Some Properties of Electrolyte Solutions in Nanoconfinement

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Some Properties of Electrolyte Solutions in Nanoconfinement Revealed by the Measurement of Transient Filtration Potential after Pressure Switch Off Andriy E. Yaroshchuk,*,† Yuriy P. Boiko,† and Alexandre L. Makovetskiy‡ F. D. Ovcharenko Institute of Biocolloid Chemistry and A. V. Dumanskiy Institute of Colloid and Water Chemistry, National Academy of Sciences of Ukraine, Vernadskiy ave. 42, 03142 Kiev, Ukraine Received April 7, 2005. In Final Form: June 8, 2005 We have demonstrated that with a composite nanoporous ceramic membrane in a batch membrane cell it is technically feasible to switch off the trans-membrane hydrostatic pressure difference within tens of milliseconds. That enabled us to resolve practically the whole time evolution of transient filtration potential. Measurements of the latter have been complemented by measurements of steady-state salt rejection by the composite membrane and by measurements of the streaming potential and hydraulic permeability of membrane supports available separately. A theory has been developed in terms of network thermodynamics for the electrical response of a bilayer membrane to a pressure perturbation. In combination with the results of salt rejection measurements, from the time transients of filtration potential we could determine the ion transport numbers within the nanoporous layer. Besides that, from the dependence of steady-state salt rejection on the trans-membrane volume flow, we have determined the diffusion permeability of and the salt reflection coefficient in the nanoporous layer. This has enabled us to estimate the contributions of Donnan and non-Donnan mechanisms to the rejection of ions by the nanoporous membrane used in this study. It has been unexpectedly found that the Donnan exclusion played only a secondary role. Our hypothesis is that the non-Donnan exclusion of ions from the nanopores might be caused by changes in water properties in nanoconfinement. Proceeding from the results of steady-state filtration experiments with the membrane and the support, we also concluded that the nanoporous layer was imperfection-free and had a quite narrow pore size distribution, which made it a suitable object for fundamental studies of ion transfer mechanisms in nanopores.

Introduction Electrostatic interactions are considered to be important for the properties of electrolyte solutions in nanoconfinement, in particular, those due to the more or less pronounced overlap of diffuse parts of double electric layers near the pore surfaces. Composite nanofiltration membranes consist of thin (usually 0.5 to 5 µm thick) nanoporous layers supported by much thicker and relatively coarse-porous supports. Because of the small thickness of nanoporous layers, their various permeabilities are relatively high and, in principle, measurable. The electrical properties of membranes and other porous media in aqueous solutions have usually been characterized through the measurements of streaming potential.1,2 However, it has been shown that the streaming potential in composite or asymmetric membranes may be strongly influenced by contributions from their supports and by the internal concentration polarization.3 Besides that, even if the streaming potential difference across the active layer could be singled out, to interpret it in useful terms in not too coarse-porous membranes, information on the electric * Corresponding author. Current address: Waste Management Laboratory, Paul-Scherrer-Institute, 5232 Villigen PSI, Switzerland. Phone: +41 (0)56 310 5316. Fax: +41 (0)56 310 4595. E-mail: [email protected].. † F. D. Ovcharenko Institute of Biocolloid Chemistry. ‡ A. V. Dumanskiy Institute of Colloid and Water Chemistry. (1) Sbaı¨, M.; Fievet, P.; Szymczyk, A.; Aoubiza, B.; Vidonne, A.; Foissy, A. J. Membr. Sci. 2003, 215, 1-9. (2) Martıń, A.; Martıńez, F.; Malfeito, J.; Palacio, L.; Pra´danos, P.; Hernańdez, A. J. Membr. Sci. 2003, 213, 225-230. (3) Yaroshchuk, A. E.; Boiko, Yu. P.; Makovetskiy, A. L. Langmuir 2002, 18, 5154-5162.

conductivity of the nanoporous layer would be needed.4 That is difficult to obtain for the active layers of composite membranes, in particular, because the former typically make only a small to moderate contribution to the membrane electrical resistance. The measurement of membrane potential is a useful membrane characterization tool, in particular, because its mechanistic interpretation does not require a knowledge of such sometimes poorly known active-layer properties as its thickness and especially porosity. However, the measurement of conventional steady-state membrane potential is not informative in the case of supported membranes because of the distribution of the applied concentration difference between the membrane layers (usually in an unknown proportion). It has been shown that non-steady-state electrochemical techniques are more useful. In this case, the state of a membrane system is rapidly perturbed, and the electrical response (usually the arising emf) to that is measured as a function of time. In general, there are three possible modes of perturbation: a concentration step, a current step, and a pressure step. The potentialities of the first mode have been explored in ref 5, and those of the second one, in ref 6. The methodological purpose of this article is to consider the pressure-step mode in some detail both theoretically and experimentally. Time-resolved measurements of the transmembrane potential difference after a pressure step have been used to determine the streaming potential coefficient (4) Yaroshchuk, A. E. Adv. Colloid Interface Sci. 1995, 60, 1-93. (5) Yaroshchuk, A. E.; Makovetskiy, A. L.; Boiko, Yu. P.; Galinker, E. W. J. Membr. Sci. 2000, 172, 203-221. (6) Yaroshchuk, A. E.; Karpenko, L.; Ribitsch, V. J. Phys. Chem. B 2005, 109, 7834-7842.

10.1021/la050917h CCC: $30.25 © 2005 American Chemical Society Published on Web 07/22/2005

Electrolyte Solutions in Nanoconfinement

of monolayer ion exchange7,8 and fine-pore track-etched9 membranes. The characteristic times encountered in refs 7-9 were several orders of magnitude longer than in this study primarily because of the much larger membranes thicknesses involved. Accordingly, no particular technical problems have been reported in the pressure switch-off. On the contrary, in this study considerable care had to be taken to make the time needed for the complete relaxation of the trans-membrane pressure difference much shorter than the characteristic time of relaxation of the transient membrane potential. A common theoretical basis for all three perturbation modes has been developed in ref 10. The approach of ref 10 first will be used to obtain an expression for the relaxation pattern of transient membrane potential for the configuration used in this study. The theory will be developed under the assumption of weak concentration polarization, which allows us to neglect the dependence of membrane transport properties on the salt concentration, thus keeping the equations linear. That will enable us to resolve them by means of Fourier transform. After that, our experimental setup will be described where much care has been taken to ensure as rapid a relaxation of the trans-membrane hydrostatic pressure difference as possible, and independent control of that difference has been implemented. In the Results and Discussion section, we shall present and interpret theoretically the results of sample measurements carried out with laboratory-made flat ceramic nanofiltration membranes and their supports in 0.03 M KCl solutions. The measurements of transient membrane potential have been complemented by simultaneous measurements of the transient trans-membrane hydrostatic pressure difference as well as by the measurements of salt rejection as a function of pressure in the steady state. The experimental time transients will be fitted theoretically, thus the initial values of membrane potential immediately after the pressure switch-off and the characteristic relaxation times will be estimated. The initial values will be used to subtract the diffusion component of the filtration potential and to single out the streaming potential. Besides that, the initial values of the membrane potential in combination with the results of measurements of the steady-state salt rejection will be used to estimate the ion transport numbers within the membrane active layer. A knowledge of ion transport numbers and the salt reflection coefficient (from the steady-state filtration data) will make it possible to draw quantitative conclusions concerning some properties of electrolyte solutions in the nanopores of the membrane used in this study. Finally, knowledge of the characteristic relaxation time will enable us to estimate some transport properties of membrane supports. Theory Basic Equations. The basic equations of linear nonsteady-state theory for the time-dependent distribution of the salt chemical potential in polarized multilayer systems have been derived in ref 10. The equations have been developed in terms of network thermodynamics and thus operate with model-independent parameters such (7) Okada, T.; Kjelstrup Ratkje, S.; Hanche-Olsen, H. J. Membr. Sci. 1992, 66, 179-192. (8) Okada, T.; Kjelstrup-Ratkje, S.; Møller-Holst, S.; Jerdal, L. O.; Friestad, K.; Xie, G.; Holmen, R. J. Membr. Sci. 1996, 111, 159-167. (9) Brendler, E.; Kjelstrup Ratkje, S.; Hertz, H. G. Electrochim. Acta 1996, 41, 169-176. (10) Yaroshchuk, A. E.; Ribitsch, V. Chem. Eng. J. 2000, 80, 203214.

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as ion transport numbers, diffusion permeabilities, chemical capacities, and so forth. Because of that, in the subsequent interpretation of experimental data we shall be able to obtain phenomenological characteristics of active and support layers without the need to specify any mechanistic model of those media. For a membrane consisting of several macroscopically homogeneous layers, the basic equation for the timedependent distribution of the salt chemical potential, µ(ξ, τ), can be written down in this dimensionless form

∂µ(ξ, τ) ∂2µ(ξ, τ) ) βk2 ∂τ ∂ξ2

(1)

where k is the layer index, the trans-membrane coordinate is scaled on the active layer thickness, ξ ≡ x/la, and the dimensionless time, τ, is obtain by scaling time by the diffusion relaxation time of the active layer defined in this way

t0 ≡

Ra 2 l χa a

(2)

Index a denotes the properties of the active layer, χ is the specific diffusion permeability of the membrane material for the salt, which is defined here as the proportionality coefficient between the salt diffusion flux and the gradient of salt chemical potential. R is the socalled specific chemical capacity defined as

R≡

( ) ∂qs ∂µ

T,P

(3)

which is a quantitative measure of how much salt (δqs) has to be added to a unit volume of a medium to change the salt chemical potential by δµ. A detailed discussion of this useful property and its estimates for various media can be found in refs 11 and 12. The coefficients βk are defined in this way:

βk2 ≡

Rkχa Raχk

(4)

By definition, βa ≡ 1. One of the boundary conditions to eq 1 is that of the continuity of the salt chemical potential at all of the boundaries (local interfacial equilibria). In the particular case of interest for this study (zero electric current), the boundary condition for the salt flux can be shown to have this form10

∂µ ∂µ χk |ξ)ξk-0 - χk+1 |ξ)ξk+0 ) -lajvcs∆σs|ξ)ξk ∂ξ ∂ξ

(5)

where ξk represents the coordinates of the boundaries between membrane layers and ∆σs ≡ σs|ξk - 0 - σs|ξk+0 represents the changes in the salt reflection coefficient at the boundaries. Despite the appearance of salt flux sources, electric charges do not arise at the boundaries because exactly as much charge is taken away from a boundary by anions as is brought to it by cations and vice versa. Specification of Equations for Our Experimental Setup. In this study, we shall consider a bilayer system (11) Zholkovskij, E. K. In Surface Chemistry and Electrochemistry of Membranes; Sorensen, T. S., Ed.; Marcel Dekker: New York, 1998; pp 793-835. (12) Jamnik, J.; Maier, J. Phys. Chem. Chem. Phys. 2001, 3, 16681678.

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r≡

x

Rsχs Raχa

(14)

jV(ω) is the Fourier transform of jV(τ), which can be shown to have this form

(

jV(ω) ) jV0 πδ(ω) Figure 1. Model and coordinates.

consisting of an active layer and an infinitely thick support. Because of vigorous stirring, the salt concentration (and thus its chemical potential) at the external active layer surface is given anytime. It can also be considered to be given anytime at the infinitely distant external surface of the support. Because the chemical potential is defined up to a constant, its preset values at the external activelayer surface and at infinity may be taken to be equal to zero. It is further assumed that the system is first brought to a quasi-steady polarization state by a constant transmembrane volume flow and after that the flow suddenly vanishes at τ ) 0. That can be formally described by a “negative unit-step function” or

jV(τ) ≡ jV0(1 - H(τ))

(6)

where jV0 is the value of trans-membrane volume flow in the initial quasi-steady state and H(τ) is the unit-step function. The model and the chosen coordinates are shown in Figure 1. The boundary conditions for this system are

µa(0, τ) ) 0

(7)

µs(∞, τ) ) 0

(8)

µa(1, τ) ) µs(1, τ)

(9) (10)

Indices a and s denote the active layer and the support, respectively. The solution to eq 1 is sought in this form

µi(ξ, τ) ≡

∫-∞+∞ dω exp(-iωτ) fi(ξ, ω)

1 2π

(

f(i ()(ξ, ω) ≡ exp ((1 - i)βi

xω2 ξ)

(12)

By applying the boundary conditions, for the complex spectral density of chemical potential difference across the active layer one obtains

fa(1, ω) ) lacs∆σs χa

(16)

where ∆t1 is the difference in the transport number of ion 1 between the active layer and the support. (It is also assumed that the ion transport numbers in the support are the same as in the bulk electrolyte solution.) While deriving eq 16, it has been assumed that both of the indicator electrodes are located in the solutions of feed concentration. If the initial membrane polarization is so long that one of the electrodes turns out to be in contact with the permeate solution, then the electrode potential difference should be added to the right-hand side of eq 16. Finally, by substituting eqs 11 and 13 into eq 16 and by taking the real part, we obtain

(

jV0 RT 1 ∆t1∆σs 1 + Im × F Pa π

(

∫-∞+∞ dω ω1x+2ωi

exp(-iωτ)

(

r + coth (1 - i)

x) ω 2

)

(17)

where we have introduced the absolute molar diffusion permeability of the active layer according to

RT χa cs l a

(18)

Experiment

where ω is the dimensionless circular frequency (scaled on 1/t0) and fi(ξ, ω) is the complex spectral density of temporal response. By substituting eq 11 into eq 1, we obtain these fundamental solutions:

-

(15)

2∆t1 µ (1, τ) F a

Em(τ) ) -

Pa ≡ (11)

)

At zero electric current and zero volume flow (i.e., after the pressure switch-off), the difference in the electric potentials between the high-pressure and the low-pressure half-cells separated by the polarized bilayer membrane for (1:1) electrolytes can be shown to be related to the difference in the salt chemical potentials across the active layer in this way

Em(τ) )

∂µa ∂µs χa |ξ)1 - χs |ξ)1 ) -lajV(τ)cs∆σs|ξ)1 ∂ξ ∂ξ

i ω

(

(1 + i)jV(ω)

(

x2ω r + coth (1 - i)

where we have denoted

x )) ω 2

(13)

Methods. All of our measurements have been carried out in a laboratory-made test cell shown schematically in Figure 2. The cell was made of organic glass. We have opted for the batch configuration because in this case it is much easier to control the cell hydraulic capacity. The latter is of primary importance for the rate of relaxation of the hydrostatic pressure difference across the membrane after the pressure switch-off. Any nonrelaxed pressure difference gives rise to a streaming potential contribution that masks the signal of interest. The hydraulic capacity is roughly proportional to the total internal volume of a cell. The latter is much easier to control and to keep at an acceptable level in the case of batch cells than in cross-flow configurations. In our design, care has also been taken to diminish the concentration polarization as much as possible. To achieve that, we used a magnetically driven disklike stirrer located close to the membrane surface (1-2 mm) and fixed at the extremity of a hollow axis supported by two four-arm catches. The liquid driven away in the radial direction due to the centrifugal action of the stirrer is replenished by the flow through the axis. This flow pattern qualitatively resembles that occurring near a rotating


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Figure 2. Schematic view of the test cell. electrode.13 Because of very good magnetic coupling, rotation rates up to 2000 rpm could be reached. The pressure difference is created by pressurized argon through a specially designed transducer without direct contact between the gas and the solution (to avoid gas dissolution). The pressure switch-off is implemented through the action of a three-way solenoid valve (Circle Seal Controls, Inc.) with a characteristic opening time of only 5 ms. The valve simultaneously cuts off the gas and opens the cell to the atmosphere. The measurements of hydrostatic pressure in the cell are performed with a piezoresistive pressure transducer (Media Nugget, SSI Technologies, Inc.). Measurements of the trans-membrane electric potential difference are carried out by a pair of bare Ag/AgCl electrodes, one of which is located in the high-pressure compartment and another in the permeate one. Because the measurements are performed in an open electrical circuit, there are no ohmic contributions to the electric potential difference, and the exact location of electrodes in the compartments is not important. All measurements have been carried out at room temperature (between 290 and 295 K). Materials. The membranes were laboratory-made ceramic nanofiltration membranes (courtesy of Laboratory of Inorganic Material Science, University of Twente, The Netherlands) with the active layer in γ-alumina and the support in R-alumina. The thickness of the active layer was around 3 µm, and that of the support was 2 mm. The average pore size in the active layer was ca. 4 nm (ref 14), and that in the support was around 0.2 µm. The active-layer porosity was around 35%, and the support porosity was about 30% (all data according to the supplier). Bare supports supplied by the same laboratory have also been used for comparison. The electrolyte was a 0.03 M solution of KCl of at pH 6.

Results and Discussion Estimates of Characteristic Relaxation Time and the Scope of Applicability of This Model. The physical picture of pressure-driven polarization of the bilayer membrane and its relaxation is rather simple. If the support has no osmotic activity (the salt reflection coefficient there is zero), then there are no concentration gradients inside it in the steady state. The whole concentration difference is localized within the active layer, (13) Levich, V. G. Physicochemical Hydrodynamics; Prentice-Hall: Englewood Clifs, NJ, 1962. (14) Nijmeijer, A.; Kruidhof, H.; Bredesen, R.; Verweij, H. J. Am. Ceram. Soc. 2001, 84, 136-140.

and if the concentration polarization may be disregarded, that difference can be experimentally determined. When the pressure is switched off, the salt concentration gradient initially confined to the active layer starts to propagate progressively into the support. At the same time, because of the vigorous stirring close to the membrane surface, the salt concentration there remains constant. The characteristic relaxation time for such a process has been estimated in ref 5 for the concentration-step mode. For these rough estimates, the difference in the mode does not matter, and one can show that about half the total concentration difference propagates into the support within this characteristic time:

t/0 ≡ t0r2 ≡ Rsχs

() la χa

2

(19)

From the definitions of chemical capacity (eq 3)) and of parameter r (eq 14)) as well as from the physics of diffusion permeabilities, it follows that in the case of relatively dense (and probably electrochemically active) active layers flanked by relatively coarse-porous supports, parameter r may be rather large. (Some estimates of this parameter are carried out below a posteriori.) In this case, the new characteristic relaxation time of eq 19 may be essentially longer than that of eq 2. One can benefit from that in two ways. First, one can enlarge the scope of applicability of our theoretical approach. Indeed, it is well known that in the pressure-driven mode the concentration profiles within the membrane generally have a nonlinear shape. The profiles are close to linear if the so-called Pećlet number is sufficiently small, but when the latter approaches and exceeds unity, the profiles progressively take a concaveup shape. In the theoretical treatment above, the chemical potential difference in the polarized state has been assumed to be sufficiently small to ensure the applicability of the linear equations. With an arbitrary salt reflection coefficient, that condition is definitely met provided only that the trans-membrane volume flow is sufficiently small so that the Pećlet number is essentially smaller than 1. However, if the reflection coefficient has only a moderate value not too close to unity (which is the case in our

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Figure 3. Theoretical transients of relative membrane potential (scaled on the initial value) calculated with eq 22: r ) 1, 2, 4, 8, and 16 (from right to left).

measurements), then it appears to be possible to use the present theory to interpret the measurements where in the initial steady state the Pećlet number is not very small and the concentration profile, accordingly, is nonlinear. Indeed, if the concentration profile linearization within the active layer occurs well before the salt concentration gradient propagates noticeably into the support, then the present theory may be applicable after some (very short) initial period of time. It can be shown that the profile linearization occurs within the characteristic relaxation time t0 given by eq 2. As discussed above, the propagation of the concentration gradient into the support is controlled by another characteristic relaxation time (t/0), which may be essentially longer. If that is the case, then the aforementioned condition is met, and the present theory may be used even if the Pećlet number in the steady state is not small. In this case, eq 17 (or, more precisely, the direct proportionality to the trans-membrane volume flow in it) is not quantitatively applicable. However, it can be shown that in the approximation of a small Pećlet number the factor ∆σs(jV0/Pa) in eq 17 is just equal to the steadystate salt rejection. By taking into account the reasoning above, we can modify eq 17 by replacing that factor with the (measurable) salt rejection, Rs, and thus obtain

Em(τ) )

(

RT 1 ∆t R 1 + Im × F 1 s π

(

∫-∞+∞ dω ω1x+2ωi

exp(-iωτ)

(

r + coth (1 - i)

)

x) ω 2

(20)

It can be shown that the term in parentheses in eq 20 tends toward 2 at very short dimensionless times. Therefore, the initial transient membrane potential is equal to

Emi ≡ Em(τ f 0) )

2RT ∆t1Rs F

(21)

The fact that parameter r may well be large can also be exploited to make the explicit dependence on it from

eqs 17 and 20 disappear. To achieve that, a new dimensionless time should be introduced according to τ* ≡ t/t/0, so eq 20 can be rewritten in this form

Em(τ*) )

(

(

RT 1 ∆t R 1 + Im × F 1 s π

∫-∞+∞ dω ω1x+2ωi

exp(-iωτ*)

(

x)

1 1 1 + coth (1 - i) r r

ω 2

)

(22)

When r f ∞, eq 22 reduces to

Em(τ*) )

( (

RT 1 ∆t R 1 + Im F 1 s π

∫-∞+∞ dω ω

exp(-iωτ*)

1 + (1 - i)

xω2

)

(23) It is seen that the explicit dependence on parameter r disappears. Of course, because this parameter enters into the definition of the characteristic relaxation time of eq 19, the dependencies plotted against dimensional time would still depend on r. However, its variation would cause only their shifts along the time axis (in logarithmic scale) without a change in shape. That simplifies the interpretation considerably. Figure 3 shows theoretical transients calculated with eq 22 for several values of parameter r. It is seen that the dependence of curve shape on that parameter practically disappears starting from r g 4. Steady-State Filtration Data. Figure 4a shows the steady-state rejection of 0.03 M KCl solution as a function of trans-membrane volume flow. The same data have been interpreted in ref 15 within the scope of the modified Spiegler-Kedem approach. The following transport parameters have been obtained: reflection coefficient σs ) 0.58, solute (diffusion) permeability Pa ) 9.1 µm/s, and parameter a (a measure of the dependence of transport properties on the salt concentra(15) Yaroshchuk, A. E. J. Membr. Sci. 2002, 198, 285-297.


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Figure 5. Time transients of filtration potential as functions of the decimal logarithm of time elapsed from the moment of pressure switch-off (corrected for the potential of asymmetry); steady-state pressure: 1 MPa (membrane), 0.3 MPa (support).

Figure 4. Steady-state nanofiltration of 0.03 M KCl solution. (a) Salt rejection against trans-membrane volume flow; (b) trans-membrane volume flow against the hydrostatic pressure difference.

tion) a ) 0.67. Figure 4b shows the trans-membrane volume flow as a function of the hydrostatic pressure difference for the same feed solution. The plot linearity is very good, and the slope is equal to 4.5 µm/(s‚MPa). However, to be able to estimate the hydraulic permeability of the nanoporous layer, we should remember that the membrane support makes a noticeable contribution to the membrane hydraulic resistance for this type of membrane.3 Because of the availability of separate support samples, we could estimate this contribution and found the hydraulic permeability of the supports to be about 17 µm/(s‚MPa). By using the model of the serial connection of two hydraulic resistances, the hydraulic permeability of the nanoporous layer alone could be estimated to be 6.1 µm/(s‚MPa). Time Transients of the Trans-Membrane Electric Potential Difference and Their Interpretation. Figure 5 shows sample time transients of the filtration potential as a functions of the decimal logarithm of time elapsed from the moment of pressure switch off. (Here the polarization was “short”; see the caption to Figure 12 for details). For comparison, it shows the data obtained for both the membrane and the support. It is seen that at short times in both cases there are several damped oscillations. They are probably due to vibrations of the test cell after the shock produced by the sudden pressure switch-off. That is confirmed by the direct measurements of the transmembrane pressure difference shown in Figure 6. It is seen that the trans-membrane pressure difference practically decays within 10 to 15 ms. After the oscillations have died out, the trans-membrane potential difference goes to zero in the case of the support but continues to change in the case of the membrane. That slow evolution is due to the transient membrane potential. Notably, the two series of measurements shown

Figure 6. Trans-membrane pressure as a function of time elapsed from the moment of pressure switch-off.

Figure 7. Transients of filtration potential obtained after the switch-off of several steady-state trans-membrane pressures: ∆P ) 0.3, 0.6, 1.0, 1.5, 2.0, 2.5 MPa (from bottom to top).

in Figure 5 have been obtained for comparable but not the same trans-membrane volume flows. Besides that, in the filtration through the membrane, the salt concentration in the support is not the same as the feed one because of the salt rejection by the nanoporous layer. Therefore, the data cannot be used for a quantitative subtraction of the contribution of the support from the trans-membrane filtration potential difference measured with the membrane. Figure 7 shows the time transients obtained after the switch-off of several steady-state trans-membrane pressures. Each curve is the average of three measurements. For better visibility, the error bars are not shown, but the

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errors were always smaller than 0.2-0.3 mV and typically less at times that were not too short. The dashed lines correspond to theoretical fits obtained with eq 23, which contains only three (explicit or implicit) fitting parameters, namely, the characteristic relaxation time, the short-time limit, and the long-time limit. The slope at intermediate times is determined in a unique way by the difference between those two limits. Therefore, there are not very many degrees of freedom in the fitting procedure, and the fact that we have succeeded in reproducing well the shape of the transients confirms the validity of our theoretical model. From the fits shown in Figure 7, we could determine the characteristic relaxation time. Theoretically, it had to be independent of pressure. Actually, there were some variations in that parameter from pressure to pressure. The average value was t0* ) 0.85 ( 0.14 s. It will be used below for estimates of some properties of the membrane supports. The steady-state value of the trans-membrane electric potential difference before the pressure switch-off, E, can be represented as the sum of the streaming potential, membrane potential, electrode potential, and potential of electrode asymmetry. (The potential of electrode asymmetry is defined as the deviation of the actually measured electrode potential difference from the theoretical value given by the Nernst equation:)

E ) Es + Em + Ee + Ea

Figure 8. Various components of the trans-membrane electric potential difference obtained through the fitting of time transients shown in Figure 7 and of steady-state salt rejection data shown in Figure 4a: [, long-time limit of the transient filtration potential; 9, electrode potential estimated from the salt rejection data; 2, short-time limit of the transient filtration potential; b, short-time limit corrected for the potential of asymmetry.

(24)

After the pressure switch-off, the streaming potential disappears simultaneously with the trans-membrane pressure difference. The electrode potential difference may be considered to be time-independent on the time scale of each single measurement because its duration is not long enough for the diffusion to change the salt concentration appreciably in the permeate compartment (especially in the case of membranes used in this study because of the very large support thickness). The same is probably true for the potential of electrode asymmetry. (However, see the discussion below.) As follows from the theoretical model as well as from the physics discussed above, in KCl solutions the transient membrane potential goes to zero with increasing elapsed time. Therefore, the variation of the trans-membrane electric potential difference between just after the pressure switch-off and the quasi-stationary state at sufficiently long times is equal to the membrane potential difference occurring across the active layer in the initial steady state. The extrapolated long-time values are equal to the sum of the electrode potential and potential of asymmetry. In Figure 8, they are compared with the values of electrode potential estimated from the salt rejection data by means of the Nernst equation. Though the qualitative trend is the same, quantitative agreement could not really be observed. In the short-time limit, the situation is even worse: although there had to be a monotone increase with the steady-state pressure difference, we have obtained almost constant values (also shown in Figure 8). At the same time, the variation of the trans-membrane electric potential difference between the short-time and the long-time limits shown in Figure 9 is a smooth monotone function of pressure that correlates quite well with the salt rejection data (see below). We believe that those findings may be explained by the instability of electrode function of one of the Ag/AgCl indicator electrodes. Indeed, at the beginning of each filtration stage the electrode located in the permeate compartment was exposed to the solution of gradually changing concentration. It is known that Ag/AgCl elec-

Figure 9. Variation of the trans-membrane electric potential difference between the short-time and long-time limits: [, measured; 2, estimated from the salt rejection data.

trodes being transferred to a solution of a different concentration come to equilibrium with it rather slowly, and even after the equilibration, the same potential of asymmetry is rarely attained. Thus, in the series of measurements presented in Figure 8, the potential of asymmetry could change from one steady-state pressure to another. At the same time, in a series of measurements carried out after the switch-off of the same pressure, the potential of asymmetry could remain reasonably constant because the electrode had enough time to equilibrate because of the long duration of filtration stages. To check that hypothesis, we have estimated the potential of asymmetry proceeding from the extrapolated long-time values and the theoretical electrode potential differences calculated from the salt rejection. After that, we have corrected the short-time values for the potential of asymmetry obtained in that way. The results are shown in Figure 8. It is seen that a curve of quite reasonable shape is obtained, which confirms our hypothesis. The fact that the shape of the transients in Figure 7 could be well reproduced by the theoretical fits also confirms the assumption that the potential of asymmetry remained reasonably constant in the course of each measurement. Variation of Transient Membrane Potential and Estimates of Ion Transport Numbers. In view of the phenomena discussed above, the most reliable value appears to be the variation of the trans-membrane electric potential difference between the short-time and the longtime limits. This quantity is plotted in Figure 9.


Langmuir, Vol. 21, No. 17, 2005 7687

Besides that, the Figure also shows the variation of transient membrane potential estimated from the salt rejection data by means of eq 21 with a single fitting value of ∆t1 for all pressures. It is seen that the fit quality is very good apart from the highest pressure where it is still satisfactory (the difference is less than 1 mV). The fitted value of the transport number of potassium ions is 0.26, which means that the membrane active layer was positively charged at pH 6. That is in agreement with the isoelectric point of γ-alumina, which is known to be around pH 8.5. Estimates of the Contribution of Donnan and NonDonnan Factors to the Salt Exclusion from Nanopores. A knowledge of ion transport numbers within the active layer and of the salt reflection coefficient enables us to estimate the contributions of the so-called Donnan and non-Donnan mechanisms16 of ion rejection by the nanopores of the membrane used in this study. For that purpose, we shall use the version of the fine-porous membrane model put forward in ref 17. However, prior to that, let us check the applicability of this model by using the available experimental data on the nanoporous layer hydraulic and diffusion permeabilities. Indeed, within the scope of the capillary model, the hydraulic permeability of the nanoporous layer can be represented by

By substituting the estimated value of the salt reflection coefficient and the bulk salt diffusion coefficient, Ds ) 1.8 × 10-9 m2/s (equal to the ion diffusion coefficients in the case of KCl), we obtain Pa ) 9.2 µm/s, whereas the experimental value is Pa ) 9.1 µm/s. Thus, the assumptions made above concerning the applicability of the fine-porous model, a narrow pore size distribution, and the absence of imperfections appear to hold. Now let us further specify the model and assume that along with the fixed electric charge giving rise to the Donnan exclusion there also is an additional non-Donnan exclusion factor having the same strength for cations and anions in the case of symmetrical electrolytes. By using the expression for the ion distribution coefficients derived in ref 17 and by assuming that the ratio of mobilities of anions and cations inside pores is the same as in the bulk, in KCl solutions (equal mobilities of cations and anions) for the transport number of co-ions, and for the salt reflection coefficient, one can derive

rp2 χh ) 8ηLeff

where X ≡ cX/2cs, cX is the concentration of fixed charges per pore volume, cs is the salt concentration in the equilibrium electrolyte solution, and γ < 1 is the nonDonnan exclusion factor. (It has been suggested that the mobility of counterions in charged nanopores may be reduced as related to the mobility of co-ions because of the electrostatic interactions with discrete surface charges. However, it can be shown that allowing for a reduced mobility of counterions would lead to an even stronger non-Donnan exclusion needed to reconcile the measured values of the co-ion transport number and the salt reflection coefficient.) Proceeding from the fitted values of transport number of potassium ions (tK+ ) 0.26) and the salt reflection coefficient (σs ) 0.58), we obtain X ) 0.26 and γ ) 0.48. From those values, one can conclude that the Donnan exclusion plays only a secondary role and that the principal contribution to the ion rejection is made by a non-Donnan exclusion factor. In view of relatively large pores as compared to the size of hydrated ions, the steric phenomena cannot explain this. Within the scope of the simple model of dielectric exclusion with the solvent inside the pores considered to be a bulklike continuum,19 the interactions with the polarized interface between the membrane matrix and the solvent turn out to be too weak for this, too. One may evoke changes in the properties of water in nanoconfinement20,21 as a possible explanation. A similar situation has also been observed for the same ceramic membrane in CaCl2 solutions in ref 3. However, in ref 3 the ion transport numbers have been determined from steady-state data, and the accuracy of their determination was not very high due to the particularities of procedure of subtraction of membrane support contribution to the trans-membrane filtration potential. In this paper, we have determined the ion transport numbers in a different and more accurate way.

(25)

where η is the bulk solution viscosity and Leff is the effective thickness of the nanoporous layer. It includes the effects of finite porosity, pore tortuosity, and possibly increased viscosity of water in nanopores. Thus, the effective thickness can be represented by

Leff ≡

θ ηp L f η

(26)

where θ is the pore tortuosity, f is the layer porosity, ηp is the solution viscosity in the nanopores, and L is the actual layer thickness. We do not have enough experimental data to determine those contributions separately. However, we can assume that approximately the same effective thickness of the nanoporous layer controls its diffusion permeability. For that assumption to hold, the principal paths have to be the same taken by the fluid in the course of its pressure-driven transfer and by the solute during its diffusion. For that, the pore size distribution has to be narrow, and the nanoporous layer ought to be imperfection-free.18 By using eq 26 and rp ) 2 nm, from χh ) 6.1 µm/(s‚MPa) we obtain Leff ) 82 µm. (Incidentally, because the actual thickness of the nanoporous layer is only about 3 µm and its porosity is around 35%, one can see that the pore tortuosity and possibly increased water viscosity in the pores play an important role.) Finally, within the scope of the fine-porous model, for the diffusion permeability one can obtain

Pa )

Ds (1 - σs) Leff

(27)

(16) Yaroshchuk, A. E. Separ. Purif. Technology 2001, 22-23, 143158. (17) Yaroshchuk, A. E.; Ribitsch, V. J. Membr. Sci. 2002, 201, 8594. (18) Yaroshchuk, A. E. J. Membr. Sci. 2004, 239, 9-15.

tc )

(

)

X 1 12 2 xX + γ2

σs ) 1 -

γ2

xX2 + γ2

(28)

(29)

(19) Yaroshchuk, A. E. Adv. Colloid Interface Sci. 2000, 85, 193230. (20) Wang, J. W.; Kalinichev, A. G.; Kirkpatrick, R. J. Geochim. Cosmochim. Acta 2004, 68, 3351-3365. (21) Floquet, N.; Coulomb, J. P.; Dufau, N.; Andre, G.; Kahn, R. Physica B: Condens. Matter 2004, 350, 265-269.

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Figure 10. Streaming potential as a function of the transmembrane pressure difference: 9, membrane; [, support; 2, support (corrected for salt rejection by the nanoporous layer).

Thus, the experimental values of the ion transport number, salt reflection coefficient, diffusion, and hydraulic permeabilities could be quantitatively reproduced within the scope of the simple model outlined above. That enables us to use the model to make predictions. In particular, we may use it to estimate the so-called electrokinetic charge density, which controls the streaming potential coefficient. Within the scope of the fine-porous model, the electrokinetic charge density can be represented in this way:3,17

Fek ) 2FcsX

(30)

By substituting the value of the dimensionless fixed charge concentration, X, estimated above, we obtain Fek ) 1.5 × 106 C/m3. To estimate the streaming potential coefficient, we also need to know the electric conductivity of the nanoporous layer, g. This property cannot be measured directly because the nanoporous layer makes only a very small contribution to the electrical resistance of the supported membrane (∼1%). However, it can be estimated in a model-independent way proceeding from the diffusion permeability and the ion transport numbers by using this relationship:4

g)

F2 Pa 2RT t+t-

(31)

From this equation, we obtain g ≈ 2700 Ω-1 m-2. Finally, by taking into account that the streaming potential coefficient of a membrane is the ratio of electrokinetic charge density to the membrane’s electric conductivity,3 we obtain

Rs ≡

Fek ) 0.56 mV/(µm/s) g

(32)

Streaming Potential. The streaming potential is another quantity that is insensitive to the instability of indicator electrodes provided that the value of the potential of the electrodes’ asymmetry remains constant during the course of one measurement. The streaming potential is defined here as the difference between the steady-state trans-membrane electric potential difference before the pressure switch-off and its short-time limit just after the switch-off. The electrode potential, the membrane potential, and the potential of asymmetry equally contribute to both of them, thus they cancel out. Figure 10 shows the streaming potential as a function of the trans-membrane pressure difference. The membranes used in this study were the same as studied in ref 3, so the contribution of membrane support

Figure 11. Transients of filtration potential obtained after the switch-off of several steady-state trans-membrane pressures (short polarization, explanations in text): ∆P ) 0.3, 0.6, 1.0, 1.5, 2.0, 2.5 MPa (from bottom to top).

to the streaming potential was considerable if not decisive. This is confirmed by the characteristic super-linear shape of the curve caused by the decrease in salt concentration in the support layer with increasing trans-membrane pressure difference due to the salt rejection by the nanoporous layer.3 For comparison, Figure 10 also shows the streaming potential measured with the membrane support. The Figure shows the support data directly obtained7 in the solution of 0.03 M KCl as well as the support streaming potential corrected for the decrease in the salt concentration with increasing trans-membrane volume flux. (Strictly speaking, the data in Figure 7 have been corrected for the potential of asymmetry.) The correction has been accomplished in this way. We have measured the streaming potential of membrane support as a function of pressure for several salt concentrations and linearly interpolated the slope as a function of concentration in log-log coordinates. From Figure 10, it is seen that the corrected streaming potential of support practically coincides with the streaming potential measured with the membrane. Moreover, the former sometimes turns out to be even slightly larger in absolute value than the latter, and the difference between them appears to be just scattered around zero without any clear trend. In our opinion, that may be caused by the uncertainties in the procedure of subtracting a relatively large support contribution. (See ref 3 for a discussion on this.) Above, we have estimated the streaming potential coefficient of the nanoporous layer to be only 0.56 mV/(µm/s). We believe that this relatively small contribution could be effectively “lost” in the background of uncertainties in the determination of the support contribution. In a sense, this confirms the validity of our estimate of the streaming potential coefficient. Besides that, we see once again that in the case of thin membranes deposited on much thicker supports the streaming potential is primarily controlled by the supports. Comparison with Another Series of Measurements. Figure 11 shows another series of membrane potential transients obtained for the same membrane sample. In this case, before the switch-off, pressure was applied for a relatively short period of time (ca. 5 min) so that the indicator electrode located downstream from the membrane was always in the solution of feed concentration. Nonetheless, that period of time was long enough for the zone of permeate concentration to penetrate sufficiently deeply into the support so that the back diffusion through the support after the pressure switch-off was negligible. It is seen that the quality of data in this case is better than in Figure 7 apparently because of the better electrode


Langmuir, Vol. 21, No. 17, 2005 7689

A Posteriori Estimates of Parameter r. Now that we have estimated a number of properties of the nanoporous layer and support, we can also estimate the value of parameter r defined by eq 14 to determine if it was sufficiently large, indeed, as it was assumed in the procedure of fitting membrane potential transients. Within the scope of the fine-porous model, for the specific chemical capacity of the nanoporous layer one can derive

Ra ) Figure 12. Variation of the trans-membrane electric potential difference between the short-time and long-time limits (short polarization, explanations in text): [, measured; 2, estimated from the salt rejection data.

stability. Besides that, owing to the relatively short filtration stages more representative statistics could easily be gathered, and each curve in Figure 11 is the average of five measurements. Remarkably, all of the transients shown in Figure 11 could be fitted with a single characteristic relaxation time of t*0 ) 1.15 s (shown in the Figure as the vertical dashed line). Figure 12 shows the variation in transient membrane potential obtained from the fitting of transients shown in Figure 11 as a function of the pressure difference in the quasi-steady state. Similar to Figure 7, in addition to the data obtained from the transients, it also shows the initial membrane potential estimated from the salt rejection with a single fitting parameter, namely, the transport number of potassium ions in the nanoporous layer. Remarkably, the best fit is achieved with exactly the same fitting value as in Figure 7, namely, tK+ ) 0.26. The fit quality is slightly worse than in Figure 7, but it is still very good. Estimates of Some Properties of Membrane Supports. Above, we have estimated the characteristic relaxation time t*0 to be 0.85 ( 0.14 and 1.15 s for the series of transients shown in Figures 7 and 11, respectively. Proceeding from the definition of the characteristic relaxation time of eq 19 and by taking into account that in the coarse-porous supports the chemical capacity is simply proportional to the porosity, one can obtain

t*0 )

fsD(eff) s Pa2

(33)

is the effective where fs is the support porosity and D(eff) s salt diffusion coefficient in the support. Because the pores in the support are relatively large and the solution inside them has mostly bulk properties, the effective coefficient may be represented as

) D(eff) s

fs D θs s

(34)

where θs is the pore tortuosity in the support. By substituting the known value of support porosity (fs ≈ 0.3) and the bulk value of the diffusion coefficient (Ds ) 1.8‚10-9 m2/s) into eqs 33 and 34, we obtain for the pore tortuosity, θ, the values of 1.7 and 2.3 for the two characteristic relaxation times estimated above. In view of a support porosity of ca. 30%, those values appear to be quite reasonable.

csfa γ2 RT x 2 γ + X2

(35)

where fa is the porosity of the nanoporous layer. By taking into account that in the case of the coarse-porous support the chemical capacity is simply proportional to the porosity and by using the definitions of the diffusion permeability of the nanoporous layer and support, one obtains

r2 )

fs2Ds θsPaLafaγ

x1 + (Xγ )

2

(36)

where La is the thickness of the nanoporous layer. By taking La ) 3 µm and the values of other parameters estimated above, we obtain r ≈ 4.5. Above, it has been demonstrated that the approximation of large r is valid for r g 4. Conclusions We have demonstrated that in a batch membrane cell it is technically feasible to switch off the trans-membrane pressure difference within tens of milliseconds. Because of that, we could determine separately the streaming potential and the membrane potential components of filtration potential in the case of thin nanoporous layers deposited on top of much thicker coarse-porous supports. The measurements of transient filtration potential have been preceded by the measurements of steady-state salt rejection by the composite membranes. Because of this, we could determine the ion transport numbers within the nanoporous layer proceeding from the measured membrane potential. Besides that, from the dependence of steady-state salt rejection on the trans-membrane volume flow, we have determined the diffusion permeability of and the salt reflection coefficient in the nanoporous layer. This enabled us to estimate the contributions of Donnan and non-Donnan mechanisms to the rejection of ions by the membrane used in this study. It has been unexpectedly found that Donnan exclusion played only a secondary role. In view of the relatively large pore size (ca. 4 nm), neither the steric hindrance nor the “classical” dielectric exclusion could explain this finding. Our hypothesis is that the nonDonnan exclusion of ions from pores might be caused by changes in water properties in nanoconfinement. Finally, from the characteristic relaxation time of membrane potential transients and the diffusion permeability of the nanoporous layer, we could estimate the specific diffusion permeability of the membrane support. This enabled us to estimate the pore tortuosity in it, which yielded values between 1.7 and 2.3. In view of support porosity of about 30%, those values appear to be quite reasonable and additionally confirm the self-consistency of our approach.

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Acknowledgment. We thank Volkswagen Stiftung (Germany) for financial support within the scope of the project “Advanced modeling of nanofiltration via improved input due to novel experimental techniques of membrane

Yaroshchuk et al.

characterization” and the Laboratory of Inorganic Materials Science, University of Twente (The Netherlands), for the membrane samples. LA050917H

Some Properties of Electrolyte Solutions in Nanoconfinement

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