Application of Electrochemical Impedance Spectroscopy to the Study

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Application of Electrochemical Impedance Spectroscopy to the Study of Dioleoyl Phosphatidylcholine Monolayers on Mercury C. Whitehouse, R. O’Flanagan, B. Lindholm-Sethson,† B. Movaghar, and A. Nelson* Center for Self-Organising Molecular Systems, School of Chemistry, University of Leeds, LS2 9JT, United Kingdom Received July 11, 2003. In Final Form: October 24, 2003 Electrochemical impedance spectroscopy has been applied to the analysis of the behavior of monolayers of dioleoyl phosphatidylcholine (DOPC) on a mercury electrode. Experiments were carried out in electrolytes KCl and NaCl (0.1 mol dm-3) and Mg(NO3)2 (0.05 mol dm-3), and the frequency dependence of the complex impedance was estimated between 65 000 and 0.1 Hz at potentials -0.4 to -1.5 V versus Ag/AgCl 3.5 mol dm-3 KCl at uncoated and coated electrode surfaces. Experiments were also carried out in the presence of gramicidin A (gA). Between the potentials of -0.4 and -0.7 V, the DOPC monolayer behaves as an almost ideal capacitor with little frequency dispersion. At more negative potentials, the impedance data show the formation of defects (-0.7 to -0.85 V), ingression of electrolyte into the layer (capacitance peak ∼ -0.935 V), reorientation of phospholipid-water structures (capacitance peak ∼ -1.0 V), and initiation of phospholipid desorption (∼-1.3 V). gA interaction with the phospholipid monolayer at -0.4 V is shown as an additional low-frequency element. A general “one capacitor model” in a RC series equivalent circuit is developed incorporating the frequency dispersion of the capacitance, distribution of the time constants of the dispersion, and a coefficient related to the interface between the solution and the coated electrode. This latter coefficient is the most robust and decreases at potentials approaching those coincident with the DOPC phase transitions.

Introduction During the last two decades, there has been an increasing interest in the development of supported membrane models.1-4 These have provided an alternative system to free-standing artificial membranes5 for investigating biological membrane function. Supported membranes have the advantage of being inherently more stable6,7 and as a consequence have been used in the development of sensors.8,9 The rationale behind this approach is that biological membranes are sensitive to the presence of specific analytes in solution, and this sensitivity can be emulated in a mimetic membrane supported on a solid surface.10 Methods for assessing the structure and properties of supported membranes include * To whom correspondence should be addressed. Fax: 44 113 343 6452. Tel.: 44 113 343 6409. E-mail: [email protected]. † Address: Department of Chemistry, Biophysical Chemistry, Umeå University, 90187 Umeå, Sweden, and Centre for Biomedical Engineering and Physics, Umea University, SE-901 87 Umea, Sweden. (1) Hong, Q.; Terrettaz, S.; Ulrich, W. P.; Vogel, H.; Lakey, J. H. Biochem. Soc. Trans. 2001, 29, 578-582. (2) Jenkins, A. T. A.; Bushby, R. J.; Evans, S. D.; Knoll, W.; Offenhausser, A.; Ogier, S. D. Langmuir 2002, 18, 3176-3180. (3) Sackmann, E. Science 1996, 271, 43-48. (4) Cornell, B. A.; Krishna, G.; Osman, P. D.; Pace, R. D.; Wieczorek, L. Biochem. Soc. Trans. 2001, 29, 613-617. (5) Koryta, J. Ions, Electrodes and Membranes; John Wiley and Sons: Chichester, 1984; pp 155-157. (6) Guidelli, R.; Aloisi, G.; Becucci, L.; Dolfi, A.; Moncelli, M. R.; Buoninsegni, F. T. J. Electroanal. Chem. 2001, 504, 1-28. (7) Sinner, E. K.; Knoll, W. Curr. Opin. Chem. Biol. 2001, 5, 705711. (8) Wiegand, G.; Arribas-Layton, N.; Hillebrandt, H.; Sackmann, E.; Wagner, P. J. Phys. Chem. B 2002, 106, 4245-4254. (9) Kiefer, H.; Klee, B.; John, E.; Stierhof, Y. D.; Jahnig, F. Biosens. Bioelectron. 1991, 6, 233-237. (10) Cornell, B. A.; BraachMaksvytis, V. L. B.; King, L. G.; Osman, P. D. J.; Raguse, B.; Wieczorek, L.; Pace, R. J. Nature 1997, 387, 580583.

optical,11 spectroscopic,12 and electrochemical13 techniques. Electrochemical methods are particularly applicable when the membrane is supported on a conducting surface.14-18 In this instance, methods of impedance can be used to study the structure and properties of the surface layers,13-17 whereas ion and electron-transfer processes in the membranes can be investigated by voltammetric,19 amperometric,20 and pulse techniques. Impedance techniques are in fact very powerful ways to study supported layer structure,21 but they can be beset by complications in the interpretation of the results. The purpose of this work is to explore the usefulness of electrochemical impedance spectroscopy in investigating the supported membrane structure and properties. The work is part of a long-term objective to utilize impedance methods to interrogate membrane interactions with analyte compounds in solution. In this work, the simplest biologically relevant supported membrane system was used. This consists of a phospholipid monolayer physically adsorbed onto a mercury electrode. A fluid phospholipid, (11) Puu, G. Anal. Chem. 2001, 73, 72-79. (12) Wenzl, P.; Fringeli, M.; Goette, J.; Fringeli, U. P. Langmuir 1994, 10, 4253-4264. (13) Gafni, Y.; Weizman, H.; Libman, J.; Shanzer, A.; Rubinstein, I. Chem.sEur. J. 1996, 2 , 759-766. (14) Lingler, S.; Rubinstein, I.; Knoll, W.; Offenhausser, A. Langmuir 1997, 13, 7085-7091. (15) Boubour, E.; Lennox, R. B. J. Phys. Chem. B 2000, 104, 90049010. (16) Boubour, E.; Lennox, R. B. Langmuir 2000, 16, 7464-7470. (17) Boubour, E.; Lennox, R. B. Langmuir 2000, 16, 4222-4228. (18) Bordi, F.; Cametti, C.; Gliozzi, A. Bioelectrochemistry 2002, 57, 39-46. (19) Marchal, D.; Boireau, W.; Laval, J. M.; Moiroux, J.; Bourdillon, C. Biophys. J. 1997, 72, 2679-2687. (20) Lindholm-Sethson, B.; Gonzalez, J. C.; Puu, G. Langmuir 1998, 14, 6705-6708. (21) Cheng, Z. L.; Yang, X. R. Chin. J. Anal. Chem. 2000, 28, 10371041.

10.1021/la035259k CCC: $27.50 © 2004 American Chemical Society Published on Web 12/03/2003

Dioleoyl Phosphatidylcholine Monolayers

dioleoyl phosphatidylcholine (DOPC), is used, which is compatible with the smooth mercury surface. This system has already been well characterized.22-28 In the following, all potentials are quoted versus the Ag/AgCl, 3.5 mol dm-3 KCl reference electrode. Around the potential of zero charge of mercury (∼-0.4 V), the monolayer capacitance is about 1.84 µF cm-2 corresponding to exactly half a bilayer of phospholipid adsorbed with the alkyl chains on the mercury surface. This structure is impermeable to ions. At more negative potentials (-0.7 V), the monolayer becomes permeable to ions, and at ∼-0.93 V, there is a well-defined phase transition represented by capacitance peak 1. This is followed by a second phase transition at ∼-1.0 V represented by capacitance peak 2. The first phase transition coincides with the monolayer becoming fully permeable to ions. It has been suggested from voltage pulsing experiments that the second phase transition proceeds by the growth of defects or inhomogeneities in the layer.25 Capacitance peak 3 is at ∼-1.3 V and coincides with the initiation of phospholipid desorption from the mercury surface and replacement by electrolyte.25 Capacitance peak 3 is broad compared to capacitance peaks 2 and 3, which are sharp. The charge structure of the coated electrode as a function of the potential shows sharp discontinuities at the potentials characterizing the three capacitance peaks.25 Previous modeling results using mean field theory showed that (i) the phase transition corresponding to capacitance peak 1 represented a rather complex reorientation whereby the monolayer reverts to two phases of thin bilayer and thin monolayer26,27 and (ii) the phase transition corresponding to capacitance peak 2 represents the conversion of the adsorbed layer system to a pored bilayer. The three consecutive transitions are the result of the increasing polar nature of the electrode with increased negative potential, giving rise to a changing affinity of the phospholipid headgroups, phospholipid hydrocarbon chains, and electrolyte for the interface.26,27 This paper looks at the impedance properties of the monolayer at all potentials between -0.4 and -1.5 V. Experimental Section (a) Electrochemical and Monolayer Techniques. The electrolytes KCl, NaCl (0.1 mol dm-3), and Mg(NO3)2 (0.05 mol dm-3) were prepared from Analar KCl and NaCl (Fisher Scientific Chemicals, Ltd.) calcined at 600 °C and Mg(NO3)2 (Merck chemicals) dissolved in 18.2 MΩ Milli-Q water. A blanket of argon gas was maintained above the fully deaerated electrolyte during all experiments. Monolayers of DOPC were prepared as described earlier23-25 by spreading 13 µdm3 of a 2 mg cm-3 solution of DOPC in pentane (HPLC grade, Fisher Scientific Chemicals, Ltd.) at the argon-electrolyte interface in the electrochemical cell.23-25 The working solution of DOPC was obtained by dilution of the stock 50 mg cm-3 solution (Avanti Lipids). A fresh mercury drop (area, A ) 0.0088 cm2) was coated with the phospholipid23-25 from the argon-electrolyte interface prior to each series of experiments. To look at the interaction of gramicidin A (gA) with the layer, aliquots of methanol solutions of gA were injected below the layer into the electrolyte, 0.1 and 0.05 mol dm-3 NaCl and Mg(NO3)2, respectively. The phospholipid monolayer with incorporated gA was then deposited on the electrode. The experi(22) Miller, I. R. In Topics in Bioelectrochemistry and Bioenergetics; Milazzo, G., Ed.; John Wiley and Sons: Chichester, 1981; Vol. 4, pp 161-224. (23) Nelson, A.; Benton, A. J. Electroanal. Chem. 1986, 202, 253270. (24) Nelson, A.; Auffret, N. J. Electroanal. Chem. 1988, 244, 99. (25) Bizzotto, D.; Nelson, A. Langmuir 1998, 14, 6269-6273. (26) Leermakers, F. A. M.; Nelson, A. J. Electroanal. Chem. 1990, 278, 53-72. (27) Nelson, A.; Leermakers, F. A. M. J. Electroanal. Chem. 1990, 278, 73-83. (28) Nelson, A.; Bizzotto, D. Langmuir 1999, 15, 7031-7039.

Langmuir, Vol. 20, No. 1, 2004 137 ments that studied gA interaction with DOPC from NaCl electrolyte have been reported previously29 when the data was subject to a multivariate analysis. The results are included here for comparison and for substantiation of the present analysis. An Autolab system, FRA and PGSTAT 30 interface (Ecochemie, Utrecht, The Netherlands), controlled with Autolab software, was used in all the electrochemical experiments. The experiments were performed in a standard three-electrode cell that was temperature-controlled at 25 °C. A Ag/AgCl, 3.5 mol dm-3 KCl reference electrode with a porous sintered glass frit separating the 3.5 mol dm-3 KCl solution from the electrolyte served as the reference, and a platinum bar served as the counter electrodes located on either side of the working electrode. A solution resistance of around 290-320 Ω was recorded for the cell irrespective of whether the mercury electrode was coated or uncoated. The diagnostic plots showed the conformity of the impedance data from a DOPC-coated electrode to that of a RC series circuit and an absence of instability at high frequencies. For this reason, the use of a fourth pseudo-reference electrode was not considered necessary at this stage. The electrochemical cell and screened cables were contained in an aluminum faraday cage. (b) Electrochemical Impedance. Measurements of the impedance (Z) of the electrode systems using frequencies (f) logarithmically distributed from 65 000 to 0.1 Hz, 0.005 V root mean square (rms) at potentials from -0.4 to -1.5 V, were carried out on uncoated and coated electrode systems. In the case of the coated electrode measurements, the structure of the DOPC layer was checked using cyclic voltammetry (CV) at 40 V s-1 immediately following deposition such that the peak current of the phase transitions 1 and 2 was larger than 3 mA cm-2 (∼75 µF cm-2). Impedance measurements at potentials (-0.9 to -0.94 V and -1.00 to -1.02 V) characterizing the capacitance peaks corresponding to the two phase transitions utilized an alternating current (ac) waveform of 0.002 V rms to cause minimum disturbance of the layer structure. Following each frequency scan of the phospholipid layer, the integrity of the layer was again checked using CV. If there was any change in the cyclic voltammogram that indicated a degradation of the phospholipid layer during the experiment, a fresh phospholipid layer was deposited on the electrode and the experiment was repeated. The experimental conditions for the measurement of impedance are listed in the following. For one measurement, one cycle was used except when the cycle was less than 1 s when the measurement time was 1 s. To reach steady state, 10 cycles were used, but when 10 cycles lasted longer than 3 s, 3 s were used. Each frequency scan took 5 min with the potential continually applied, commencing with the highest frequency. These times are a compromise in providing sufficient time to carry out the measurement and to reach steady state but enabling all experiments to be done within a specified time on one DOPC layer without any change in the layer’s structure. No significant difference in the spectra was noted when longer equilibration periods were used before each experiment. The impedance data were transformed to the complex capacitance plane, and the complex capacitance axes were expressed as Re Yω -1 and Im Yω-1. This was done using the Excel (Microsoft) spreadsheet. In the development of the models, equations expressing the admittance of the systems were solved for real and imaginary components using Maple (Waterloo Maple, Inc.). Curve fitting was carried out using Igor (Wavemetrics). (c) Data Treatment. As a result of the absence of any electroactive component, the simplest equivalent circuit model is the uncompensated solution resistance (Ru) of the cell and the capacitance (C) of the working electrode in series.8 Ru can be determined by extrapolating the Im Z versus Re Z plot to the Re Z axis.30 In the complex capacitance plane, values of Re Yω-1 were plotted against Im Yω-1 for all values of the frequency.14,31,32 For a series RC circuit, the Re Yω-1 versus Im Yω-1 plots give (29) Lindholm-Sethson, B.; Nystrom, J.; Geladi, P.; Nelson, A. Anal. Bioanal. Chem. 2003, 375, 350-355. (30) Janek, R. P.; Fawcett, W. R.; Ulman, A. J. Phys. Chem. B 1997, 101, 8550-8558. (31) Lindholm-Sethson, B. Langmuir 1996, 12, 3305-3314. (32) L. Strasˇa´k, L.; Dvorˇa´k, J.; Hason, S.; Vetterl, V. Bioelectrochemistry 2002, 56, 37-41.

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Figure 1. Plots in the complex capacitance plane of impedance data derived from the DOPC monolayer adsorbed on mercury in 0.1 mol dm-3 KCl at the potentials indicated on the diagram. Data are represented by crosses and eq 2 fits as the solid lines. R and β values with errors (SD) and χ2 from the fits are indicated on the diagram. Numbers on the plots indicate frequencies of adjacent data points expressed as and representing values of log(ω, rad s-1) as follows: 1, 5.61; 2, 4.42; 3, 3.24; 4, 2.05; and 5, 0.87. a single semicircle for a RC element where the capacitor has no frequency dispersion. The extrapolation of this semicircle to the Im Yω-1 axis at low frequency gives the capacitance (C)14,31,32 in the RC circuit. When applied to the phospholipid-coated electrode, any additional elements to the RC semicircle at lower frequencies will correspond to properties of the phospholipid layer. Further, if the semicircle representing the RC element is not perfect,31 the nonideality of the capacitor is indicated, which can be due to dielectric relaxations coupled to the RC charging process and to additional circuit elements at the interface between the capacitor and the solution resistance.31,33 In this study, the extrapolated semicircle to the Im Yω-1 axis at a low frequency is termed the zero frequency capacitance (C). C is, therefore, only an empirical quantity and is the capacitance that forms part of the RC element. Its bulk dielectric properties are determined by effects superimposed on or in addition to the RC element. In a series RC network, where the connection between the capacitor and resistor is ideal and the capacitor has no frequency dispersion, the values Re Y/(Im Yω) ) Re Z/(Im Zω) ) RC34 and Re Y/(Im YωRC) ) 1. Deviation from unity in an electrochemical system indicates additional elements in the circuit. The positive deviation of the plot of Re Y/(Im YωRC) versus log ω from unity at lower frequencies can be due to low-frequency relaxations in (33) Peng Diao, P.; Jiang, D.; Cui, X.; Gu, D.; Tong, R.; Zhong, B. J. Electroanal. Chem. 1999, 464, 61-67. (34) Starzak, M. E. The Physical Chemistry of Membranes; Academic Press: Orlando, 1984; pp 141-173.

the dielectric as well as complications at the interface between the coated electrode and the solution. The negative deviation of this plot from unity at high frequencies is indicative of a higher frequency relaxation of the capacitor. In this study, Re Y/(Im YωRuC) has been calculated and plotted versus frequency. The plots are displayed from frequencies greater than 100 Hz, where all the plots approach unity and any deviations can be clearly seen. In all diagrams, the frequency scale is expressed as the log of angular frequency, log(ω ) 2πf).

Results (a) Monolayer between -0.4 and -0.7 V. The phospholipid layer on mercury represents a near ideal capacitor between these potentials. This is evidenced by the near perfect semicircle of the complex capacitance plot (see Figure 1) and an almost absence of a lower frequency element. Further, the Re Y/(Im YωRuC) versus log ω plot conforms to unity at medium frequencies and at the highest frequency decreases slightly (see Figure 2a). The zero frequency capacitance (∼1.8 µF cm-2) is constant between the potentials -0.4 and -0.7 V (see Figure 1). At applied potentials equal to and more negative than -0.7 V, an additional element at lower frequencies on the plots in the complex capacitance plane becomes significant in the form of a “tail” (see Figure 1). (b) Monolayer between -0.7 and -0.935 V. At the onset of the first phase transition from -0.7 to -0.85 V,

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the same potential (Figure 3F). At -1.3 and -1.4 V, a second element fused to the RC semicircle is observed together with a significant “tail” at -1.3 V (Figure 3D,E). Such deviations from ideal capacitative behavior would be expected here because the potentials of -1.3 and -1.4 V are coincident with the phospholipid desorption process. (e) gA Interaction with the DOPC-Coated Electrode. Figure 4 displays (a) plots in the complex capacitance plane and (b) Re Y/(Im YωRuC) versus log ω plots of the DOPC-coated electrode in the presence of gA added to NaCl electrolyte. The interaction of gA with the DOPC monolayer at -0.4 V gives rise to the presence of lowfrequency relaxations and a positive deviation from unity of the Re Y/(Im YωRuC) versus log ω plots. The same effect is observed in the Mg (NO3)2 electrolyte, although the slope of the low frequency “tail” is decreased. Model

Figure 2. Re Y/(Im YωRuC) versus log ω plots derived from impedance data of the DOPC monolayer adsorbed on mercury in 0.1 mol dm-3 KCl. (a) -0.4 V (open squares), -0.85 V (open triangles), -0.985 V (closed circles), and -1.050 V (crosses); (b) -1.1 V (open circles), -1.2 V (crosses), and -1.5 V (closed triangles) and uncoated mercury in 0.1 mol dm-3 KCl at -1.0 V (squares).

the low frequency “tail” increases in significance (Figure 1). At more negative potentials (-0.85 V), a large positive deviation of the Re Y/(Im YωRuC) versus log ω plot from unity at low frequencies is evident (see Figure 2a) and correlates with the increasing significance of the lowfrequency relaxations on the plot in the complex capacitance plane. At potentials of about 5 mV, on either side of the potential characterizing the first phase transition (∼-0.935 V), only a single semicircle is observed on the complex capacitance plot corresponding to the RC charging of the monolayer undergoing the transition (Figure 1). (c) Monolayer between -0.95 and -1.050 V. At all potentials more negative than those characterizing the first phase transition, the form of the Re Y/(Im YωRuC) versus log ω plot approaches that derived from the Hg/ electrolyte interface impedance data (Figure 2b). The zero frequency capacitance has a value of ∼10 µF cm-2. A lowfrequency component in addition to the RuC element is apparent (Figure 1). At more negative voltages (-1.00 V) coincident with the second phase transition, this element develops into the first part of a semicircle that at even more negative potentials disappears (Figure 3A). It is interesting that the zero frequency capacitances of the coated electrode at potentials positive to and negative to the second phase transition are almost identical. Further, the Re Y/(Im YωRuC) versus log ω plots (Figure 2a), derived from the impedance data of the coated electrode at potentials positive (-0.985 V) and negative (-1.05 V) to potentials characterizing the second phase transition, are the same. (d) Monolayer between -1.1 and -1.5 V. The zero frequency capacitance increases at more negative potentials (Figure 3), and at -1.5 V, the value (∼17 µF cm-2) is equal to that of the mercury-electrolyte interface at

One way of developing a model that better explains the experimental data can be obtained when the ideal doublelayer capacitance is replaced by a so-called constant phase element, CPE ) 1/A(iω)β, which is originally an empirical formula.31,33,35-43 A is a frequency-independent real constant, and β is fractional with an experimental value between 1/2 (for an ideally porous electrode) and 1 (for a perfectly smooth electrode). The closer the β value approaches to 1, the smoother is the electrode surface. As a result, β values have been calculated in previous work for the various experimental plots of Re Y versus Im Y for supported monolayers under differing conditions. When applied to a RC network, the term (iωC) is replaced by (iω)βA. Alternatively, this can be rewritten as (iω)βω01-βC, where ω01-β is a dummy constant that corrects for units. If ω0 is set to unity, C is numerically equal to A. The β value gives a general idea of the nonideality of the RC circuit element. This treatment only considers the interface between the resistor and the capacitor. This can be perceived in the term (iω)βω01-βC, where, if β is less than unity, the capacitor does not sense the full ac frequency applied. This is clearly exhibited with a rough or porous electrode where the interface is complicated to various degrees.31,33,42 In the present work, the dielectric properties of the capacitor are also considered to be important especially when phospholipid layers on the electrodes are considered. For a simple RC series circuit, the admittance is given by34

Y)

1 R + (1/iωC)

(1)

In eq 1, the capacitance has no frequency dispersion. Equation 1 can be modified in several ways. First, the idea derived from the CPE is introduced, whereby the frequency term, iω, in eq 1 is put to the power β, where β is 1 or less.31,33,37-43 The additional dummy constant, ω01-β, is added, which corrects for units and is numerically (35) Lorenz, W.; Moeckel, F. Z. Elektrochem. 1956, 60, 507. (36) Armstrong, R. D.; Rice, W. P.; Thirsk, H. R. J. Electroanal. Chem. 1968, 16, 517-529. (37) Sadkowski, A. J. Electroanal. Chem. 2000, 481, 222-226. (38) Zoltowski, P. J. Electroanal. Chem. 1998, 43, 149-154. (39) La’ng, G.; Heusler, K. E. J. Electroanal. Chem. 1998, 457, 257260. (40) Berthier, F.; Diard, J.-P.; Michel, R. J. Electroanal. Chem. 2001, 510 , 1-11. (41) Schiller, C. A.; Strunz , W. Electrochim. Acta 2001, 46, 36193625. (42) Kerner, Z.; Pajkossy, T. Electrochim. Acta 2000, 46, 207-211. (43) Macdonald, J. R. J. Chem. Phys. 2002, 116, 3401-3409.

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Figure 3. Plots in the complex capacitance plane derived from impedance data of the DOPC monolayer adsorbed on mercury in 0.1 mol dm-3 KCl (A) at potentials (a) -0.998, (b) -1.0, (c) -1.006, (d) -1.012, and (e) -1.016 V and at potentials (B) -1.1, (C) -1.2, (D) -1.3, (E) -1.4, and (F) -1.5 V. Data are represented as crosses and eq 2 fits as the solid lines. R and β values with errors (SD) and χ2 from the fits are indicated on the diagram. Numbers on plots indicate frequencies of adjacent data points expressed as and representing values of log(ω, rad s-1) as follows: 1, 5.61; 2, 4.42; 3, 3.24; 4, 2.05; and 5, 0.87.

set to unity. Second, a capacitor with frequency dispersion is introduced with a capacitance, Cinf, at frequencies where dipole or charge movements are not active, a capacitance, Cs, which corresponds to the real dielectric in the direct current limit, and a time constant,τ, for the relaxation.34 Finally, if the relaxation time of the dielectric is not discrete it can be represented as a series of times distributed around a most-probable value.34,44,45 The width of the distribution is given by R, where R is 1 or less. In this “one capacitor model”, therefore, R modifies the properties of a classical Debye dielectric, whereas β characterizes the ideality of the interface between the capacitor and the resistance or, in this case, the interface between the coated electrode and the solution:

{

Y) R+ β

(iω) ω0

[

1-β

1 Cs - Cinf

1 + (iωτ)R

]}

-1

+ Cinf

(2)

All the experimental data derived from the monolayer on the electrode were fitted to eq 2, and the fitting procedure was carried out as follows. The Re Y/Im Y versus log ω plot was fitted using initial guesses for the

parameters estimated from observation of the data plots. The experimental Ru that is equivalent to R in eq 2 was inputted as a fixed parameter. Subsequently, the Re Yω-1 versus log ω plot was fitted. A value for the zero frequency capacitance, C, equivalent to Cinf in eq 2, was obtained. A second fitting was carried out holding Cinf constant because it is an experimental value. This was the final fit and provided the χ2 estimate of fitting and a standard deviation (SD) value for each parameter. The overall goodness of fit was examined by comparing the experimental and model values of Re Yω-1, Im Yω-1, and Re Y/(Im YωRuC) versus log ω plots and the Re Yω-1 versus Im Yω-1 plots. An example of the result of such a fit to the data using eq 2 is displayed in Figure 5. Table 1 shows values of the coefficients together with their errors derived from the fitting for representative experiments. The models were applied to the data in the following way. Equation 2 was fitted to the impedance data derived from a monolayer of DOPC at -0.4 V. Cinf in eq 2 was (44) Fruebing, P.; Kruger, H.; Goering, H.; Gerhard-Multhaupt, R. Polymer 2002, 43, 2787-2794. (45) Fruebing, P. Advanced lab experiments; Institute of Physics, University of Potsdam: Potsdam, Germany, 2002, pp 1-21.

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Figure 5. Plots (crosses) of (a) Re Yω-1, (b) Im Yω-1, and (c) Re Y/(Im YωRuC) versus log ω derived from impedance data of the DOPC-coated electrode in 0.1 mol dm-3 KCl at -1.050 V together with fits (solid line) using eq 2. Figure 4. (a) Plots in the complex capacitance plane and (b) Re Y/(Im YωRuC) versus log ω plots derived from impedance data of the DOPC monolayer adsorbed on mercury in 0.1 mol dm-3 NaCl (open circles), in 0.05 mol dm-3 Mg (NO3)2 (closed triangles), in 0.1 mol dm-3 NaCl with 12.7 nmol dm-3 gA (open squares), and in 0.05 Mg (NO3)2 with 12.7 nmol dm-3 gA (crosses). Equation 2 fits are shown as solid lines. R and β values with errors (SD) and χ2 from fits to data are indicated on the diagram next to data symbols. Measurements were carried out at -0.4 V. Numbers on the plots in part a indicate frequencies of adjacent data points expressed as and representing values of log(ω, rad s-1) as follows: 1, 5.61; 2, 4.42; 3, 3.24; 4, 2.05, and 5, 0.87.

identified with the zero frequency capacitance. The very small low-frequency relaxation outside the RuC semicircle was fitted by the model, but because of the large error of τ and Cs, the parameters are quoted as insignificant (NS). Equation 2 was also fitted to data derived from monolayers of DOPC at potentials more negative than -0.7 V, which showed the occurrence of significant low-frequency relaxations. Cinf in eq 2 was again identified with the zero frequency capacitance. The low-frequency relaxations corresponded to additional elements in the phospholipid layer with a capacitative element (Cs - Cinf ) and a long time constant (τ). The model can be used to distinguish between interfacial effects (β < 1) and dielectric effects (R < 1) on the admittance data. Where β is