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Streaming Potential in Cylindrical Pores of Poly(ethylene terephthalate) Track-Etched Membranes: Variation of Apparent ζ Potential with Pore Radius Philippe De´jardin,*,† Elena N. Vasina,† Vladimir V. Berezkin,‡ Vladimir D. Sobolev,§ and Vitaly I. Volkov| European Membrane Institute, UMR 5635 (CNRS, ENSCM, UMII), Universite´ Montpellier II, CC047, 2 Place Euge` ne Bataillon, F-34095 Montpellier Cedex 5, France, A. V. Shubnikov Institute of Crystallography of the Russian Academy of Sciences, Leninsky Prospect 59, 119333 Moscow, Russia, Institute of Physical Chemistry of the Russian Academy of Sciences, Leninsky Prospect 31, 119991 Moscow, Russia, and Federal Research Center, Karpov Institute of Physical Chemistry, Vorontsovo Pole 10, 103064 Moscow, Russia Received December 15, 2004. In Final Form: March 4, 2005 Streaming potential variation with pressure measured through poly(ethylene terephthalate) tracketched membranes of different pore sizes led to the determination of an apparent interfacial potential ζa in the presence of 10-2 M KCl. The variation of ζa with the pore radius r0 is interpreted by (i) the electric double layer overlap effect and (ii) the presence of a conductive gel layer. We propose a method which integrates both effects by assuming a simple model for the conductive gel at the pore wall. We observed the following three domains of pore size: (i) r0 > 70 nm, where surface effects are negligible; (ii) ∼17 nm < r0 < 70 nm, where the pore/solution interface could be described as a conductive gel of thickness around 1 nm; (iii) r0 < ∼17 nm, which corresponds to the region strongly damaged by the ion beam and is not analyzed here. The first one (ζ ) -36.2 mV) corresponds to the raw material when etching has completely removed the ion beam predamaged region, which corresponds to the second intermediate domain (ζ ) -47.3 mV). There the conductance of the gel layer deduced from the treatment of streaming potential data was found to be compatible with the number of ionic sites independently determined by the electron spin resonance technique.
Introduction Well-structured porous materials can be useful in nanotechnology1 and quantitative separation of proteins or particles from mixtures. Manufacturing of polymeric membranes by means of swift ion beam irradiation followed by chemical etching allows very precise control of the pore size and shape.2 The final surface chemical composition and the pore shape depend on the polymer chemical structure, the conditions of irradiation (heavy ion, energy), and etching (temperature, time, UV exposure,3 alkali,4 surfactant5). Investigations were performed on different materials, such as polyolefins,6 bisphenol A polycarbonate,7 polyimide,8 and poly(ethylene terephthalate) (PET).8-10 * To whom correspondence may be addressed: tel, +33 467 14 91 21; fax, +33 467 14 91 19; e-mail,
[email protected]. † European Membrane Institute, UMR 5635 (CNRS, ENSCM, UMII), Universite´ Montpellier II. ‡ Shubnikov Institute of Crystallography RAS. § Institute of Physical Chemistry RAS. | Federal Research Center, Karpov Institute of Physical Chemistry. (1) Toulemonde, M.; Trautmann, C.; Balanzat, E.; Hjort, K.; Weidinger, A. Nucl. Instrum. Methods Phys. Res., Sect. B 2004, 216, 1-8. (2) Fleischer, R. L. Tracks to Innovation; Springer: Berlin, 1998. (3) Zhu, Z. Y.; Duan, J. L.; Maekawa, Y.; Koshikawa, H.; Yoshida, M. Radiat. Meas. 2004, 38, 255-261. (4) Samoilova, L. I.; Apel, P. Y. Radiat. Meas. 1995, 25, 717-720. (5) Apel, P. Y.; Blonskaya, I. V.; Orelovitch, O. L.; Root, D.; Vutsadakis, V.; Dmitriev, S. N. Nucl. Instrum. Methods Phys. Res., Sect. B 2003, 209, 329-334. (6) Apel, P. Y.; Didyk, A. Y.; Salina, A. G. Nucl. Instrum. Methods Phys. Res., Sect. B 1996, 107, 276-280. (7) Dehaye, F.; Balanzat, E.; Ferain, E.; Legras, R. Nucl. Instrum. Methods Phys. Res., Sect. B 2003, 209, 103-112.
In the case of PET, etching with NaOH creates carboxy ionizable groups at the pore surface.11 The negative charge density depends on pH through the chemical equilibrium between protonated carboxylic and deprotonated carboxylate groups. The chain scissions produce also phenolic groups.12 Hence the membranes may interact ionically with solutes.13 The membrane-solution interface can be modified by different means, as desired. To improve the biocompatibility, coverage by protein-repellent polymers, for instance poly(ethylene glycol), can be performed, as has been done for types of membranes,14,15 micromachines, and microfluidic devices.16,17 Specific functions can also be added, as for similar nanoporous mineral membranes.18 Electrostatic interactions between charged proteins and (8) Siwy, Z.; Apel, P.; Dobrev, D.; Neumann, R.; Spohr, R.; Trautmann, C.; Voss, K. Nucl. Instrum. Methods Phys. Res., Sect. B 2003, 208, 143148. (9) Apel, P. Y.; Didyk, A. Y.; Kravets, L. I.; Kuznetsov, V. I. Nucl. Tracks Radiat. Meas. 1990, 17, 191-193. (10) Apel, P.; Angert, N.; Bruchle, W.; Hermann, H.; Kampschulte, U.; Klein, P.; Kravets, L. I.; Oganessian, Y. T.; Remmert, G.; Spohr, R. Nucl. Instrum. Methods Phys. Res., Sect. B 1994, 86, 325-332. (11) Pasternak, C. A.; Alder, G. M.; Apel, P. Y.; Bashford, C. L.; Edmonds, D. T.; Korchev, Y. E.; Lev, A. A.; Lowe, G.; Milovanovich, M.; Pitt, C. W. Radiat. Meas. 1995, 25, 675-683. (12) Steckenreiter, T.; Balanzat, E.; Fuess, H.; Trautmann, C. Nucl. Instrum. Methods Phys. Res., Sect. B 1997, 131, 159-166. (13) Vasina, E. N.; Dejardin, P. Biomacromolecules 2003, 4, 304313. (14) Thomas, M.; Valette, P.; Mausset, A. L.; Dejardin, P. Int. J. Artif. Organs 2000, 23, 20-26. (15) Yan, F.; Dejardin, P.; Mulvihill, J. N.; Cazenave, J. P.; Crost, T.; Thomas, M.; Pusineri, C. J. Biomater. Sci., Polym. Ed. 1992, 3, 389402. (16) Popat, K. C.; Mor, G.; Grimes, C.; Desai, T. A. J. Membr. Sci. 2004, 243, 97-106. (17) Popat, K. C.; Desai, T. A. Biosens. Bioelectron. 2004, 19, 10371044.
10.1021/la046913e CCC: $30.25 © 2005 American Chemical Society Published on Web 04/06/2005
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charged membranes may be used to improve membrane separation properties.19,20 Therefore, it is of great interest to characterize the electrical state of the membranesolution interface by its zeta (ζ) potential obtained, for instance, via streaming potential measurements. The same technique is also used to study protein adsorption at interfaces as this phenomenon alters the electric interfacial state.21-28 Measuring the streaming potential through the membrane helps to characterize the electrical state of the solution/solid interface in pores by determining the electric potential at the shear plane, called ζ potential.29 However, as mentioned in very early studies, as well as in more recent ones, the classical Smoluchowski relation for deducing ζ is not applicable when surface effects are important.30 For pores of track-etched membranes, a loosened layer, located at the pore wall, arises from the polymer damage by the swift heavy ion beams in the first step of membrane preparation. In the presence of water, this layer can be considered to be a conductive gel. The “surface” effect may be considerable when the gel layer conductance is not negligible with respect to the bulk pore conductance. Obviously, the smaller the pore radius, the larger the surface effect. To describe the peculiar nature of such a conductive wall, it was proposed31 that the pore structure is of fractal nature with a surface scaling exponent higher than 2. The specific state of the interface is also illustrated by the electric current fluctuations for very small pores (∼2 nm), which were attributed to the dangling carboxylate end groups.8 We consider here the analysis of streaming potential measurements in the presence of 10-2 M potassium chloride aqueous solution, as a function of pore radius, through PET32 track-etched membranes. Materials and Methods The PET films were irradiated by Xe ions (1 MeV/nucleon) at the accelerator U-400 (Flerov Laboratory of Nuclear Reactions, Joint Institute of Nuclear Research, Dubna, Russia) and then submitted to chemical etching (5 M NaOH; T ) 65 °C) to produce pores. The fluence of irradiation was 108-109 cm-2, depending on the pore size. The pore diameters of the samples varied in the range 20 nm up to 580 nm. The thickness of the membranes was 10 µm. The streaming potential measurements were carried out in a standard (Millipore) dead-end-type filtration cell supplied with two Ag/AgCl electrodes mounted over and under the membrane pressed between two cell parts via a silicone rubber gasket (Figure 1a). The electrodes enabled us to measure the (18) Leary Swan, E. E.; Popat, K. C.; Desai, T. A. Biomaterials 2005, 26, 1969-1976. (19) Kohli, P.; Harrell, C. C.; Cao, Z.; Gasparac, R.; Tan, W.; Martin, C. R. Science 2004, 305, 984-986. (20) Lee, S. B.; Mitchell, D. T.; Trofin, L.; Nevanen, T. K.; Soderlund, H.; Martin, C. R. Science 2002, 296, 2198-2200. (21) Norde, W.; Rouwendal, E. J. Colloid Interface Sci. 1990, 139, 169-176. (22) Zembala, M.; Dejardin, P. Colloids Surf., B 1994, 3, 119. (23) Ethe`ve, J.; Dejardin, P. Langmuir 2002, 18, 1777-1785. (24) Vasina, E. N.; Dejardin, P. Langmuir 2004, 20, 8699-8706. (25) Elgersma, A. V.; Zsom, R. L. J.; Lyklema, J.; Norde, W. Colloids Surf. 1992, 65, 17-28. (26) Kim, K. J.; Fane, A. G.; Nystrom, M.; Pihlajamaki, A. J. Membr. Sci. 1997, 134, 199-208. (27) Lettmann, C.; Mockel, D.; Staude, E. J. Membr. Sci. 1999, 159, 243-251. (28) Werner, C.; Jacobasch, H. J.; Reichelt, G. J. Biomater. Sci., Polym. Ed. 1995, 7, 61-76. (29) Hunter, R. J. In Colloid Science; Ottewill, R. H., Rowell, R. L., Eds.; Academic Press: London, 1981; p 81. (30) Erickson, D.; Li, D. Q.; Werner, C. J. Colloid Interface Sci. 2000, 232, 186-197. (31) Grzywna, Z. J.; S. Liebovitch, L.; Siwy, Z. J. Membr. Sci. 1998, 145, 253-263. (32) Cail, J. I.; Stepto, R. F. T. Polymer 2003, 44, 6077-6087.
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Figure 1. (a) Experimental cell for streaming potential measurements through pores of track-etched membranes. Gas pressure is applied while potential is obtained via two silversilver chloride electrodes. (b) Example of ∆Es versus ∆P for a pore radius of 17 nm. potential difference, ∆Es, arising during the electrolyte flow through the membrane under pressure difference, ∆P (Figure 1b). T ) 21 °C. A rigid porous PTFE support of thickness 5 mm (Figure 1a) helped to maintain the membrane without strong deformation at high pressures. Pictures which illustrate the structure of such membranes can be found elsewhere.33 The KCl solution was prepared with deionized water, without any buffer adjustment. Because of the equilibrium with the atmospheric carbon dioxide, the pH is slightly acid, pH 6.5. An order of magnitude of the fraction of deprotonated groups could be estimated from the pKa values of carboxylic acids. For PET, the model of phthalic acid (pKa ) 3.7) provides an upper boundary for the ionizable fraction near 1. We can expect however a priori a lower experimental value for the etched material, as the successive ionizations are interdependent, but it is difficult to assess a priori the ionization state of the gel.
Results and Discussion The presence of a gel layer at the pore wall created by chemical etching after irradiation of materials by swift ion beams was already mentioned.33,34 However, the conclusion, that the charge density at the shear plane varies quite strongly with the pore size, remains surprising. We can expect that the state of the pore wall, submitted to progressive destruction by etching, will attain a steady state when the pore becomes large enough. This apparent discrepancy may originate from the etched material conductance which was not taken into account properly. The problem is analogous to the surface conductance effect which becomes increasingly important when the bulk solution conductivity diminishes, especially for small pores, as the area/volume ratio of a cylinder varies as the inverse of its radius. (33) Berezkin, V. V.; Volkov, V. I.; Kiseleva, O. A.; Mitrofanova, N. V.; Sobolev, V. D. Adv. Colloid Interface Sci. 2003, 104, 325-331. (34) Berezkin, V. V.; Kiseleva, O. A.; Nechaev, A. N.; Sobolev, V. D.; Churaev, N. V. Colloid J. 1994, 56, 258-263.
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Theoretical Derivation of Apparent Zeta Potential Variation with Pore Radius. The apparent zeta potential of the interface is deduced from the linear variation of the streaming potential ∆Es as a function of pressure differential ∆P via the classical Smoluchowski relation29
σSmol )
dEs ηλ0 dP 0r
(1)
where λ0 is the electrolyte solution conductivity, η is the solution viscosity, 0 is the vacuum dielectric permittivity, and r is the relative permittivity. Applying this relation whatever the material (etched or unetched, surface treated or not) and the pore radius r0, we get an “apparent” zeta potential, denoted as ζa ) ζSmol. In the pore size range examined, we assume that the interfacial zeta potential ζ, which may differ from ζa, does not vary. We wish to provide a model relating the apparent to the interfacial potential. Two simultaneous effects have to be taken into account: (i) the curvature and the double layer overlap and (ii) the back current due to the gel layer. The first effect was analyzed many years ago by Rice and Whitehead for the circular cylindrical geometry.35 For a pore of radius r0, the final expression relating ζa and ζ takes the following form29 (see Appendix for details about the derivation):
[
1 - β F(κr0) 1 1 ) σa σ
]
I12(κr0) I02(κr0)
(2)
F(κr0)
with
F(κr0) ) 1 -
2I1(κr0)
(3)
κr0I0(κr0)
and
β)
(0rσκ)2 ηλ0
(4)
I0 and I1 are the zero-order and first-order modified Bessel functions of the first kind,36 respectively, and κ-1 is the double layer thickness or Debye screening length (see eq A4 in Appendix). Now let us consider the effect of the back current due to the gel layer. Kgel is the gel layer conductance (r > r0) and K0 is the bulk pore conductance (r < r0). Then eq 2 becomes
1+ 1 1 ) σa σ
[
]
Kgel I12(κr0) - β F(κr0) - 2 K0 I (κr ) F(κr0)
0
0
(5)
At this stage, we have to choose a model for the gel layer conductivity distribution. For a gel layer of thickness ∆rc and constant conductivity λgel, Kgel/K0 is φc[2(∆rc/r0) + (∆rc/r0)2] with φc ) λgel/λ0, where λ0 is the solution conductivity. This model, however, is probably unrealistic. More likely, the gel conductivity λgel(r) is a decreasing function of distance to the pore axis. If we assume the exponential distribution λgel(r) ) λ0,gel exp[-(r - r0)/∆rexp] (35) Rice, C. L.; Whitehead, R. J. Phys. Chem. 1965, 69, 4017-4024. (36) Abramovitz, M.; Stegun, I. A. Handbook of Mathematical Functions; Dover Publications, Inc.: New York, 1972; p 378.
Figure 2. (a) Functions F(u) (solid line) and f(u) (dashed line) plotted versus u-1. (b) Illustration of the accuracy of their power series expansions in 1/u, truncated to order 2 (dash); 3 (dashdot); 4 (solid), by their ratios over the exact values.
and define φexp ) λ0,gel/λ0, then Kgel/K0 ) φexp[2(∆rexp/r0) + 2(∆rexp/r0)2]. Anyway, both distributions lead to the same effect with the correspondence ∆rexp ) ∆rc/2 and φexp ) 2φc. We shall assume the exponential distribution in what follows with the notations φ and ∆r, instead of φexp and ∆rexp, respectively. Moreover, we will assume the same mobility for the counterions in the diffuse layer and those adsorbed on the material. The number of ionic sites N in the gel around one pore was recently measured by electron spin resonance spectroscopy (ESR),33 so we can check the accuracy of our model, as φ ∆r is directly linked to N (see Appendix eq A11). For the function F(u), with u ) r0κ which appears in eq 5, and f(u) ) F(u) - [I12(u)/I02(u)], we can write the following power series expansions about the point u-1 ) 0:
F(u) ) 1 - 2u-1 + u-2 + 1/4u-3 + 1/4u-4 + O(u-5) (6) f(u) ) -u-1 + u-2 + 3/8u-3 + 1/4u-4 + O(u-5) (7) In the range of our measurements (κ-1 ) 3.0 nm; r0 > 10 nm; u-1 < 0.30), the accuracy of the approximations for F(u) and f(u) (Figure 2a) by these power series expansions truncated to second, third, and fourth order is illustrated in Figure 2b. Using as an approximation of f(u), its power series expansion truncated to the second order in u-1 leads to the following approximation of eq 5:
[
]
F(κr0) 1 1 1 ) 1 + (2φ∆r + βκ-1) + (2φ∆r2 - βκ-2) 2 σa σ r0 r 0
(8) Fitting a second-order polynomial to the data F(κr0)/ζa as a function of 1/r0, one can get ζ-1 as the intercept at the
Streaming Potential in Nanopores
origin of the abscissas. Then β can be evaluated (eq 4), and φ and ∆r obtained from the coefficients of r0-1 and r0-2. Experimental Results and Discussion. For the present material, the back current due to “surface” conductance originates, in the presence of 10-2 M KCl, from the counterions present in the volume of the etched material at the pore wall as well as in the diffuse double layer. Although this volume conductance may be considered as a surface effect in the general framework of the streaming potential technique, to avoid confusion with the surface conductance of nonporous materials, we will use below the term of gel conductance. However, this term includes the double layer surface effect, relative to the boundary between the gel and the bulk solution. At such high salt concentration and in the pore size range examined, we expect that the volume pore conductance can be estimated as a first approximation from the bulk conductivity, while there is a priori a non-negligible contribution of the walls to the back current compared to the pore volume one. That would not be the case for highly diluted solutions where the pore conductance becomes independent of the solution conductivity.37 According to our previous analysis, F/ζa experimental data are plotted as a function of the inverse of the pore radius (Figure 3). We found the existence of three domains (Figure 3a): (i) r0 > 70 nm, where F/ζa is constant; (ii) ∼17 nm < r0 < 70 nm, where a linear variation of F/ζa with 1/r0 is observed; (iii) r0 < ∼17 nm, with a possible different behavior, but data are only few and dispersed. In the domain r0 > 70 nm the interfacial potential is equal to -36.2 mV (Figure 3a, dashed line). In the second domain the data can be treated according to the foregoing model (eq 8) and the intercept with the ordinate axis provides ζ-1, say ζ ) -47.3 mV (Figure 3a, solid line intercept). From eqs A4 and 4, with 0 ) 8.854 × 10-12 C V-1 m-1, r ) 80; η )10-3 Pa s, λ0 ) 1276 µS cm-1, T ) 21 °C, we obtain κ-1 ) 3.0 nm and β ) 0.86. The product φ ∆r is then deduced from the slope. Assumption of the null curvature leads to an estimation of both φ and ∆r. The three parameters ζ, φ, and ∆r are given in Table 1. We provided also the values of these parameters when treating the complete data (Figure 3b) and the data restricted to r0 < 70 nm (Figure 3c): the quality of the fits worsens (Table 1, regression coefficients). Moreover, the inclusion of the very small pore sizes may be not justified as (i) the approximation of the function f by the first two terms of eq 7 becomes too rough at large values of 1/r0, (ii) previous work38 on the pore structure showed the existence of a strongly damaged zone around the track, and (iii) the bulk conductivity may differ significantly from λ0. The pore surface can also be characterized by paramagnetic probes such as Cu2+, which form chelate complexes with two carboxylic groups. The total amount and local concentration of carboxylic groups may be calculated from the electron spin resonance (ESR) spectra.39 ESR studies have shown that the number of copper ions trapped in the gel around one pore, NCu, was not proportional to the pore diameter and presented a nonmonotonic variation with a minimum.33 The existence of the strongly predamaged zone was emphasized by this technique, which demonstrated the very high number of (37) Stein, D.; Kruithof, M.; Dekker: C. Phys. Rev. Lett. 2004, 93. (38) Vilensky, A. I.; Larionov, O. G.; Gainutdinov, R. V.; Tolstikhina, A. L.; Kabanov, V. Y.; Zagorski, D. L.; Khataibe, E. V.; Netchaev, A. N.; Mchedlishvili, B. V. Radiati. Meas. 2001, 34, 75-80. (39) Hong, Y.-S.; Cho, C.-H.; Mitrofanova, N. V.; Nechaev, A. N.; Volkov, V. I.; Mchedlishvili, B. V.; Lee, C.-H. Magn. Reson. Imaging 2001, 19, 588.
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Figure 3. F/ζa versus the inverse of the pore radius r0, where F is the Rice-Whitehead factor F(κr0) ) 1 - 2(κr0)-1I1(κr0)/I0(κr0). ζa is the apparent potential according to the Smoluchowski relation. Consideration of (a) three domains: r0 g 70 nm (O, dashed line), mean value of ζ is -36.2 mV; 17 nm e r0 e 70 nm (b, solid line, ∆r ) 0.53 nm; dashed line, ∆r ) 1.0 nm; dashdotted line, ∆r ) 3.0 nm); r0 < 17 nm (0); (b) the full range of data; (c) two domains: r0 g 70 nm (O, dashed line); r0 e 70 nm (b, solid line).
available ionic sites, 2NCu, per pore at very small pore sizes. A regular increase of the number of such sites per pore is suggested from two observations above a critical radius around 25 nm. Assuming (i) the same mobility for counterions in gel and ions in the bulk electrolyte solution and (ii) a large excess of those gel counterions with respect to “free” ions K+ and Cl- in gel, the ratio of the maximal gel conductance Kgel over the pore conductance K0 is (Kgel/ K0)ESR ) 2NCu/(2πr02LCbNAv), where L is the length of the pore, Cb is the bulk electrolyte concentration (10-2M), and NAv is the Avogadro number. According to our model Kgel ∼ r0∆r, while K0 ∼ r02, thus Kgel/K0 ∼ r0-1, more precisely Kgel/K0 ) 2φ∆rr0-1 (see eq A11 in Appendix). Figure 4 provides (Kgel/K0)ESR as a function of the inverse of the radius for two ESR data. The linear variation is plausible with a slope of 21 nm, which is indeed of the same order of magnitude as 2φ∆r ) 17.4 nm, determined from the streaming potential data (Table 1). The lower value from streaming potential data can be connected to the partial ionization of the gel: the fraction of charged groups is 17.4/21 ) 0.83. With pH ) 6.5, this would correspond to 〈pKa〉 ) 5.7 for the carboxylic groups. These few data show the feasibility of the method. The previous analysis considered the sum of the contributions of (i) the diffuse layer in the pore, and (ii) the ions located inside the gel, the latter being not submitted to the hydrodynamic convection forces. A
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Table 1. ζ Potential of the Pore Wall/10-2 M KCl Solution Interface of PET Track-Etched Membranes, Parameter β (see Eq 4), and Product O ∆r, with ∆r the Gel Layer Thickness and O the Ratio of Conductivities (exponential model), According to the Fit of Eq 8 to the Experimental Data as F(Kr0)/ζa ) b0 + b1r0-1 + b2r0-2 a data range examined
ζ (mV)
β
φ ∆r (nm)
∆r (nm)
φ
regression coefficient
figure
17 nm e r0 e 70 nm r0 g 70 nm full data range r0 e 70 nm r0 g 70 nm
-47.3 -36.2 -38.0 -37.6 -36.2
0.86
8.7
0.53
16.5
0.98
3a
0.55 0.54
3.7 3.6
0.66 0.67
0.81 0.75
3b 3c
a
5.65 5.44
The assumption b2 ) 0 leads to the estimation of φ and ∆r.
Figure 4. Ratio of the gel conductance over the bulk electrolyte conductance in pores of PET track-etched membranes. Calculation from the number of ionic sites determined by ESR,33 as 2NCu, over the pore conductance assuming a homogeneous bulk conductivity through the whole pore section. The slope is 21 nm.
possible discrimination might be obtained by complementary experiments of streaming current,40 or taking advantage of the extrapolated ζ potential. A previous observation41 of the surface current linear variation with the pore diameter is in accordance with the present interpretation. After having examined the density of charges in the gel as a whole, let us look now at their distribution at the interface. Whatever the data range fitted (Table 1), the thickness ∆r of the gel layer remains of the same order of magnitude. ∆r was estimated from the assumption of a null second-order term in eq 8. Thus the precision is not high, but it remains of the order of 1 nm: Figure 3a illustrates the deviation from data when we assume ∆r ) 1.0 and 3.0 nm. With the specific mass of PET, F ) 1.397 g cm-3, and the molar mass of the polymer unit, M ) 192 g/ mol, the bulk polymer corresponds to the concentration 7.3 M. We can compare it with the ionic volume concentration at the gel edge: Csurf(r0) ) 2Cbφ ) 0.33 M, which corresponds to 4.5% of the bulk polymer. The total charge density is then σ0 ) eNAvCsurf(r0)∆r ) -0.017 C m-2. From ζ ) -47.3 mV, the electrokinetic density of charges is given by σek ) 4neκ-1 sinh(eζ/2kBT) ) -0.012 C m-2, where n ) CbNAv. The quite high value for the ratio σek/σ0 ≈ 0.7 might be linked to the flexibility of the chains bearing the end carboxylate charges, which are positioned inside some volume, and then diminishing the mean intercharge distance, compared to a charged flat surface bearing the same number of charges. In addition, it is remarkable that the extended monomer length (1.095 nm)32 is comparable with ∆r: it suggests the absence of long dangling chains in the etching process. Our data analysis in which three domains of pore size were considered is supported by other work. Previously,34 (40) Werner, C.; Korber, H.; Zimmermann, R.; Dukhin, S.; Jacobasch, H. J. J. Colloid Interface Sci. 1998, 208, 329-346. (41) Wolf, A.; Reber, N.; Apel, P. Y.; Fischer, B. E.; Spohr, R. Nucl. Instrum. Methods Phys. Res., Sect. B 1995, 105, 291-293.
Figure 5. Schematic representation of the interfacial potential as a function of the radius of track-etched membranes. After etching out the core of radius 10-20 nm, the etching of the cross-linked material leads to a constant interfacial potential ζ ) -47.3 mV, until a radius around 70 nm, where the interface corresponding to the etched raw material is slightly less negative with ζ ) -36.2 mV.
for more dilute electrolyte concentrations, the ratio of the measured conductance in the pores to that expected from uniform bulk conductivity (Figure 5 in ref 34) was assumed to be linear with 1/r02. The linearity was supposed to be valid at quite large values of 1/r02 (r0 < 25 nm). However, for smaller 1/r02, the conductances ratio is more likely to vary as a fractional power of 1/r02, and possibly as (1/r02)1/2 ) 1/r0 as in our present model (see Appendix, eq A11). That study suggests exclusion of data for r0 < 25 nm from our present treatment. The lowest value of 17 nm in our treatment via eq 8 (Figure 3a) is near this limit. The track structure38 was also extensively studied by means of higheffective liquid chromatography and atomic force microscopy, as well as the etching rate.42 These three studies34,38,42 emphasized the very special nature of the third domain (very small pores) which is not taken into account in our model. Our choice of the exponential conductivity distribution is simple but arbitrary; more complex conductivity profiles may exist with more complex dependence on the pore size. Chemical and physical processes, such as cross-linking, influence the structure of PET up to the region 100 nm thick.38 Then the change in the F/ζa behavior observed at r0 ) 70 nm corresponds probably to the passage from the cross-linked material to the raw one with increasing r0. In the domain r0 > 70 nm, the ionic site density of the etched material is too small to be properly detected, thus the observed plateau within the experimental error (Figure 3a,c). The results are summarized by the scheme in Figure 5. (42) Apel, P. Y. Nucl. Tracks Radiat. Meas. 1991, 19, 29-34.
Streaming Potential in Nanopores
Langmuir, Vol. 21, No. 10, 2005 4685
Conclusion We have provided an analysis of the variation of the apparent potential ζa with pore radius of PET track-etched membranes in an intermediate range of pore size, taking into account the partial double layer overlap, curvature, and the presence of a surface thick conductive gel. The model appears to be coherent with the determination of the ionic sites number per pore, obtained by electron spin resonance spectroscopy.33 In addition, our model emphasizes a change in the structure of the polymer with increasing pore radius. This change can be related to the damage produced by swift ion beams.38 We provided a description of the interface with a more negative interfacial potential for small pores (r0 < 70 nm; ζ ) -47.3 mV) than for larger ones (r0 > 70 nm; ζ ) -36.2 mV). The method presented can also be useful for the electrokinetic characterization of pores of any material covered with polyelectrolytes. Acknowledgment. This work was partly supported by the French Program “Re´seau National des Technologies pour la Sante´” (Contract 032906020) and performed within the framework of program 12889 of DRI-CNRS. Appendix Let us consider a circular cylinder of radius r0. r is the distance to the axis. The Poisson equation relating electric potential ψ(r) and charge density F(r) is
F 1 d dψ r )r dr dr
( )
(A1)
where ) 0r, the dielectric permittivity, and 0 is the dielectric permittivity of vacuum and r is the relative permittivity of the aqueous medium. For a 1-1 electrolyte, the charge density is given by the Boltzmann equation
F(r) ) -2ne sinh
eψ (kT )
(A2)
where n is the molar concentration of each ion (or the molar concentration of the 1-1 electrolyte), e is the (negative) elementary charge, k is the Boltzmann constant, and T is the temperature. For small enough absolute values of ψ, typically below 50 mV for a plane,35 the linear approximation sinh(u) ≈ u leads to a potential distribution close to the exact one. Within this approximation, the combination of eqs A1 and A2 leads to the linearized Poisson-Boltzmann equation
1 d dψ r ) κ2ψ r dr dr
( )
( ) 2ne2 kT
1/2
(A4)
The solution is35
ψ)σ
I0(κr) I0(κr0)
vz(r) )
(A5)
where I0 is the zero-order modified Bessel function of the first kind.
[
]
σ∆Es I0(κr) ∆P (r02 - r2) 14ηL ηL I0(κr0)
(A6)
The streaming potential is ∆Es ) Es(entrance) - Es(exit), z is the flow direction, η is the viscosity, and I0 is the zero-order modified Bessel function of the first kind. The current i ) i1 + i2, where i1 is the current due to transport of charge by the fluid and i2 is the back conduction current
i1 ) 2π
∫0r
0
vz(r)F(r)r dr )
[
-ΩA
]
∆Es ∆P F(κr0) + Ωηκ2f(κr0) (A7) L L
where A is the cross-sectional area of the pore, Ω ) ζ/η, F(u) ) 1 - [2I1(u)/(uI0(u))] and f(u) ) F(u) - [I1(u)/I0(u)]2, where I1 is the first-order modified Bessel function of the first kind. In conditions of streaming potential measurements, i1 + i2 ) 0. The conduction current i2 is the sum of the currents in the electrolyte solution, on one hand, and in the gel, on the other hand. The electric field is ∆Es/L. Assuming an exponential variation of the conductivity
λgel(r) ) λ0,gel exp[-(r - r0)/∆r]
(A8)
r > r0 with λ0,gel ) λgel(r0), the conductance of the gel layer is given by
Kgel )
∫r∞ 0
(
)
λ0,gel r0 λgel(r) 2πr dr ) 2π (∆r)2 1 + L L ∆r
(A9)
while the conductance of the electrolyte solution of conductivity λ0 is
K0 )
πr02 λ L 0
(A10)
then
[ ( )]
Kgel λ0,gel ∆r ∆r )2 + K0 λ0 r0 r0
(A3)
where κ-1 is the double layer thickness or Debye screening length.
κ)
The solution of the equation of motion gives the velocity profile vz(r), for a pressure differential ∆P ) P(entrance) - P(exit) between the extremities of the pore of length L. It is the sum of the Poiseuille profile and the electrokinetic correction.
2
i2 ) ∆EsK0(1 + Kgel/K0)
(A11) (A12)
Writing i1 + i2 ) 0 using (A7) and (A12) without and with gel contribution leads to eq 2 and to eq 5 in text, respectively. It was mentioned29,43 that this derivation was not strictly correct as the bulk conductivity in the pore is not λ0 near the wall: eq A10 does not take into account the variation of the ionic concentration in the double layer. But we are taking into account this contribution by including it, as a first approximation, in the gel conductance term. LA046913E (43) Dukhin, S. S.; Deryaguin, B. V. In Surface and Colloid Science; Matijevic, E., Ed.; John Wiley: New York, 1974; Vol. 7, pp 52-72.
