pubs.acs.org/Langmuir © 2009 American Chemical Society
Ion-Rejection, Electrokinetic and Electrochemical Properties of a Nanoporous Track-Etched Membrane and Their Interpretation by Means of Space Charge Model Andriy Yaroshchuk,*,† Yuriy Boiko,‡ and Alexandre Makovetskiy§ † ICREA and Department of Chemical Engineering, Polytechnic University of Catalonia, Carrer de Jordi Girona, 08034, Barcelona, Spain, ‡Institute of Biocolloid Chemistry, National Academy of Science of Ukraine, 42 Acad. Vernadskoho Boulevard, 03142 Kiev, Ukraine, and §National Technical University of Ukraine, 37 Prospect Peremogy, 03056 Kiev, Ukraine
Received March 2, 2009. Revised Manuscript Received May 21, 2009 Due to their straight cylindrical pores, nanoporous track-etched membranes are suitable materials for studies of the fundamentals of nanofluidics. In contrast to single nanochannels, the nano/micro interface, in this case, can be quantitatively considered within the scope of macroscopically 1D models. The pressure-induced changes in the concentration of dilute KCl solutions (salt rejection phenomenon) have been studied experimentally with a commercially available nanoporous track-etched membrane of poly (ethylene terephthalate) (pore diameter ca. 21 nm). Besides that, we have also studied the concomitant stationary transmembrane electrical phenomenon (filtration potential) and carried out time-resolved measurements of the electrical response to a rapid pressure switch-off (within 5-10 ms). The latter has enabled us to split the filtration potential into the streaming potential and membrane potential components. In this way, we could also confirm that the observed nonlinearity of filtration potential, as a function of the transmembrane volume flow, was primarily caused by the salt rejection. The results of experimental measurements have been interpreted by means of a space charge model with the surface charge density being a single fitting parameter (the pore size was estimated from the membrane hydraulic permeability). By using the surface charge density fitted to the salt rejection data, the results of electrical measurements could be reproduced theoretically with a typical accuracy of 10% or better. Taking into account the simplifications made in the modeling, this accuracy appears to be good and confirms the quantitative applicability of the basic concept of space charge model to the description of transport properties of dilute electrolyte solutions in nanochannels of ca. 20 nm.
Introduction In microfluidics, of great importance is the electromechanical coupling due to the separation of electrical charges at interfaces (responsible for the electrokinetic phenomena like electroosmosis). Recently, ever more attention has been paid to nanofluidics.1 There is a rather broad range of dimensions where a system already becomes nanofluidic (e.g., due to the finite thickness of diffuse parts of double electric layers as compared to the channel dimensions), but some of the macroscopic approaches (e.g., the continuous description of the solvent) may well remain quantitatively applicable because the system dimensions are still much larger than the molecular scale. Within this range of dimensions, important qualitatively new phenomena to be expected in nanofluidics, as compared to microfluidics, are related to the concentration polarization and concentration gradientinduced phenomena due to mechanochemical and electrochemical couplings. A number of recent studies have been devoted to electrochemical and electrokinetic properties of nanochannels in aqueous electrolyte solutions. The most relevant to the topic of this study are the direct or indirect observations of current-induced con*Corresponding author:
[email protected].
(1) Eijkel, J. C. T.; van den Berg, A. Microfluidics & Nanofluidics 2005, 1, 249. (2) Pu, Q. S.; Yun, J. S.; Temkin, H.; Liu, S. R. Nano Lett. 2004, 4, 1099. (3) Datta, A.; Gangopadhyay, S.; Temkin, H.; Pu, Q.; Liu, S. Talanta 2006, 68, 659. (4) Huang, K. D.; Yang, R. J. Electrophoresis 2008, 29, 4862. (5) Huang, K. D.; Yang, R. J. Microfluidics & Nanofluidics 2008, 5, 631. (6) Dhopeshwarkar, R.; Crooks, R. M.; Hlushkou, D.; Tallarek, U. Anal. Chem. 2008, 80, 1039.
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centration polarization,2-8 the studies of surface conductivity9,10 and streaming current,11 and nanochannel permselectivity (exclusion/enrichment effect in the authors’ terms) studied with cationic and anionic fluorescent tracers 12. It is useful to differentiate between the phenomena controlled by the “bulk” of nanochannels and those governed by the nano/micro interfaces. The former comprise the surface conductivity,9,10 streaming current,11 and exclusion/enrichment effect12 as well as several further phenomena (e.g., concentration potential or capillary osmosis; capillary osmosis, or diffusion-osmosis in the authors terms, has been studied theoretically in ref 13) that have not yet been studied experimentally with engineered nanochannels. The experimental data of this kind usually could be rather easily interpreted in terms of relatively simple models. The phenomena of the second group are much more difficult to interpret quantitatively in systems with single nanochannels due to the fact that the structure of nano/micro interfaces is essentially non-1D in this case. Because of the complexity of transport phenomena at the nano/micro interface, in the reports on the experimental and theoretical studies of concentration-polarization phenomena with engineered nanochannels, the analysis has (7) Mani, A.; Zangle, T. A.; Santiago, J. G. Langmuir 2009, 25, 3898. (8) Kim, S. J.; Wang, Y.-C.; Lee, J. H.; Jang, H.; Han, J. Phys. Rev. Lett. 2007, 99, 044501. (9) Stein, D.; Kruithof, M.; Dekker, C. Phys. Rev. Lett. 2004, 93, 035901. (10) Schoch, R. B.; Renaud, P. Appl. Phys. Lett. 2005, 86, 253111. (11) van der Heyden, F. H. J.; Stein, D.; Dekker, C. Phys. Rev. Lett. 2005, 95, 116104. (12) Plecis, A.; Schoch, R. B.; Renaud, P. Nano Lett. 2005, 5, 1147. (13) Qian, S.; Das, B.; Luo, X. J. Colloid Interface Sci. 2007, 315, 721.
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mostly been 1D both macroscopically and microscopically,6,14,15 and only in 7,16,17 the microscopic heterogeneity of ion distribution inside nanochannels was taken into account (but the model remained macroscopically 1D). In fact, in refs 16 and 17, the nano/micro interface was not considered, and the concentration-polarization phenomena were assumed to be caused by changes in the sign of surface charge along the nanochannel. This made quantitatively applicable a macroscopically 1D model. To our knowledge, only current-induced ionic concentration polarization was studied experimentally and theoretically with engineered nanochannels. We are unaware of published experimental studies of pressure-induced changes in ion concentrations at nano/micro interfaces (mechano-chemical coupling) in such systems. In nanoporous track-etched membranes (TEM), numerous identical “nano-channels” are put in parallel. Therefore, on average, the transport through their ensemble can be considered 1D. For understanding the fundamentals of polarization phenomena at nano/micro interfaces, this one-dimensionality is a clear advantage. TEM are obtained through irradiation and subsequent etching of thin polymer films. In this way, systems are obtained having numerous identical straight cylindrical pores. The pore size may range from 10-15 nm to 5 μm depending primarily on the etching conditions. Recently, Han et al.18 have noted the importance of TEMs as model objects for understanding the fundamentals of pressure-induced concentration polarization (“molecular sieving” in the authors terms) in nanofluidics. Nanoporous TEMs have been extensively studied by Bohn et al. as the layer separators in laminated multilayer microanalytical systems.19-22 The pressure-driven rejection of various salts (e.g., LiCl, NaCl, KCl, NaF, KNO3, and so on; a total of 18 various salts) and a salt mixture (KCl/LiCl) by thermally compacted nanoporous (pore diameter ca. 8-10 nm) TEMs of polyethylene terephtalate was reported in refs 23 and 24. It was found that (1:1) salts were noticeably rejected (up to ca. 60%) from moderately dilute solutions (10 mM), and the limiting rejections (at sufficiently high transmembrane volume flows) correlated with the bulk ion mobilities in quantitative agreement with the electrostatic mechanism of salt rejection. Salts with double-charge co-ions (Na2SO4, K2SO4) were rejected even more (80-85%), while a salt with double-charge counterions (CaCl2) was rejected essentially worse (ca. 20%) than the (1:1) salts,25 again in agreement with the electrostatic mechanism. The surface charge density, fitted to the salt rejection data by means of a space-charge model for (1:1) electrolytes, was found to be ca. -16 mC/m2, which corresponds to ca. 0.1 surface charges per nm2 of pore surface. We shall see below that this value is in agreement with the findings of this paper. Electrical measure(14) Hughes, B. T.; Berg, J. M.; James, D. L.; Ibraguimov, A.; Liu, S.; Temkin, H. Microfluid Nanofluid 2008, 5, 761. (15) Plecis, A.; Nanteuil, C.; Haghiri-Gosnet, A.-M.; Chen, Y. Anal. Chem. 2008, 80, 9542. (16) Daiguji, H.; Yang, P.; Majumdar, A. Nano Lett. 2004, 4, 137. (17) Daiguji, H.; Oka, Y.; Shirono, K. Nano Lett. 2005, 5, 2274. (18) Han, J.; Fu, J.; Schoch, R. B. Lab Chip 2008, 8, 23. (19) Kemery, P. J.; Steehler, J. K.; Bohn, P. W. Langmuir 1998, 14, 2884. (20) Kuo, Tzu-Chi; Sloan, L. A.; Sweedler, J. V.; Bohn, P. W. Langmuir 2001, 17, 6298. (21) Kuo, Tzu-Chi; Cannon, D. M., Jr.; Chen, Y.; Tulock, J. J.; Shannon, M. A.; Sweedler, J. V.; Bohn, P. W. Anal. Chem. 2003, 75, 1861. (22) Cannon, D. M., Jr.; Kuo, Tzu-Chi; Bohn, P. W.; Sweedler, J. V. Anal. Chem. 2003, 75, 2224. (23) Yaroshchuk, A. E.; Dukhin, S. S. J. Membr. Sci. 1993, 79, 133. (24) Yaroshchuk, A. E. Adv. Colloid & Interface Sci. 1995, 60, 1. (25) Nechaev, A. N. 1993, personal communication.
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ments were not carried out in refs 23 and 24. Besides that, all the measurements of rejection of salts were carried out at a single transmembrane pressure (5 MPa). In the context of nanoporous media, the space-charge model was introduced by Dresner,26 further developed by Osterle,27-30 and extensively used for the description of ion and solvent transport through microporous charged membranes, namely, osmosis,31-34 ion diffusion,31,34-37 membrane (diffusion) potential,38-40 electrokinetic phenomena, and surface conductivity.41-50 The pressure-driven salt rejection was theoretically studied in refs 23, 24, 34, and 51-53. The socalled electroviscosity was studied in refs 42,50, and 51. The finite ion size was accounted for in ref 54. Various approximate approaches to the description of diffuse parts in the double electric layers structure in fine pores have been developed in many papers. A brief summary of these approaches can be found, for example, in ref 55 where the author developed an approximate analytical approach to the use of the space-charge model in nanochannels with slightly overlapped diffuse parts of double electric layers. In the context of nanofluidics, the model (sometimes also called PoissonBoltzmann model), has been recently used, for example, in the description of surface conductance9 and steaming current11 and in the analysis of capillary electrophoresis and chromatography in nanochannels.56-58 Despite this considerable effort, the model has not been used in the past for a quantitative interpretation of simultaneous measurements of salt-rejection, electrokinetic and electrochemical phenomena in well-defined nanopores in an unsupported membrane, where all these phenomena are controlled by the same (26) Dresner, L. J. Phys. Chem. 1963, 67, 1365. (27) Morrison, F. A.; Osterle, J. F. J. Chem. Phys. 1965, 43, 2111. (28) Gross, R. J.; Osterle, J. F. J. Chem. Phys. 1968, 49, 228. (29) Fair, J. C.; Osterle, J. F. J. Chem. Phys. 1971, 54, 3307. (30) Osterle, J. F.; Pechersky, M. J. Phys. Chem. 1971, 75, 3015. (31) Selegny, E.; Boyd, J. D.; Gregor, H. P. In Charged Gels and Membranes, Part 1; Springer: New York, 1976, p 255. (32) Sasidhar, V.; Ruckenstein, E. J. Colloid Interface Sci. 1981, 82, 439. (33) Sasidhar, V.; Ruckenstein, E. J. Colloid Interface Sci. 1982, 85, 332. (34) Hijnen, H. J. M.; van Daalen, J.; Smit, J. A. M. J. Colloid Interface Sci. 1985, 107, 525. (35) Martinez, L.; Hernandez, A.; Tejerina, F. Sep. Sci. Technol. 1988, 23, 243. (36) Mafe, S.; Manzanares, J. A.; Pellicer, J. J. Membr. Sci. 1990, 51, 161. (37) Hijnen, H. J. M.; Smit, J. A. M. J. Membr. Sci. 1995, 99, 285. (38) Hijnen, H. J. M.; Smit, J. A. M. J. Membr. Sci. 2000, 168, 259. (39) Fievet, P.; Aoubiza, B.; Szymczyk, A.; Pagetti, J. J. Membr. Sci. 1999, 160, 267. (40) Westermann-Clark, G. B.; Christoforou, C. C. J. Electroanal. Chem. 1986, 198, 213. (41) Koh, W. H.; Anderson, J. L. AIChE J. 1975, 21, 1176. (42) Anderson, J. L.; Koh, W. H. J. Colloid Interface Sci. 1977, 59, 149. (43) Koh, W. H. J. Colloid Interface Sci. 1979, 71, 613. (44) Westermann-Clark, G. B.; Anderson, J. L. J. Electrochem. Soc. 1983, 130, 839. (45) Szymczyk, A.; Fievet, P.; Aoubiza, B.; Simon, C.; Pagetti, J. J. Membr. Sci. 1999, 161, 275. (46) Szymczyk, A.; Aoubiza, B.; Fievet, P.; Pagetti, J. J. Colloid Interface Sci. 1999, 216, 285. (47) Labbez, C.; Fievet, P.; Szymczyk, A.; Aoubiza, B.; Vidonne, A.; Pagetti, J. J. Membr. Sci. 2001, 184, 79. (48) Fievet, P.; Szymczyk, A.; Labbez, C.; Aoubiza, B.; Simon, C.; Foissy, A.; Pagetti, J. J. Colloid Interface Sci. 2001, 235, 383. (49) Szymczyk, A.; Fievet, P.; Aoubiza, B. Desalination 2003, 151, 177. (50) Sbai, M.; Fievet, P.; Szymczyk, A.; Aoubiza, B.; Vidonne, A.; Foissy, A. J. Membr. Sci. 2003, 215, 1. (51) Neogi, P.; Ruckenstein, E. J. Colloid Interface Sci. 1981, 79, 159. (52) Smit, J. A. M. J. Colloid Interface Sci. 1989, 132, 413. (53) Yaroshchuk, A. E. J. Membr. Sci. 2000, 167, 163. (54) Cervera, J.; Garcia-Morales, V.; Pellicer, J. J. Phys. Chem. B 2003, 107, 8300. (55) Petsev, D. N. J. Chem. Phys. 2005, 123, 244907. (56) Pennathur, S.; Santiago, J. G. Anal. Chem. 2005, 77, 6772. (57) Xuan, X. Electrophoresis 2008, 29 3737. (58) Baldessari, F.; Santiago, J. G. J. Colloid Interface Sci. 2008, 325, 526.
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nanoporous medium, and consequently, can be interpreted by using a single set of model parameters. In ref 50, where a monolayer (ceramic) membrane was also used and an integrated set of measurements (streaming potential, membrane potential, electric conductivity, and hydraulic permeability) were carried out, the salt rejection was probably too low (due to the larger pore size of ca. 55 nm) to be reliably measurable, even in the most dilute solutions used. Besides that, the pore geometry in the membrane prepared via sintering of a ceramic powder evidently was much more complex than in TEMs, which made the outcome of modeling, assuming a simple pore geometry (identical straight cylindrical capillaries), less reliable. Information on the electrochemical and electrokinetic properties of TEMs found in the literature is rather limited. In most cases, only the streaming potential was measured.59-63 We are aware of only one attempt to carry out integrated measurements of streaming potential and membrane electric conductivity with them.62 The phenomenon of electroviscosity with TEMs was studied in ref 61. Quite recently Zhou et al. have studied the current-induced concentration polarization of a single pore of a TEM.64 In ref 65, the electrochemical coupling (current-induced concentration polarization) in nanoporous TEMs was studied by the current switch-off technique. In this way, one could obtain information on the electrochemical perm-selectivity and diffusion permeability of these membranes. The principal purpose of this paper is to study the mechanochemical coupling (pressure-driven concentration-polarization phenomenon) and its effects on the electro-mechanical coupling (electrokinetics). In parallel, we shall check the validity of spacecharge model and show it to be applicable to the description of nanofluidics in aqueous electrolyte solutions in ca. 20 nm pores (at screening lengths up to ca. 6 nm). It is worth mentioning that, recently, doubts have been expressed concerning the applicability of this model to the description of nanochannels with overlapped diffuse parts of double electric layers.66-70 This issue has been addressed theoretically by Baldessari and Santiago,58 who came to the conclusion, that the space-charge model (or the PoissonBoltzmann approach in the authors terms) is applicable to DELoverlapped nanochannels provided that the center-line ion concentrations are properly defined, proceeding from the thermodynamic equilibrium with reservoirs. In this context, an additional experimental verification of the model applicability to nanopores, of several tens of nanometers in diameter, appears to be useful and timely. The pressure switch-off technique was introduced and described in detail in ref 71, where it was used for the studies of electrochemical and electrokinetic properties of an active layer of a nanoporous ceramic membrane. The measurements of steady(59) Kim, K. J.; Stevens, P. V. J. Membr. Sci. 1997, 123, 303. (60) Brendler, E.; Ratkje, S. K.; Hertz, H. G. Electrochim. Acta 1996, 41, 169. (61) Huisman, I. H.; Pradanos, P.; Calvo, J. I.; Hernandez, A. J. Membr. Sci. 2000, 178, 79. (62) Berezkin, V. V.; Kiseleva, O. A.; Nechaev, A. N.; Sobolev, V. D.; Churaev, N. V. Colloid J. 1994, 56, 258. (63) Dejardin, P.; Vasina, E. N.; Berezkin, V. V.; Sobolev, V. D.; Volkov, V. I. Langmuir 2005, 21, 4680. (64) Zhou, K.; Kovarik, M. L.; Jacobson, S. C. J. Am. Chem. Soc. 2008, 130, 8614. (65) Yaroshchuk, A. E.; Zhukova, O. V.; Ulbricht, M.; Ribitsch, V. Langmuir 2005, 21, 6872. (66) Taylor, J.; Ren, C. L. Microfluid Nanofluid 2005, 1, 356. (67) Ren, C. L.; Li, D. Anal. Chim. Acta 2005, 531, 15. (68) Conlisk, A. T.; McFerran, J.; Zheng, Z.; Hansford, D. Anal. Chem. 2002, 74, 2139. (69) Zheng, Z.; Hansford, D. J.; Conlisk, A. T. Electrophoresis 2003, 24, 3006. (70) Choi, Y. S.; Kim, S. J. J. Colloid Interface Sci. 2009, 333, 672. (71) Yaroshchuk, A. E.; Boiko, Yu.P.; Makovetskiy, A. L. Langmuir 2005, 21, 7680.
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state salt rejection and transient filtration potential after pressure switch-off enable one to determine the membrane, electrokinetic and electrochemical properties in a single series of measurements. This is an important advantage because of the considerable sampleto-sample variability of properties of nanoporous TEMs. Besides that, the availability of information on several thermodynamically independent properties79 presents a challenge for the modeling. Indeed, it is relatively easy to fit the results of measurements of a single property, especially by using models containing several adjustable parameters. Fitting several sets of experimental data, obtained in parallel by using a single model, is much more difficult and, if successful, confirms the validity of the model.
Theoretical Section Macroscopic Transport Equations. Our theoretical analysis will be based on the continuous version of irreversible thermodynamics. In the approximation of no direct coupling between ionic flows80 (see ref 24 for discussion of the scope of its applicability), the system of transport equations for the ion flows can be written in this way: ! ðeÞ Pi dμi ð1Þ þ Jv τi ji ¼ ci RT dx where Pi is the membrane permeability with respect to i-th ion, μ(e) i is its electrochemical potential, x is the dimensionless transmembrane macroscopic coordinate (scaled on the membrane thickness, L), JV is the transmembrane volume flow density, ci is the local virtual (or corresponding) concentration of i-th ion,24 and τi is the ion transmission coefficient at zero voltage (one minus corresponding reflection coefficient), which is defined as the proportionality coefficient between the ion flux and the volume flux at zero differences of electrochemical potentials for all the solutes.24 Under filtration conditions, the transmembrane electric current is usually zero, which means that X Zi ji ¼ 0 ð2Þ i
where Zi are ion charges (in elementary charge units). By applying this condition to eq 1 and taking into account the definition of electrochemical potential: ðeÞ
ðcÞ
μi μi
þ FZi φ
ð3Þ
where μ(c) i is the ion chemical potential, F is the Faraday constant, and φ is the electric potential in virtual solution (see ref 24 for the definition of virtual quantities); we obtain this for the local virtual electric field: dφ 1 X ti dμi F ¼ þ Jv dx F i Zi dx g ðcÞ
ð4Þ
where ti is the ion transport number in the membrane phase defined as Z 2 ci Pi ti P i 2 j Zj c j P j
ð5Þ
F is the electrokinetic charge density: FF
X
Zi c i τ i
ð6Þ
i
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g is the membrane electric conductivity per unit area: g
F2 X 2 Z ci Pi RT i i
ð7Þ
In binary electrolytes, there is just one independent equation for the salt transfer, which can be written in this way: js ¼ -Ps ðcs Þ
dcs þ Jv cs Τs ðcs Þ dx
where, in the second term on the right-hand side, we have transformed to the integration over the virtual concentration by using eq 8. The first term on the right-hand side of eq 15 corresponds to the membrane potential, and the second term describes the streaming potential. In binary electrolytes, the definitions of electrokinetic charge density and membrane electric conductivity per unit area (eqs 6 and 7) take this form:
ð8Þ
where cs is the virtual salt concentration; the local diffusion permeability of membrane to salt is defined in this way: g
νþ þ νPs ðcs Þ ν þ νP þ ðcs Þ þ P - ðcs Þ
ð9Þ
where ν+ and ν- are the stoichiometric coefficients, and the local salt transmission coefficient is defined as Τs ðcs Þ τ þ ðcs Þt - ðcs Þ þ τ þ ðcs Þt - ðcs Þ
ð10Þ
Under conditions of stationary filtration: js ¼ cs 00 Jv
ð11Þ
where c00s is the salt concentration in the permeate. With eq11, for a macroscopically homogeneous membrane, eq can be solved in quadratures to yield Zcs 0 Jv ¼ cs 00
Ps ðcs Þdcs cs 00 -cs Τs ðcs Þ
ð16Þ
F2 Z þ ν þ cs ðZ þ P þ -Z - P - Þ RT
ð17Þ
Below, we shall see that in our experiments, the so-called external concentration polarization, near the upstream membrane surface, could not be completely eliminated despite vigorous stirring. Therefore, it has to be accounted for in the modeling. Within the scope of the film model for the description of the external concentration polarization, one considers the unstirred layer as a membrane with special properties, namely zero electrokinetic charge density and a salt transmission coefficient equal to 1. Besides that, one can take into account that, within the unstirred layer, the salt diffusion coefficient only slightly depends on the salt concentration and can be considered constant. By doing so, the salt transport equation within the unstirred layer can be integrated to obtain Rint Jv δ ¼ exp Ds Robs
ð12Þ
where c0s is the salt concentration at the upstream membrane surface. This is a transcendental equation for the determination of salt concentration in the permeate, c00s . It can also be used to calculate explicitly the transmembrane volume flow as a function of permeate concentration. In binary electrolytes, eq 4 reduces to this
F FðZ þ ν þ Þcs ðτ þ -τ - Þ
ð18Þ
where δ is the thickness of unstirred layer, Ds is the salt diffusion coefficient within it, and Rint is the intrinsic salt rejection defined as Rint 1 -
cs 00 cs 0
ð19Þ
Robs is the observable salt rejection defined as dφ θs ðcs Þ dμs Fðcs Þ ¼ þ Jv dx F dx gðcs Þ
ð13Þ
Robs 1 -
cs 00 csf
ð20Þ
where t þ ðcs Þ t - ðcs Þ þ ð14Þ Zþ ZFor a macroscopically homogeneous membrane, eq 13 can also be integrated to obtain this for the transmembrane electric potential difference:72 θs ðcs Þ
RT Δφ φ0 -φ00 ¼ F RT F
Zcs 0 cs 00
Zcs 0 cs 00
θs ðcs Þ dcs -Jv cs
θs ðcs Þ dcs cs
Zcs 0 cs 00
Z1 0
Fðcs ðxÞÞ dx gðcs ðxÞÞ
where csf is the feed salt concentration. Since the permeate concentration is known, to estimate the salt concentration at the upstream membrane surface, c0s, it is sufficient to know (or to fit to experimental data) the thickness of unstirred layer, δ, or its diffusion permeability, Ds/δ. For the unstirred layer, we can also use eq 13, and by taking into account that the electrokinetic charge is equal to zero and the ion transport numbers can be considered reasonably constant for the electric potential difference across the unstirred layer, we obtain
Ps ðcs ÞFðcs Þ=gðcs Þ dcs cs 00 -cs Τs ðcs Þ ð15Þ
Δφδ φf -φ0 ¼
RT csf θsδ ln 0 F cs
ð21Þ
where ðbÞ
(72) Yaroshchuk, A. E.; Boiko, Yu.P.; Makovetskiy, A. L. Langmuir 2002, 18, 5154.
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θsδ
tþ tðbÞ þ Zþ Z-
ð22Þ
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t(b) ( are the ion transport numbers in the unstirred layer (assumed to be identical to those in bulk electrolyte solution). Since the indicator electrodes are practically always located outside the (quite thin) unstirred layers, the contribution given by eq 21 has to be added to the transmembrane electric potential difference defined by eq 15. However, in our experiments we used KCl solutions where θsδ ≈ 0. Therefore, in the subsequent analysis, the electric potential difference across the unstirred layer will be neglected. In our experiments, we used bare Ag/AgCl electrodes to measure the electric potential difference. The electric potential of such electrodes depends on the concentration of chloride ions. Therefore, in the interpretation, we have to account for the difference in the electrode potentials described by this equation Δφe φe 0 -φe 00 ¼ -
RT asf ln 00 F as
ð23Þ
where asf and as00 are the salt activities in the feed and permeate solutions. Since the salt concentrations in these solutions are known from experiment, the corresponding salt activities can be calculated in a model independent way by using experimental data on the activity coefficients. Specification of Membrane Transport Properties. Until now, we have not specified the microscopic model for the description of membrane transport properties appearing in eqs 12 and 15. In our experiments, we used nanoporous TEMs having identical straight cylindrical pores. Therefore, for the modeling, we shall use the capillary model. If the length of a capillary is much larger than its diameter, then the ion and volume flows can be considered 1D far away from the capillary edges. This simplifies the mathematics considerably and enables one to derive general expressions for the membrane phenomenological coefficients in terms of surface forces without specifying their nature. Although often just capillaries with a circular cross-section are considered, in ref 24 it has been shown that general expressions for a capillary of an arbitrary cross-section can also be derived. In the approximation of no direct coupling between the ion flows for an ideal solution, they look this way (the membrane ionic permeabilities, Pi, here are defined at zero transmembrane volume flow): Pi ¼
f ÆDi Γi æ L
ð24Þ
In the particular case of circular cross-section, eqs 24 and 25 read this way: Pi ¼
4 τi ¼ 4 Rp
ÆΓi F½1æ ÆF½1æ
ð25Þ
where Di is the ion diffusion coefficient inside the pores, Γi is the ion distribution coefficient, F [] is a linear operator transforming the right-hand side of Stokes equation into its solution, L is the membrane thickness, and the brackets < > mean averaging over the pore cross-section. The finite membrane porosity is accounted for by the factor f in eq 24. Notably, while using eq 24, we neglect the so-called osmotic corrections to the ionic permeabilities defined at zero transmembrane volume flow. In ref 23, it was shown that, in not too large pores (as compared to the screening length), these corrections are quite small. Most probably, they are essentially smaller than the typical experimental error of our measurements. Langmuir 2009, 25(16), 9605–9614
ZRp drrDi ðrÞΓi ðrÞ
ð26Þ
drrðR2p -r2 ÞΓi ðrÞ
ð27Þ
0
ZRp 0
where Rp is the pore radius. From eqs 24 and 25, it is seen that the phenomenological coefficients are essentially controlled by the ion distribution coefficients. To obtain the latter, we shall use the space-charge model within the scope of the Poisson-Boltzmann approach. It is assumed that the quasi-equilibrium distributions of ions, within the diffuse parts of double electric layers, are determined solely by the quasi-equilibrium electrostatic potential, so Γi ðrÞ ¼ expð -Zi ψðrÞÞ
ð28Þ
which follows from the local equilibrium (constant electrochemical potentials over the pore cross-section) in the approximation of an ideal solution. In eq 28, we have introduced the dimensionless quasi-equilibrium electrostatic potential according to this ψ Fψ~/RT, where ψ~ is the dimensional electrostatic potential. By substituting eq 28 into the Poisson equation for the cylindrical pore geometry, (1:1) electrolytes and the radial space variable scaled on the pore radius, we obtain the PoissonBoltzmann equation in this form: 1 ψ00 þ ψ0 ¼ λ2 sinh ψ r
ð29Þ
where λ κRp, and the reciprocal Debye (screening) length, κ, is defined as sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2F 2 I K RTεε0
ð30Þ
I is the solution ionic strength, εε0 is the dielectric permittivity of the solvent. The boundary conditions to eq 28 are ψ0 ð0Þ ¼ 0;
τi ¼
2f R2p L
ψ0 ðRp Þ ¼ -
σF εε0 RT
ð31Þ
where σ is the surface charge density. Equation 29 does not have solution in quadratures. There are approximate analytical solutions in several limiting cases (low and high potential, small potential variation in radial direction, weak overlap of diffuse parts of double electric layers, see ref 55 for a brief overview). None of them is applicable to the case of moderately overlapped diffuse parts of double electric layers and rather strongly charged surfaces, characteristic of nanoporous TEMs and dilute electrolyte solutions used in this study. Therefore, we integrated eq 29 and calculated the phenomenological coefficients (as functions of salt concentration in the reference solution) by means of eqs 26 and 27 numerically. These concentration-dependent phenomenological coefficients were used to model the salt rejection as well as the electrokinetic and electrochemical phenomena by means of eqs 12, 15, and 18-20. DOI: 10.1021/la900737q
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Figure 2. Steady-state rejections of KCl against transmembrane volume flow and their theoretical fits.
Figure 1. Schematics of the experimental setup.
For the estimates of pore size from the membrane hydraulic permeability, we shall use the Hagen-Poiseuille equation in this form: Jv ¼
πR4p np ΔP 8ηL
ð32Þ
where ΔP is the hydrostatic pressure difference, η is the dynamic water viscosity (assumed to be equal to the viscosity of bulk water), np is the pore density (number of pores per unit area). With this, we disregard the phenomenon of electroviscosity.42,49,61 The rational for this is discussed below.
Experimental Section Methods. The batch cell-type experimental setup used in this study and schematically shown in Figure 1 was the same as described in ref 71. The stirrer is driven by an external motor via magnetic coupling. Stirring rates of up to 1200 rpm can be achieved. The pressure difference is created by pressurized argon through a specially designed transducer without direct contact between the gas and the solution (to avoid the gas dissolution). The pressure switch-off is implemented with a 3way 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 piezo resistive pressure transducer (Media Nugget, SSI Technologies, Inc.). The TEM was supported by a 2 mm thick layer of porous polyfluorethylene. The pure water permeability of this material is 7.10-3 m/(s 3 MPa), which is more than 2 orders of magnitude higher than that of nanoporous TEM (ca. 310-5 m/(s 3 MPa). Therefore, the relative pressure drop across this support layer is very small. At the same zeta potential of the pore surface, the contribution of a layer in a composite structure to streaming potential is proportional to the hydrostatic pressure drop across this layer.73 Therefore, the contribution of support layer to the measured filtration potential could be neglected. To improve the homogeneity of distribution of volume flow through the TEM,81 we put a high-porosity coarse porous Millipore filter between the membrane and the support. Its pure water permeability is ca. 710-1 m/(s 3 MPa), and its contribution to the filtration potential is also obviously negligible. The electric potential difference was measured by a pair of bare Ag/AgCl electrodes. The data acquisition was performed by a personal computer with aid of dedicated software. The rejection of KCl was determined from the salt concentrations in the feed, and permeate solutions deduced from their electric conductivities measured in a thermostatted conductivity cell. (73) Labbez, C.; Fievet, P.; Szymczyk, A.; Aoubiza, B.; Vidonne, A.; Pagetti, J. J. Membr. Sci. 2001, 184, 79.
9610 DOI: 10.1021/la900737q
Materials. The membrane was a commercially available RoTrac (polyethylene terephthalate) TEM (Oxyphen, Switzerland) with the nominal pore size (see below for the discussion on the actual pore size, which was experimentally found to be 20.8 nm) of 30 nm, the thickness of 8 μm, and the track density of 7 1013 m-2 (according to the manufacturer). The mixed cellulose ether filters (SSWP) that were supplied by Millipore had a nominal pore size of 3 μm, porosity of ca. 80% and wet thickness of 145 μm. KCl solutions of two concentrations (2.5 mM and 10 mM) and two pH values (pH 5.7 and pH 7.5) were used.
Results and Discussion Figure 2 shows the rejection of salt from KCl solutions, as a function of transmembrane volume flow. In more dilute solutions, the rejection is about 40-50%, and it goes down to ca. 1015% with an increase in the salt concentration. This is in agreement with the well-known decrease in the thickness of diffuse parts of double electric layers with growing electrolyte ionic strength. From Figure 2, it is also seen that the rejections first increase, pass through maxima at a moderate transmembrane volume flow, and gradually decline with its further increase. This is caused by the accumulation of the rejected solute near the upstream membrane surface (external concentration polarization) due to the existence (despite vigorous stirring) of a stagnant layer near the membrane surface, where the back transport of solute occurs predominantly via diffusion. With increasing transmembrane volume flow, this solute accumulation becomes more pronounced, and the back diffusion is less capable of keeping the concentration at the membrane surface close to the feed one. This is a universal phenomenon in pressure-driven membrane processes. Nevertheless, in the case of conventional reverse osmosis or nanofiltration membranes, the drop in the rejection at higher transmembrane volume flows is often (over)compensated by the increase in the so-called intrinsic rejection (defined with the solute concentration at the membrane surface instead of the feed one). According to ref 74, with the pore size of ca. 20 nm, the intrinsic rejection ceases to increase with the transmembrane pressure at ΔP ≈ 0.2 MPa. At the hydraulic permeability of ca. 30 μm/(s 3 MPa) (see below), this corresponds to the transmembrane volume flow of ca. 6 μm/s, which is just the approximate location of maxima in Figure 2. The dependences of transmembrane volume flow on the applied hydrostatic pressure difference were strictly linear (in dilute electrolyte solutions used in this study, the contribution of osmotic corrections is negligible) and passed through the origin, and their slopes (ca. 30-32μm/(s 3 MPa) only slightly (74) Yaroshchuk, A. E. J. Membr. Sci. 2000, 167, 149.
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Figure 3. Transmembrane + electrode potential against transmembrane volume flow.
depended on the feed composition. In particular, at pH 5.7, the permeability was slightly lower (29.6 μm/(s 3 MPa) in the case of 10 mM solution than with the 2.5 mM solution (31.7μm/(s 3 MPa). This difference can be explained by the difference in the temperatures (290 K for the 2.5 mM and 287 K for the 10 mM measurements) and, consequently, in the water viscosity. Besides that, we observed a somewhat lower hydraulic permeability (30.5μm/(s 3 MPa) at pH 7.5 than at pH 5.7 in 2.5 mM solutions. A part of this dependence on the feed pH can probably be explained by the electroviscosity, which is caused by electro-osmosis in the field of streaming potential.42,61 With TEMs due to the known pore density, the variance in the pore size, estimated from the hydraulic permeability by means of eq 32, is just one-fourth of the variance in the permeability, that is ca. 1.5%. Nevertheless, the electroviscosity gives rise to a systematic correction to the permeability, which does not strongly depend on the solution concentration and pH within the studied range of these parameters. However, it is known that this phenomenon usually brings about only relatively small corrections (10-20 %)82 to the hydraulic permeability.23 As discussed above, due to the known pore density, the corresponding correction to the pore size is just one-fourth of the correction to the hydraulic permeability. This small correction was taken into account in this way. First, the pore size was estimated from the hydraulic permeability by neglecting the electroviscosity. With this approximate pore size, the surface charge densities were estimated by fitting the salt rejection data as described below. With the information on the pore size and surface charge density available, we could estimate the extent of electroviscosity, which was found to be ca. 15%. This was used to correct the estimate of the pore size (which increased approximately 4%) and the surface charge density. The electroviscosity was estimated again for the refined values of pore size and surface charge density, but its magnitude practically did not change anymore. The refined pore diameter estimated in this way is 20.8 ( 0.2 nm, which is in some disagreement with the nominal pore size (30 nm) indicated by the membrane manufacturer but in a reasonable agreement with the pore size (24 ( 4 nm) of the same membrane, estimated in65 by means of gas-liquid perm-porometry, and supported by the successful interpretation of results of current switch-off measurements by using this pore size. Figure 3 shows the measured stationary transmembrane electric potential difference as a function of transmembrane volume flow. The dependences are clearly sublinear (especially in the case of more dilute solutions). The nonlinearity is caused by the rejection of salt and the related appearance of membrane and electrode potential differences. Similar nonlinearity was observed Langmuir 2009, 25(16), 9605–9614
Figure 4. Time transients of transmembrane hydrostatic pressure difference.
in the behavior of filtration potential across the barrier layer of nanoporous ceramic membrane. 72 However, in that study, the sublinear pattern originating from the nanoporous layer was masked by the superlinear pattern related to the microporous membrane support under conditions of salt rejection by the nanoporous layer. Accordingly, a special subtraction procedure had to be applied to single out the sublinear pattern characteristic of nanoporous systems. This kind of sublinear behavior was also predicted theoretically in ref 75. In their nonlinearity, our results are in disagreement with the findings of,63 where a linear dependence of transmembrane potential on the applied hydrostatic pressure difference was reported for a similar membrane (polyethylene terephthalate) TEMs with the pore size of 17 nm) and feed solution (10 mM KCl, pH 6.5). A possible explanation is that in ref 63 the pressure difference was applied just for short periods of time, and the concentration difference across the membrane did not have time to develop. In the pressure switch-off technique, it is very important to have as quick a relaxation of transmembrane hydrostatic pressure difference as possible. Otherwise, a noticeable part of relaxation of transient membrane potential occurs before the streaming potential component of filtration potential disappears completely. This makes less direct the procedure of splitting of filtration potential into the streaming and membrane potential components because a theoretical extrapolation to the initial state is needed. Figure 4 shows a series of transients of transmembrane hydrostatic pressure difference. It decays practically to zero within as short as 7-10 ms, whatever the applied stationary pressure difference. Very similar results were obtained in other series of measurements. Figure 5 shows a set of measurements of time transients of transmembrane electric potential differences corresponding to the pressure switch-offs shown in Figure 4. Initially, there is a very rapid change in the transmembrane potential synchronous to the change in the transmembrane pressure difference. However, in contrast to the latter, the transmembrane potential difference does not drop to zero when the hydrostatic pressure difference disappears completely. The remaining transmembrane potential is the sum of electrode and transient membrane potential. The latter approaches the initial membrane potential (being a (75) Fievet, P.; Sbai, M.; Szymczyk, A. J. Membr. Sci. 2005, 264, 1.
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Figure 5. Time transients of transmembrane + electrode electric potential difference corresponding to the pressure transients shown in Figure 4. Stationary pressure differences (in MPa) are indicated on the graph.
component of stationary filtration potential) at very short times immediately after the pressure switch-off. The electrode potential remains constant throughout the measurement provided that the latter is not excessively long.83 If the characteristic time of relaxation of transient membrane potential is much longer than the characteristic time of relaxation of transmembrane pressure difference, then the initial value of transient membrane potential can be directly observed experimentally. In refs 76 and 77, it was shown that for the system of interest (membrane flanked by a coarse porous filter from one side and a bulk electrolyte solution from the other) the characteristic relaxation time of transient membrane potential is given by this relationship trel
4Db 2 Pm
! -2 1 1 þ pffiffiffiffiffiffiffiffiffiffi γs Θs
ð33Þ
where Pm is the diffusion permeability of membrane, Db is the bulk diffusion coefficient of salt, γs is the porosity of support, and Θs is the salt diffusivity reduction factor in the support (including the effects of finite porosity and pore tortuosity and defined as the ratio of effective salt diffusion coefficient in the support to that in the bulk solution). All these parameters of TEMs and Millipore filter used in this study were determined in ref 65. The membrane diffusion permeability was found to be 3.1 μm/s in a 2.5 mM KCl and 5.7 μm/s in a 10 mM KCl solution. The Millipore filter porosity was γs ≈ 0.8 and the diffusivity reduction factor was Θs ≈ 0.5. By using these data, we can estimate the characteristic time of relaxation of transient membrane potential to be ca. 110 s in the 2.5 mM solution and ca. 30 s in the 10 mM solution. This, indeed, is much longer than the 7-10 ms characteristic of the relaxation of transmembrane pressure difference. Nevertheless, from Figure 5, it is seen that after the complete decay of transmembrane pressure difference the transmembrane potential still exhibits a sort of decaying irregular fluctuations, making difficult its precise determination exactly at the point where the transmembrane pressure is already zero. These (76) Yaroshchuk, A. E.; Makovetskiy, A. L.; Boiko, Yu.P.; Galinker, E. W. J. Membr. Sci. 2000, 172, 203. (77) Yaroshchuk, A. E.; Ribitsch, V. Chem. Eng. J. 2000, 80, 203.
9612 DOI: 10.1021/la900737q
Figure 6. Experimentally determined components of stationary transmembrane electric potential difference and their theoretical fits against transmembrane volume flow: SP is the streaming potential and MP is the initial membrane potential.
fluctuations faded out completely within ca. 100 ms following the pressure switch-off in the 2.5 mM solution and within ca. 30 ms in the 10 mM solution.84 By using the model developed in refs 76 and 77, it can be shown that due to the very broad dispersion of this relaxation process, even at times ca. 3 orders of magnitude shorter than the characteristic relaxation time, the magnitude of transient membrane potential is already ca. 3-4% smaller than its initial value. Therefore, one should keep in mind that the absolute values of initial transient membrane potential estimated below by using the immediately “post-fluctuation” data may be underestimated by ca. 3-4%. Figure 6 shows the streaming and membrane components of filtration potential (separated by means of pressure switch-off) as functions of transmembrane volume flow. The electrode potential (defined by eq 23) was subtracted by using experimental data on the feed and permeate concentrations and the empirical equation for the activity coefficient in KCl solutions. Langmuir 2009, 25(16), 9605–9614
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Article Table 1. Fitted Properties of Membrane and Unstirred Layer
feed composition 2.5 mM KCl, pH 5.7 2.5 mM KCl, pH 7.5 10 mM KCl, pH 5.7
conc. of surface charges (nm-2) (this study)65
conc. of surface charges (nm-2)34
thickness of unstirred layer (μm)
0.09 0.10 0.16
0.07 n.d. 0.16
62 46 84
The streaming potential component is a largely linear function of transmembrane volume flow, while the membrane potential component is strongly nonlinear. This confirms our assumption that the concentration polarization is the principal cause of nonlinearity of dependences of stationary transmembrane potential difference on the transmembrane volume flow. Interpretation within the Scope of Space-Charge Model. The pore size of 20.8 ( 0.2 nm has already been estimated above from the membrane hydraulic permeability. The salt rejection data shown in Figure 1 were fitted by using the model described in the Theoretical Section (the space-charge model plus continuous version of irreversible thermodynamics for the description of ion transfer inside the membrane pores and the Nernst model for the stagnant layer) and the value of pore size estimated from the hydraulic permeability. In this way, we could estimate the membrane surface charge density and the diffusion permeability of the stagnant layer. The obtained parameters are listed in Table 1. It is seen that the surface charge density noticeably increased with the feed salt concentration and also somewhat increased with the feed pH value. Both trends correspond to the mechanism of surface charge formation due to the dissociation of weakly acidic surface groups.38,65 The obtained concentrations of surface charges are in very good agreement with the results obtained in ref 65 for the same membrane by means of current switch-off technique and also shown in Table 1 for comparison. Besides that, our data are in agreement with the results reported in refs 23 and 24, where a TEM of the same chemistry but smaller pore size (ca. 8 nm, obtained via a thermal compaction) was used. In fact, the surface charge concentration estimated in the present study for the 10 mM solution is ca. 50% higher than in refs 23 and 24. However, it can be expected that in smaller pores the degree of dissociation of surface groups is somewhat smaller due to the better overlap of diffuse parts of double electric layers. On the other hand, rather unexpectedly, the fitted thickness of stagnant layer turns out to be not the same in different series of measurements. This, probably, was caused by the fact that in our experimental setup the distance between the membrane and the stirrer disk was not fixed precisely. Therefore, the effective thickness of unstirred layer could vary from one series of measurements to another. The fitted values of surface charge density and thickness of unstirred layer were used to calculate theoretically (“reproduce”) the streaming potential and stationary membrane potential (assumed to be equal to the initial membrane potential immediately after the pressure switch-off) and to compare them with the corresponding experimental data. The results are shown in Figure 6. The “reproduced” values of initial membrane potential were obtained in two ways. First, it was assumed that the initial membrane potential was controlled by the salt concentration at the membrane surface, which was higher than the feed concentration due to the concentration polarization. From Figure 6, it is seen that in the more dilute solution, the “reproduced” values obtained in this way turn out noticeably larger (up to ca. 50%) than the experimentally observed ones. Second, it was assumed that the initial membrane potential was controlled by the feed salt concentration. Figure 6 shows that in the more dilute Langmuir 2009, 25(16), 9605–9614
solution, this approach leads to considerable improvements in the agreement between the “reproduced” and the measured values of initial membrane potential. An explanation for this could be that immediately after the pressure switch-off, the near-membrane stagnant layer with increased salt concentration is “shakenoff” due to a rapid membrane movement caused by the mechanical relaxation of the cell part supporting the membrane. Due to that, the near-membrane solution zone is vigorously stirred, and the salt concentration at the membrane surface acquires a closeto-feed value. In the case of more concentrated solution, the use of feed concentration in the procedure of “reproduction” of membrane potential did not really improve the agreement with the experiment. Although we do not know the reason for this for the moment, it can be noted that the magnitude of membrane potential in this case is fairly small, which means potentially larger relative errors, for example, due to the potentials of electrode asymmetry. As is seen from Figure 6, the relative difference between the measured and “reproduced” values of streaming potential always does not exceed 10% and is typically smaller. From Table 1, it is seen that the surface charge density depends on the salt concentration and pH value. However, we do not have enough data to be able to reproduce these dependences in a quantitative way useful for the modeling. Therefore, in the fits of rejection data as well as in the process of “reproduction” of results of electrical measurements, these dependences were disregarded despite the fact that the salt concentration did change across the membrane. Non-negligible pH changes across membrane could also occur.78 In view of these approximations, the accuracy of “reproduction” of results of electrical measurements appears to be quite good. The fact that we have managed to describe the volume transfercontrolled (streaming potential), the diffusion-controlled (membrane potential) and “mixed” (salt rejection) properties within the scope of a single (space-charge) model by using single values of (concentration- and pH-dependent) surface charge densities confirms the quantitative applicability of this model to the description of dilute electrolyte solutions in nanopores of as little as ca. 20 nm in diameter. (78) Yaroshchuk, A. E.; Vovkogon, Yu.A. J. Colloid Interface Sci. 1995, 172, 324. (79) Within the scope of irreversible thermodynamics, thermodynamically independent properties are quantified by different coefficients in the Onsager matrix of phenomenological coefficients. (80) In this approximation, it is assumed that the flux of a solute relative to the center of mass is proportional to the gradient of electrochemical potential of this solute alone. Generally, for example, due to the solution nonideality, there are some contributions of gradients of electrochemical potentials of other solutes. (81) Because of the relatively low porosity of polyfluoroethylene support, a considerable proportion of pores of TEM could otherwise be seated against impermeable zones of this material. (82) Of course, the electroviscosity depends on the pore size, surface charge density and the electrolyte concentration. The indicated correction magnitude corresponds to the properties of membranes used in this study. (83) The electrode potential is controlled by the salt concentrations in the feed and the permeate compartments. After pressure switch-off, the permeate concentration ultimately changes due to the salt diffusion through the membrane. However, in our setup, this takes quite a long time (at least 1 h) primarily due to the presence of relatively thick porous support beneath the membrane. Our measurements were essentially shorter (up to 100 s). (84) The fact that the fluctuations decayed more rapidly in the more concentrated solution (lower electrical resistance) can be an indication of their appearance due to the presence of some parasitic inductivities and capacities in our system.
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Conclusions It has been demonstrated that the pressure-driven transfer of electrolyte solutions through charged nanopores of ca. 20 nm in diameter gives rise to noticeable rejection of salts from sufficiently dilute electrolyte solutions. It has also been shown that this mechano-chemical coupling between the volume and solute transfer also influences the pattern of electro-mechanical coupling (electrokinetic phenomena), by causing pronounced nonlinearity in the dependences of transmembrane electric potential on the transmembrane volume flow not observed in microfluidic systems. The use of rapid transmembrane pressure switch-off made possible the separation of stationary transmembrane potential difference into a streaming and a concentration gradient-induced component and confirmed that the nonlinearity was primarily caused by the concentration polarization.
9614 DOI: 10.1021/la900737q
The whole set of experimental data could be quantitatively interpreted within the scope of space-charge model (for the description of nanopores) combined with the film model for the description of external concentration polarization. This confirms the applicability of this continuous model to the description of transport properties of (1:1) electrolyte solutions in nanopores (nanochannels) of several tens of nanometers in diameter. Acknowledgment. This study was partially supported by the Spanish Ministry of Science and Innovation (MICINN) within the scope of the project “Integraci on de procesos de extraccion reactiva y procesos de membranas en la eliminacion de compuestos indeseados en etapas de potabilizacion de aguas superficiales y de regeneracion de aguas tratadas (PERMEAR)”.
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