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Filtration Potential across Membranes Containing Selective Layers Andriy E. Yaroshchuk,*,†,‡ Yuriy P. Boiko,† and Alexandre L. Makovetskiy§ F. D. Ovcharenko Institute of Bio-Colloid Chemistry, National Academy of Sciences of Ukraine, Kiev, Ukraine, Institute of Chemistry, Karl-Franzens University Graz, Heinrichstrasse 28, 8010 Graz, Austria, and A. V. Dumanskiy Institute of Colloid and Water Chemistry, National Academy of Sciences of Ukraine, Kiev, Ukraine Received January 3, 2002. In Final Form: March 29, 2002
The filtration potential in the membranes containing selective layers is studied both theoretically and experimentally. A general thermodynamic expression is obtained for the pressure-induced potential difference across a membrane with an arbitrary number of macroscopically homogeneous layers. That expression is specified for the model of straight cylindrical capillaries in each of the layers. The limiting cases of very wide and very fine pores are considered for the supports and active layers, respectively. In membranes containing selective layers, the dependencies of filtration potential on the transmembrane volume flow are usually nonlinear. That occurs because the concentration differences arising in the membrane depend nonlinearly on the transmembrane volume flow. The concentration gradients cause nonlinearities due to the occurrence of membrane potential and because the streaming potential coefficient in coarseporous layers (e.g., membrane supports) depends on electrolyte concentration. The membrane potential gives rise to a sublinear behavior while the concentration dependence of support electrokinetic properties causes a superlinearity. Therefore, the pattern of nonlinearity essentially depends on the relative contribution of constituent layers into the filtration potential. That contribution is shown to be roughly proportional to the layer’s hydraulic resistance and to the effective zeta-potential in its pores and inversely proportional to the layer electric conductivity. The filtration potential measured with plane ceramic nanofiltration membranes shows a slightly superlinear dependence on the transmembrane volume flow. After the subtraction of support contribution (carried out with the due account of the dependence of support electrokinetic properties on the concentration) the sub-linear pattern characteristic of the active layer contribution is restored. That pattern allows for an extrapolation from which information can be obtained on the ion transport numbers in active layers.
Introduction The measurement of streaming potential is considered a technique of choice for accessing the zeta-potential of liquid/solid interfaces. In the case of relatively coarseporous media, with the average pore size far exceeding the thickness of double electric layer, the well-known Smoluchowski formula is applicable, and the interpretation of streaming potential measurements in terms of zetapotential is straightforward. It has also been attempted to use the measurements of transversal [As opposed to the so-called tangential streaming potential measured in thin channels made of two membranes (cf. refs 1 and 2).] streaming potential to access the electrokinetic properties of fine-porous media, in particular, of active layers of RO, NF, and UF membranes.3-6 In this case, the interpretation in terms of Smoluchowski formula becomes impossible because of overlap of diffuse parts of double electric layers * To whom correspondence should be addressed. Tel: +43 316 380 5443. Fax: +43 316 380 9850. E-mail: andriy.yaroshchuk@ kfunigraz.ac.at. † F. D. Ovcharenko Institute of Bio-Colloid Chemistry. ‡ Institute of Chemistry. § A. V. Dumanskiy Institute of Colloid and Water Chemistry. (1) Childress, A. E.; Elimelech, M. J. Membr. Sci. 1996, 119, 253. (2) Childress, A. E.; Elimelech, M. Environ. Sci. Technol. 2000, 34, 3710. (3) Elmarraki, Y.; Persin, M.; Sarrazin, J.; Cretin, M.; Larbot, A. Sep. Purif. Technol. 2001, 25, 493. (4) Condom, S.; Chemlal, S.; Chu, W.; Persin, M.; Larbot, A. Sep. Purif. Technol. 2001, 25, 545. (5) Benavente, J.; Jonsson, G. J. Membr. Sci. 2000, 172, 189. (6) Huisman, I. H.; Pradanos, P.; Hernandez, A. J. Membr. Sci. 2000, 178, 55.
and due to the strong deviations of electrical conductivity of pore solution from its bulk value (see below for more detail). Nevertheless, those phenomena, in principle, can be taken into account within the scope of a mechanistic model,7 and thus, the measurements can be useful for making less ambiguous the determination of model adjustable parameters (for example, in combination with the measurements of salt rejection carried out in parallel). However, in this case one needs to be able to single out the contribution of layer of interest into the measured electric potential difference. Besides that, it should be remembered that in the case of membranes with selective layers the streaming potential is not the only pressureinduced component of electric potential difference arising under nanofiltration conditions. The other components are the membrane and (possibly) electrode potentials (see, e.g., ref 8). Therefore, we use the more general term of filtration potential instead of streaming potential, which by definition should be measured at no concentration difference. The principal purpose of this paper is to critically assess the informativeness of measurements of transversal filtration potential in the case of membranes containing selective layers. The paper is organized in this way. First, we shall derive a general thermodynamic (in terms of phenomenological coefficients) expression for the pressure-induced electric potential difference arising under nanofiltration conditions across a membrane containing arbitrary number of macroscopically homogeneous layers. Further on, that (7) Bowen, W. R.; Cao, X. W. J. Membr. Sci. 1998, 140, 267. (8) Benavente, J.; Jonsson, G. Colloids Surf. A 1999, 159, 431.
10.1021/la025503s CCC: $22.00 © 2002 American Chemical Society Published on Web 06/01/2002
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expression will be specified for the model of straight cylindrical capillaries within each of the layers. To conclude the theoretical part, this formula will be further specified for the limiting cases of very wide and very fine pores relevant to the situations encountered in the supports and active layers of nanofiltration membranes. In the Experimental Section the design of our setup and the measurement protocol will be described. In the Results and Discussion section, first, we shall consider what properties control the relative contributions of various layers to the overall electric potential difference. Further on, we shall discuss two kinds of phenomena, namely, the membrane (concentration) potential and the coupling between the concentration polarization due to selective layers and the dependence of electrokinetic properties of nonselective layers (e.g., supports) on the salt concentration. Both of them will be shown to cause nonlinearities in the dependence of filtration potential on the transmembrane volume flow. Finally, the results will be presented of sample measurements of filtration potential and salt rejection carried out with fine-porous composite ceramic membranes as well as their supports. It will be shown that for those membranes the contribution of supports was dominant, and the dependence of their electrokinetic properties on the salt concentration needed necessarily to be accounted for to obtain correct information on the electrokinetic properties of active layers.
potential (µei ≡ µci + FZiφ), we obtain this for the derivative of local virtual electric potential φ,
dφ
)-
dx
ti dµci
1
∑ F i Z
i
dx
+ JV
Fek
(2)
g
where Zi is the ion charge, µci is the ion chemical potential, and ti is the ion transport number in the membrane phase at zero transmembrane volume flow defined in this way:
ti )
Zi2ciPi
∑j
(3)
Zj2ciPj
Fek is the so-called electrokinetic charge density defined in this way:
∑i Ziciτi
Fek ) F
(4)
As discussed below, the electrokinetic charge density is the proportionality coefficient between the streaming current (an electrokinetic phenomenon) and the transmembrane volume flow. g is the specific (per unit area and length) membrane electric conductivity at zero transmembrane volume flow defined in this way:
Theory Our theoretical analysis is based on the continuous version of irreversible thermodynamics. In the approximation of no direct coupling between ionic flows (see ref 9 for the discussion of the scope of its applicability) the system of transport equations for ion flows can be written down in this way:
(
Pi dµei + JVτi ji ) c i RT dx
)
(1)
where Pi is the specific (per unit area and length) ionic membrane permeability at zero transmembrane volume flow, µei is the ion electrochemical potential, x is the transmembrane macroscopic coordinate, JV is the transmembrane volume flux, τi is the ionic entrainment coefficient at zero electric potential difference (one minus corresponding reflection coefficient, see ref 9 for the definition), and ci is the local virtual concentration of ith ion. The first term in the right-hand side of eq 1 describes the electro-diffusional ion transfer, and the second one describes the coupling between the ion flows and the volume flow. The virtual quantities are defined as those in a bulk electrolyte solution that could be in thermodynamic equilibrium with a given point inside the membrane. Introducing such a solution is always possible if a local thermodynamic equilibrium occurs. In the absence of slow chemical reactions, that is usually a very reasonable assumption in the case of porous membranes in electrolyte solutions (see ref 9 for the further details on the virtual quantities). It should be stressed that, in contrast to real quantities, the virtual ones remain continuous at phase boundaries (e.g., at the interfaces between the membrane constituent layers). By applying the condition of zero electric current (∑iZiji ) 0) to eq 1 and by using the definition of electrochemical (9) Yaroshchuk, A. E. Adv. Colloid Interface Sci. 1995, 60, 1.
g≡
F2
∑i Zi2ciPi
(5)
RT
In the case of a single salt (binary electrolyte), eq 2 reduces to the following:
(
)
tc dµs Fek 1 tg dφ )+ JV + dx F Zg Zc dx g
(6)
where we have used indices “g” and “c” to denote the counterions and coions, respectively, and introduced the salt chemical potential through µs ≡ (νgµcg + νcµcc)/(νg + νc); νg,νc are the salt stoichiometric coefficients. To obtain an expression for the overall electric potential difference Ef, we should integrate eq 6 over each layer and sum the results up over all the layers. If each layer is macroscopically homogeneous (i.e., its transport properties do not explicitly depend on the macroscopic transmembrane coordinate) the result can be written down in this form:
Ef ≡ φ′ - φ′′ )
1
N
∑ ∫µ (x )
Fk)1
µs(xk+1) s
k
( ) t(k) g Zg
+
t(k) c Zc
dµs N
JV
F(k) xk+1 ek
∑ ∫x
k)1
k
dx (7) g(k)
where the prime and double prime denote the feed and permeate, respectively. The coordinate xk corresponds to the left boundary of the kth layer. The integrands may be functions of virtual salt concentration, but are considered to be not explicitly dependent on the transmembrane space variable within each layer. The volume flow could be taken outside of both integration and summation signs because it is constant in the case of steady-state one-dimensional transport.
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To make use of eq 7, one also needs to know the profile of virtual salt concentration inside the membrane. It can be found through the integration of this transport equation within each layer:10
c′′sJV ) -Ps
dcs + JVcsΤs dx
(8)
where c′′s is the salt concentration in the permeate, Ps is the salt diffusion permeability at zero transmembrane volume flow defined as
νg + νc νg νc + Pg Pc
Ps ≡
(9)
and Ts is the salt entrainment coefficient (one minus reflection coefficient) defined this way:
Ts ≡ τgtc + τctg
(10)
Equation 8 can be solved in quadratures within each macroscopically homogeneous layer to yield this (the continuity of virtual salt concentration at the layer boundaries is taken into account):
JVlk )
P(k)(c ) dc
∫cc(x(x ) )c′′ -s c Τs (k)(cs ) s
s
k
k+1
s
s
s
(11)
s
where lk is the thickness of kth layer. The boundary condition is a given feed concentration at the outermost upstream membrane interface. [That does not necessarily mean that the concentration polarization is disregarded. An unstirred layer (within the scope of Nernst model) may be formally included into consideration as one of membrane layers.] It is easy to show that for an N-layer membrane there are N equations for N unknown concentrations at the layer boundaries. [Consider that the permeate concentration is also unknown in advance.] Finally, if the concentrations at all boundaries (including the downstream one, hence, the permeate) have been found, the concentration profiles within each layer can be obtained through the solution of this equation very similar to eq 11:
JVx )
∫c (x)
cs(xk) s
P(k) s (cs) dcs
c′′s - csΤ(k) s (cs)
Is )
∫0R dr rF(r)v(r)
2p R2
(13)
where p is the membrane porosity and v(r) is the convective solution velocity. Now, let us use the Poisson equation to relate the volume electric charge density inside the capillary to the electrostatic potential φ (it is assumed that the relative dielectric permittivity is a constant independent of both micro-coordinate and electric field strength):
F(r) d2φ 1 dφ )+ 2 r dr 0 dr
(12)
The concentration profiles within the layers are needed to calculate the integrals in the second term in the righthand side of eq 7. Those in the first term depend only on the concentrations at the boundaries. The first term in eq 7 is controlled by the concentration gradients and, thus, can be considered a concentration (membrane) potential contribution. The second term is directly proportional to the transmembrane volume flux and, thus, is roughly equivalent to the classical streaming potential. It should be noted, however, that this second term is essentially modified (as compared to the classical streaming potential) due to the variation of salt concentration throughout the membrane. In multilayer membranes, both the concentration potential contributions and the concentration-related changes in the “streaming potential” term may arise not only within selective layers, but also within nonselective ones (like membrane sup(10) Yaroshchuk, A. E. J. Membr. Sci. 2000, 167, 167.
ports) because of concentration polarization occurring due to the selective layers. Now, let us specify the integrands in eq 7 within the scope of model of identical straight cylindrical capillaries. The first term in the right-hand side, in fact, does not need much specification since its physical meaning is sufficiently clear anyway. It is roughly proportional to the logarithm of ratio of salt concentrations at the boundaries of a layer and to the difference between transference numbers [The transference number is defined as the transport number devided by the ion valency.] of counter- and co-ions within it. In charged capillaries that difference increases with increasing surface charge density and decreasing capillary size. The ratio of concentrations cannot be deduced from the properties of a given layer alone since it is controlled by all the layers (including nonselective ones). The specification of the second term is more meaningful. To carry it out, note that the electrokinetic charge density is defined as the proportionality coefficient between the streaming current density and transmembrane volume flux. To estimate this coefficient, let us consider a bunch of identical straight cylindrical capillaries of radius R with an internal distribution of volume electric charge [Note that this distribution need not necessarily satisfy the Poisson-Boltzmann equation and, thus, possible nonelectrostatic interaction of ions with the membrane matrix are included in this derivation.] F(r), where r is the microscopic radial coordinate inside the capillary. The streaming current is defined as the convective flow of electric charge at zero electric potential difference. Hence, for the streaming current density Is, we obtain the following:
(14)
where and 0 are the relative dielectric permittivity and the dielectric permittivity of vacuum, respectively. By substituting eq 14 into eq 13 and by taking the integral two times by parts, we obtain the following
Is )
[
20p 2
R
φ(R)R
|
dv dr
r)R
(
2
)]
∫0R dr rφ(r) ddrv2 + 1r dv dr
-
(15)
Now, let us make use of the Stokes equation, which for this geometry becomes
η
(
)
dP d2v 1 dv + ) dx dr2 r dr
(16)
where η is the solution viscosity amd P is the hydrostatic pressure. Under the streaming current conditions there is no electric field, so the pressure gradient is the only “driving force” in the right-hand side of Stokes equation. By using eq 16 with the conventional boundary condition
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of zero velocity derivative at the pore axis, from eq 15 we obtain the following:11
0p dP (ζ - φ j) η dx
Is )
(17)
where we have denoted ζ ≡ φ(R) (zeta-potential), and
φ j≡
∫0R dr rφ(r)
2 R2
(18)
is the average electrostatic potential over the capillary cross section. Exactly the same expression for the streaming current density (of course, with a different definition of φ j ) can be obtained for a slitlike capillary.12 Probably, that is a general relationship valid for straight capillaries of any shape. The transmembrane volume flux is directly proportional to the gradient of hydrostatic pressure:
JV ) -χ
dP dx
(19)
where χ is the local specific hydraulic permeability. Thus, for the electrokinetic charge density, we finally obtain
Fek ) -
0p ζ ηχ eff
(21)
An estimate of specific hydraulic permeability of a bunch of straight cylindrical capillaries can be obtained through the Hagen-Poiseuille formula [This is only an estimate because we neglect osmotic corrections and electroviscosity here.]:
χ)
pR2 8η
80 ζeff R2
σR 40
(24)
where σ is the surface electric charge density. In this limiting case of good overlap of diffuse parts of double electric layers, it is often more convenient to deal with a fixed charge density per unit pore volume. By “smearing out” the surface charge over the pore interior, it is easy to obtain this very simple relationship
(23)
In wide pores where the diffuse parts of double electric layers are much thinner than the pore, the average electrostatic potential is practically zero, and the effective zeta-potential reduces to the zeta-potential and becomes independent of the pore size. Thus, in wide pores the electrokinetic charge density is inversely proportional to the square of pore size. It should be stressed that in our analysis the streaming current is scaled on the transmembrane volume flow. In the practice of streaming potential measurements the transmembrane pressure difference is usually used instead to calculate the streaming potential coefficient. At a given hydrostatic pressure difference, the volume flux increases in direct proportion to the square of pore size (at a given membrane porosity). Thus, the inverse proportionality of electrokinetic charge density to the square of pore size cancels out, and the (11) Pastor, R.; Calvo, J. I.; Pradanos, P.; Hernandez, A. J. Membr. Sci. 1997, 137, 109. (12) Yaroshchuk, A.; Ribitsch, V. Langmuir 2002, 18, 2036.
(25)
where FX ≡ 2σ/R is the volume fixed charge density. Equation 25 is one of the basic relationships in the socalled fine-porous membrane model.14,15 Thus, in the limiting case of very fine pores, the electrokinetic charge density becomes explicitly independent of pore size. Finally, let us substitute eq 23 into the relationship for filtration potential of eq 7:
Ef )
N
1
∑ ∫µ (x )
µs(xk+1)
Fk)1
s
k
( ) t(k) g
+
Zg
t(k) c Zc
dµs + N
JV‚80
∑
1
ζ(k) xk+1 eff
∫x
k)1(R(k))2
(22)
By substituting this into eq 20, we obtain
Fek ) -
ζeff )
Fek ) -FX (20)
where we have introduced an effective zeta-potential through
j ζeff ≡ ζ - φ
streaming potential coefficient becomes independent of pore size, which is well-known from the Smoluchowski’s analysis. In fine pores, there are two competing trends. On one hand, the first factor in eq 23 increases with decreasing pore size. On the other hand, the average electrostatic potential becomes ever closer to the potential of pore surface due to the overlap of diffuse parts of double electric layers, and the effective zeta-potential decreases. In ref 13 the Poisson-Boltzmann equation has been solved analytically just in this limiting case of small variation of electrostatic potential inside cylindrical pores. By using the results of that paper, in the first approximation in effective dimensionless reciprocal screening length, the effective zeta-potential can be shown to be
k
dx (26)
g(k)
Now, let us consider a bilayer membrane consisting of a fine-porous active layer and a coarse-porous (nonselective) support (concentration polarization near the upstream membrane surface is disregarded). In the steadystate pressure-driven membrane processes, there are no solute concentration gradients across nonselective supports. Therefore, only the active layer contributes into the first term in eq 26. Both layers make contributions to the second term. To specify them, let us use the limiting expressions for the electrokinetic charge density derived above for very fine and very coarse pores. Thus, we obtain
Ef )
∫µ′
1 F
µ′′ s s
(
) (∫
t(a) t(a) g c + dµs + Zg Zc JV -
(a) l(a)FX 0 (a)
g
dx +
)
80ζ(s)(c′′s)l(s) (R(s))2g(s)(c′′s)
(27)
where the indices “a” and “s” denote the active layer and support, respectively; l(a) and l(s) are their thicknesses. The zeta-potential and specific electric conductivity of the (13) Yaroshchuk, A. E.; Dukhin, S. S. J. Membr. Sci. 1993, 79, 133. (14) Schmid, G.; Schwarz, H. J. Membr. Sci. 1998, 150, 197. (15) Schmid, G. J. Membr. Sci. 1998, 150, 159.
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support should be taken at the permeate concentration. To our knowledge for the first time, the last observation was made by Benavente and Jonsson.5 For the numerical analysis below, to describe the active layer we shall use the classical fine-porous membrane model complemented by a single non-Donnan exclusion factor as described in ref 16. Within the scope of that approach all the non-Donnan ion exclusion factors are “assembled” in a single distribution coefficient, γ, common for cations and anions and independent of concentration. The Pe´clet number will be defined this way
Τ(a) s (c′s) Pes ≡ JV (a) Ps (c′s)
(28)
Above, we have used the model of identical straight cylindrical capillaries to specify the phenomenological coefficients. In most membranes, the pores have essentially more complex geometry. However, many of the conclusions drawn above (and below) remain valid in this case, too. First of all, all the qualitative conclusions made before the introduction of capillary model, evidently, remain valid. Moreover, even the model equations may well remain applicable, at least semiquantitatively. Indeed, in the limiting case of very fine pores, the pore geometry does not matter much (of course, the pore size distribution should not be too broad, so that all the pores remain sufficiently fine). The same is true, in a sense, in the opposite limiting case of coarse pores, where the Smoluchowski formula is applicable. Indeed, it can be shown that the streaming potential scaled on the hydrostatic pressure difference is independent of geometry of the pores if they are sufficiently large. That can be exloited to derive a version of eq 27 less explicitely dependent on the pore properties. To achieve that, the last term in the right-hand side of eq 27 should be expressed through the pressure difference across the support by means of eq 22. If it is further assumed that the porosity and pore tortuosity are the same for the volume flow and for the electric current, this relationship is obtained:
Ef )
∫µ′µ′′
1 F
s
s
(
)
t(a) t(a) g c + dµs - JV Zg Zc
∫0l
(a)
F(a) X
dx + g(a) 0ζ(s)(c′′s) (s) ∆P (29) ηλ(c′′s)
where λ is the specific electric conductivity of solution in the pores, η is its viscosity, and ∆P(s) is the hydrostatic pressure difference across the support. If separate samples of supports are available, the latter quantity, in principle, can be estimated from experimental data. Equation 29 appears to be less model-dependent than eq 27. Experimental Section Materials. The membranes used in this study were laboratorymade disk ceramic NF membranes and their supports (diameter 39 mm, courtesy of Laboratory of Inorganic Material Science, University of Twente, The Netherlands). The active layers made of γ-alumina had the thickness of about 1.5 µm, the average pore size of 4 nm, and porosity of about 30%. The supports made of R-alumina had the thickness of about 2 mm, the average pore size of 0.1 µm and porosity of about 30%. All the chemicals were high purity grade; water was bidistilled. (16) Yaroshchuk, A. E.; Ribitsch, V. J. Membr. Sci. 2002, 201, 85.
Experimental Setup. The measurements of filtration potential and salt rejection have been carried out in a batch test cell (volume 150 cm3) made of Plexiglas and equipped with a pair of Ag/AgCl indicator electrodes connected to a high-resistance voltmeter. Due to a robust design, the cell withstands hydrostatic pressures up to 50 bar. Vigorous stirring is ensured by a specially designed disklike stirrer situated very close to the membrane surface (about 1 mm) and fixed at the tip of a hollow glass axis driven by a permanent magnet fixed at its opposite end. A pair of four-arm catches made of Plexiglas holds up the axis. Due to the very good magnetic coupling with the external magnet, pretty high rotation speeds (up to 2000 rpm.) can be achieved, and thus, a concentration polarization can be thoroughly eliminated. Because of the relatively large thickness of supports in combination with their monolayer structure and rather small pore size, the ion-exchange capacity of supports is pretty large. Therefore, much care had to be taken to equilibrate the membranes with solutions of given pH values. To accelerate the process, large amounts of corresponding solutions were filtered through the membrane until the permeate pH value stabilized in time. The salt rejection was determined through the measurements of feed and permeate electrical conductivities in a separate thermostated cell.
Results and Discussion Let us start from identifying the properties that control the relative contribution of membrane layers into the filtration potential. The first term in eq 26 is governed by the ion transport numbers and the ratios of virtual salt concentrations at the layers’ boundaries. Generally speaking, the electrochemical permselectivity of active layers is usually larger than that of supports, especially if the principal rejection mechanism is the Donnan exclusion. Besides that, in the traditional orientation of bilayer composite or asymmetric membranes (active layers facing the feed solution) under steady-state nanofiltration/ reverse osmosis conditions, there are no concentration gradients (and, accordingly, no concentration potential) across supports. Therefore, typically, the first term in eq 26 is controlled by the active layers. Nevertheless, there may be situations (e.g., a membrane with predominantly non-Donnan rejection mechanism used in the inverse orientation in an electrolyte solution with nonequal mobilities of cations and anions) where the support makes a contribution to it. In the second (“streaming potential”) term, for rough estimates, one can neglect the variation of virtual salt concentration within the layers and take the integrals. Thus, we obtain that the contribution of a particular layer is controlled by this combination of its properties: l/R2(ζ/g). One can also say that the layers’ contributions into the total filtration potential are controlled by their contributions into the membrane hydraulic resistance.5,17,18 The pore size in membrane supports is much larger than in their active layers. At the same time, the supports are much thicker. The typical ratio of thicknesses of supports to those of active layers is ca. 100-1000. The typical ratio of their pore sizes is about 100. Thus, the “geometrical” factors are “in favor” of active layers by one to 2 orders of magnitude. That is conform, e.g., with the finding of ref 5 where it is reported that the hydraulic permeability of a composite RO membrane was about an order of magnitude smaller than that of a microfiltration membrane believed to be similar to the support used in the RO membrane. (17) Labbez, C.; Fievet, P.; Szymczyk, A.; Aoubiza, B.; Vidonne, A.; Pagetti, J. J. Membr. Sci. 2001, 184, 79. (18) Szymczyk, A.; Labbez, C.; Fievet, P.; Aoubiza, B.; Simon, C. AIChE J. 2001, 47, 2349.
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Figure 1. Theoretically estimated filtration potential in a bilayer membrane against the Pe´clet number in the active layer. The inscriptions indicate the layer, which makes a dominant contribution as well as the rejection mechanism in the active layer; “active layer + support” means equal contributions of each layer. The dashed straight line shows the extrapolation of linear part to zero volume flow. The parameters for the active layer are: the concentration of fixed charges is 5 times the feed concentration (for the Donnan exclusion); the distribution coefficient is γ ) 0.3714 (for the non-Donnan rejection mechanism).
The “electrical” factors, namely, the ratio of effective zeta-potential to the specific electric conductivity, clearly “favor” the supports. Indeed, the effective zeta-potential may be much smaller than the surface potential due to the overlap of diffuse parts of double electric layers (DEL), and the specific electric conductivity in fine charged pores is known to be essentially higher than that in wide pores due to the electrostatic adsorption of counterions. Besides that, under conventional NF conditions, the salt concentration in the supports may be several times lower than in the active layers. The latter may bring about a 5- to 10-fold increase in the relative contribution of supports due to the decrease in the electric conductivity (roughly directly proportional to the salt concentration) and the typical increase in zeta-potential with decreasing salt concentration. Thus, it is impossible to tell a priori which layer in a composite bilayer membrane makes a decisive contribution to the filtration potential.17 Figure 1 shows examples of theoretically calculated dependencies of filtration potential on the transmembrane Pe´clet number for several different relative contributions of active layer/support to it. It is seen that the curves are typically nonlinear. The pattern of nonlinearity (sub- or superlinearity) depends on the relative contribution of constituent layers. There may also occur almost linear plots, but that is rather exceptional since an approximate linearity can only be caused by the accidental compensation of nonlinearities due to the active layers and supports. Of course, there also is a trivial linear case, namely that of nonselective membrane. However, that can be easily checked experimentally through the measurements of salt rejection. It is also seen from the figure that the nonlinearity due to the support is qualitatively insensitive to the rejection mechanism in the active layer, namely the plots have quite similar shapes in the two limiting cases of purely Donnan and purely non-Donnan rejection mechanisms (at the same limiting rejection). Whatever the pattern of nonlinearity, all the curves sooner or later tend to become linear. However, because
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of initial nonlinearity, the continuations of their linear parts usually do not pass through the origin. Since both active layers and supports generally control the nonlinearity, not much useful information can be obtained from the intercept. However, if the support contribution has been subtracted, the nonlinearity can be exploited to obtain information on the electrochemical properties of active layer. Indeed, from eq 27, it is seen that with increasing transmembrane volume flow, the first term in the righthand side tends to saturation while the second one continues to increase linearly. The saturation value of the first term can be determined from the intercept of continuation of this linear part of dependence with the ordinate.19 From that (and the limiting rejection) one can estimate the ion transport numbers within the active layer. [Strictly speaking, a dependence of streaming potential coefficient of active layer on the virtual salt concentration also leads to a contribution into this “saturation” value”. However, in ref 11 this contribution has been shown to be relatively small.] To be able to single out the active layer contribution from the measurements carried out for membranes as a whole, first of all, one needs separate samples of membrane supports, which often are not easy to obtain. However, even if the samples are available, the task is not that easy as it appears at first glance. The subtraction of support contribution from the overall filtration potential should be made with the due account of the fact that this contribution is controlled by the permeate concentration. Since the latter essentially depends on the transmembrane volume flow, the reference measurements with supports have to be carried out at several concentrations ranging from the limiting permeate concentration (at high volume flows) to the feed one. Figure 2 shows some examples of experimental data obtained for ceramic NF membranes in CaCl2 solutions. Figure 2a shows the filtration potential directly measured for the membrane and the support as well as the corrected support contribution. Figure 2b shows the corresponding salt transmission data, and Figure 2c shows the streaming potential coefficient of support measured at several salt concentrations. The curves shown in Figure 2b have been obtained by means of modified Spiegler-Kedem procedure developed in ref 20. The fitted transport coefficients are given in the Table 1. They have been used to calculate the Pe´clet number from the transmembrane volume flux by eq 28. Expectedly, the support streaming potential coefficient increases considerably with decreasing salt concentration. At the same time, at almost all the transmembrane volume flows involved, the salt concentration in the permeate (and, thus, in the support) was noticeably lower than in the feed. Therefore, an attempt to estimate the active layer contribution through a simple subtraction of the support streaming potential from the membrane filtration potential would lead to quite erroneous results. The correct procedure is to determine the streaming potential coefficient of support at each permeate concentration involved and use those values for the subtraction. They can be either directly measured or estimated through the interpolation between the data obtained at several salt concentrations (the latter procedure has been used to calculate the corrected support contribution in Figure 2a)). Figure 3 shows two examples of the outcome of correct subtraction procedure. From the comparison of Figure 3 (19) Minning, C. P.; Spiegler, K. S. In Charged Gels and Membranes; Selegny, Ed.; D. Reidel: Dordrecht, 1976; Vol. 1, p 277. (20) Yaroshchuk, A. E. J. Membr. Sci. 2002, 198, 285.
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Figure 2. Experimental data obtained for the bilayer ceramic membrane and/or its support in CaCl2 feed solutions of pH7.3: (a) filtration potential (corrected for the electrode potential) and the support contribution corrected for the variation of salt concentration in the permeate as functions of transmembrane volume flow. (b) salt transmission (permeate concentration scaled on the feed one) as a function of Pe´clet number; feed concentrations are indicated in the figure. The curves are approximations obtained by the modified Spiegler-Kedem method (explanations in text). (c) streaming potential coefficient in the support as a function of electrolyte concentration. Table 1. Membrane Transport Coefficients Fitted to the Experimental Salt Rejection Data by the Modified Spiegler-Kedem Procedure20 concentration coefficient
0.011M
0.1M
Ts Ps (µm/s) aa
0.27 2.66 1.15
0.56 5.07 0
a Parameter a is a quantitative measure of the rate of dependence of transport coefficients on salt concentration.20
and Figure 2a, it is seen that the “simple” subtraction would overestimate the active layer contribution about three times for the 0.011 M feed solution. Still a more serious error could, obviously, be made if one disregarded the presence of support at all and ascribed the whole measured values of filtration potential to the active layer alone. That, probably, was the reason for the observation made by Huisman, Pradanos et al.6 for an asymmetric UF membrane; namely, that the values of zeta-potential obtained from streaming potential measurements were systematically larger than those calculated from the salt rejection and electroviscosity. Notably, for monolayer track-etched membranes the opposite was the case.21 It is also interesting to compare the two curves shown in Figure 3. It is seen that, despite an order of magnitude difference in the feed concentrations, the difference in (21) Huisman, I. H.; Pradanos, P.; Calvo, J. I.; Hernandez, A. J. Membr. Sci. 2000, 178, 79.
Figure 3. The active layer contribution to the filtration potential vs Pe´clet number obtained through the subtraction of support contribution: bilayer ceramic membrane, CaCl2 solutions of pH7.3, feed concentrations are indicated in the figure; the dashed straight lines show the extrapolations of linear parts to zero transmembrane volume flow.
their slopes is not very large. At the same time from Figure 2c, it is seen that for the support the dependence on the salt concentration is essentially stronger. Indeed, the slopes in Figure 3 are different about 3 times, while the streaming potential coefficients taken at 0.01 and 0.1 M concentrations in Figure 2c are different almost 10 times. That is in agreement with the different factors controlling the streaming potential in fine and wide pores. In fine
Filtration Potential across Membranes
pores, both the electrokinetic charge density and the electric conductivity are mainly controlled by counterions. If the salt concentration is sufficiently low, the concentration of counterions in pores is almost equal to the volume concentration of fixed charges, and practically does not depend on the salt concentration.13 Incidentally, from the fact that the dependence of the slope on the salt concentration is not very strong in the interval between 0.01 and 0.1 mol/L, one can conclude that in the active layer the fixed charge concentration lies somewhere within that same range. The reasons for the strong dependence of streaming potential coefficient on the salt concentration in wide pores have already been discussed above. Due to the different rates of dependence of contributions of active layers and supports on the salt concentration, the relative contribution of active layers into filtration potential generally increases with increasing salt concentration. Another feature of filtration potential data shown in Figure 2a for the membrane is their superlinearity with respect to the transmembrane volume flow. From the same figure, it is seen that the data obtained for the membrane support are strictly linear. For our membrane with its thick and relatively fine-porous support, the nonlinearity was mainly caused by the dependence of support streaming potential coefficient on the permeate salt concentration. That is confirmed by the similar pattern of nonlinearity in the corrected support contribution shown in Figure 2a. As already discussed above, another source of nonlinearity is the contribution of membrane potential. In Figure 2a, it is “masked” by the superlinearity arising due to the support contribution. From Figure 3, it is seen that the correct subtraction procedure “restores” the pattern of nonlinearity characteristic of active layers (compare with Figure 1a). Moreover, the extrapolation of linear parts yields more or less reasonable intercepts, including a smaller value in the more concentrated solution. However, a more quantitative analysis shows that the values of counterion (chloride) transport numbers obtained from the intercepts [From Figure 3 it is seen that at very small Pe´clet numbers both the plots tend not exactly to zero but to a common slightly negative value. Most probably, the reason for such behavior was a potential of asymmetry. In the estimates of transport numbers above that was assumed to be the case, and a potential of asymmetry of 3 mV was added to both the intercept values.] (0.79 in 0.011 M and 0.69 in 0.1 M solutions as compared with 0.56 in the bulk) are too small to be compatible with the experimentally observed salt rejections within the scope of the simple Donnan exclusion model. Indeed, the chloride transport number corresponding to the fitted entrainment coefficient Ts ) 0.27 (in the 0.011 M solution) is equal to 0.95 and that corresponding to Ts ) 0.56 (in the 0.1 M solution) is equal to 0.87. There are several possible explanations for that, the most obvious being the assumption that a non-Donnan mechanism (e.g., the dielectric exclusion22) contributes to the salt rejection. However, Figures 2a and 3 show that, for the membrane used in this study, the active layer contribution was estimated as a moderate difference between two relatively large quantities. Accordingly, it was hardly possible to single it out with a high accuracy, and it is not clear whether much meaning can be ascribed to the extrapolated values. One more source of uncertainity in the subtraction procedure should also be noted. As stated in the Experimental Section, it was very difficult to completely equilibrate a membrane (and a support) with a solution of a given pH value. Therefore, one could not be absolutely (22) Yaroshchuk, A. E. Adv. Colloid Interface Sci. 2000, 85, 193.
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sure that the supports used to obtain information needed for the subtraction were equlibrated with the solutions of exactly the same pH value as used in the measurements with the membranes. In more favorable situations, e.g., that of ref 5 [According to ref 3 the relative support contribution to the filtration potential was typically below 3%. The problem, however is that the authors of ref 3 used not exactly the membrane support, but a membrane they believed to be sufficiently similar to it. Nevertheless, if the membrane was indeed somewhat similar to the support, and its relative contribution was only as small as a couple of percent, that did not matter much.], the accuracy might be higher. From the intercepts in Figure 2 of ref 5 and the information on the salt rejection obtained from the authors,23 we could estimate the sodium ion transport number within active layers of HR95 membrane to be 0.88 in 0.005 M NaCl and 0.65 in 0.02 M NaCl. Conclusions In membranes containing selective layers, the dependencies of filtration potential on the transmembrane volume flow are usually nonlinear. The nonlinearities occur due to the fact that the concentration differences arising in the membrane depend nonlinearly on the transmembrane volume flow. There are basically two causes for the nonlinearity: the membrane (concentration) potential and the dependence of streaming potential coefficient in coarseporous layers (e.g., membrane supports) on electrolyte concentration. The patterns of nonlinearity caused by those two factors are qualitatively different. The membrane potential gives rise to a sub-linear behavior while the concentration dependence of support electrokinetic properties causes a superlinearity. Therefore, the pattern of nonlinearity occurring for a composite bilayer membrane essentially depends on the relative contribution of constituent layers into the filtration potential. That contribution is roughly proportional to the layer’s hydraulic resistance and to the effective zeta-potential in its pores and is inversely proportional to the electric conductivity there. Typically, the “hydraulic” factor favors fine-porous active layers, while the “electrical” factors favor coarseporous supports. Therefore, it is impossible to tell a priori which layer makes a dominant contribution. For understanding the membrane performance, of primary interest are the electrokinetic properties of active layers. To single them out, calibration measurements with supports have to be carried out. Moreover, since the streaming potential in the supports occurs at the permeate concentration, the calibration measurements should be carried out at several electrolyte concentrations ranging from the feed to the limiting permeate one. Such a protocol has been experimentally implemented for laboratory-made plane ceramic membranes. It has been shown that for the membranes used in this study, the support made a dominant contribution to the filtration potential. Accordingly, the dependencies of filtration potential on the transmembrane volume flow were slightly superlinear. Measurements of streaming potential performed with separately available membrane supports at several electrolyte concentrations enabled us to correctly carry out the procedure of subtraction of support contribution. That restored the pattern of nonlinearity (namely, sublinearity) characteristic of active layer contribution and enabled us to extrapolate the linear parts occurring at larger transmembrane volume flows down to the (23) Benavente, J. Personal communication.
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interception with the ordinate. From the intercepts, reasonable values for the ion transport numbers in the active layer could be obtained, though they were too small to be compatible with the observed salt rejections within the scope of simple Donnan exclusion model. One of explanations for that is a contribution of (a) non-Donnan rejection mechanism(s). However, it should be noted that for the studied membrane, the active layer contribution was determined as a moderate difference of two relatively large quantities. Accordingly, the precision of the sub-
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traction procedure could not be high, and one should be cautious about ascribing too much meaning to its results. Acknowledgment. The financial support of Volkswagen Stiftung (Germany) within the scope of research project “Advanced modeling of nanofiltration via improved input due to novel experimental techniques of membrane characterization” is gratefully acknowledged. LA025503S
