Electrochemical and Other Transport Properties of Nanoporous Track

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Langmuir 2005, 21, 6872-6882

Electrochemical and Other Transport Properties of Nanoporous Track-Etched Membranes Studied by the Current Switch-Off Technique Andriy Yaroshchuk,*,† Olga Zhukova,† Mathias Ulbricht,† and Volker Ribitsch‡ Lehrstuhl fu¨ r Technische Chemie II, Universita¨ t Duisburg-Essen, Universita¨ tsstrasse 5, 45141 Essen, Germany, and Institut fu¨ r Chemie, Karl-Franzens-Universita¨ t, Heinrichstrasse 28, 8010 Graz, Austria Received February 24, 2005. In Final Form: May 6, 2005 Measurements of transient membrane potential after current switch-off have been used to study the electrochemical and other transport properties of nanoporous track-etched membranes (pore diameter of 24 nm) in KCl solutions of various concentrations. Due to their identical straight cylindrical pores within the nanorange, those membranes are a suitable object for the studies of fundamentals of nanofluidics. We have developed a theory, which enabled us to interpret the transients of membrane potential in terms of such properties as the ion transport numbers, the membrane diffusion permeability, and the specific chemical capacity. The fitted values have been further interpreted within the scope of the space-charge model and revealed good self-consistency. The dependence of fitted values of the surface charge density on salt concentration was in agreement with the mechanism of fixed charge formation due to the dissociation of weakly acidic groups, and the surface charge density corresponded well to the measurements by an independent method. Thus, our measurements have revealed a quantitative applicability of standard space-charge model to the description of electrochemical phenomena and electrolyte diffusion in straight cylindrical pores of tens of nanometers in diameter. Proceeding from our data, we could estimate the limiting current densities for our nanofluidic system. They have turned out to be more than 2 orders of magnitude lower than usually encountered in microfluidic systems with electro-osmotic fluid delivery. That finding may point to a considerable handicap in the application of such nanofluidic elements in microsystems.

Introduction In microfluidics, the electromechanical coupling due to the electrical charge separations at interfaces is of great importance. In practical applications, this kind of coupling is exploited, for example, in the electro-osmotic reactant delivery in the total microanalysis systems.1 Currently, ever more attention is paid to nanofluidics. There is a rather broad range of dimensions where a system already becomes nanofluidic (e.g., due to the noticeable thickness of double electric layers as compared to the channel dimensions), but in simple fluids, some of the macroscopic approaches (e.g., the continuous description of the solvent) remain applicable because the system dimensions are still much larger than the molecular scale. Within that dimension range, the qualitatively new phenomena to be expected in the nanofluidics as compared to the microfluidics are the so-called surface conductivity due to the electrostatically adsorbed counterions and the appearance of concentration polarization due to electro-chemical and mechanochemical couplings. Those phenomena manifest themselves in full in the transport properties of fine-porous track-etched membranes. They are obtained through the irradiation of thin polymer films with accelerated heavy ions followed by their etching usually with a concentrated alkali. Typical film thicknesses are between 6 and 35 µm, depending on the pore size to be prepared. When the irradiation and etching conditions are properly selected, almost identical * To whom correspondence should be addressed. 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]. † Universita ¨ t Duisburg-Essen. ‡ Karl-Franzens-Universita ¨ t. (1) Reyes, D. R.; Iossifidis, D.; Auroux, P. A.; Manz, A. Anal. Chem. 2002, 74, 2623-2636.

straight cylindrical pores are obtained.2 Depending on the etching time and mode, a wide spectrum of isoporous membranes with pore sizes ranging from 15 nm to 5 µm is obtained. Due to the well-defined pore geometry within the nanorange, the fine-porous track-etched membranes should be a suitable object for the studies of fundamentals of nanofluidics. Moreover, those membranes have already found some applications in the design of laminated 3D microfluidic systems, for example, as separators of crossed channels located in the neighboring layers.3-6 The transfer of analytes from one channel to another is implemented electrically so that electro-osmosis is an important phenomenon in this context. Besides that, as it will be shown below, the electrochemical properties of track-etched membranes are of paramount importance, too, since they control the concentration polarization of membranes due to the passage of direct current. In electro-osmosis, the so-called zeta potential, which controls the electromechanical coupling, is of fundamental importance. In coarse-porous media, the zeta potentials are obtained through the measurements of streaming potential. In membranes with fine pores (as compared to the Debye screening length), those measurements alone cannot be straightforwardly interpreted in terms of the zeta potential due to the overlap of diffuse parts of double electric layers and because of the noticeable contribution of electrostatically adsorbed counterions to the electric conductivity of the pore solution. Therefore, a meaningful (2) Apel, P.; Schulz, A.; Spohr, R.; Trautmann, C.; Vutsadakis, V. Nucl. Instr., Methods Phys. Res, Sect. B 1998, 146, 468-474. (3) Kemery, P. J.; Steehler, J. K.; Bohn, P. W. Langmuir 1998, 14, 2884-2889. (4) Tzu-Chi Kuo, Sloan, L. A.; Sweedler, J. V.; Bohn, P. W. Langmuir 2001, 17, 6298-6303. (5) Tzu-Chi Kuo, Cannon, D. M.; Jr., Chen, Y.; Tulock, J. J.; Shannon, M. A.; Sweedler, J. V.; Bohn, P. W. Anal. Chem. 2003, 75, 1861-1867. (6) Cannon, D. M.; Jr., Tzu-Chi Kuo, Bohn, P. W.; Sweedler, J. V. Anal. Chem. 2003, 75, 2224-2230.

10.1021/la050499g CCC: $30.25 © 2005 American Chemical Society Published on Web 06/16/2005

Electrochemical Properties of Nanoporous Membranes

interpretation of streaming potential in nanopores calls for additional information on the membrane electric conductivity. Besides that, the passage of electric current through nanopores is usually accompanied by concentration polarization. As discussed below, its extent is controlled by the ion transport numbers and diffusion permeability of pores. The information on the transport, electrokinetic, and electrochemical properties of track-etched membranes found in the literature is not extensive and incomplete for our purposes. In most cases, only the streaming potential has been measured.7-10 We are aware of only one attempt to carry out integrated measurements of streaming potential and membrane electric conductivity with tracketched membranes.11 Though the measurements of electric conductivity of track-etched membranes have been routinely used to control the etching process,2 they have practically never been combined with the streaming potential measurements. The measurements of the membrane potential have been performed only in ref 12 and those of salt rejection only in ref 13. In this study, we have used the RoTrac brand of tracketched membranes manufactured by Oxyphen. The particularity of this kind of track-etched membranes is that due to a very high track density (7 × 1013m - 2, i.e., more than 1 order of magnitude larger than in other commercialised track-etched membranes) even their nanoporous varieties have relatively high porosity (e.g., 5% in the case of 30 nm pore diameter grade according to the manufacturer). Since the thickness of 30 nm membranes is only 8 µm, the high porosity makes them attractive for various applications where a high permeability is called for. At the same time, that makes the membrane diffusion permeability and electric conductivity too high to be reliably determined by the classical steady-state techniques essentially based on the determination of membrane contribution to the diffusion or electrical resistance of a test cell. In the case of membranes of interest, this contribution turns out too small to be reliably determined on the background of diffusion resistances of unstirred layers or the electrical resistances of layers of electrolyte solutions between the membrane and the indicator electrodes. However, as it is demonstrated below, the relevant information may be obtained through the interpretation of results of non-steady-state measurements. The purpose of this study is to use the extensive potentialities of a non-steady-state electrochemical technique (measurements of transient membrane potential after current switch-off) to obtain detailed information on the electrochemical and other transport properties of a nanoporous track-etched membrane. As it will be shown below, the membrane polarization by direct electric current is essentially influenced by the hydrodynamic conditions near the membrane surfaces, namely, by the thickness of unstirred layers. In electrochemistry, it is known that even in the absence of forced convection the unstirred layers in free solutions cannot be considered infinitely thick but usually they have (7) Lettmann, C.; Mockel, D.; Staude, E. J. Membr. Sci. 1999, 159, 243-251. (8) Kim, K. J.; Stevens, P. V. J. Membr. Sci. 1997, 123, 303-314. (9) Brendler, E.; Ratkje, S. K.; Hertz, H. G. Electrochim. Acta 1996, 41, 169-176. (10) Huisman, I. H.; Pradanos, P.; Calvo, J. I.; Hernandez, A. J. Membr. Sci. 2000, 178, 79-92. (11) Berezkin, V. V.; Kiseleva, O. A.; Nechaev, A. N.; Sobolev, V. D.; Churaev, N. V. Colloid J. 1994, 56, 258-266. (12) Canas, A.; Benavente, J. Separ., Purif. Technol. 2000, 20, 169175. (13) Berezkin, V. V.; Nechaev, A. N.; Mchedlishvili, B. V. Radiat. Meas. 1995, 25, 703-707.

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effective thickness of some 400 µm.14 That probably occurs due to a natural convection caused by a number of reasons (for example, temperature and density gradients). However, the extent of this natural convection evidently depends on the details of cell geometry near the membrane, on the cell orientation with respect to gravity, and, most importantly, on the current density. Because of that, the effective thickness of unstirred layers may considerably change from one measurement to another giving rise to a poor reproducibility of results and to artificial correlations. To resolve this problem, we have set up reproducible hydrodynamic conditions by putting the track-etched membrane between two coarse-porous membrane filters and by setting up vigorous stirring near the external surfaces of the latter. Electrical capacities in electrolyte solutions charge very rapidly, namely, at characteristic times of diffusion relaxation of the diffuse part of double electric layers. Even in very dilute solutions, due to the very small thickness of diffuse part of double electric layers (at most several tens of nanometers), that time does not exceed ∼10-6 s. Therefore, for our purposes, the electric current can always be considered continuous, which simplifies the interpretations remarkably. The concentration polarization of boundaries between layers with different electrochemical properties gives rise to the appearance of diffusion components of the transmembrane electric potential difference even if there is no external composition difference. This concentration polarization cannot disappear immediately. The cause for a time lag is the fact that the distributed chemical capacity (see below for the definition) of a vicinity of the boundary has to “discharge”. The chronopotentiometry after stepwise changes in the polarizing current (both switch-on and switch-off) has been used in refs 14-16 to study the polarization of monolayer ion-exchange membranes. Though the basics of the current switch-off technique used in the present study are the same as in refs 14-16, the profound differences in the structure of the membranes involved (very thin and relatively porous in this study vs relatively thick and dense in refs 14-16) made the characteristic time scales typical of this study several orders of magnitude shorter than those encountered in refs 14-16. That has given rise to considerable differences in the approaches to the data acquisition and data treatment. Besides that, the nonsteady-state theory developed in this paper is modelindependent (network thermodynamics), whereas the mechanistic Nernst-Planck approach was used in refs 14-16. In refs 17 and 18, the chronopotentiometry has been applied to study the ion transport in bilayer bi-polar membranes. Their structure is closer to that of the systems of interest for this study. Nevertheless, the typical thicknesses of layers in bi-polar membranes are more than 1 order of magnitude larger than the thicknesses of tracketched membranes. Accordingly, the relaxation processes studied in refs 17 and 18 were essentially slower than in this study. Besides that, the principal emphasis of refs 17 and 18 was on the differentiation between the reversible and irreversible contributions to the trans-membrane potential difference and on the understanding of ion (14) Pismenskaia, N.; Sistat, P.; Huguet, P.; Nikonenko, V.; Pourcelly, G.; J. Membr. Sci. 2004, 228, 65-75. (15) Sistat, P.; Pourcelly, G. J. Membr. Sci. 1997, 123, 121-131. (16) Kontturi, K.; Mafe´, S.; Manzanares, J. A.; Sundholm, G.; Vapola, R.; Electrochim. Acta 1997, 42, 2569-2575. (17) Wilhelm, F. G.; van der Vegt, N. F. A.; Wessling, M.; Strathmann, H.; J. Electroanal. Chem. 2001, 502, 152-166. (18) Wilhelm, F. G.; van der Vegt, N. F. A.; Strathmann, H.; Wessling, M. J. Appl. Electrochem. 2002, 32, 455-465.

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transfer in the so-called over-limiting mode. The latter made the system too complex for the development of a quantitative theory. In the mechanistic interpretation of membrane properties below, we shall employ the so-called space-charge model. The basic assumptions of this model are the continuous description of the solvent, a fixed charge homogeneously smeared over the pore surface, pointlike ions, and the applicability of the mean-field approximation within the scope of Poisson-Boltzmann equation. This model has been initially put forward by Osterle and coworkers.19-21 Thereafter, there have been numerous attempts to obtain analytical solutions in various limiting cases (for example, low or high surface potentials, weak or strong overlap of diffuse parts of double electric layers); however, the modern numerical routines enable one to solve the non-linearized Poisson-Boltzmann equation and to perform all of the necessary integrations over the pore cross-section numerically with high precision. Just this approach has been adopted below. The corresponding procedures are described in ref 22. Theory Basic Equations. The basic equations of linear nonsteady-state theory for the time-dependent distribution of salt chemical potential of binary electrolyte in polarized multilayer systems have been derived in ref 23 proceeding from the standard expressions of continuous version of irreversible thermodynamics for the ionic fluxes and the concept of chemical capacity. For a membrane consisting of several macroscopically homogeneous layers, the basic equation for the time-dependent distribution of the salt chemical potential, µ, can be written down in this dimensionless form

∂2µ ) 2 ∂τ ∂ξ

∂µ β2k

(1)

where k is the layer index, the trans-membrane coordinate is scaled on the membrane thickness, lm, i.e., ξ ≡ (x/lm), and the dimensionless time, τ, is obtained by scaling time by the diffusion relaxation time of the membrane defined in this way

t0 ≡

Rm 2 l χm m

(2)

Index “m” denotes the properties of membrane, χk is the specific diffusion permeability of material of kth layer to the salt. The coefficients βi are defined in this way

β2k ≡

Rkχm Rmχk

(3)

R is the so-called specific chemical capacity defined as

R≡

( ) ∂qs ∂µ

T,P

(4)

That is a quantitative measure of how much salt (δqs) has to be added to a unit volume of a medium to change the (19) Morrison, J. C.; Osterle, J. F. J. Chem. Phys. 1965, 43, 21112115. (20) Gross, R. J.; Osterle, J. F. J. Chem. Phys. 1968, 49, 228-234. (21) Fair, J. C.; Osterle, J. F. J. Chem. Phys. 1971, 54, 3307-3316. (22) Yaroshchuk, A. E. Adv. Colloid Interface Sci. 1995, 60, 1-93. (23) Yaroshchuk, A. E.; Ribitsch, V. Chem. Eng. J. 2000, 80, 203214.

Figure 1. Model and coordinates.

salt chemical potential there by δµ. A detailed discussion of this useful property and its estimates for various media can be found in.24,25 By definition, βm ≡ 1. Further, we shall consider the deviations of salt chemical potential from its equilibrium value small so eq 1 is a linear differential equation in partial derivatives. In a multilayer system with macroscopically homogeneous layers, the coefficients in eq 1 are simply layer-specific constants. One of the boundary conditions to eq 1 is the continuity of salt chemical potential at all of the boundaries between the layers (local interfacial equilibria). At zero transmembrane volume flow, the boundary condition for the salt flux can be shown to have this form23

I‚lm ∂µ ∂µ χk |ξ)ξk - χk+1 |ξ)ξk ) ∆t | ∂ξ ∂ξ FZ1ν1 1 ξ)ξk

(5)

where ξk are the coordinates of the boundaries, ∆t1 ≡ t1|ξk-0 - t1|ξk+0 are the changes of ion transport number at the boundaries, I is the electric current density, F is the Faraday constant, Z1 is the charge of ion “1” (in proton charge units), and ν1 is its stoichiometric coefficient. Thus, the boundaries between the layers with different electrochemical properties are sources of salt flux. Despite the appearance of salt flux sources, electric charges do not arise at the boundaries since exactly as much charge is taken away from a boundary by anions as it is brought to it by cations and vice versa. Model Specification for Our Experimental Setup. For the interpretation of measurements of the transient membrane potential after current switch-off, we shall consider a symmetrical three-layer system consisting of a membrane flanked from two sides by identical filters. It is assumed that due to vigorous stirring, the salt concentrations (and, thus, its chemical potentials) at the external surfaces of the filters are given anytime. The trans-membrane volume transfer (electro-osmosis) will be disregarded. In accordance with our experimental setup, the salt concentrations will be taken as the same at both external filter surfaces. Since the chemical potential is defined up to a constant, its preset constant value at the external surfaces may be taken equal to zero. The model and the chosen coordinates are shown in Figure 1. (24) Zholkovskij, E. K. In Surface chemistry and electrochemistry of membranes; Sorensen, T. S., Ed.; Marcel Dekker: New York, 1998; pp 793-815. (25) Jamnik, J.; Maier, J. Phys. Chem. Chem. Phys. 2001, 3, 16681678.


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If the origin of the abscissa is chosen at the middle of the membrane as in Figure 1, it can be shown that the distribution of salt chemical potential is antisymmetrical with respect to this point (in the linear approximation of small deviations from equilibrium). The boundary conditions for this system look this way

If the current is switched off in a stepwise manner (which is the case in our experiments) its Fourier transform looks this way

µm(0,τ) ) 0;

It can be shown that, at zero electric current (i.e., after the current switch-off), the difference in the electric potentials between the half-cells separated by a polarized three-layer system for a (1:1) electrolyte is related to the difference in the salt chemical potentials across the membrane in this way

(6)

(21 + h,τ) ) 0; 1 1 µ ( ,τ) ) µ ( ,τ); 2 2

(7)

µf

m

(8)

f

Em(τ) )

I(τ) ‚lm ∂µm ∂µf |ξ)1/2 - χf |ξ)1/2 ) ∆t | χm ∂ξ ∂ξ FZ1ν1 1 ξ)1/2

(9)

where the indices “m” and “f” denote the membrane and the filter, respectively, and I(τ) is the current density as a function of time. The conditions at the left boundary between the membrane and the filter are the same for the chemical potential, which must be continuous here, too. For its spatial derivative, the condition is basically the same as eq 9 with the only difference being that the jump of ion transport number here has the opposite sign to that at the right boundary. The solution to eq 1 is sought in this form (Fourier transform)

µi(ξ,τ) ≡

xω2 ξ)

(

xω2 ξ) + A

(-) i

(

exp - (1 -

xω2 ξ) (12)

i)βi

where Ai(()are the integration constants. By applying the boundary conditions, we obtain this

1 1 fm ,ω - fm - ,ω ) (1 + i) 2 2 2 I(ω)∆t1lm 1 ω Fχm 1-i ω coth + r coth (1 - i)b 2 2

( )

x

(

)

( x)

(

xω2 )

(13)

where I(ω) is the Fourier transform of I(τ)

x

lf b ≡ βf lm

(15)

Rfχf Rmχm

(16)

2∆t1 ∆µm(τ) F

(17)

( )(

(

I0(∆t1)2 la 1 1-i + {Re} χa 1 + F-1 π F2 exp(- iωτ)

( x)

1-i coth 2

F≡

∫-∞+∞

(

ω + r coth (1 - i)b 2

dω

xω2

ω

x) ω 2

)

(18)

2Pm 2b 2χm lf ≡ ‚ ≡ r χf lm Pf

(19)

Pm and Pf are the molar diffusion permeabilities of membrane and filter, respectively, defined as

Pm,f ≡

RT χm,f cs lm,f

By using the definition of chemical capacity of eq 4, one can show that the specific chemical capacity of ionexchange media with respect to the salt is essentially smaller than that of bulk electrolyte solutions (the latter is simply equal to cs/RT). From that, the definition of parameter r (eq 14) and the physics of diffusion permeabilities, it follows that in the case of relatively dense and electrochemically active nanoporous membranes flanked by coarse-porous filters, parameter r may be pretty large.1 (Some estimates of this parameter are carried out below a posteriori.) In this case, it is useful to introduce a new characteristic relaxation time according to

t0* ≡ t0r2

(20)

If a new dimensionless time is introduced according to τ* ≡ (t/t0*), eq 18 can be rewritten in this form

E ˜ m(τ*) ) (14)

r≡

Em(τ) )

(11)

Accordingly, the general solutions for the spectral densities have this form

fi(ξ,ω) ) Ai(+) exp (1 - i)βi

)

By substituting eqs 13 and 16 into eq 17 and by taking the real part, for the transient membrane potential, we obtain this

(10)

where ω is the dimensionless circular frequency (scaled on 1/t0), fi(ξ,ω) is the complex spectral density of temporal response. By substituting eq 10 into eq 1, we obtain these fundamental solutions

(

i ω

where we have denoted

∫-∞+∞ dω exp(-iωτ)fi(ξ,ω)

1 2π

fi( ()(ξ,ω) ≡ exp ((1 - i)βi

(

I(ω) ) I0 πδ(ω) -

dω

2I˜0(∆t1)2

(

Pm + {Pf}/{2}

(

1+ 1+

(

1-i 1 {Re} F π

)

exp(-iωτ*)

(

)∫

+∞

-∞

xω2 coth((1 - i) 2F xω2 ) + 1r coth(1 2- i 1r xω2 )

ω

)

(21)

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where E ˜ m ≡ (FEm/RT) is the dimensionless transmembrane electric potential difference, and the reduced current density I˜0is defined as I˜0 ≡ I0/2Fcs. With eq 21, it is easy to consider the limiting case of very large r. Indeed, when r f ∞, eq 21 reduces to this

E ˜ m(τ*) 9 8 rf∞

(

(

( )

2I˜0(∆t1)2 1 1 + 1 Im 1+π F Pm + Pf/2

∫-∞+∞ dω ω

exp(-iωτ*) 1+1-i 2

(

ω coth (1 - i) F 2 2

x

x

ω 2

)

)

(22)

Since parameter r enters into the definition of characteristic relaxation time of eq 20, the dependencies plotted against the dimensional time would still depend on r. However, in logarithmic scale, its variation would only cause their shift along the time axis without change of shape. That simplifies the interpretation considerably so it is important to know what values of parameter r may be considered sufficiently large in this context. Sample calculations have revealed that the accuracy of approximation of infinitely large r depends on the ratio of layer diffusion permeabilities, F. Thus for instance, when F f 0 (the relaxation controlled by the diffusion through the filter), the difference between the relaxation patterns is still noticeable even when r g30. When the ratio of diffusion permeabilities of membrane to that of filter increased, the situation improved, and already at F ) 0.5 the dependence of the shape of transients on parameter r practically disappeared at r g10. As it follows from the experimental data presented below, the limiting case F f 0 is unrealistic for the system used in this study. Therefore, in the interpretation of experimental data, we shall use the limiting case r f ∞ and check the value of this parameter a posteriori. Experimental Section Materials. The membrane was a commercially available RoTrac (poly(ethylene terephthalate)) track-etched membrane (Oxyphen, Switzerland) with the nominal pore size (however, see below the discussion on the actual pore size, which was experimentally found to be 24 nm) of 30 nm, the thickness of 8 µm and the track density of 7 × 1013 m-2 (according to the manufacturer). The cellulose nitrate filters were supplied by Millipore, had the nominal pore size of 3 µm and the wet thickness of 145 µm. The filter porosity is not specified by the manufacturer, so we have determined it ourselves gravimetrically and obtained the value of 81% in distilled water. The electrolyte was KCl of various concentrations. The pH value was kept at 5.5. The membrane and filters were equilibrated with working solutions overnight. Methods. All of the measurements have been carried out in a two-compartment test cell schematically shown in Figure 2. The membrane/filter sandwich system was polarized by a pair of disk-shape Ag/AgCl electrodes of a large area. The exposed membrane area was 1.13 cm2. Vigorous stirring was ensured by propeller-like stirrers located parallel to the membrane surface and driven by external motors. Another pair of stirrers was located near the current-supply electrodes to diminish their polarization. The trans-membrane potential difference was measured by a pair of bare Ag/AgCl electrodes. Since our measurements have been carried out after current switch-off (i.e., at zero electric current), in principle, each of the indicator electrodes could be located anywhere in the corresponding half-cell. However, in test measurements without the membrane system, we have noticed that when the electrodes were exposed to the current during the membrane polarization there were some poorly

Figure 2. Test cell: 1, membrane sample; 2, propeller-like stirrers; 3, polarizing disk-shape Ag/AgCl electrodes; 4, indicator Ag/AgCl electrodes; 5, openings for the solution delivery; 6, motors.

Figure 3. Relative trans-membrane potential scaled on the initial value against dimensionless time, r f ∞: F ) 0 (left curve); Ff∞ (right curve). reproducible responses to the current switch-off. We have interpreted that as a result of current passage through the nonisolated tips of Ag/AgCl electrodes. Due to the corresponding concentration polarization, a local distribution of electrolyte concentration developed in the vicinity of each electrode. Because of inevitable differences in the shape of electrodes and in the hydrodynamic conditions in their vicinity, those distributions could not be identical, hence a transient potential difference had been observed. The phenomenon disappeared when the electrodes were receded into the corresponding bores so that they were not exposed to the current. The whole test cell was enclosed in a Faraday cage, and all of the cables were thoroughly screened. The data acquisition has been done by a personal computer with the aid of dedicated software. Pore size distribution analyses by gas-liquid perm-porometry (based on liquid displacement from the liquid-filled pores) were carried out using a PMI Capillary Flow Porometer (Porous Materials, Inc., Ithaca, NY). Membrane samples with a diameter of 3.1 cm were characterized by measuring the gas flow as a function of the trans-membrane pressure, first through the dry membrane, second after wetting the membrane with 1,1,2,3,3,3hexafluoro-propene (“Galwick”, PMI, surface tension 16 dyn‚cm-1). The pore size distribution was then estimated based on the Laplace equation by using the PMI software.

Results and Discussion Figure 3 shows two relaxation patterns calculated at r f ∞ for the limiting modes of membrane-controlled (F f ∞) and filter-controlled (F f 0) relaxation. It is seen that in the former case, the dispersion is essentially


Langmuir, Vol. 21, No. 15, 2005 6877

Figure 4. Transients of membrane potential after current switch-off measured at various current densities (linearity check): (a) cs ) 0.00125 M; I ) 0.27; 0.44; 0.88 A/m2; (b) cs ) 0.0025 M; I ) 0.88; 1.33; 1.77 A/m2; (c) cs ) 0.005 M; I ) 1.77; 2.65; 4.42 A/m2; (d) cs ) 0.01 M; I ) 5.31; 8.85; 13.27 A/m2; (e) cs ) 0.02 M; I ) 17.70; 26.55; 35.40 A/m2.

broader. It can be shown that in terms of shape, the relaxation patterns corresponding to the intermediate (mixed) modes are always situated between those two extremes. The noticeable dependence of the dispersion breadth on the ratio of layer diffusion permeabilities will enable us to reliably fit the parameter F in the interpretation of experimental data below. Checks of Linearity of Dependences on the Density of Polarizing Current. Figure 4 shows several examples of series of membrane potential transients obtained for the same salt concentration at various densities of polarizing current. To better visualize the linearity of the dependence on the current density (or the lack thereof), along with the directly measured transients, the figure also shows the results of a linear extrapolation of data obtained at the lowest current density to the densities used in the other measurements. If the extrapolated data coincide with the measured ones, the dependence on the current density is perfectly linear. It is seen that, at lower salt concentrations, this is the case up to

the highest current densities used in the measurements. At higher salt concentrations, there are some mismatches at the highest current densities, but the linearity is still very good at the intermediate ones. From that, we have concluded that at the lowest current densities the linear theory presented above was definitely applicable and could be used for the quantitative fitting of transients. That is not especially surprising in view of the largest signal magnitude of only a couple of mV, which corresponds to a pretty weak concentration polarization. (Thus for instance, for the transport number of cations approaching one, a 2-fold change in the salt concentration at the interface between the membrane and the filter would give rise to a signal with the largest magnitude of about 16 mV.) Theoretical Fitting of Time Transients and Determination of Electrochemical and Transport Properties of Membrane. Figure 5 shows the results of theoretical fitting of membrane potential transients obtained after current switch-off.

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Figure 5. Theoretical fits (dashed lines) of membrane potential transients, the vertical segments represent the error bars: (a) cs ) 0.00125 M; I ) 0.27 A/m2; (b) cs ) 0.0025 M; I ) 0.88 A/m2; c) cs ) 0.005 M; I ) 1.77 A/m2; (d) cs ) 0.01 M; I ) 5.31 A/m2; (e) cs ) 0.02 M; I ) 17.70 A/m2; (f) cs ) 0.05 M; I ) 88.50 A/m2. Table 1. Summary of Results of Fitting the Transients of Membrane Potential concentration (mol/dm3)

E ˜ mi/I˜(s µm-1)

t0*(s)

Pm/Pf

t0*F2(s)

0.00125 0.0025 0.005 0.01 0.02 0.05

0.15 0.135 0.098 0.068 0.042 0.021

97.6 74.8 39.6 24.4 20 19.8

0.45 0.5 0.7 0.85 0.98 1.03

79.1 74.8 77.6 70.5 76.1 83.2

It is seen that the fit quality is rather good. From the fits, we could determine the following three parameters: the initial value of transient membrane potential immediately after the current switch-off, Emi, the characteristic relaxation time, t0*, and the ratio of diffusion permeabilities of membrane and filter, Pm/Pf ≡ F/2. The values of those parameters obtained at various concentrations are gathered in Table 1.

To allow for a better comparison of initial transient membrane potentials obtained in solutions of various concentrations, the initial values are scaled on the reduced current densities used in the corresponding measurements. By using eq 22, for the initial membrane potential one can obtain this

4(∆t1)2 8(∆t1)2 E ˜ mi ) ≡ I˜ Pm + Pf/2 Pf(1 + F)

(23)

Since the ratio of diffusion permeabilities of membrane and filter can be determined from the fitting of time transients, the right-hand side of Eq(23) contains only two unknowns, namely, ∆t1 and Pf. Now, we can additionally use the information on the characteristic relaxation time also available from the fits. Indeed, by


Figure 6. Difference of ion transport numbers between the membrane pores and the filter as a function of salt concentration.

Figure 7. Diffusion permeabilities of membrane and filter against the salt concentration.

using the definitions of eqs 19 and 20, one can obtain this

Rf t0*F2 ≡ 4 lf2 χf

(24)

Remarkably, the right-hand side of eq 24 contains only properties of the filter. Since the latter is quite coarseporous, it can be expected that its properties are independent of salt concentration. From Table 1, it is seen that the combination in the left-hand side of eq 24 indeed is much less dependent on the salt concentration than the characteristic relaxation time itself. The remaining variation does not show any clear trend with changing salt concentration and, probably, is caused by experimental errors and/or fitting uncertainties. By using the definitions of specific chemical capacity of eq 4, and of absolute molar diffusion permeability as well as by taking into account that in the case of coarse-porous filters the distribution coefficient is equal to the filter porosity, we obtain this

γflf t0*F2 ≡ 4 Pf

(25)

where γf is the filter porosity. Finally, by combining eq 23 and 25, one obtains this

∆t1

xγf

)

x

1 F

E ˜ milf(1 + F) 2I˜t0*

(26)

The right-hand side of eq 26 contains only fitted or known quantities. The filter porosity has been determined gravimetrically in a separate measurement and was found to be 0.81. Thus, with eq 26, we can determine the difference of ion transport number between the membrane

Langmuir, Vol. 21, No. 15, 2005 6879

Figure 8. Membrane electric conductivity as a function of salt concentration.

and the filter. The results are plotted in Figure 6 as a function of salt concentration. It is seen that with decreasing concentration, ∆t1 approaches 0.5, which corresponds to ideal perm-selectivity (due to an ever better overlap of diffuse parts of double electric layers). With increasing concentration ∆t1 decreases in a monotone way. Both features are in quantitative agreement with the space-charge model. The knowledge of ion transport numbers in the membrane enables us to determine the absolute values of diffusion permeabilities of membrane and filter by using eq 23. The results are shown in Figure 7. It is seen that the filter diffusion permeability is practically independent of salt concentration, whereas that of the membrane noticeably increases with the latter. That is also in agreement with the predictions of the spacecharge model. The knowledge of membrane diffusion permeability and of ion transport numbers enables us to estimate the membrane electric conductivity by using this equation derived in ref 18 in approximation of no direct coupling between the ion fluxes

gm ≡

χm(FZ1ν1)2 t1t2

(27)

The results are shown in Figure 8. It is seen that the dependence on the salt concentration may be well approximated by a straight line in log-log coordinates. In not too concentrated bulk (1:1) electrolyte solutions, the slope of such a line is close to 1, whereas that in Figure 8 is around 0.7. That again can be explained by the effect of electrostatically adsorbed counterions whose contribution to the membrane electric conductivity progressively increases with decreasing salt concentration. From the data shown in Figure 8, one can also conclude that the membrane’s electrical resistance was approximately equal to that of a solution layer of only about 100 µm in thickness. As already discussed in the Introduction, the electrical resistance of such highly conducting films may be very difficult to measure precisely by the difference method due to the much larger contribution of the resistances of solution layers inevitably occurring between the membrane and the indicator electrodes. Checks of Quantitative Applicability of SpaceCharge Model. Now let us validate the space-charge model by checking the quantitative correlation between the data on the cation transport number and the membrane diffusion permeability. To do so, we need accurate information on the membrane pore size. The nominal pore size provided by the manufacturer is 30 nm, but in measurements by gas-liquid perm-porometry with this

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Figure 9. Estimated salt diffusion coefficient in the membrane pores scaled on the bulk diffusion coefficient of KCl.

Figure 10. Estimated salt diffusion coefficient in the filter pores scaled on the bulk diffusion coefficient of KCl.

membrane, we have systematically found the average pore size of 24 nm and a narrow pore size distribution ((4 nm). By using this pore diameter, we have estimated the surface charge density in the membrane pores proceeding from the fitted ion transport numbers. Further, we used this density to estimate the electrostatic correction to the membrane diffusion permeability with respect to the salt, kel. After that, we have estimated the salt diffusion coefficient in pores according to this

Dm )

Pmlm γmkel

(28)

where we have used the value of membrane porosity (3.2%) corresponding to the track density of 7 × 1013 m-2 and to the pore diameter of 24 nm. Finally, the salt diffusion coefficient in pores has been scaled on the concentrationdependent bulk diffusion coefficient of KCl.26 The results are shown in Figure 9. It is seen that the values are just moderately scattered around unity with a slight trend to decrease with increasing concentration. From comparison of Figure 7 and Figure 9, one can see that the principal part of dependence of membrane diffusion permeability on the salt concentration is accounted for by the electrostatic correction, kel. The remaining slight variation may well be a consequence of experimental errors or of inadequacies in the space-charge model as applied to this specific membrane (for example, because of not precisely cylindrical pore geometry and/or overlap of some of the pores giving rise to a pore size distribution). In our opinion, that observation confirms the quantitative applicability of standard space charge model as well as the fact that the water viscosity in this kind of pores is very close to that of bulk water.

Figure 11. Fixed electric charge density at the membrane pore surface estimated from the ion transport numbers by means of space-charge model as a function of salt concentration.

Figure 10 shows the results of the same procedure carried out for the Millipore filter by using the measured values of thickness (145 µm) and porosity (81%). (Since the pores in the filters have been very large compared with the thickness of diffuse parts of double electric layers, the electrostatic factor for the filters was assumed to be unity.) It is seen that the salt diffusion coefficient is independent of salt concentration and somewhat smaller than the bulk one. The most probable explanation is just the pore tortuosity. Figure 11 shows the surface charge density estimated from the fitted ion transport numbers within the scope of the space-charge model. It is seen that the surface charge concentration increases roughly in direct proportion to the square root of salt concentration. Such behavior is typical for weakly acidic (or basic) ionogenic groups at pH values not too far away from their pK. It has been reported that for track-etched membranes from poly(ethylene terephthalate) the pK is around 4.5,13 which is really not far away from our working pH5.5. Due to the particularities of manufacturing process, in particular of the etching step, there is a significant fraction of carboxylic end groups on the pore surface. In another study, the concentration of carboxylic groups on the pore surface of poly(ethylene terephthalate) membranes from Oxyphen has been estimated by using a chemical/photometric method at 50 pmol/cm2 (see ref 27). This value corresponds to a maximum surface concentration of fixed charges of 0.3 nm-2, which is in good agreement with the results shown in Figure 11. A Posteriori Estimates of Parameter r. Now, let us estimate a posteriori the values of parameter r to check whether it really was sufficiently large, as it was assumed in the interpretation of transients of membrane potential. By using the definition of eq 14, one can obtain this

r2 ≡

1 γfPflf fel γmPmlm

(29)

where fel is the electrostatic correction to the specific chemical capacity of membrane. It can be estimated within the scope of space-charge model and can be shown to vary from ca. 0.35 to almost 1 within the salt concentration range used in this study. All of the other parameters entering in eq 29 have been estimated above. By using their values, one can show that parameter r exceeded 15 in the most concentrated solution and increased in (26) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd revised ed.; Dover Publications: Mineola, NY, 2002; p 571. (27) Geismann, Ch.; Ulbricht, M. Macromol. Chem. Phys. 2005, 206, 268-281.


Figure 12. Limiting current density as a function of salt concentration.

monotone way with decreasing concentration. Thus, the use of approximation of large r has been justified. Estimates of Limiting Current Density. Finally, the data obtained above can be used to estimate the so-called limiting current density for this membrane/filter sandwich system. Above, we have shown that when electric current passes through an interface between two media with different ion transport numbers, the salt concentration at this interface either decreases or increases depending on the current direction. If the concentration decreases, the electric potential difference across the membrane grows super-linearly due to both the build-up of membrane potential and the increase in the electrical resistance of membrane and unstirred layer. When the concentration goes to zero, the voltage goes to infinity and the current density cannot increase anymore. This current density is called a limiting one. Actually, the situation is not that simple, and higher currents (the so-called over-limiting currents) do occur in membrane systems. However, in the over-limiting mode, there occurs water splitting, thermoconvection, electroconvection and, probably, other complicated phenomena, which makes the optimization of system operation much more difficult than in the underlimiting mode. Therefore, the limiting current density is an important property of a nano-fluidic system. If the dependencies of properties of nanoporous system on the salt concentration are taken into account, rigorous results for the limiting current density are available only in the form of a system of transcendential equations containing integrals. An approximate solution can be obtained if the variation of electrochemical, electrokinetic, and transport properties on the salt concentration within the nanoporous systems are disregarded. The corresponding expression for the three-layer system used in this study is

Pm + (Pf/2) ∆t1

I˜lim ∼

(30)

where Pm and Pf are the diffusion permeabilities of membrane and filter, respectively. The properties appearing in the right-hand side of eq 30 should be taken at the salt concentration in the reservoirs bathing the external surfaces of the filters. Since in our experiments the salt concentration inside the membrane/filter sandwich system deviated only slightly from that value, we can just substitute the values estimated above into eq 30. The results are shown in Figure 12. For comparison, we have also plotted the typical current densities occurring in microfluidic systems with electroosmotic fluid delivery. In those estimates, we have proceeded from the typical voltage of 30 kV/m. It is seen that the limiting current density for our system is systematically more than 2 orders of magnitude lower

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than the typical current densities encountered in microfluidic systems. The gap gradually closes with increasing salt concentration because of somewhat different slopes of the lines, but it would remain very large even in unrealistically concentrated electrolyte solutions. High voltages are applied in microfluidics to get noticeable electro-osmotic flows, which gives rise to large current densities. If even remotely similar current densities cannot be achieved in nanofluidic systems because of concentration polarization phenomena, that is a major problem. Of course, we should keep in mind that the polarization phenomena are essentially controlled by the effective thickness of unstirred layers. In this study, by using the coarse-porous filters, we have on purpose set up rather thick stagnant zones near the surface of the nanoporous membrane to better control the hydrodynamic conditions there. At the same time, by doing so, we have aggravated the concentration polarization. If there were no filters, the limiting currents would be essentially larger especially if vigorous stirring occurred immediately near the surface of the nanoporous membrane. However, it is notoriously difficult to make the effective thickness of stagnant layers in water-like fluids smaller than 10 µm even if vigorous turbulent stirring is set up. And it is virtually impossible to have turbulent mixing in microsystems. Finally, even if the thickness of unstirred layers decreased by 1 order of magnitude, that still would leave a gap of more than 1 order of magnitude between the limiting current densities and those needed to have electro-osmotic flows of reasonable magnitude. What can be done? Working in the over-limiting mode is a potential option, but one should keep in mind that in this mode the electro-osmosis will definitely be a nonlinear function of applied voltage. Besides that, considerable changes (and large gradients) in the local pH values are to be expected, which may essentially influence the electrophoretic mobility of amphoteric molecules. All that may well make the picture too complex to allow for an optimization. In addition, pH changes may be detrimental to pH-sensitive bio-samples. The use of larger pores (channels) appears to be a more reliable option. Indeed, it can be shown that in not too narrow pores the difference in ion transport numbers between the pore interior and the bulk solution decreases in direct proportion to the reciprocal pore radius. Besides that, the diffusion permeability slightly increases with the pore radius, too. Therefore, using the pores of some 300 nm in diameter in combination with more favorable hydrodynamic condition could, in principle, help make the limiting current densities approach the current densities needed in microfluidics. Nevertheless, even in this case one would still remain dangerously close to the limiting currents. Accordingly, in a quantitative optimization of system performance, it would be necessary to take into account the electrochemical phenomena. This will be the subject matter of the next study. Conclusions Measurements of transient membrane potential after current switch-off have been used to study the electrochemical and other transport properties of nanoporous track-etched membranes. Due to their identical straight cylindrical pores within the nanorange, those membranes are a suitable object for the studies of fundamentals of nanofluidics.

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During the membrane polarization by direct current, owing to the difference in the electrochemical properties between the membrane pores and the adjacent media there appear concentration gradients which give rise to diffusion potentials. After the current switch-off the polarization disappears not immediately, and its relaxation can be followed electrically. We have developed a linear theory, which enables one to interpret the transients of membrane potential in terms of phenomenological properties such as the ion transport numbers, the diffusion permeability and the specific chemical capacity. The measurements have been carried out with a track-etched membrane having the pores of 24 nm in diameter and the porosity of 3.2% in KCl solutions of various concentrations ranging from 0.00125 to 0.05 M. To better define the hydrodynamic conditions near the membrane surface, the membrane was put between two coarse-porous Millipore filters (pores size 3 µm), and vigorous stirring was set up at the external surfaces of the filters. From the quantitative interpretation of membrane potential transients, we have estimated the ion transport numbers in the membrane pores as well as the diffusion permeabilities of membrane and filters. Those data have been further interpreted within the scope of space-charge model and revealed good self-consistency. The concentration dependence of fitted values of surface charge density was in agreement with the mechanism of fixed charge formation due to the dissociation of weakly acidic groups, which is believed to be the principal mechanism in the

Yaroshchuk et al.

case of studied membranes. Thus, to sum up, our measurements have revealed a quantitative applicability of standard space-charge model to the description of electrochemical phenomena and electrolyte diffusion in straight cylindrical pores of tens of nanometers in diameter. At the same time, proceeding from our data, we could estimate the so-called limiting current densities for our membrane/filter sandwich system. They have turned out more than 2 orders of magnitude lower than the current densities usually encountered in microfluidic system with electro-osmotic fluid delivery. That finding may, unfortunately, point to a considerable handicap in the application of nanofluidic elements in microsystems with electroosmotic fluid delivery. A partial remedy to that might be provided by an increase in the pore dimensions toward hundreds of nanometers as well as improved mass transfer near polarized interfaces. Nevertheless, even in channels measuring several hundreds of nanometers in diameter the concentration polarization phenomena will be far from completely negligible and have to be taken into account to allow for a quantitative optimization of system performance. Acknowledgment. The financial support of Deutsche Forschungsgemeindschaft (DFG) via project du 128/14-1 is gratefully acknowledged. LA050499G

Electrochemical and Other Transport Properties of Nanoporous Track

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