Journal of Membrane Science 302 (2007) 1–9 Characterization of ion transport in thin films using electrochemical impedance spectroscopy I. Principles and theory Viatcheslav Freger ∗ , Sarit Bason Zuckerberg Institute for Water Research and Department of Biotechnology and Environmental Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel Received 7 May 2007; received in revised form 10 June 2007; accepted 21 June 2007 Available online 26 June 2007 Abstract The paper presents the principle and theoretical basis for application of electrochemical impedance spectroscopy (EIS) to study ion transport and partitioning in thin films or membranes supported by a solid electrode and exposed to an electrolyte solution. It is shown that the equivalent circuit is in general composed of two parallel branches: one corresponds to transient charge transfer processes, in which all ions (both electrochemically active and inactive) participate, and the other to the diffusion and reaction of electroactive ions only. Among the experimentally accessible parameters three appear to be of particular relevance to ion transport: (a) the high-frequency resistance of the film directly related to the sum of permeabilities of all mobile ions in the membrane including counter-ions bound to the fixed charges, (b) the diffusion impedance of the electroactive ion that is capable of separately retrieving the values of diffusion and partitioning coefficient of the specific ion and (c) dielectric capacitance of the film, which may yield the effective thickness of the film, particularly interesting if the membrane is not homogeneous. Such information may be highly relevant to analysis of ion exclusion mechanisms in the film and provide inputs to computational models of ion transport in membranes. Experimental examples involving thin polyamide films are provided to partly illustrate the use of equivalent circuit, data analysis and possible artifacts. © 2007 Elsevier B.V. All rights reserved. Keywords: Thin films; Ion difusion and partitioning; Diffusion impedance; Electrochemical impedance spectroscopy 1. Introduction Many membrane processes involve selective transport or retention of electrolytes. This includes both transport of individual ions in electro-membrane processes such as electrodialysis and transport of neutral ion mixtures (salts) in dialysis or pressure-driven processes, such as reverse osmosis (RO) and nanofiltration (NF). Despite wide commercial use, many details of the mechanism of electrolyte transport and structure of the membranes and, in particular, of the ultra-thin active polyamide layer for RO and NF are still obscure and motivate search for new characterization techniques. So far, the greatest experimental challenge has been determination of individual ion transport parameters required for predictive computational modeling of salt separations in RO/NF ∗ Corresponding author. Tel.: +972 8 6479316; fax: +972 8 6472960. E-mail address: vfreger@bgu.ac.il (V. Freger). 0376-7388/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2007.06.046 [1–8]. Some electrochemical techniques, such as membrane potential combined with diffusion or resistance measurements, allow retrieval of individual ionic transport parameters in films or membranes. In these methods, the film has to be placed between two solutions for measurements [9]. Unfortunately, this precludes their use for very thin films, since free-standing sub-micron films are usually insufficiently robust, while measurements involving supported films (e.g. commercial polyamide RO and NF membranes) are difficult due to the large and unknown diffusional and electrical resistance of the support [10,11]. Up to date, the most promising approach was to employ supported membranes in non-steady-state (transient) modes focusing on short time scales [11–14]. An alternative approach theoretically analyzed in this paper is the use of thin films directly supported by a solid electrode. In this case the mechanical strength of the film is not a limitation as long as the film remains intact and firmly adheres and completely covers the electrode. A number of electrochemical methods used to study diffusion may be applied to characterize transport in 2 V. Freger, S. Bason / Journal of Membrane Science 302 (2007) 1–9 such films [15–17]. This approach has been used for studying diffusion in solids [17], Nafion [18], electropolymerized thin films [19,20] and in self-assembled lipid bilayers [21] and surfactant monolayers [22]. The present publication generalizes the approaches proposed earlier and adds some aspects important for relatively permeable and not entirely homogeneous thin films, such as the ones employed in polyamide RO/NF composites. This study focuses on the electrochemical impedance spectroscopy (EIS), which is perhaps the most versatile and powerful method [15–17]. Although EIS experiments are run in a quasisteady-state mode whereby oscillations of voltage and current become reproducible, in fact it is an inherently non-steady-state technique that allows simultaneous analysis of many charge transfer phenomena on different time scales. The method gained much popularity in studies of coatings and corrosion on metals [23–25], self-assembled layers [21,22,26], polymer degradation [27], analytical chemistry [28], etc. A whole separate branch of EIS is studies of ion channels in biological and synthetic lipid membranes [21,29,30]. It has also been applied to study separation membranes using a variety of approaches [31–34], as well as ion-selective membranes and membrane-covered (modified) electrodes and sensors [35–37]. A critical step in understanding and quantitative analysis of EIS data is construction of the so-called equivalent circuit (EC), which must address all relevant physical phenomena and incorporate them as correctly connected elements. This paper precedes an experimental study of ion transport in ultra-thin electrode-supported films [38] and presents a detailed explanation and theoretical basis of the method that shows how different physical phenomena are combined into a general EC that is subsequently used for analysis of ion transport. 2.1. Ion transport in membranes: basic relations The kind of experiment considered in this paper involves only the diffusive ion transport thus both for steady and non-steady experiments the transport at any location in the membrane may be described by a set of regular Nernst–Planck equations without a convection term written for each ion in solution [1–8]: ci dϕi , RT dx ωi = Pi D i Ki = , δ δ (2) where δ is the thickness of the membrane. Note that all steadystate experiments and many non-steady-state experiments may only measure ωi or their combinations. The resulting ωi have a clearly defined value, even if the properties vary across the membrane in an unknown manner. In the latter case however both Pi and δ should be viewed as effective or average parameters, often dependent on interpretation or availability of additional information from experiments of a different type. Diffusive transport of mixed electrolytes through the membrane may be satisfactorily modeled, if experiment could produce ωi and their compositional dependence for all ions involved. Unfortunately, standard filtration or salt diffusion experiments are inherently unable to produce individual ionic permeabilities due to electroneutrality condition and coupling of ion fluxes. For instance, for a single 1:1 salt (e.g. NaCl) it is only possible to measure the lump “salt” permeability given by [9]: ωs = 2ω+ ω− . ω+ + ω− (3) Retrieval of the individual permeabilities of cation and anion (ω+ and ω− ) requires additional experiments. EIS may supply such information, since for a single salt it is capable of measuring the ac (or high frequency) membrane resistance given by the relation [17,35]: 2. Theory Ji = −Pi Ki is the partitioning coefficient. Thermodynamic relations are necessary to link Ki hence Pi to the local solution composition for specific exclusion mechanisms—steric, Donnan and/or dielectric. It is seen that Pi is essentially a material property and thus is not directly measurable, if the film thickness is unknown; the directly measurable parameter is usually the absolute diffusion permeability: (1) where Ji is the flux of ion i, x the distance from the feed side, Pi the intrinsic diffusion permeability of the membrane to the ion, ci the local concentration of the ion and ϕi is its electrochemical potential adding up both chemical and electrical potential terms. The ion fluxes are coupled via electroneutrality. In this paper, by definition, all concentrations are defined as ones in solution, which corresponds to the true solution concentrations outside the membranes and the so-called virtual (or corresponding) concentrations inside the membrane [39]. In such a way all parameters are determined for well-defined and measurable conditions. The intrinsic diffusion permeabilities may be expressed as Pi = Di Ki , where Di is the diffusivity of ion i in the membrane and Rm = F 2 Ac RT , s (ω+ + ω− ) (4) where A is the membrane area and cs is the salt concentration. It is easily seen that, once both ωs and Rm are available, both ω+ and ω− may also be calculated. It must be stressed that during the measurements Rm nearly always needs to be separated from several other interfering impedances and in fact EIS is the most reliable way to do that. Although knowledge of all ωi for given conditions may suffice for simple phenomenological modeling, Di and Ki known separately provide a much better insight into the separation mechanism, which has been a debated issue over the last decade [40,41]. Unfortunately, neither filtration or diffusion nor many EIS experiments can split ωi to the diffusion and partitioning factors for thin films. Nevertheless, for some ionic species EIS allows splitting of Pi to Di and Ki . This feature of EIS is unique and, along with splitting ωs to ω+ and ω− , it will be analyzed below and explored in the subsequent experimental work. V. Freger, S. Bason / Journal of Membrane Science 302 (2007) 1–9 Fig. 1. (a) Experimental setup and (b) concentration of oxidized species at the electrode surface as a function of applied potential. 2.2. Principle of measurements The proposed setup is schematically shown in Fig. 1a. The conditions at the solution side of the film are determined by the bulk solution composition, while at the other side they may be controlled by the potential applied to electrode. From the experimental viewpoint the species present in the bulk solution may be of two types: • Electroactive or redox species (i.e., oxidizable or reducible), which may be both ionic and neutral and are convertible to their counterparts due to electrode reactions bringing about faradaic charge transfer processes. • Electrochemically inactive species, which are not affected by potential variations and may only be associated with nonfaradaic processes, i.e., transient charging or discharge of capacitive elements, such as interfacial double electric layer. Fig. 1b schematically shows typical compositional changes occurring at the electrode surface upon potential variation for a redox couple. At low potentials the oxidized form is completely converted to the reduced form and the opposite occurs at high potentials. In the middle narrow range the concentration of reduced/oxidized species is linearly related to the applied potential and is the most sensitive to the potential variation. An inactive (supporting) electrolyte does not participate in reaction and is usually added in large excess to eliminate potential gradients and migration of redox species, thus the faradaic current becomes a pure diffusion current of rapidly and reversibly reacting electroactive species. This allows measuring uncoupled diffusion of individual ions. The extreme potentials may be used to set up a zero concentration of a redox species at the electrode, thereby their total concentration difference across the film is exactly known and the rate of steady-state diffusion may be measured as a faradaic current. This is done by performing chronoamperometry (CA) [15] at an appropriate potential until a steady-state current is obtained. The film permeability to the redox species is then found as [19,42]: ωi = I Ji = , ci nFAcro (5) where I is the measured steady-state diffusion current, n the number of electrons transferred in a single reaction and cro is the bulk concentration of the redox species. The advantage of 3 this method is that it may be used for any redox-active species, including relatively slowly or irreversibly reacting, provided a potential may be applied that completely eliminates the relevant species at the electrode surface. Electrochemical impedance spectroscopy (EIS) utilizes the middle potential range. A suitable dc potential (bias) is perturbed with a small amplitude ac signal (E ≪ RT/F ≈ 50 mV, usually 5–10 mV) and the resulting current perturbation is analyzed as a function of frequency ignoring the dc component. By probing a wide range of frequencies, usually, 10−3 to 106 Hz, EIS allows simultaneous analysis of various charge transfer phenomena at different time scales, both faradaic and non-faradaic. Remarkably, EIS allows separately measuring D and K for redox-active ions [18], whereas CA and EIS involving only inactive ions can only yield P, their product. Also, working in a specific narrow range of potentials focuses on a specific redox species, and minimizes interferences from other redox species or impurities, which could be a nuisance in CA [42]. Overall, compared to steady-state CA, EIS potentially yields a more comprehensive picture of ion transport in a membrane. It however imposes more severe restrictions on the choice of redox couples that must react rapidly and reversibly and its analysis is more complex (see next sections). 2.3. Equivalent circuit and its elements The EIS results are usually presented as the complex impedance Z versus frequency f, i.e., impedance spectrum, fully represented by the frequency dependence of two real quantities, the absolute value |Z| and the phase lag φ [15–17]. A customary way of presenting the spectra used here is the Bode plot that shows log|Z| and φ versus log f. Impedance spectra are customarily analyzed by means of equivalent circuits (EC). Each element in EC and the way it is connected reflect an underlying physical phenomenon. The relevant elements may be of three basic types [15–17]: (a) Resistors (R) (|Z| = R and φ = 0) that appear as plateaus in the Bode plot. (b) Capacitors (C) (|Z| = 1/2πfC and φ = 90◦ ), |Z| appearing as a straight line with a −1 slope. (c) Elements associated with diffusion of redox-active species, which are various modifications of the Warburg impedance. The latter has φ = 45◦ , while |Z| appears as a −1/2 slope and is given by |Z| = ZW = 1 √ , YO 2πf (6) where YO is a parameter characteristic of the medium. For a reversible redox couple, in which both reduced and oxidized forms have similar D and K and are at the same concentration cro /2 in solution, YO is given by YO = √ F2 cro AK D. 4RT (7) More general expressions for YO may be found in [15–17]. 4 V. Freger, S. Bason / Journal of Membrane Science 302 (2007) 1–9 paints and coatings [23], lipid bilayers containing ion channels [29,30], self-assembled monolayers [26], as well as synthetic separation membranes [43–45]. For a planar film Rm is given by Eq. (4) for a single salt; a more general formula for a multi-component mixture is [17,35]: Rm = RT δ 1 1 RT  = 2  , 2 F A ω i ci F A D i K i ci (8) where the summation is only over mobile ions and does not include fixed charges in the film. The expression for film capacitance Cm depends on the polarization mechanism. In the case of a planar homogeneous dielectric film subject to pure dielectric polarization it is given by the classic formula of electrostatics [17,35]: Cm = εε0 Fig. 2. The generic equivalent circuit for a setup shown in Fig. 1a. The faradaic elements (absent without redox species) are depicted with a dotted line. The curved arrows show the two types of current. RE and WE designate reference and working (solid) electrodes. The principal difference between the redox-active and inactive ions is that after setting a fixed potential the former brings about a current driven by variations of their concentration at the electrode surface (cf. Fig. 1b), whereas the latter—a finite variation of charge driven by the need to attain the equilibrium in the double electric layer. The former process is therefore of a resistive nature, while the latter of a capacitive one. Since the two currents are carried in parallel by different carriers, being driven by the same overall potential difference, the corresponding impedances form two parallel branches of the EC, faradaic and non-faradaic. The general EC proposed in this work for the setup shown in Fig. 1a is presented in Fig. 2. The upper parallel branch represents the non-faradaic impedance contributed by the inactive ions (supporting electrolyte) and the lower one—the faradaic impedance contributed by the redox species (not necessarily charged). Both branches together are connected in series with a resistance Rs , which accounts for the potential drop in the solution and wires that reduces the actual potential across each parallel branch compared to the potential maintained between the working and reference electrodes. Due to the presence of a film separating the solution and electrode, each branch includes several elements. In the nonfaradaic branch, the total potential drop is divided between the film and the double electric layer at the electrode surface connected in series, with the film behaving as a capacitor and a resistor connected in parallel. The film capacitance originates from its properties (e.g. dielectric) different from those of the surrounding medium. A potential applied across the film causes its polarization by a dielectric and other mechanisms, which induces transient accumulation of opposite charges at its two interfaces. However, since the film has a finite resistance, the excess charges may eventually diffuse through the film and recombine. The film is then expected to behave as a “leaky” capacitor Cm connected in parallel with a resistor Rm and in series with the double layer capacitance Cdl . This or similar electric representations have been used in impedance analysis of A , δ (9) where ε is the dielectric constant of the film. This relation is widely used in analysis of paints and coatings on metals [23] and has been applied to the active layer of composite separation membranes [43,44]. For non-uniform films, such as the polyamide layer of RO and NF membranes, where only a fraction of the overall polyamide thickness serves as the real barrier to ion transport [46–48], it could estimate the effective thickness δef . Apparently, this effective thickness rather than the total polyamide thickness would have to be used in Eqs. (2) and (9). However, it must be stressed that such estimate requires that the validity of dielectric interpretation be ensured and the value of ε be known for the hydrated polymers, which could significantly differ from its dry value (e.g. [37]). The use of faradaic branch has been much rarer in membrane science. The structure of this branch reflects the fact that the faradaic current, i.e., the overall rate of the electrode reaction, is controlled by two resistances-in-series, diffusion and reaction kinetics. The reaction kinetics is represented by a charge transfer resistance Rct connected in series with diffusion impedance [15–17]. Its is inversely proportional to the concentrations of redox species and is relatively small for fast reversible reactions, such as Fe(CN)6 3− ↔ Fe(CN)6 4− . The novel point of this paper is the representation of the diffusion impedance, which accounts in a general way for two aspects specific for thin films, particularly, non-homogeneous or rough. The first aspect, the finite thickness of the film, has been well known. It requires that the regular Warburg (valid for an infinitely thick layer) be replaced with the so-called porous Warburg or Oelement [17,49]. This modified element has been successfully applied for analysis of homogeneous films tightly attached to a solid electrode [17,18]. Analytically, O-element is described by a formula involving the complex hyperbolic tangent function [49]: ZO =  1 √ tanh(B 2πfj), YO 2πfj (10) with YO is given by the same√expression as for the regular Warburg (Eq. (7)) and B = δ/ D. The parameter B2 has the meaning of the characteristic time of diffusion through the film. It is easily seen that at high frequencies (f ≫ B−2 ) ZO becomes V. Freger, S. Bason / Journal of Membrane Science 302 (2007) 1–9 identical with the regular Warburg impedance ZW (Eq. (6)). At low frequencies (f ≪ B−2 ) ZO will be proportional to the steady-state diffusion resistance of the film to the redox species: RO = B 4RT δ 4RT 1 1 = 2 = 2 . YO F A Dro Kro cro F A ωro cro (11) Importantly, at high √ frequencies ZO ≈ ZW is determined by the quantity Kro Dro (cf. Eq. (7)), while at low frequencies ZO ≈ RO is determined by the absolute permeability ωro = Dro Kro /δ. The whole spectrum of ZO then allows determination of both Dro and Kro provided δ is known. The second aspect considered here for the first time is the imperfect attachment of the film to the electrode, so that a thin gap may exist between the film and electrode. Such a situation may arise not only for poorly prepared or deteriorated samples, but may also be an intrinsic feature of the film. For instance, a significant amount of solution may always be trapped between the electrode and a film due to film roughness. A similar situation may occur, if the film is not fully homogeneous and its innermost parts touching the electrode are more permeable and loose than the outermost layer facing the solution. The film and the gap may be viewed as a planar bilayer film. General and rather cumbersome formulas for such case were developed recently [50–52]. They however simplify when one layer (solution) is much more resistant than the other (a film). For instance, they show that a layer of small resistance added at the solution side of the film, such as unstirred solution layer, may be ignored. In contrast, a gap layer sealed between a much more resistant film and electrode will behave simply as the so-called bounded Warburg or a T-element ZT connected in parallel to the film impedance ZO [52]. The formula for ZT is similar to ZO but the hyperbolic cotangent function replaces hyperbolic tangent [17,49]: ZT =  1 √ coth(B 2πfj). YO 2πfj current (cf. Fig. 2) and make the whole ZO unobservable. Note that for any reasonable values of parameters the resistance of the gap will be negligible compared with the film and will not have any effect on the non-faradaic branch, however the faradaic capacitance CT of the gap may strongly interfere with film characterization. Awareness of its possible presence is then crucial for successful data analysis and rationalizes the need to minimize the gap between the film and electrode. In the context of composite membranes, this essentially precludes the use of whole supported membranes for measurements employing redox species. Along with the high resistance of the support that is known to severely interfere with non-faradaic currents as well [10,11], this provides yet another incentive for using the setup shown in Fig. 1a. 3. Experimental The sample preparation procedures and setup for experiments presented below are fully described elsewhere [38]. Briefly, a thin polyamide film was attached to a PEEK-shrouded glassy carbon rotating disc electrode (Metrohm, 0.07 cm2 ) either by dissolving away the supporting layer of a commercial polyamide membrane NF-200 (Dow-Filmtec, kindly supplied by Dr. J.-P. DeWitte) or by a direct interfacial polymerization synthesis using monomers employed in manufacturing of fully aromatic RO membranes. The electrode with the sample was placed in a N2 -purged cell containing appropriate solution and connected as a working electrode to a potentiostat/impedance analyzer (Gamry). The counter electrode and reference were, respectively, a Pt plate and Ag/AgCl/3N KCl electrode with a Vicor plug. The supporting pH buffers were 1 M hydrogen/sodium citrate (pH 4), 0.67 M mono-/diphosphate (equimolar sodium/potassium, pH 7) and 1 M hydrogen/sodium borate (pH 9). (12) The expressions for YO and B are identical to Eq. (10) with the parameters K, D and δ pertinent to the gap layer. Analytic solutions are also known for analogues of O- and T-elements for cylindrical and spherical geometries [49]. They may be important for analysis of diffusion through defects in a thin film, e.g. cracks or pinholes [53], however such cases are not considered here. Straightforward calculations show that for a gap of solution (K = 1) thinner than about 0.1–1 ␮m, which is usually the case (otherwise the film may detach and fall off), the crossover frequency BT−2 = D/δ2T of the gap approaches the upper instrumental limit (105 to 106 Hz) and in the whole f range ZT behaves essentially as a capacitance: F2 AδT cro . (13) 4RT This capacitance results from the redox species trapped in the gap. A problem arises when their amount exceeds the one that may diffuse through the film during one potential cycle or, in the EIS language, the gap impedance |ZT | = 1/2πfCT becomes smaller than ZO . In this case ZT will take up most of the faradaic CT = BYO = 5 4. Results and discussion The above EC was initially devised for characterization of ion transport in free-standing ultra-thin polyamide films either isolated from composite RO and NF membranes or directly synthesized on the electrode. The full results will be presented in a follow up publication [38], then only two representative examples are given here to illustrate the use of the general EC for interpretation and analysis of different elements. The first example in Fig. 3 shows the impedance spectrum of a thick polyamide film. The film was prepared on a disc electrode from a fully aromatic polyamide using a reaction used for manufacturing of polyamide RO membranes. The spectra were obtained for the same film placed in different pH buffers of similar ionic strength in absence of any redox species, thereby the lower branch in Fig. 2 was missing and the effect of pH on different non-faradaic elements was directly observed. Preliminary experiments with a bare electrode determined that Rs ∼ 10–30 , but due to relatively large thickness hence large Rm and small Cm , this Rs was unobservable within the instrumental frequencies limits, which left EC with only three non-negligible elements Cm , Rm and Cdl . 6 V. Freger, S. Bason / Journal of Membrane Science 302 (2007) 1–9 Fig. 3. Equivalent circuit and impedance spectra of a synthetic PA film in different pH buffers: working electrode glassy carbon rotating disc electrode, rotation speed 1000 rpm. The resulting spectra as well as the relevant EC are presented in Fig. 3. Notably, the above interpretation of the two capacitances suggests that they should be weakly dependent on the solution used, since all buffers have similar ionic strengths of the order 1 M. Indeed, at low and high frequencies the spectra tend to merge into the same sloped lines corresponding to respectively, Cdl ∼ 20–30 ␮F/cm2 (typical of Cdl ) and Cm ∼ 0.1 ␮F/cm2 . The latter may be used to estimate the effective thickness of the film by adopting a dielectric interpretation (Eq. (10)) and assuming ε ≈ 3 for polyamide [38]. This yields δef ∼ 30 nm. This value is significantly smaller than the total superficial thickness of synthetic films, which was of the order of 1 ␮m, as determined by atomic force microscopy using the method of Ref. [47]. Presently, it is not clear whether this discrepancy reflects the complex structure of the film [46–49] or is a result of the inadequacy of the dielectric interpretation that has shown some inconsistencies for thin polyamide films [38] and thin-film composite membranes [45]. Fig. 3 also shows that, unlike the capacitances, the value of Rm largely varies for the three solutions, the neutral buffer producing the highest resistance and the acidic buffer the lowest one. Considering Eq. (8) and similar concentrations in all solutions, it may be concluded that the ionic permeabilities ωi of the most permeable ions in the buffers were largely different. Indeed, acidic and basic solutions contained significant amounts of highly mobile H+ and OH− ions contributing large ωi in Eq. (8) and thus largely reducing Rm , whereas the neutral buffer contained only much slower and larger alkali cations and phosphate anions having smaller values of ωi . The second example illustrates the significance of the elements building up the lower faradaic branch in Fig. 2, particularly, the diffusion impedances. Fig. 4 shows EIS spectra of a sample measured consecutively in three different buffers (pH 4, 7 and 9) containing 0.1 M of equimolar mixture of K3 Fe(CN)6 and K4 Fe(CN)6 . The sample was a glassy carbon disc electrode covered with a film of polyamide ca. 30 nm thick taken from a NF-200 membrane. Preliminary experiments with this system using bare electrode (which also verified negligible diffusion resistance of the Fig. 4. Impedance spectra of a NF-200 film in different buffers containing 0.1 M of equimolar mixture of K3 Fe(CN)6 and K4 Fe(CN)6 . Working electrode rotating glassy carbon disc, 0.07 cm2 ; rotation 1000 rpm. unstirred layer) and a film in the same buffers without redox species (not shown here) indicated that elements Rm , Cm , Cdl and Rct contributed impedances that were comparable or smaller than Rs ∼ 10–30 at frequencies higher than about 10 kHz. Therefore the spectral patterns observed in that region of the spectra, largely dominated by Rs , did not allow reliable parameter estimation and were then uninteresting. However, addition of redox species did allow evaluation of diffusion parameters through the much larger faradaic diffusion impedance ZO . A plateau seen in Fig. 4 at low frequencies was absent in similar experiments without Fe(CN)6 species added and then was obviously attributed to the resistance of the film to diffusion of ferro- and ferricyanide given by Eq. (11). The height of the plateau was independent of the type and pH of supporting electrolyte, which was another indication that it was a faradic resistance RO of the film related to the presence of the same amount of Fe(CN)6 in solution, which was also found independent of the pH and buffer. The value of RO was in good agreement with the values of permeability ωi of this film to Fe(CN)6 4− and Fe(CN)6 3− deduced separately from CA results, i.e., steadystate diffusion current under, respectively, extreme oxidation or reduction potentials (cf. Fig. 1b) using Eq. (5) [42]. Note that the similar permeability to both ions measured by CA justifies the use of simple Eq. (7) in place of a more general relation. Note that only RO , the low-frequency part of the diffusion impedance ZO of the film, is observed in Fig. 4. The high-frequency Warburg part was apparently hidden (i.e., shortcircuited) by a smaller capacitive impedance showing up as a line with a −1 slope that follows the plateau and could be easily mistaken for a Cdl . A striking feature of this capacitance is that it strongly varied between the samples, being the smallest for the first taken spectrum (pH 4) and largest for the last one (pH 9). Its magnitude increased from ca. 40 to 500 and then to 1800 ␮F/cm2 , as deduced from the intercept of the slopes with f = 1/2π = 0.169 Hz. For pH 4 (freshly prepared sample) it could be attributed to Cdl , yet experiments with a film in absence of Fe(CN)6 species showed that the actual Cdl was smaller. This rules out its assignment to any physical phenom- V. Freger, S. Bason / Journal of Membrane Science 302 (2007) 1–9 ena, such as Cdl or Cm , particularly, for pH 7 and 9 and means that this capacitance was simply an artifact resulting from gradual film detachment, i.e., pockets of solution trapped under the film bringing about the faradaic gap capacitance CT . Using Eq. (13) with K = 1 (a gap of free solution), we could estimate that the average thickness of the gap δT was, respectively, 8, 100 and 360 nm for the three spectra. This increasing gap could be at least partly explained by the known fact that the film becomes more swollen hence more hydrophilic at higher pH [47], which progressively worsened its adhesion to the rotating electrode. Indeed, after further increasing pH to 10 the film fell off. This example illustrates that the use of redox species for characterization of diffusion in thin films, which is potentially the most powerful and informative approach, may have its own disadvantages associated with possible presence of CT that imposes stricter requirements on sample preparation. These can be particularly severe for loosely attached, inhomogeneous or supported films. Evaluation of these and other parameters from EIS spectra is accomplished by fitting a model circuit to the whole measured spectrum using appropriate software, subject to judgment as to which real physical elements or artifacts such as CT show up in the spectrum. Implementation of this approach will be demonstrated in a subsequent publication dedicated to EIS experiments employing the ultra-thin active layer of RO composite polyamide membranes [38]. Acknowledgements The authors are indebted to Prof. Yoram Oren and Prof. Ora Kedem for numerous discussions and to Dr. Jean-Paul DeWitte at Liquid Separations, Dow for kindly supplying membrane samples. Financial support from German-Israeli Foundation (Young Scientists Grant 1-2035.1102.05/2001) and Ministry of Science of Israel (Project N 01-01-01496, Program for Scientific and Technological Development for Quality of Environment and Water) is gratefully acknowledged. 5. Conclusions To summarize, the following important information on ionic transport in the film could potentially be retrieved from EIS spectra using the setup in Fig. 1a: 1. Electrical resistance of the film Rm that belongs to the nonfaradaic branch and thus may be measured for any neutral salt solution. In combination with salt permeability ωs that may be deduced from standard RO/NF filtration, it may yield individual ionic permeabilities. For a charged membrane Rm lumps together the constant amount of counter-ions bound to fixed charges and the free salt in the membrane. The variation of this parameter with concentration may thus point to the actual ratio between the counter-ions bound to fixed charges and the free salt inside the membrane, which is known to be the main factor that determines the strength of Donnan exclusion. 2. Diffusion and partitioning coefficients of redox-active ions from fully resolved spectrum of ZO , i.e., both ZW and RO . The general EC and the presented example indicate that availability of the ZW (high frequency) part of ZO is not guaranteed and, depending on the values of other impedances in the system, may be precluded by either double layer or gap (CT ) capacitances. However, RO that is not blocked by any capacitance will always show up at sufficiently low frequencies and, similar to Rm , may be used to examine variation of ion exclusion or permeability with concentration. Unlike Rm , it is focused on a specific ion. 3. The effective film thickness δef could be estimated from Cm , yet this estimate is heavily dependent on the dielectric interpretation of the film capacitance and on the knowledge of the dielectric constant. As the authors are not aware of any method that allows direct measurements of δef of heterogeneous films, such as the active layer of polyamide RO and NF membranes, this approach warrants examination, which is carried out in the subsequent experimental study [38]. 7 Nomenclature A B ci cro C Cdl Cm CT D Di Dro E f F I j Ji Ki Kro n Pi R Rct Rm RO Rs T x YO Z surface area of membrane or electrode (m2 ) time constant (s1/2 ) ion concentration in solution (mol/m3 ) redox ions concentration (mol/m3 ) capacitance (F) double electric layer capacitance (F) membrane capacitance (F) low-frequency capacitance of T-element (F) diffusion coefficient (m2 /s) ionic diffusion coefficient (m2 /s) redox ions diffusion coefficient (m2 /s) potential (V) frequency (Hz) Faraday’s constant (C/mol) electric current (A) imaginary unit ionic flux (mol/(m2 s)) ionic partition coefficient redox ions partition coefficient number of electrons transferred in a single reaction intrinsic ionic permeability coefficient of film (m2 /s) universal gas constant (J/(mol K)) charge transfer resistance ( ) membrane resistance ( ) low-frequency resistance of O-element ( ) solution resistance ( ) absolute temperature (K) distance from feed side across membrane (m) admittance parameter ( −1 s1/2 ) impedance ( ) 8 V. 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