia membrane has been estimated from nitrogen adsorption–desorption experiments (Section 2.2.2.2 of [10]) to be rH ≈ 0.75 nm, the effective pore radius, rp , is found from Eq. (1) to be approximately equal to 1 nm. In [15] we have explained in detail how the cylindrical pore hindered transport theory can be used to estimate the limiting rejection of neutral solutes at high volume flux provided that the capillary pore radius is taken equal to rp . For granular porous membranes, the neutral solute limiting rejection is, therefore, assumed to be approximately equal to the reflection coefficient, σf , of a hard sphere solute particle in a cylindrical capillary pore with radius rp . In comparing theory with experiment, we adopt the following assumptions: 1. At a working pressure of 15.5 bar, the relatively high volume flux densities of jV ∼ 30 l/(h m2 ) observed imply that we are in the high Péclet number limit and the observed rejections are equal to the limiting ones. This can be seen by estimating the Péclet number in the membrane, Pem = jV `/D¯ s , where ` ∼ 0.150 m is the membrane thickness. Since the ratio of effective solute diffusivity in the membrane to the one in bulk solution, D¯ s /Ds , is generally of order ∼10−4 –10−3 (due to tortuosity and enhanced hindered diffusion effects) with Ds ∼ 10−5 cm2 /s, Pem is clearly much bigger than one [27]. Since, for the hafnia nanofilters studied here, the solute rejection was not measured over a range of volume fluxes, in this particular case we can only

247

present indirect evidence for the validity of the high Péclet number assumption. We do this as follows: the hafnia nanofilter has a structure and rejection properties closely resembling those of the zirconia nanofilter studied in [27]. If, as usual, the solute flux densities are based on total membrane area, then the effective solute diffusivity D¯ s is given by D¯ s = φp τ −1 H (rs /rp )Ds , where φp is the membrane porosity, τ the tortuosity, and H (rs /rp ) the hydrodynamic hindered transport coefficient [30]. If we use the estimate φp ∼ 0.3 [29], typical of ceramic nanofilters, and use the hydrodynamic hindrance coefficient calculated for neutral spheres in cylindrical pores [30], the vitamin B12 (rs ≈ 0.74 nm) rejection versus volume data presented in [27] lead to the theoretical estimate τ ∼ 30. This value for the tortuosity is greater than the values, τ ∼ 5–10, typically found for macroporous granular membranes. Although the theoretical explanation for this low value of tortuosity has not yet been completely elucidated, we have found that it is typical of the ceramic nanofilters studied in our laboratory. The experimental and theoretical results for the rejection of vitamin B12 presented in [27] for nanofilters of two very different average thicknesses and the same effective pore size (rp ≈ 1.2 nm) provide direct evidence that this high value of tortousity (τ ∼ 30) is linked to the nanofiltration layer, and not an artifact of the mesoporous support: the experimental rejection versus volume flux curves change in just the way expected theoretically when the membrane thickness increases from ` ≈ 45 to 125 nm (see Fig. 11 of [27]). The tortuosity of 30 has been obtained by using the estimated value of D¯ s /Ds of 2 × 10−4 [27] and λ = rs /rp ≈ 0.62 in the definition of τ τ = φp H (λ)

Ds . D¯ s

(2)

Using the value of tortuosity (τ = 30) obtained above for zirconia nanofilters, we can simulate the neutral solute rejection versus volume flux curves for the hafnia nanofilters (rp ≈ 1 nm) under study here. In Fig. 2 we plot the theoretical predictions for the rejection of sucrose (rs ≈ 0.47 nm) as a function of volume flux for six different values of tortuosity. The results presented in Fig. 2 clearly

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Fig. 2. Simulated sucrose rejection as a function of volume flux density for rp = 1 nm and different values of tortuosity (τ = 5, 10, 15, 20, 25, and 30 in order of increasing rejection).

show that if we use the tortuosity of 30 typical of ceramic nanofilters, even for the smallest neutral solute studied [Ds ≈ 0.52 × 10−5 cm2 /s, λ ≈ 0.47 and H (λ) ≈ 0.055], we have very nearly attained the asymptotic limiting rejection at the observed volume flux of jV ∼ 30 l/(h m2 ) since Pem ≈ 4.3. Since for the bigger neutral solutes we will be even deeper into the asymptotic regime, we can, therefore, safely assume that the measured rejection is the limiting one when the volume flux takes on the observed value of jV ∼ 30 l/(h m2 ). 2. At the experimental working pressure and volume flux, concentration polarization effects are negligible, which implies that the observed rejections are the intrinsic membrane ones. This can be seen by comparing the Péclet number, Pecp = jV δ/Ds , in the laminar concentration polarization layer (CP), located at the feed–membrane interface, with the Péclet number, Pem , in the membrane. Under typical tangential cross-flow operating conditions, the ratio of CP layer to membrane thickness, δ/`, is in the order of ∼10. Using the values of D¯ s /Ds given in item 1 above, we see that the ratio of Péclet numbers obeys the inequality Pecp /Pem 1. The observed rejection can, therefore, closely approach the intrinsic limiting value at the experimental volume flux (Pem large), while CP effects remain unimportant (Pecp small). The validity of the above arguments has been confirmed by the experimental and theoretical results presented in [27] for the zirconia nanofilters cited earlier. Further evidence for weak CP effects comes from the experimen-

Fig. 3. Neutral solute Stokes radii, rS , as a function of molecular weight (MW) (Da): experimental data for PEG 1500 (MW = 1500), PEG 4000 (MW = 3000), PEG 6000 (MW = 7500); solid curve, theoretical extrapolation used to estimate rS for PEG 400 (MW = 400), PEG 600 (MW = 600), PEG 1000 (MW = 1000).

tally observed stability of the solute rejection as a function of time (see [31] and Fig. 14 of [10]). 3. A solute molecule can be modeled as a hard sphere whose effective size can be estimated from the solute Stokes radius. 4. The Stokes radii of the PEG series of molecules used here can be estimated from extrapolation of results for a heavier PEG series presented in [32]: the approximate relation used is rs (PEG) ≈ 0.095(MW)1/3 nm (see Fig. 3), where MW is the PEG molecular weight. This type of relation arises from modeling a roughly spherical organic molecule as a sphere of uniform density, which implies that mass ≈ density ×4π rs3 /3, and therefore rs ∼ (MW)1/3 . Recent direct measurements [33] of the free diffusivities of PEG 400, 600, and 1000, when interpreted using the Stokes–Einstein relation for solid spherical solutes, indicate that the estimates presented in Table 1 are only about 10% too low . This in turn implies that our estimate of the effective pore radius (1 nm) is also only about 10% too low. In Fig. 4 we present the theoretical limiting rejection (filtration reflection coefficient, σf ) as a function of λ = rs /rp for the capillary pore hindered transport model [30]. Also presented in this figure are the observed rejections of four neutral solutes measured on the hafnia membrane [10]. The theoretical predictions [15], which do not contain any adjustable parameters, are in excellent agreement with the experimental data. The sucrose rejection simulations presented in Fig. 2 for τ = 30 suggest that the small difference

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249

Table 1 Hydraulic radius and pore size of the hafnia membrane and Stokes radii of neutral solutes (sucrose and PEG series) used in the rejection measurementsa rH b

rp c

0.75

1

Stokes radius, rS Sucrose

PEG 400d

PEG 600d

PEG 1000d

PEG 1500e

PEG 4000e

PEG 6000e

0.47

0.7

0.8

0.95

0.94

1.3

2.0

a

All lengths in nm. Determined from nitrogen adsorption–desorption measurements at 77 K. c Determined using Eq. (1). d Estimated by extrapolation from experimental results presented in [32]. e Experimental results presented in [32]. b

of about 7% seen in Fig. 4 between the theoretical (0.54) and experimental (0.58) rejections arises in this particular case because at the observed volume flux of jV ∼ 30 l/(h m2 ), we are not quite yet completely in the limiting flux regime. (Fig. 2 also shows that if τ had taken on a value typical of macroporous granular porous media, τ ∼ 5–10, then at an observed volume flux of jV ∼ 30 l/(h m2 ), we would have been very far from the limiting flux regime.) These neutral solute results provide strong evidence that the hafnia membrane does indeed behave as a granular porous nanofilter with a characteristic pore radius of ∼1 nm and lends credence to our use of a continuum hydrodynamic transport theory at the micropore level. Once the solute size, rs , is known, only one structural measurement (for rH ) is needed to characterize the effective membrane pore size, rp , and predict the limiting rejection of neutral solutes.

Fig. 4. Neutral solute rejection as a function of λ = rS /rp : hafnia nanofilter experimental data for sucrose (MW = 342), PEG 400 (MW = 400), PEG 600 (MW = 600), and PEG 1000 (MW = 1000); spherical solute rejection coefficient, σf (solid curve), predicted by the capillary pore hindered transport theory.

3. Electrophoretic mobility and membrane charge In order to apply the electro-transport theory developed in [15] to the hafnia nanofilters studied in [10], we need to estimate the effective membrane charge density, Xm ≡

σw , FrH

(3)

(in moles per unit pore volume), which depends only on the membrane hydraulic radius, rH , and the effective surface charge density on the pore walls, σw . At high volume flux (Pem 1) where ionic diffusion can be neglected, the limiting rejection of a binary electrolyte depends on the effective membrane charge density, σw [c¯1 (0+ ), c¯2 (0+ ), (pH)mf ], just inside the membrane at the feed interface (at x = 0+ ), where c¯1 (0+ ) and c¯2 (0+ ) are the average (macroscopic) counter-ion and co-ion concentrations and (pH)mf the pH just inside the membrane (see Fig. 4 of [15]). As already stated (Section 2), rH can be estimated from nitrogen adsorption–desorption measurements. As we now discuss, σw can be estimated using a working assumption developed in [15] relating the pore wall surface charge to the electrokinetic surface charge density, σek , obtained by measuring the electrophoretic mobility of the hafnia powder particles used to prepare the membranes in a bulk solution having the same pH and ionic concentrations as the feed solution used in the filtration experiments. Our working assumption, therefore, implies that, despite the possible differences between the local electrolyte environment at the point x = 0+ just within the membrane and the one at the shear plane of the agglomerate, we estimate the pore wall charge

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density just inside the membrane by the electrokinetic charge density of the powder particles under the same bulk conditions σw [c¯1 (0+ ), c¯2 (0+ ); (pH)mf ]≈σek [C1s , C2s ; (pH)s ], (4) where C1s , C2s , and (pH)s are, respectively, the ionic concentrations and pH at the shear plane of the ‘particles’, or agglomerates, used to measure the electrophoretic mobility. If the membrane is charged, then the powder particle zeta-potential and the feed interface Donnan potential are non-zero. Due to the strongly overlapping double layers in the pores of nanofiltration membranes, these two potentials take on, in general, different values (see Section 4). Our working assumption (Eq. (4)) is then clearly an approximation because the pH and ionic concentrations at the shear plane (beginning of the diffuse part of the electrical double layer) are different from both their bulk values and their values just inside the membrane. With the above assumption (Eq. (4)) the pore wall surface charge density and, therefore, also the effective normalized charge density, |Xm | (5) ξm = f , c appearing in the electro-transport theory (Appendix A) can be independently estimated from electrophoretic mobility measurements. Consequently, the dependence of the effective membrane charge density Xm on the pH and ionic concentrations in the membrane at x = 0+ need not be explicitly specified. As shown in Section 4, this working assumption (Eq. (4)) allows us to account in a reasonable way for almost all of the data obtained by filtering five binary electrolytes through a ceramic hafnia nanofilter at three different pH values. Extensive granulometric and electrophoretic mobility measurements [35] on the powders used to prepare ceramic NF membranes have shown that the size of the powder ‘particles’ used to characterize the grain surface charge are generally a factor of 50–100 bigger than the size of the individual grains (∼5–10 nm in diameter) making up the granular porous structure of an NF membrane. We are thus led to conclude that the powder particles whose mobility is measured are in reality agglomerates, composed of many individual grains. These agglomerates have effective sizes,

denoted by the radius, Ragg , in the order of several hundred nanometers. This observation suggests that we make the plausible simplifying assumption that the agglomerates can be modeled as impermeable, although not necessarily spherical, particles with smooth surfaces and surface charge densities equal to the surface charge density of the individual hafnia grains (cf. [36]). Due to large sizes of the powder particles and the relatively high electrolyte concentrations involved the thin double layer approximation is valid (λD Ragg , whereP λD = (RT/2F 2 I)1/2 is the Debye length, I = (1/2) i zi2 Ci the ionic strength, and Ci the ionic concentration in the bulk solution). Moreover, the surface charge density on each individual hafnia grain turns out to be sufficiently weak that the weak ζ -potential approximation also holds; i.e. the reduced zeta potential, ζ˜ = (F ζ /RT), satisfies z1 ζ˜ < 2. Accordingly, for interpreting the electrophoretic mobility measurements, the combined thin double layer and weak ζ -potential approximations allow us to use the classical Smoluchowski equation [34], µE ≈

ζ , η

(6)

to relate the ζ -potential to the electrophoretic mobility, µE , and to adopt the Debye–Hückel approximation in an open plane geometry to relate the electrokinetic surface charge density, σek , to the ζ -potential σek ≈

ζ . λD

(7)

For numerical estimates at room temperature, the above results can be used to derive the following convenient formulae relating the ζ -potential and electrokinetic surface charge density to the measured electrophoretic mobility: ζ˜ ≈

[µE ] µE = , µ0 1.8

(8)

where [µE ] is measured in (m/s)/(V/cm) with µ0 ≡ (RT/ηF ) ≈ 1.8(m/s)/(V/cm) and p (9) [σek ] ≈ 3.0[µE ] [I], where [σek ] is measured in C/cm2 and [I] in mol/l. Now that we have the desired relation (Eq. (9)) between the measured electrokinetic quantity, µE , and

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Fig. 5. Membrane pore wall surface charge density, σw , as a function of the bulk (feed) solution pH (estimated from the electrokinetic surface charge density, σek , deduced from the measured electrophoretic mobilities).

the sought after surface charge density, σek , we can finally find a formula, based on our working assumption (Eq. (4)) and the above approximations, that permits us to translate electrophoretic mobility data into normalized effective membrane charge densities (Eqs. (3) and (5)): √ [µE ] [I] [σek ] Xm ≈ 0.30 , (10) ≈ 0.10 cf [rH ][cf ] [rH ][cf ] with [I] in mol/l; [µE ] in (m/s)/(V/cm); [rH ], the membrane hydraulic radius in nm; and [cf ], the electrolyte feed concentration used in the rejection experiments (in mol/l). Armed with these results we now turn to comparing the theoretical predictions for electrolyte rejection with experiment. Using Eq. (9) and the working assumption Eq. (4), the electrophoretic mobility data obtained in [10] can be translated into pore wall surface charge as a function of feed pH for each of the electrolytes studied (see Fig. 5).

4. Electrolyte rejection: comparison of theory with experiment In this section we compare the theoretical predictions of the homogeneous electro-transport model developed in [15] with the experimental results obtained by filtering three symmetric (NaCl, NaNO3 , and CaSO4 ) and two asymmetric electrolytes (CaCl2 and Na2 SO4 ) through a hafnia nanofilter [10]. The homogeneous approximation to the space–charge electro-transport theory is valid when there is strong overlap of electrical double layers in the pores. In this

251

case local quantities such as the pore-level electric potential and ionic concentrations differ only slightly from their spatial average values, and therefore the 3D space–charge model reduces to a macroscopic extended Nernst–Planck (ENP) electro-transport theory (see Appendix A). We have previously [15] demonstrated that the homogeneous approximation used here for 3D granular porous membranes is valid when the dimensionless membrane surface charge density, σ∗ ≡

|z1 |F 4RT

|σw | rp

=

1 2

rH rp

rp λm

2 , (11)

which determines the normal derivative of the electric potential at the pore wall via Gauss’s law, is smaller than one. This condition for the validity of the homogeneous approximation can be reformulated in terms of the ratio of the effective pore size, rp , and an effective Debye length, λm , based on the membrane fixed charge density: rp /λm < 1.7 where λm = (2RT/|z1 |F 2 |Xm |)1/2 . The theoretical predictions for each electrolyte are presented separately in Fig. 6a–e along with the experimental rejection results obtained in [10] for a fixed electrolyte feed concentration, cf = 10−3 M, and different values of feed pH. For a given salt and feed pH, the value of the effective membrane charge associated with the each measured rejection exhibited in Fig. 6a–e has been estimated using Eq. (10). Along with the exact homogeneous theory results for the limiting rejection, R, we plot both the weak membrane charge approximation (Eq. (B.1), Appendix B), which is valid near the IEP, and the GCE approximation, which is valid for large ξm (Eq. (B.4), Appendix B) [15]. On the one hand, the weak charge approximation, although quantitatively accurate only in a small region near the IEP, allows us to gauge the importance of negative rejection. The GCE approximation, on the other hand, is quantitatively accurate for ξm > 5–10 depending on the electrolyte. The utility of the simple GCE approximation is evident from Fig. 6a–e, since a majority of the experimental data points are accurately given by this approximation. For certain electrolytes there are relatively large errors in the estimates of the experimentally determined electrokinetic surface charge density near the IEP where Xm = 0, due to the sensitive dependence of

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Fig. 6. Theoretical predictions for the limiting rejection, R, as a function of normalized membrane charge density, Xm /cf : exact homogeneous theory results (solid curves), weak charge homogeneous results (dot-dash curves), homogeneous good co-ion exclusion (GCE) results (dashed curves); data points are the experimental results for 10−3 M electrolytes obtained with the hafnia nanofilter: (a) NaCl, (b) NaNO3 , (c) CaSO4 , the data point × (pH 9.3) uses the renormalized membrane charge density predicted by the specific counter-ion adsorption model (Section 4.1.2), (d) CaCl2 , and (e) Na2 SO4 .

σw ≈ σek on pH (Fig. 5). Near the IEP the steepness of the slope of the measured electrophoretic mobility versus pH curves (Figs. 5–7 and Table 2 of [10]) implies that a small error in the pH measured in the filtration experiments can lead to relatively large errors in the µE values obtained by interpolation of the experi-

mental data. It is important to note that it is impossible to define a unique IEP for the hafnia membrane: the variation of the effective pore wall surface charge density as a function of pH, which was deduced from the electrophoretic mobility measurements in Section 3 (Fig. 5), clearly shows that the IEP, if it exists, depends

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Fig. 7. Membrane pore wall surface charge density, σw , for NaCl as a function of the bulk (feed) solution pH (estimated using the electrokinetic surface charge density, σek , deduced from the measured electrophoretic mobilities): 10−3 M NaCl bulk (feed) concentration (solid curve); 10−2 M NaCl bulk (feed) concentration (dashed curve).

on the choice of electrolyte (and the bulk electrolyte concentration, see below). Nevertheless, by way of example, we can say that the IEP for electrolytes not containing the divalent sulfate ion is between 6 and 7. For electrolytes containing the sulfate ion, however, the effective surface charge density is negative or zero over the whole range of pH studied, and therefore no IEP exists. At low pH, where one would normally expect a positive membrane charge, this behavior is presumably due to the strong specific sulfate adsorption on the protonated amphoteric groups. In order to simplify the comparison between the theoretical predictions and the experimental results, we have adopted the following approximations [15]: 1. At the working pressure of ∼15 bar and measured volume fluxes of ∼30 l/(h m2 ), the observed rejections are the limiting ones obtained at high Péclet number where ionic diffusion is negligible with respect to ionic convection and electrical migration. Since, for the hafnia nanofilters studied here, the salt rejection was not measured over a range of volume fluxes, we can only present indirect

Table 2 Ratios of cation to anion bulk (free solution) ionic diffusion coefficients for the different electrolytes studied [39]

Dcation /Danion

NaCl

NaNO3

CaSO4

CaCl2

Na2 SO4

0.65

0.76

0.70

0.37

1.25

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evidence for the validity of the high Péclet number assumption. The justification for assuming that we are observing the limiting rejections comes from the experimental evidence presented in, for example, [24,27,28]. The experimental results presented in these references for salt rejection by similar ceramic nanofilters show that these membranes are typically operating in the limiting rejection mode already at a volume flux of ∼30 l/(h m2 ). Within the frame-work of electro-transport theory [27] these rejection versus volume flux data imply that the ratio of effective salt diffusivity in the membrane to that in the bulk, D¯ s /Ds , is generally of order ∼10−4 –10−3 . These values are typical of the ceramic nanofilters studied in our laboratory [37]. The experimental and theoretical results for the rejection of Na2 SO4 presented in [27] for nanofilters of two very different average thicknesses and the same effective pore size (rp ≈ 1.2 nm) provide direct evidence that these unexpectedly low values of D¯ s /Ds are linked to the nanofiltration layer, and not an artifact of the mesoporous support: the experimental rejection versus volume flux curves change in just the way expected theoretically when the membrane thickness increases from ` ≈ 45 to 125 nm (see Fig. 7 of [27]). Although the theoretical explanation for these extremely low values of D¯ s /Ds for salts is not yet clear, the answer may be found in an enhanced coupling of electrical, tortuosity and hindered diffusion effects in granular charged porous nanofilters. Using the above values for D¯ s /Ds and Ds ∼ 10−5 cm2 /s, we see that the Péclet number, Pem = jV `/D¯ s , should be much bigger than one for salt tranport in our hafnia nanofilters at a volume flux of ∼30 l/(h m2 ). 2. Concentration polarization effects are negligible, which allows us to interpret the observed rejections as the true limiting intrinsic rejections. This approximation can be justified by employing the same arguments used for neutral solutes (see Section 2). 3. Only electrostatic interactions (ion-charged membrane and ion–ion) are important, allowing us to neglect steric and dielectric interactions. 4. Ionic concentrations are assumed to be sufficiently dilute so that activities can be replaced by concentrations.

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5. In the electro-transport model the macroscopic ionic electrochemical potentials are normalized in such a way that in the absence of membrane charge, i.e. Xm = 0, the ionic rejections vanish; in other words the standard chemical potential is assumed to be constant across the external solution–membrane interface, and therefore only rejection by charge is included in the model. 6. Although in charged granular porous membranes D¯ i may be much smaller than Di , we assume that the ratio of ionic diffusion coefficients is approximately the same inside and outside the membrane; i.e. D¯ 1 /D¯ 2 ≈ D1 /D2 , where Di are the bulk, dilute solution diffusion coefficients (Table 2). This approximation should be valid in the absence of strong steric hindrance effects. Approximations 3–6, although physically plausible, are difficult to justify in detail, for this entails taking into account the finite size and/or correlations of the ions in restricted micropores. Considering that such a theory has not been fully developed, this is difficult to do in physically transparent way. For this reason these approximations should be considered as working assumptions whose validity will be tested by comparing the theory (without any adjustable parameters) with experiment. Due to the presence of the acid and bases used to adjust the pH of the bulk solutions, we should in principle take into account the multi-electrolyte transport effects [38,39] arising from the presence of the protons, H+ , and hydroxide ions, OH− . We expect these multi-electrolyte effects to be small for pH values of 9 and 6, due to the low concentrations of protons and hydroxide ions involved. On the other hand, these effects could be important at the acidic pH of 3, since at this pH the proton concentration is equal to cf . In the present work, due to the complicated nature of multi-electrolyte transport, we neglect the influence of the acid on the salt transport at all the pH values studied (the problem of multi-electrolyte transport will be addressed in a future article [40]). If we neglect the influence of the acid or base, we can use the rejection results established in [15] for binary electrolytes (see Appendix B). By examining Fig. 6a–e we see that our working assumption (Eq. (4)) allows us to account at least qualitatively for almost all the data obtained by filtering five binary electrolytes at three different pH values (pH 3, 6.2, and 9.3) [10].

Two data points, that of CaSO4 at pH 9.3 and NaCl at pH 6.2, cannot, however, be accounted for, even qualitatively, by this simplified theory. Possible explanations for these discrepancies will be presented below. A perusal of Figs. 5 and 6a–e shows that, as was already remarked for other inorganic materials and membranes [4,34], for the same values of feed concentration, cf , and pH, the membrane charge density, Xm , depends strongly on the nature of the electrolyte. For the symmetric (1:1) electrolytes NaCl and NaNO3 , the charge is positive at a pH of 3, close to zero at a pH of 6, and negative at a pH of 9. Electrolytes possessing divalent ions, however, give rise to a very different behavior. For the electrolytes possessing the divalent sulfate anion SO4 2− (CaSO4 and Na2 SO4 ), specific anion adsorption apparently leads to a negatively charged membrane over the whole range of pH studied in [10]. For the asymmetric electrolyte CaCl2 , specific adsorption of the divalent calcium cation, Ca2+ , tends to lead, at least over the studied pH range, to membranes strongly skewed towards a positive charge polarity. The generally good agreement obtained (Fig. 6a–e) without any adjustable parameters suggests that the electro-transport model employed here, although not without some limitations, can be a useful first approximation for predicting the filtration performance of charged ceramic membranes. To better understand the theoretical predictions and their limitations, we tabulate in Tables 3–5 for each data point the values of the important quantities involved in salt electro-tranport. Firstly, for each of the three experimental pH values, we present for each salt the measured rejections and electrophoretic mobilities, Rexpt and µE [10]. Secondly, we present the following electrokinetic quantities derived from the measured electrophoretic mobility, µE , and the homogeneous electro-transport theory: σw is the pore wall surface charge density, deduced using Eq. (9) from the measured electrophoretic mobilities and the working approximation, Eq. (4); |Xm | the effective membrane charge density, deduced, using Eq. (10), from the measured hydraulic radius, rH , and the estimated pore wall surface charge density, σw ≈ σek ; ξm is equal to |Xm |/cf , the normalized charge density; 1Φ˜ Df the dimensionless feed interface Donnan potential (see [15] and Appendices A and B) calculated from the estimated normalized charge

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density, ξm ; ζ˜ the reduced zeta-potential, calculated from the measured electrophoretic mobilities using Eq. (8); kif is equal to c¯i (0+ )/cif = exp(−zi 1Φ˜ Df ), the feed interface Donnan partition coefficients, calculated from the dimensionless Donnan potentials; kis is equal to Cis /Ci = exp(−zi ζ˜ ), the shear plane ionic partition coefficients derived from the ζ -potential; (pH)mf is equal to pH − 1Φ˜ Df / ln(10), the pH in the membrane at x = 0+ induced by the Donnan potential; (pH)s is equal to pH − ζ˜ / ln(10), the pH at the shear plane induced by the zeta-potential and finally σ ∗ is the dimensionless surface charge density (Eq. (11)) that delimits the range of validity of the homogeneous approximation. From Fig. 5 and Tables 3–5 we can get a general overall idea of the magnitudes of the important quantities describing the membrane charge. Since the estimated pore wall surface charge density falls in the range 0 < |σw | < 0.2 C/cm2 , we see that the effective membrane charge density falls in the range 0 < |Xm | < 3 × 10−2 M (or ξm < 30). The dimensionless charge density, σ ∗ , therefore always obeys the inequality σ ∗ 1, which confirms the validity of the homogeneous approximation to the space charge electrotransport theory [15]. Since the bulk concentration used in the electrophoretic mobility measurements was the same as the feed concentration (10−3 M) used in the rejection measurements, the partition coefficients, kis and kif , can be directly compared to assess the differences in ionic concentrations between the point just inside the membrane at x = 0+ and the agglomerate shear plane. We now discuss some general qualitative conclusions that can be drawn from the electro-transport theory and ask whether they are also exhibited by the experimental data. We would like to stress that since our comparison between theory and experiment does not depend on any adjustable parameters and is built on a series of approximations outlined earlier, we do not expect there to be perfect agreement between the theoretical predictions and the experimental measurements. What we are looking for is evidence that our approximate method [15] of predicting ionic rejections by ceramic nanofilters is reasonably accurate. We would also like to identify under what conditions we can expect our approximate method for predicting the filtration properties of ceramic NF membranes to work.

4.1. Symmetric electrolytes Due the asymmetry of the streaming potential under interchange of co-ion and counter-ion when the membrane charge changes sign, for the same normalized membrane charge density, |Xm |/cf , the limiting rejection, R, will be higher for the membrane charge polarity for which D¯ 2 < D¯ 1 [15]. This leads to a relatively weak asymmetry of the limiting rejection about the IEP. Unfortunately the agreement between the theoretical and experimental results for NaCl and NaNO3 is not good enough to illustrate this theoretical prediction in an unambiguous way. Furthermore, for CaSO4 the effective membrane charge density is always negative over the pH range studied, and therefore in this case only one branch (Xm < 0) of the limiting rejection curve is accessible. Due to a more effective screening of membrane charge, for the same ionic diffusivities and normalized membrane charge density, the limiting rejection of a (z:z) symmetric electrolyte decreases as the valence z increases [15]. If, for the moment, we put aside the CaSO4 data point at ξm ≈ −20 (Fig. 6c), this screening effect can be seen by comparing the rejection results for CaSO4 with the results for NaCl and NaNO3 (Fig. 6a–c). These results show that the experiments are in qualitative agreement with the theoretical predictions showing that for roughly similar values of D¯ cation /D¯ anion , CaSO4 should have a much lower limiting rejection than either NaCl or NaNO3 at the same value of normalized membrane charge density, Xm /cf . We now attempt to provide an explanation for the two large discrepancies between theory and experiment observed for CaSO4 at pH 9.3 and NaCl at pH 6.2. 4.1.1. Proximity of the IEP: NaCl near pH 6 The explanation for the discrepancy between our approximate theoretical prediction for the limiting rejection and the measured value is very probably that in an NaCl bulk solution at pH 6.2 the hafnia electrokinetic surface charge density is very close to zero (the IEP). In this case the surface charge density is a very sensitive function of both the pH and the NaCl concentration. To demonstrate this, we compare (Fig. 7) the electrokinetic surface charge density used in the theoretical predictions, which was deduced from electrophoretic mobility measurements at a bulk

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electrolyte concentration of 10−3 M, with the surface charge density deduced from electrophoretic mobility measurements at a higher bulk electrolyte concentration (10−2 M) [41]. We see that at the higher bulk NaCl concentration, the IEP moves to higher pH (∼6.8 instead of ∼6.2) and for pH < 6.8 the electrokinetic surface charge density is now substantially higher in absolute value than it was at 10−3 M. Although it is beyond the scope of the present article to provide a quantitative demonstration, the changes in the electrokinetic surface charge density exhibited in Fig. 7 as a function of bulk pH and electrolyte concentration near pH 6 can plausibly explain why our approximate theory becomes inaccurate near the bulk IEP. The increase in counter-ion concentration and the slight decrease in pH in the positively charged pore (with respect to the pH at the shear plane) will lead to a pore wall surface charge density substantially higher than the electrokinetic surface charge density deduced from electrophoretic mobility measurements (even though the feed and bulk solutions in the two cases are the same). The upshot of this discussion is that the effective membrane surface charge density associated with the measured rejection at a feed pH of 6.2 should be substantially higher than the value used in Table 4 and Fig. 6a, which tends to bring theory into closer agreement with experiment. 4.1.2. Specific ion adsorption model: CaSO4 at pH 9.3 The data point for CaSO4 at Xm /cf ≈ −20 (obtained for pH ≈ 9.3), which lies far below the theoretical curve, is clearly inconsistent with the macroscopic electro-transport theory prediction (Fig. 6c). This anomalous result leads us to question the validity of our working approximation (Eq. (4)) in this particular case. That the hafnia surface charge characteristics be complicated in the presence of a divalent cation and a divalent anion is not surprising, once we review what happens when these divalent ions appear in a mono-divalent electrolyte. As already noted, σek as a function of pH is highly skewed towards positive values for CaCl2 and highly skewed towards negative values for Na2 SO4 over the whole pH range studied (Fig. 5). We hypothesize that for CaSO4 , where both divalent co-and counter-ions are present, strong specific SO4 2− adsorption at low pH keeps the surface

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charge negative for all pH values, and, at least for pH ≈ 9.3, specific Ca2+ adsorption renders the surface charge density a sensitive function of the counter-ion (Ca2+ ) concentration. If, following our working assumption (4), we use the naive value of 20.2 for ξm (Table 5) in the electro-transport theory, we find a theoretical limiting rejection of 0.76, which is much higher than the observed value of 0.14. The problem is that the feed interface Donnan potential induced by this high hypothetical membrane charge density (ξm = 20.2) is high (1Φ˜ Df ∼ −1.2); such a high value for the Donnan potential would lead to a counter-ion concentration in the membrane (at x = 0+ ) fives times greater than the counter-ion concentration at the shear plane, i.e. k1f ≈ 5k1s (Table 5). In the particular situation under examination, we conjecture that this high counter-ion concentration cannot truly exist in the membrane, because it would tend to strongly screen the membrane charge and in the process reduce its own value. It is by now well known from membrane potential, rejection, and titration measurements that the effective charge density in both organic and inorganic membranes depends on the ionic concentrations and/or pH [16–18,34,39,42]. Despite the rather sophisticated theories developed to model this behavior for metal oxide surfaces and explain the simultaneous presence of high titrated surface charge densities and relatively low ζ -potentials and electrokinetic surface charge densities [34], it seems safe to say that the mechanisms of surface charge generation are in general not completely understood. Consequently, in order to explain qualitatively the anomalously low rejection observed for CaSO4 at pH 9.3, we will adopt a simple Langmuir model of localized counter-ion adsorption [42,43]. In this model the membrane surface charge density is assumed to depend on the pH and ionic concentrations in the following way: σw = σm0 +

z1 a1 c¯1 z2 a2 c¯2 + . 1 + b1 c¯1 1 + b2 c¯2

(12)

We associate σm0 with the titrated intrinsic surface charge (this would be the true effective surface charge in the absence of specific ion adsorption). The intrinsic surface charge density, σm0 , and the Langmuir coefficients, ai and bi , most likely depend on the pH. This model for the membrane surface charge density (Eq. (12)) can then be used to calculate the concen-

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tration dependent normalized membrane charge density (via Eqs. (3) and (5)) for a negatively charged membrane in the presence of CaSO4 : ξm (k1f , k2f ) = ξm0 −

2α1 k1f 1 + β1 k1f

+

2α2 k2f 1 + β2 k2f

,

(13)

where for simplicity ξm0 , αi , and βi are all assumed to be given approximately by their values at a fixed pH, that of the feed. Eq. (13) leads to a membrane charge density, Xm , that, although remaining negative, decreases in absolute value due to specific counter-ion adsorption as the counter-ion concentration increases. In the absence of the experimental data necessary for elucidating the ionic concentration dependence of the electrophoretic mobility and the electrokinetic surface charge density, σek , we are obliged to adopt the following simplifying approximations: 1. For the particular case under study (pH of 9.3), it is only the counter-ion term in Eq. (13) that is changing significantly over the range of concentrations encountered. This is plausible because at basic pH the amphoteric sites at the metal oxide surface are heavily de-protonated, and therefore, although specific counter-ion (Ca2+ ) adsorption may be important, co-ion (SO4 2− ) adsorption probably is not. 2. The difference in pH between (pH)mf and (pH)s is not important in Eq. (13). 3. We are far from counter-ion saturation, and therefore the Langmuir form can be replaced by a simpler Henry law, linear in the counter-ion concentration; indeed, if this were not the case, ξm (k1f , k2f ) could not be a strong function of the counter-ion concentration. These approximations lead to the following approximate forms for the ionic concentration dependent surface charge density: σ (c1 ) ≈ σeff − 2a1 c1 ,

(14)

where c1 = c¯1 (0+ ) when σ (c1 ) refers to the pore wall surface charge density, σw , and c1 = C1s when σ (c1 ) refers to the electrokinetic surface charge density, σek . In the case of the pore wall surface charge density, Eq. (14) leads the following expression for the concentration dependent normalized membrane charge density, ξm = |σw |/(FrH cf ): ξm (k1f ) ≈ ξeff − 2α1 k1f .

(15)

In Eqs. (14) and (15) σeff and ξeff take into account both the intrinsic charge and the specific anion (SO4 2− ) adsorption. (For reasons of notational simplicity, from now on we define α ≡ α1 .) In our model of specific counter-ion adsorption, there are two unknown parameters, σeff and α. Since we do not have sufficient electrophoretic mobility data to determine the two parameters unambiguously, we adopt the following strategy: we fix the value of σeff , and therefore ξeff = |σeff |/(FrH cf ), as a function of α by using the only experimental data available to us, namely the electrokinetic surface charge density, σek , deduced from the value of the electrophoretic mobility measured at 10−3 M and pH 9.3. This is equivalent to requiring that if k1f were to take on the value k1s ≈ 2.41, then ξm would be equal to 20.2 (see the CaSO4 entry in Table 5). With this constraint we now try to determine if a physically reasonable choice of α can lead to a theoretical prediction for the limiting rejection in line with the measured value. Using the above constraint, we obtain a ‘counter-ion concentration dependent’ normalized membrane charge that is parameterized by α alone: ξm (k2f ; α) = ξeff (α) −

2α k2f

,

(16)

where ξeff (α) = 20.2 + 4.82α and we have used k2f = 1/k1f , valid for a symmetric electrolyte. This result can then be substituted into the equation (Eq. (A.9)) obeyed by the co-ion partition coefficient, yielding a quadratic equation of the form [k2f ]2 + 21 k2f ξeff (α) − (1 + α) = 0.

(17)

This equation can easily be solved for k2f as function of α, and the solution, " #1/2 2 (α) ξeff (α) 1 ξeff f , (18) + +4(1+α) k2 (α)= − 4 2 4 can be substituted into the expression (B.6) (Appendix B) for the salt rejection, yielding R(α) = 1 −

k2f (α) t1 + (1 − t1 )[k2f (α)]2

.

(19)

R(α) is a renormalized rejection, parameterized by α, that takes into account specific counter-ion adsorption. The renormalized rejection, R(α) assumes the

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unrenormalized value of R(0) = 0.76 in the absence of specific adsorption (α = 0), as required. It also decreases with increasing α, i.e. increasing counter-ion adsorption. The choice α ∼ 50 leads to negative effective membrane charge density with ξm (50) ∼ 5 and a renormalized rejection, R(50) ∼ 0.2, close to the observed value of 0.14. This renormalized rejection, which takes into account specific counter-ion adsorption, is denoted by a cross in Fig. 6c. This choice for α also leads to a physically reasonable value for σeff (α) in the order of −2 C/cm2 . If we imagine that at the basic pH of 9.3 under consideration, σeff is dominated by the (negative) intrinsic surface charge density arising from the de-protonated amphoteric groups, then this result is in qualitative agreement with the known result that intrinsic surface charge densities measured for metal oxides by titration tend to be an order of magnitude greater than the surface charge densities measured by electrokinetic techniques. It should be kept in mind that the simple model presented here to explain the self-consistent screening of membrane surface charge by specific counter-ion adsorption is only supposed to illustrate the feasibility and physical soundness of the mechanism; a more sophisticated theory is surely needed to account for all the complexity of specific ion adsorption on metal oxide surfaces. Within the scope of more sophisticated models, it might even be possible for the renormalized effective membrane charge to change sign due to an overcompensation created by strong specific counter-ion adsorption. 4.2. Asymmetric electrolytes Due to the sensitive dependence of the co-ion feed side Donnan partition coefficient on the co-ion valence, the limiting rejections of asymmetric binary electrolytes are, unlike those for symmetric electrolytes, highly asymmetric about the IEP [15]. The limiting rejection of (1:2) and (2:1) binary electrolytes has been studied experimentally in [10] and theoretically in [15]. These results, which are presented in Fig. 6d and e, show that there is good agreement between theory and experiment. The electrochemical valence asymmetry in co-ion Donnan partitioning leads to the strongly skewed, S-shaped, theoretical rejection curves for the two asymmetric electrolytes presented in Fig. 6d and e. We see

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that for CaCl2 (Fig. 6d) and a positively charged membrane there is a high limiting rejection, even at relatively modest effective membrane charge densities, due to the presence of a divalent co-ion that experiences strong Donnan exclusion and a monovalent counter-ion that is only weakly attracted by the membrane. On the other hand, for a negatively charged membrane, there is a monovalent co-ion that is only weakly excluded from the pores, and a divalent counter-ion that is strongly attracted, leading to low limiting rejections, even at relatively high effective membrane charge densities. The limiting rejection in this case can even become moderately negative for small values of ξm . Although the S-shaped curve is clearly visible in the experimental results for CaCl2 (Fig. 6d), only the high limiting rejection (1:2) branch is seen for Na2 SO4 ; this is because the strong specific adsorption of the sulfate ion renders the membrane charge negative over the whole pH range studied here (see Fig. 5). The similarities between the experimental limiting rejection data for CaCl2 for positive membrane charge and the limiting rejection data for Na2 SO4 for negative membrane charge are particularly striking and lend credence to the electro-transport theory adopted here (incorporating electrostatic interactions, but neglecting steric and dielectric ones). The small quantitative differences between the CaCl2 theoretical limiting rejection curve, when mirror reflected about zero membrane charge, and the Na2 SO4 rejection curve (presented in Fig. 6d and e) can be traced to the difference in the ratio |z1 |D¯ 1 /(|z2 |D¯ 2 ) in the two cases (see Table 2). For asymmetric electrolytes, if |z1 |D¯ 1 6= |z2 |D¯ 2 , a small part of the asymmetry of R about the IEP is a consequence of the asymmetry of the streaming potential upon interchange of co-and counter-ion diffusion coefficients, just as in the case of a symmetric electrolytes. If the ratio |z1 |D¯ 1 /(|z2 |D¯ 2 ) were the same in the two cases, the limiting rejection curves, as predicted by homogeneous electro-transport theory, would be mirror images of each other, reflected about zero membrane charge (ξm = 0). 5. Discussion and conclusion For neutral and charged species, we have compared the experimentally observed limiting rejections

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obtained using hafnia nanofilters with the theoretical predictions of suitable continuum membrane transport theories. Using nitrogen adsorption/desorption and electrophoretic mobility measurements, the hafnia nanofilters were found to possess, respectively, pore radii ∼1 nm and pH/electrolyte dependent surface charge densities in the range 0 ≤ |σw | ≤ 0.2 C/cm2 . Since the pore radius can be estimated from the relation (1), only one membrane structural measurement, for the hydraulic radius, is needed to characterize the limiting rejection of neutral solutes. The theoretical predictions of the hydrodynamic hindered transport theory were found to be in excellent agreement with the experimental results for four neutral solutes. These results support our hypothesis that the hafnia ceramic nanofilter can be modeled as a granular porous medium possessing an effective pore radius approximately equal to about 1 nm, with the solute transport, which takes place in the liquid-filled intergranular spaces, capable of being described by pore level continuum theories. For charged species, we have attempted to predict the observed limiting rejection of five different binary electrolytes at a fixed feed concentration of 10−3 M for a range of membrane charge densities (obtained at pH values of 3, 6, and 9). The comparison between theory and experiment has been carried out by introducing an independently measured electrokinetic surface charge density, σek , in the macroscopic electro-transport theory via the working approximation Eq. (4). Our comparison between theory and experiment reveals that our method provides a reasonable approximate starting point for estimating ionic rejection from electro-transport theory, using as input charge and structural parameters measured directly on the membrane material. The serious breakdown of our working approximation for estimating the pore wall surface charge density (Eq. (4)) in the case of NaCl at pH 6 and CaSO4 at pH 9 shows that a more comprehensive theory of charge generation at the pore wall– solution interface is needed for establishing a fully quantitative theory of ionic rejection by ceramic NF membranes without any adjustable parameters. The theoretical results for limiting electrolyte rejection presented in Fig. 6a–e can be considered as the signature of an amphoteric membrane that rejects ions by electrostatic interactions alone. The reasonably good overall agreement between theory and

experiment shows that the ceramic nanofilter studied here follows in a reasonable way the signature expected for an amphoteric membrane. These results also suggest indirectly that, at least for the hafnia nanofilters with rp ≈ 1 nm studied here, steric and dielectric interactions do not play an important role in ion transport. The essentially zero rejection observed for Na2 SO4 near the IEP (Fig. 6e) implies that steric effects are unimportant even for the sulfate ion, whose hydrated radius is usually estimated to be in the order of 0.25–0.38 nm, about twice the size of a water molecule. If the sulfate ion did indeed act as a hard sphere of this radius, the hindered transport theory (Section 2) would predict a steric rejection ∼20–40% (for rp ≈ 1 nm) near the IEP, a result that is clearly not seen in the experimental data for the hafnia nanofilter. This observation implies that there are perhaps important gaps in both our understanding of the effective size of ionic species in micropores and our understanding of the role of steric rejection for these species. Serious complications may therefore arise in any attempt to introduce hindered transport effects into the homogeneous electro-transport theory (cf. [17,21]). Extensive electrolyte rejection and electrophoretic mobility measurements over a wide range of electrolyte concentrations and pH values are clearly needed to map out, for the different electrolytes, the dependence of σw and σek on ionic concentration and pH. Moreover, measurements of other electrokinetic quantities, such as the membrane electrical conductivity and streaming potential (measured across the membrane thickness), would also allow us to investigate more directly and in more detail how the effective membrane charge density changes with salt concentration and pH. Such measurements would also allow us to test the validity and predictive powers of the macroscopic electro-transport theory adopted here. (The membrane electrokinetic measurements probe the true membrane charge density in the pores, whereas the powder particle electrophoretic mobility probes the effective surface charge density at the agglomerate shear plane (Section 3).) In tight UF and probably all NF ceramic membranes prepared from metal oxides, the surface charge density is sufficiently weak and the pore radius sufficiently small to ensure the validity of the simplified homogeneous approximation to the space–charge model. For the hafnia membranes studied here, the

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parameter controlling the validity of the homogeneous approximation, σ ∗ , is always very small (< 4×10−2 ), and therefore the homogeneous theory should give an accurate description of ionic transport. Although here we have studied only a few neutral solutes and simple binary electrolytes, we would like to end by outlining a general strategy that can be adopted for estimating the limiting solute rejection rates for an arbitrary feed mixture of neutral solutes and salts using ceramic porous granular nanofilters. 1. Estimate the pore hydraulic radius, rH , using standard nitrogen adsorption/desorption measurements performed on the membrane material. 2. If the ceramic membrane has a structure resembling a sintered packing of spherical, nearly monodisperse, grains, use the grain consolidation model to estimate the effective pore radius, rp (Eq. (1), Section 1). Then use the 1D cylindrical pore (with radius rp ) hindered transport theory to predict the limiting rejection of the neutral solutes. 3. Using a bulk solution with the same pH and ionic concentrations as the feed solution to be filtered, measure the electrophoretic mobility of the ceramic powder used to prepare the membrane. 4. Use the measured electrophoretic mobility to estimate the electrokinetic surface charge density appropriate for the bulk solution under study. 5. Use the estimated electrokinetic surface charge density in the working assumption (4) to estimate the pore wall surface charge density, σw . 6. Use the measured pore hydraulic radius, rH , and the estimated pore wall surface charge density, σw , to estimate the effective membrane charge density, Xm (Eq. (10)), appropriate for the feed solution under study. 7. Check to see if the homogeneous electro-transport theory is valid (σ ∗ 1?). If so, find the limiting ionic rejections by solving (at high membrane Péclet number) the extended Nernst–Planck (ENP) equations (presented in Appendix A, but now generalized to take into account all the distinct ionic species present), using as input the effective membrane charge density estimated as described above (item 6) for the feed solution under study (assuming also that the ratio of ionic diffusivities is the same inside and outside the membrane). Note that certain ␥-alumina membranes appear to possess structures made up of plate-shaped grains,

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stacked in such a way as to give rise to slit-shaped pores [1,5]. Although we have not studied this case in detail, we expect that the type of theory developed in Section 2 should still apply, but with the following two changes: the constant of proportionality relating the effective pore radius to the pore hydraulic radius will most likely be different from that found in Eq. (1) and the slit-shaped pore hindered transport theory should be used to calculate the neutral solute rejection. On the other hand, the macroscopic homogeneous electro-transport theory used here does not depend on the grain shape at all; therefore, provided that the pore hydraulic radius and the ceramic powder electrophoretic mobility can be measured, our electro-theory can be applied to any ceramic nanofilter, independent of the grain shape. This method for estimating the rejection of any mixture of neutral molecules and salts could be very useful, because the NF membrane parameters entering the theory are estimated from standard measurements performed on the membrane material, and not other more time-consuming transport measurements. More detailed applications of the transport theory developed in [15] to the rejection of single salt and multi-electrolyte mixtures by ceramic nanofilters are currently in progress. Acknowledgements We would like to the thank our collaborators, M. Lopez and L. Cot, who participated in the experimental studies reported in Part I [10]. One of the authors (J.P.) would like to thank C. Guizard, R. Vacassy, A. Yaroshchuk, R. Takagi, A. Nechaev, M. Persin, and G. Rios for helpful discussions on electro-transport phenomena.

Appendix A. Macroscopic electro-transport theory: binary electrolyte [15] The macroscopic ion flux densities (moles per unit time per unit membrane area) are given by the macroscopic extended Nernst–Planck (ENP) equations (i = 1 (counter-ion), i = 2 (co-ion)): dc¯i F dΦ − zi c¯i D¯ i + c¯i jV ; ji ≈ −D¯ i dx RT dx

(A.1)

264

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where x, the macroscopic transverse position coordinate, varies between 0 and ` (the membrane thickness); jV is the macroscopic volume flux density (volume of solution per unit time per unit membrane area, nearly equal to the average mass-centered flow at low electrolyte concentration); c¯i (x) is the averaged ionic concentration in the membrane (moles per unit pore volume); D¯ i is the effective ionic diffusion coefficient in the membrane and Φ(x) is the averaged macroscopic electric potential in the membrane. The flux densities ji and jV are constant across the system. The ENP equations are solved under the following conditions — approximate macroscopic charge electroneutrality: |z2 |c¯2 (x) − |z1 |c¯1 (x) + |Xm | ≈ 0;

(A.2)

The Donnan co-ion partition coefficient at the feed interface, k2f , obeys the equation [k2f ]−|z1 |/|z2 | − k2f −

ξm = 0, |z1 |ν1

(A.9)

where ξm ≡

|Xm | cf

(A.10)

is the normalized (dimensionless) membrane charge density.

Appendix B. Macroscopic electro-transport theory predictions [15]

zero net macroscopic electric current density: B.1. Limiting rejection, R, of a binary electrolyte

2 X zi ji = 0; I =F

(A.3)

i=1

filtration boundary condition: p ci

= ji /jV ;

(A.4)

(A.5)

where the feed ionic partition coefficients are defined by kif ≡

c¯i (0+ ) cif

,

(A.6)

and 1Φ˜ Df = F1ΦDf /(RT) ≡ F [Φ(0+ ) − Φ f ]/(RT) is the dimensionless feed interface Donnan potential. Donnan equilibrium at the permeate interface: [k1 ]1/z1 = [k2 ]1/z2 = exp[−1Φ˜ D ], p

p

p

(A.7)

where the permeate ionic partition coefficients are defined by p ki

≡

c¯i (`− )

,

ti =

|zi |D¯ i , |z1 |D¯ 1 + |z2 |D¯ 2

R(GCE) = 1 −

f k2(GCE)

t1

,

(B.3)

|z2 |D¯ 2 ξm −|z2 /z1 | .(B.4) R(GCE) = 1 − 1 + ν1 |z1 | |z1 |D¯ 1 Arbitrary membrane charge: R=1−

and 1Φ˜ D = F1ΦD /(RT) ≡ F [Φ(`− ) − Φ p ]/(RT) is the dimensionless permeate interface Donnan potential.

R=1−

p

(B.2)

is the effective ionic transport number in the membrane (i = 1, 2). High membrane charge (good co-ion exclusion (GCE), ξm 1):

(A.8)

p

ci

(B.1)

where

Donnan equilibrium at the feed interface: [k1f ]1/z1 = [k2f ]1/z2 = exp[−1Φ˜ Df ],

Weak membrane charge (ξm 1): ξm |z1 | , Rweak ≡ t1 − |z1 | + |z2 | ν1 |z1 |

p

(ξm /(|z1 |ν1 )) + k2f t1 (ξm /(|z1 |ν1 )) + k2f

! k2f ,

(B.5)

.

(B.6)

k2f t1 + (1 − t1 )[k2f ]1+(|z1 /z2 |)

J. Palmeri et al. / Journal of Membrane Science 179 (2000) 243–266

B.2. Symmetric electrolyte (z:z) Feed interface co-ion partition coefficient: s 2 ξ ξm m f + + 1. k2(z:z) = − 2z 2z

(B.7)

Limiting symmetric electrolyte rejection: −1 s 2 1 ξm ξm + +1 . (B.8) Rz:z =1− t1 − 2 z 2z B.3. Asymmetric electrolyte (2:1) |z1 | = 2 and |z2 | = 1, f (ξm )= − k2(2:1)

ξm 21/3 ξm2 Γ (ξm /2) , + + 6 12Γ (ξm /2) 3 × 21/3

(B.9)

where the function Γ is defined by i1/3 h p . Γ (s) ≡ 27 − 2s 3 + −4s 6 + (27 − 2s 3 )2 ) (B.10) For a 1:2 binary electrolyte (i.e. |z1 | = 1 and |z2 | = 2), f f (ξm ) = [k2(2:1) (−ξm )]−2 . k2(1:2)

(B.11)

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