3910
Langmuir 2006, 22, 3910-3919
Transport Properties and Electrokinetic Characterization of an Amphoteric Nanofilter Anthony Szymczyk,* Mohammed Sbaı¨, Patrick Fievet, and Alain Vidonne UniVersity of Franche-Comte´ , Laboratoire de Chimie des Mate´ riaux et Interfaces, 25030 Besanc¸ on Cedex, France ReceiVed July 12, 2005. In Final Form: February 8, 2006 Transport properties of a tubular nanofilter with amphoteric properties have been investigated by means of the SEDE (steric, electric, and dielectric exclusion) homogeneous model. Within the scope of this 1D model, the separation of solutes results from transport effects (described by means of extended Nernst-Planck equations) and interfacial phenomena including steric hindrance, the Donnan effect, and dielectric exclusion (expressed in terms of (i) the Born dielectric effect, which is connected to the lowering of the dielectric constant of a solution inside nanodimensional pores, and (ii) the interaction between ions and the polarization charges induced at the dielectric boundary between the pore walls and the pore-filling solution). The effective volume charge density of the membrane has been determined from tangential streaming potential experiments coupled with conductance experiments in a potassium chloride solution at various pH values ranging from 2 to 11. The inferred values have been used in the SEDE model to compute the ion rejection rates with the dielectric constant of the solution inside the pores as a single adjustable parameter. The model provides a relatively good description of experimental data even at extreme pH values for which a ternary system has been considered (K+, Cl-, and H+ or OH- depending on the pH). The fit to experimental data at the membrane isoelectric point indicates that the confinement effect decreases the dielectric constant inside the pores only slightly (with respect to its bulk value). However, the (pH-dependent) ionization of surface sites has been found to lead to a substantial lowering of the dielectric constant inside the pores.
1. Introduction Nanofiltration (NF) is the most recently developed pressuredriven membrane process in the liquid phase. This promising technique has attracted increasing attention because of its potential application in several industrial areas such as the plating, textile, paper, and food industries. NF membranes are polymeric or ceramic porous materials, most of which develop a surface electric charge when brought into contact with a polar medium. This electric charge may arise through several mechanisms including the dissociation of functional groups and/or the adsorption of charged species from the solution onto the pore walls. The combination of pore diameters around a few nanometers with electrically charged materials makes the prediction of NF membrane performance extremely difficult. Indeed, the separation of solutes results from complex mechanisms that may include steric hindrance and Donnan, dielectric, and transport effects.1 NF membranes are usually described as a bundle of capillaries with effective structural features (pore size and thickness-toporosity ratio) and electrical properties such as their effective volume charge density (defined as the number of moles of fixed charges per unit of pore volume).2,3 Modern approaches to the description of transport through charged porous membranes are based upon the so-called space charge model originally developed by Osterle and co-workers.4-6 Within very narrow pores such as those of NF membranes, the electrical double layers cannot fully develop and overlap strongly. As a result, both the local electric potential and ion concentrations are almost radially * Corresponding author. E-mail:
[email protected]. Tel: +33-3-81-66-20-32. Fax: +33-3-81-66-20-33. (1) Oatley, D. L.; Cassey, B.; Jones, P.; Bowen, W. R. Chem. Eng. Sci. 2005, 60, 1953. (2) Tsuru, T.; Nakao, S. I.; Kimura, S. J. Chem. Eng. Jpn. 1990, 23, 604. (3) Lefebvre, X.; Palmeri, J.; David, P. J. Phys. Chem. B 2004, 108, 16811. (4) Morrison, F. A.; Osterle, J. F. J. Chem. Phys. 1965, 43, 2111. (5) Gross, R. J.; Osterle, J. F. J. Chem. Phys. 1968, 49, 228. (6) Fair, J. C.; Osterle, J. F. J. Chem. Phys. 1971, 54, 3307.
constant (provided the electric charge carried out by the pore walls is not too high7). Neglecting these variations yields the so-called homogeneous approximation. This approximation considerably decreases computational effort and is currently used in most studies devoted to transport modeling through NF membranes.1-3,8-14 Within the scope of the homogeneous model, membrane separation is described as being the result of the following steps: first, a distribution of species at the membrane/ feed solution interface, followed by solute transport through the pores by a combination of convection, diffusion, and electromigration (for charged solutes only) and finally a distribution of species at the membrane/permeate interface. Until now, the homogeneous model has been essentially used by considering the exclusion mechanism at membrane solution interfaces as being the result of Donnan-type exclusion only2,10,13 or a combination of both steric and Donnan effects.1,3,8,9,12-14 Some attempts have been made to improve the description of ion transport through NF membranes by taking into account the Born (solvation) dielectric effect in partitioning equations at membrane/solution interfaces,15,16 which is justified by recent studies carried out with various halide salts.17 Recently, Yaroshchuk18 pointed out the relevance, in NF pores, of the interaction (7) Szymczyk, A.; Aoubiza, B.; Fievet, P.; Pagetti, J. J. Colloid Interface Sci. 1999, 216, 285. (8) Bowen, W. R.; Mohammad, A. W.; Hilal, N. J. Membr. Sci. 1997, 126, 91. (9) Wang, X. L.; Tsuru, T.; Nakao, S. I.; Kimura, S. J. Membr. Sci. 1997, 135, 19. (10) Palmeri, J.; Blanc, P.; Larbot, A.; David, P. J. Membr. Sci. 1999, 160, 141. (11) Yaroshchuk, A. E. Sep. Purif. Technol. 2001, 22-23, 143. (12) Labbez, C.; Fievet, P.; Szymczyk, A.; Vidonne, A.; Foissy, A.; Pagetti, J. J. Membr. Sci. 2002, 208, 315. (13) Garcia-Aleman, J.; Dickson, J. M. J. Membr. Sci. 2004, 235, 1. (14) Lefebvre, X.; Palmeri, J. J. Phys. Chem. B 2005, 109, 5525. (15) Hagmeyer, G.; Gimbel, R. Desalination 1998, 117, 247. (16) Bowen, W. R.; Welfoot, J. S. Chem. Eng. Sci. 2002, 57, 1121. (17) Diawara, C. K.; Loˆ, S. M.; Rumeau, M.; Pontie´, M.; Sarr, O. J. Membr. Sci. 2003, 219, 103.
10.1021/la051888d CCC: $33.50 © 2006 American Chemical Society Published on Web 03/09/2006
Amphoteric Nanofilter Properties/Characterization
between ions and the polarization charges induced at the dielectric boundary between the pore walls and the pore-filling solution. This additional dielectric effect is also known as the dielectric exclusion via image charges. (The concept of image charge is sometimes used to construct an equivalent system where the net interaction is the same.19) Until now, very few attempts have been made to assess the ability of transport models accounting for the dielectric effect via image charges to describe experimental data such as ionic rejection rates.20 In a recent work, we proposed the SEDE model, which is an improved version of the homogeneous model including steric, electric (i.e., Donnan), and dielectric effects (in term of both Born and image charge effects) in the exclusion mechanism of NF membranes.21 The SEDE model was shown to provide a rather good description of the transport properties of a negatively charged polyamide membrane for symmetric and asymmetric binary electrolytes. This work also demonstrated that dielectric exclusion phenomena cannot be neglected in the complex exclusion mechanism of such an NF membrane. The aim of the present work is to go further in the assessment of the SEDE model abilities by investigating the filtration performances of an amphoteric ceramic nanofilter with respect to a millimolar potassium chloride solution at various pH values ranging from 2 to 11. Although the feed solution can be described reasonably as a binary system over a wide pH range, protons (or hydroxide ions) cannot be neglected at the lowest (or highest) pH values, and a ternary system therefore has to be considered for extreme pH values. (In this case, three different ionic rejection rates have to be computed simultaneously instead of a single value, the salt rejection rate, when binary systems are considered.) The effective pore size as well as the membrane (more rigorously, the active layer) thickness-to-porosity ratio can be estimated from neutral solute rejection rates whereas the membrane volume charge density is inferred (for various pH values under consideration) here from tangential streaming potential experiments coupled with conductance measurements. The single model fitting parameter is the effective dielectric constant of the solution inside the pores, the membrane charge dependence of which is put into evidence. 2. Steric, Electric, and Dielectric Exclusion (SEDE) Model. We shall give here a brief description of the SEDE model. (More details can be found elsewhere.21) The active layer of the membrane is modeled as a bundle of identical cylindrical ion-selective capillaries of length ∆x and radius rp (rp , ∆x) separating the feed solution from the permeate one (Figure 1). The presence of the coarse-porous support layer is disregarded here because it does not affect the ionic retention properties of the membrane (but this does not mean that the pressure drop through the membrane occurs through the active layer only; see the discussion below). The distribution of ions at both 0+|0- and ∆x-|∆x+ interfaces is usually modeled by assuming local thermodynamic equilibrium. Within the scope of the SEDE model, partitioning equations take the form21
ci(0+) ci(0-)
γi(0-)
) φi exp(-zi∆Ψ(0+|0-)) exp(-∆W′i,Born) × γi(0+) exp(-∆W′i,im(0+|0-)) (1a)
ci(∆x-) ci(∆x+)
Langmuir, Vol. 22, No. 8, 2006 3911
where ci(x) is the concentration of ion i at position x, φi is its steric partitioning coefficient that is defined as the ratio of the accessible section to the pore cross-section (for cylindrical pores, φi is equal to [1 - (ri,Stokes/rp)]2 where ri,Stokes is the Stokes radius of ion i and rp is the effective pore radius), γi is its activity coefficient (calculated here according to the extended law of Debye-Hu¨ckel theory), zi is its charge number, ∆Ψ is the normalized Donnan potential (∆Ψ ) (e/kT)∆ψDonnan where e is the elementary charge, k is the Boltzmann constant, and T is the temperature), and ∆W′i,Born and ∆W′i,im denote the increase in the interaction energy (scaled on kT) due to Born and image force dielectric effects, respectively. The variation of the interaction energy resulting from the Born effect is given by21
∆W′i,Born )
)
(2)
where 0 is the vacuum permittivity, ri,cav is the radius of the cavity formed by ion i in the solvent (ri,cav is calculated according to the procedure proposed by Rashin and Honig22), p is the effective dielectric constant of the solution inside the pores (this unknown feature is used as a fitting parameter in the SEDE model), and b the dielectric constant of the external bulk solution. Yaroshchuk has derived an approximate relation for the interaction energy due to image forces in charged cylindrical pores.11,18 Within the scope of the SEDE model, this reads as follows:21
∆W′i,im )
2Ri π
∫0∞
K0(k)K1(ν) - β˜ (k)K0(ν)K1(k) I1(ν)K0(k) + β˜ (k)I0(ν)K1(k)
dk (3)
with
Ri )
(ziF)2 8π0pRTNArp
(4)
ν ) xk2 + µ2
µ ) κbrp
x
∑i
zi2cbi φi
γbi
γm i
(5)
exp(-zi∆Ψ - ∆W′i,Born - ∆W′i,im) 2Ib
κb )
(6)
( ) 0bRT
-1/2
2F2Ib 1 Ib ) cbi zi2 2 i
(7)
∑
β˜ )
(
k
xk2 + µ2
(8)
)
p - m p + m
(9)
In eqs 3-9, I0, I1, K0, and K1 are modified Bessel functions, k is the wave vector, F is the Faraday constant, R is the ideal gas constant, NA is Avogadro’s number, and m is the dielectric constant of the membrane. Superscript b denotes bulk solution (18) Yaroshchuk, A. E. AdV. Colloid Interface Sci. 2000, 85, 193. (19) Jo¨nsson, B.; Wennerstro¨m, H. J. Chem. Soc., Faraday Trans. 1983, 79,
γi(∆x+)
) φi exp(-zi∆Ψ(∆x-|∆x+)) × γi(∆x-) exp(-∆W′i,Born) exp(-∆W′i,im(∆x-|∆x+)) (1b)
(
(zie)2 1 1 8π0kTri,cav p b
19. (20) Bandini, S.; Vezzani, D. Chem. Eng. Sci. 2003, 58, 3303. (21) Szymczyk, A.; Fievet, P. J. Membr. Sci. 2005, 252, 77. (22) Rashin, A. A.; Honig, B. J. Phys. Chem. 1985, 89, 5588.
3912 Langmuir, Vol. 22, No. 8, 2006
Szymczyk et al.
follows
ji )
JVci(∆x+) Ak
(12)
where Ak is the porosity of the membrane active layer. Substitution of eq 12 into eq 11 allows the ionic concentration gradients within pores to be derived:
dci JV ziFci dψ ) (Ki,cci - ci(∆x+)) dx Ki,dDi,∞Ak RT dx
Figure 1. Schematic representation of a cylindrical pore separating a feed solution (high-pressure side) from a permeate (low-pressure side).
(feed or permeate) properties. External solutions are assumed to be homogeneous so that the concentration polarization is disregarded in the present study. Consequently, ci(0-) and ci(∆x+) are given by the bulk-phase concentrations (cbi ). The electroneutrality conditions in external solutions and inside the membrane pores are expressed as follows
∑i zici(0-) ) 0 ∑i zici(∆x
e x e ∆x
∑i ziji ) 0
(14)
Equation 11 along with eqs 12 and 13 allow the following expression for the axial electric potential gradient to be derived:
(10b) -
dci ziciKi,dDi,∞F dψ + Ki,cciV dx RT dx
dψ dx
(10c)
)
dci
(11)
where ji is the molar flux of ion i through the pores, Di,∞ is its diffusion coefficient in the external solution (value at infinite dilution), ψ is the local electric potential inside the pores, V is the solvent velocity inside the pores, Ki,d is the hindrance factor for diffusion inside the pores, and Ki,c is the factor accounting for the effect of pore walls on the solute convective flux. The expressions for Ki,d and Ki,c are given in the Appendix. The three terms on the right-hand side of eq 11 represent the diffusive, electromigrative, and convective components of the molar flux ji, respectively. The ionic molar fluxes ji are related to the permeate volume flux JV (based on membrane area) as (23) Schlo¨gl, R. Q. ReV. Biophys. 1969, 2, 305. (24) Dresner, L. Desalination 1972, 10, 27. (25) Deen, W. M. AIChE J. 1987, 33, 1409. (26) Garcia-Aleman, J.; Dickson, J. M.; Mika, A. J. Membr. Sci. 2004, 240, 237. (27) Hoffer, E.; Kedem, O. Desalination 1967, 2, 25.
JV
∑i ziKi,dDi,∞ dx + A ∑i ziKi,cci k
F RT
where X is the effective volume charge density of the membrane (i.e., the mole number of fixed charges per unit of pore volume2,3). The extended Nernst-Planck equation (ENP) forms the basis for the description of solute transport through charged porous membranes.23,24 It describes ion transport in terms of diffusion under the action of the solute concentration gradient, migration under the action of a spontaneously arising electric field, and convection due to solvent flow. In the case of NF membranes for which steric hindrance effects are likely to play a nonnegligible role in the partitioning at membrane/solution interfaces, the ENP is frequently modified by hydrodynamics coefficients accounting for the effect of finite pore size on the various components of the ionic transport.1,3,8,9,12-16,20,21,25,26 Assuming that all partitioning effects are located at interfaces16,20 and that no direct coupling between ionic fluxes occurs (i.e., mutual friction between solutes is neglected27), the modified ENP for ion i reads as follows
ji ) -Ki,dDi,∞
F
-
))0
∑i zici(x) + X ) 0 for 0
In a steady-state filtration process, no net electric current flows through the pores, so the electric current density is zero:
(10a)
+
+
(13)
(15)
∑i zi ciKi,dDi,∞ 2
This electric potential gradient arising to fulfill the zero electric current condition at steady state plays a substantial role in the transport of charged solutes through pores because it acts differently on cations and anions. Indeed, the spontaneously arising electric field enhances the molar flux of coions in the permeate direction whereas it diminishes that of counterions so as to allow the transfer in stoichiometric proportions of both species into the permeate compartment. Equation 15 clearly shows that the electric potential gradient through the membrane pores can be split into diffusive and convective (i.e., proportional to JV) contributions.3,14 The choice of the permeate volume flux (JV) as an independent variable in eqs 13 and 15 instead of the pressure difference between both feed and permeate sides is frequently made in NF modeling. The main reason is that commercially available nanofilters have a multilayer structure, so the total pressure drop through the membrane is the sum of individual pressure drops occurring through each layer. Because support layers separated from the skin layer are almost never available, it is difficult to quantify the effective pressure drop through the selective skin layer. It is therefore more convenient to work with the volume flux JV because it is identical in each layer and constant (at steady state) throughout the membrane. The drawback is that electroviscous3,14,28,29 and osmotic effects3,14 cannot be explicitly expressed in the model. (This could be done, provided support layer contributions be neglected, by using an appropriate relation between the volume flux and pressure drop such as the averaged Stokes equation.3,14) The numerical scheme is as follows: I. First, knowing ci(0-) for the p ionic species in the feed solution, eq 1a is introduced into eq 10c, which is numerically solved together with eq 6 by considering eq 3. It allows both (28) Huisman, I. H.; Pradanos, P.; Hernandez, A. J. Membr. Sci. 2000, 178, 55. (29) Huisman, I. H.; Pradanos, P.; Calvo, J. I.; Hernandez, A. J. Membr. Sci. 2000, 178, 79.
Amphoteric Nanofilter Properties/Characterization
Figure 2. Schematic representation of the membrane cross section. Intermediate layers have been omitted for clarity.
∆Ψ(0+|0-) and ∆W′i,im(0+|0-) values to be computed. The p values of ci(0+) are then calculated by means of eq 1a. II. Next, an initial estimate of ci(∆x+) is made for p - 1 components (with the value for the pth ion being deduced from eq 10b). Introducing eq 1b into eq 10c and solving numerically together with eq 6 by considering eq 3 yields both ∆Ψ(∆x-|∆x+) and ∆W′i,im(∆x-|∆x+) values. Introducing these values into eq 1b gives the ci(∆x-) initial values for the p components. III. Equations 3 and 15 are solved simultaneously using the estimates of ci(∆x-) for p - 1 components so as to determine the axial concentration profiles (for 0+ e x e ∆x-) of these p - 1 components (with the concentration of the pth ion being deduced from eq 10c). Finally, the p values of ci(0+) obtained from transport equations (step III) are compared with those deduced from partitioning equations (step I). A new set of ci(∆x+) values is fixed, and steps II-IV are repeated until the error function is minimized. 3. Experimental Section 3.1. Membrane and Chemicals. The membrane used in this work is a three-channel tubular membrane (cutoff: 1 kD) manufactured by Tami Industry (Nyons, France). It has a multilayer structure with a titania (anatase) active layer and an alumina/titania/ zirconia support and intermediate layers. The external diameter of the membrane is 10 mm, and its length is 600 mm. Each channel has a hydraulic diameter (dh) of 3.6 mm and a cross-sectional area of 11 mm2. A schematic representation of the membrane is given in Figure 2. Experiments have been carried out with potassium chloride of pure analytical grade. Millimolar solutions at various pH values (2.0, 3.0, 6.2, 8.0, 9.0, and 11.0) have been prepared from milliQ-quality water (conductivity
