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Threshold for Spontaneous Oscillation in a Three-Phase Liquid Membrane System Involving Nonionic Surfactant Ben Nanzai,* Tomohisa Funazaki, and Manabu Igawa Department of Material and Life Chemistry, Faculty of Engineering, Kanagawa UniVersity, 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama 221-8686, Japan ReceiVed: May 6, 2010; ReVised Manuscript ReceiVed: August 3, 2010
This study of self-oscillation was conducted using a new three-phase liquid membrane system of ethanol aqueous solution, benzyl alcohol solution with nonionic surfactant, and pure water. Relations of the initial ethanol concentration to the oscillation amplitude and frequency, and to the induction period before oscillations were investigated. The oscillation amplitude is independent of the initial ethanol concentration, but the frequency and the induction period are related to it. The oscillation frequency increased concomitantly with the increased ethanol initial concentration, but the induction period before the electrical oscillations decreased with increasing concentration. To estimate the influence of ethanol diffusion on the electrical oscillations, the ethanol concentration in each phase was measured using separate experiments after different durations of oscillation. The diffusion coefficient was calculated using Fick’s second law. Results show successful estimation of the threshold for oscillations. The threshold is defined in terms of the ethanol concentration at the interface between the benzyl alcohol phase and the pure water phase. 1. Introduction In biological systems, rhythmic oscillations are used in transmission systems such as nerves, synapses, and taste and olfactory senses.1-7 Such oscillations are phenomena resulting from the direct conversion from chemical energy to mechanical or electrical energy. Various rhythmic oscillations of electrical potential and convection flow are apparent in systems with a clear boundary. Dupeyrat and Nakache first suggested the fundamental mechanism of liquid-phase oscillation phenomenon in a two-phase system (water and oil).8 They reported that the contact between a two-phase liquid, i.e., cation surfactant (octadecyl-trimethyl ammonium chloride) aqueous solution and I2 nitrobenzene solution saturated with KI, produced the liquidphase oscillation phenomenon. They reported that interfacial tension variations and the Marangoni effect9 induced this phenomenon. Yoshikawa and Matsubara described a similar oscillation phenomenon occurring in a three-phase liquid membrane system10 consisting of the phase of nitrobenzene containing picric acid imposed between two aqueous phases, one containing a cation surfactant such as cetyl-trimethyl ammonium bromide (CTAB). They concluded that the electrical potential oscillation in that study resulted from the repetitive formation and destruction of a monolayer of CTA+ at the water-oil interface. Many subsequent reports have described similar three-phase experimental systems. Yoshikawa et al. also reported results of widely various investigations related to (1) the oscillations of pH and electrical potential using computer simulation,11 (2) the influence of alcohol concentration on the oscillations,12 and (3) the oscillations with the four basic species of different taste categories, such as salty, sweet, bitter, and sour.13 The Gohoshi group conducted experiments elucidating the effect of a three-phase apparatus design on electrical oscillations.14,15 Yoshidome et al. attempted to apply the oscillation system to the analysis of inorganic compounds.16,17 * To whom correspondence should be addressed. Tel.: (+81)-45-4815661 × 3880. Fax: (+81)-45-413-9770. E-mail:
[email protected].
In addition, our group previously reported the oscillation phenomenon from the viewpoint of metal ion transport.18 Recently, Szpakowska et al. explained oscillation phenomena in terms of the adsorption and the desorption of ion pairs.19,20 Pimienta et al. observed the relations between the electrical potential oscillations and interfacial tension oscillations.21 Kovalchuk and Vollhardt simulated the various model system behaviors and confirmed them from interfacial tension oscillations.22-24 One study in particular24 was performed using a system involving a nonionic surfactant. Ogawa et al. compared experimental and calculated results of electrical oscillations.25 Nevertheless, details of the rhythmic oscillations of electrical potential and the convection flow at three-phase liquid membrane remain unclear. As described in this work, we proposed a new experimental system using a three-phase liquid membrane to characterize oscillation phenomena more simply. This new three-phase system consists of an aqueous solution of ethanol, benzyl alcohol solution containing nonionic surfactant, and pure water. As characteristic points of this system, (1) the oscillation phenomenon has not been investigated with these three-phase components, (2) the surfactant is nonionic, (3) the surfactant is included in the organic phase, and (4) the glass tube is straight. Condition (1) was chosen for several reasons. First, as a transport solute, ethanol is an eminently treatable material with high water solubility. Furthermore, ethanol is well-known as an antifoam agent that changes its interface composition in the presence of surfactant. Second, as an organic phase, benzyl alcohol is also treatable; it has low toxicity and little odor. Third, as a surfactant, polyethylene glycol mono-4-nonylphenyl ether (POE) has no electrical charge as described above. Almost all oscillation systems in previous studies involved ionic surfactants. As the reason for condition (2), this ionic property was considered as a principal reason for the oscillation phenomena described in previous reports. Consequently, adoption of the nonionic surfactant is important to establish a new theory of oscillation phenomena: we propose that a dipole of nonionic surfactant
10.1021/jp104116q 2010 American Chemical Society Published on Web 08/23/2010
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Figure 1. Schematic of the three-phase oscillation system. The scale ratio of the figure differs from the actual. Glass tube inner diameter, 1.5 mm; length, 10 mm. The respective distances of the Pt needle to interface I and interface II are 15 and 20 mm, respectively.
generates the electrical potential. Jones et al. and Sugawara et al. reported that an electrode with a barium and calcium salt sensor shows a stable response to nonionic surfactants.26-28 In addition, Kojima et al. discovered that the nonionic surfactants show charge separation only at the water-oil interface instead of in the liquid membrane.29 Furthermore, Dudnik and Lunkenheimer estimated the surface potential of nonionic surfactant at the air-water interface.30 Regarding condition (3), it is a vestige of previous reports18 that the surfactant is used as an ion carrier. In biological systems, surface active solutes also exist in the membrane instead of outside. However, in this study, the surfactant location has important meaning for oscillation phenomena: the surfactant exists at both liquid-liquid interfaces. A reason for condition (4) is that most previous studies were performed using a U-shaped glass tube. However, we selected a straight-type glass tube for this study to provide improved reproducibility and ease of understanding.18 In the U-shaped glass tube, the distances between liquid-liquid interfaces and the electrical oscillations are different according to their position on the interface (i.e., inside corner or outside corner).14,31 However, the distances between interfaces shows little difference in the straight glass tube. Additionally, we investigated the effect of ethanol concentration on the electrical potential variations to treat this phenomenon systematically. Most previous reports are presented as qualitative discussions because of the complexity of this phenomenon. 2. Experimental Section Ethanol (99.5%), benzyl alcohol (Wako Pure Chemical Industries Ltd.), polyethylene glycol mono-4-nonylphenyl ether (POE n ) 2; Tokyo Chemical Industry Co. Ltd.) were used for this study in addition to ethanol aqueous solutions with concentrations of 1.0-4.5 M prepared using Milli-Q water. The concentration of the surfactant POE was adjusted at 100 mM in the benzyl alcohol solution. A three-phase liquid membrane (60/20/60 µL) was formed in a linear glass tube (100 mm) with 1.5 mm inner diameter. Although water and benzyl alcohol have high mutual solubility, we used pure liquid solvents for the setup of each oscillation experiment in this study. POE n ) 2 is only slightly dissolved in water. Using a microsyringe, these phases were put in the order corresponding to the pure water phase (phase III) first, the benzyl alcohol phase (phase II) second, and the ethanol aqueous solution phase (phase I). Then the platinum needle was placed in each aqueous phase as an electrode. For high reproducibility, the distances from each water-oil interface to the electrode were optimized empirically, respectively, at 15 and 20 mm. We performed some preliminary experiments and confirmed better reproduc-
ibility in this condition. Figure 1 presents details of the threephase setup. Electrical potential measurements were conducted using an electrometer (3120A; Toho Technical Research Co., Ltd.). Considering the electrostatic effect, antistatic treatment was performed on the entire system. Electrical potential measurements were performed for 720 min. To determine the amount of ethanol in each phase, samples were removed after 3, 6, 9, 12, 15, 30, 60, 90, 120, 240, 480, 600, and 720-min experiments and measured. Each operation was performed independently as a different trial. Visual observations of the liquid-liquid interface in the three-phase system were made using a digital microscope (BS-D8000II; Sonic Co. Ltd.). The ethanol concentration in each phase was determined using gas chromatography (GC-9A; Shimadzu Corp.) with a flame ionization detector and a packed column (3 mmø × 1.0 m), Porapack N 80/100 mesh. Methanol was used as an internal standard substance. 3. Results and Discussion 3.1. Amplitude and Frequency of Oscillation. The variations of electrical potential show some baseline wandering. These behaviors are apparently derived from some composition changes in the triple phase, although they are unclear. In this study, for simplicity of discussion, this broad baseline wander is omitted from the oscillation results by curve fitting of highdimensional function. Figure 2 shows the representative electrical oscillations in some different initial concentrations of ethanol in phase I. Electrical oscillation was observed in each case of the initial concentration of ethanol, although the surfactant has no charge. In a preliminary study, steady electrical oscillations were confirmed as being absent from the system without ethanol or POE. The oscillation does not occur solely because of the electrical charge of the surfactant. Furthermore, this electrical oscillation has sharp and periodic spikes. The amplitude of oscillations decreased gradually with time. No significant relation was found between the amplitude and the initial concentration of ethanol. However, the oscillation frequency showed a certain dependence on the initial concentration of ethanol. Figure 3 presents the relation between the initial concentration of ethanol and the oscillation frequency. The frequency of oscillations was measured during stable oscillations for 10 min after each induction period. Experiments were performed three times in all cases. They show high reproducibility. Results show that the frequency of oscillations increased concomitantly with increasing initial concentration of ethanol. Along with the results of oscillation amplitude, this result of oscillation frequency agrees well with the result of the report of Yoshikawa and Matsubara.12 They concluded that the addition
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Figure 2. Oscillation examples of the electrical potential difference from each initial concentration of ethanol in phase I (2.0, 3.0, and 4.0 mol L-1). The baseline has been pretreated for predigestion. Other trials correspond with these tendencies in the case of the other initial concentration of ethanol.
Figure 3. Relation between the initial concentration of ethanol in phase I and the frequency of electrical oscillations. Each value is the average of at least three trials. The error bar shows the standard deviation.
of ethanol affects the critical micelle concentration and explained the existence of a flexion point in the correlation between the initial concentration of ethanol and the oscillation frequency. A flexion point is noted at the 2.0-2.5 M initial concentration of ethanol in Figure 3. Because details of this mechanism remain unclear, much remains to be investigated. 3.2. Visual Observations. For this study, we also conducted visual observations of both liquid-liquid interfaces. We were able to observe and count each single oscillation movement at interface II using a digital microscope. Visual observations show that the time span of electrical oscillations matched with that of interfacial oscillations at interface II. Figure 4 presents a representative comparison between the frequency of interfacial oscillations and that of electrical oscillations in the case of 4.0 M of ethanol. Results verified good correspondence between the oscillation frequencies. On the basis of these results, it can be considered that the electrical oscillations occurred because of interfacial oscillations at interface II, which agrees well with values reported by Szpakowska.19,20 Meanwhile, the interfacial
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Figure 4. Frequency of the electrical oscillations and interfacial oscillations between phases II and III at each time after the three-phase formation. Each value of electrical oscillations is the average of at least three trials. The error bar shows the standard deviation. The initial concentration of ethanol in phase I was 4 M.
oscillations at interface I were fast and disordered. Consequently, they were not periodic. From these results, the interfacial oscillations seemed to depend on the ethanol concentration at each interface. The adsorption and desorption or passage of ethanol at the interface induced the interfacial oscillations. The ethanol concentration at interface I was so high that the stable oscillation is difficult to form. In contrast, the ethanol concentration at interface II increased gradually with the ethanol diffusion. Therefore, the interfacial oscillation at interface II and electrical oscillation were periodic and increased with time, as shown in Figure 4. Details of the oscillation mechanism will be discussed in a later section. However, some differences are apparent between the electrical oscillations and the interfacial oscillations in the start and depression of oscillation. In addition, the generation of submicrometer bubbles at interface II was observed. Further consideration is necessary to elucidate the mechanism of these phenomena, but they are apparently induced by the transfer of ethanol. 3.3. Induction Period of Electrical Oscillations. As Figure 2 shows, an induction period exists before the electrical oscillations in each case. Furthermore, the induction period length seems to correlate with the increasing initial concentration of ethanol. Figure 5 portrays the relation between the initial concentration of ethanol and the induction period length. The average values were calculated from experiments performed at least five times. The induction period length decreased concomitantly with the increasing initial concentration of ethanol. The ethanol concentration has great relevance to the electrical oscillation, as described above. In other words, the existence of ethanol induces oscillation phenomena; the abundance of ethanol in a certain region is important. From this viewpoint, it can be presumed that the diffusion of ethanol molecule from phase I to phase II strongly influences the phenomena. Therefore, we describe estimation of the amount of ethanol diffusion to phase II and phase III in the next section. In addition, from Figure 5, the flexion point is around 2.0 M of ethanol, which shows similarity with results related to the oscillation frequency depicted in Figure 3. Some relation exists between the existence of ethanol molecules and the behavior of surfactant molecules, as discussed in the previous section.
Spontaneous Oscillation in a Liquid Membrane System
Figure 5. Relation between the initial concentration of ethanol in phase I and the length of induction period before electrical oscillations. Each value is the average of at least five trials. The error bar shows the standard deviation.
Figure 6. Concentration change of ethanol in each phase. Each symbol represents the initial concentration of ethanol in phase I. ∆:4 M, ]:3 M, 0: 2 M, O:1 M.
3.4. Amount of Ethanol Diffusion. For each phase, the concentration of ethanol over time is depicted in Figure 6 (1.0, 2.0, 3.0, 4.0 M are noteworthy examples). The reproducibility of these data is high, but the trial is performed once for one plot independently in Figure 6. The concentration of ethanol in phase I decreased gradually with time. Simultaneously, the phase II concentration increased. Subsequently, that in phase III increased. These behaviors show the same tendency as that of common diffusion. The diffusion rate increased concomitantly with increasing initial concentration of ethanol. To clarify the inception of oscillations, the ethanol concentration of phase II after the induction period is estimated from the
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Figure 7. Oscillation of electrical potential difference in the two-phase system with each initial concentration of ethanol in benzyl alcohol solution (0.20, 0.25, 1.0 mol L-1). The baseline has been pretreated for predigestion.
approximation presented in Figure 6. Results show that the concentrations of phase II are dependent on the initial concentration of phase I, although we presumed that a certain critical concentration induces the oscillation. On the basis of the discussion presented above, the electrical oscillation seemed to arise at interface II. In addition, considering the diffusion of ethanol molecule, the concentration of phase II is inhomogeneous. For this reason, the measured concentrations of phase II can be interpreted as the average value. In other words, it is necessary to consider the ethanol concentration at interface II. Therefore, we performed oscillation experiments using two phases. The first phase is benzyl alcohol including ethanol and nonionic surfactant; the second phase is pure water. 3.5. Two-Phase Experiment. We conducted two-phase experiments using benzyl alcohol solutions of 0.20, 0.25, and 1.0 M ethanol. The ethanol in the benzyl alcohol solution is distributed homogeneously; the concentration can be regarded as equal to the ethanol concentration of the liquid-liquid interface. Figure 7 shows electrical oscillations in each twophase system. The oscillations were observed in the case of initial ethanol concentrations of more than 0.25 M. Additionally, the induction period is short: less than 2 min, which is less than any other value of the three-phase experiment presented in Figure 5. That short induction period means that the ethanol concentration is homogeneous in the benzyl alcohol solution. The oscillations start soon after the two-phase system forms and stabilizes. In other words, the induction period is the time necessary for the diffusion of ethanol from interface I to interface II in the three-phase system. These results show that the electrical oscillations are generated when the ethanol concentration of interface II is greater than 0.20-0.25 M. This value is interpreted as the threshold to induce the electrical oscillations. When the interfacial concentration of ethanol attains this value, the balance of interfacial condition is disrupted and the convection flow is generated because of the Marangoni effect.9 To support additional discussions of ethanol diffusion, the diffusion coefficient of ethanol in phase II is estimated next. 3.6. Diffusion Coefficient. To estimate ethanol diffusion, the diffusion coefficient was calculated using Fick’s law. Because the concentrations of three-phase system are inhomogeneous and do not achieve equilibrium in this study, we adopt Fick’s second law, as described below (eq 1). It is solved for concentration c (eq 2). This equation is known as an “error-
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Figure 8. Pseudodiffusion coefficient in each case of initial concentration of ethanol in phase I (1.0, 2.0, 2.5, 3.0, and 4.0 mol L-1). These values are calculated using the equation derived using Fick’s second law. The error bar is attributable to the variability of the induction period that is assigned to the equation. A dashed line indicates the diffusion coefficient when there are no interfacial oscillations.
function integral”; it is therefore inverted using the error function as follows (eq 3). The solution to Fick’s second law is based on Crank’s solution.32
∂c ∂2c )D 2 ∂t ∂x c)
c0 2√πDt
[
(1)
∫ e-x /4Dtdx
(2)
( )]
(3)
2
c ) c0 1 - erf
x 2√Dt
Therein, c signifies the concentration at a given point after the diffusion, D represents the diffusion coefficient, and x denotes the distance of diffusion. Additionally, c0 stands for the concentration of the diffusion source, and t represents the time of diffusion. In this study, c is the concentration of interface II. It is presumed to be 0.25 M from the results of the preceding section. The distance x between interface I and interface II is 10 mm. The source concentration c0 is assumed to become homogeneous and equal to the concentration at interface I when the concentration change of ethanol becomes small. The concentration of the phase II side interface is calculated from the partition coefficient of ethanol between water and benzyl alcohol. The coefficient was obtained as 0.85 from an experiment of partition equilibrium. The time of diffusion t is the induction period before the electrical oscillations. From substitution of these values, the diffusion coefficient in each case is calculated as 4.5-5.2 mm2 min-1. These values are several dozen times larger than the level of free diffusion (10-1 mm2 min-1 or less).33 These calculated diffusion coefficients are therefore considered as pseudodiffusion coefficients. Figure 8 presents the pseudodiffusion coefficients in respective cases of initial ethanol concentration. These results suggest that these large values are derived from the flow in phase II, as induced
by the violent turbulent flow of interface I. Consequently, ethanol molecules are transported not only by diffusion but also by convection flow. Actually, in the case of 0.10 M initial ethanol concentration with no interfacial oscillations, the diffusion coefficient is calculated as 0.30 mm2 min-1 from the average concentration of ethanol in phase II after certain diffusion. This value is shown as a dashed line in Figure 8 and closely approximates the level of the free diffusion coefficient described above. 3.7. Oscillation Mechanism. We examined the mechanism of oscillation phenomena in a liquid membrane system. First, the role of ethanol is considered. Ethanol, which is necessary to induce the Marangoni effect at the interface, is known as an antifoaming agent; it decreases the interface tension where a surfactant is present. The surfactant molecules move from the lower to the higher interface tension involving the surrounding solution. The presence of a certain level of ethanol in this study triggers this effect at both liquid-liquid interfaces. The benzyl alcohol solution includes the surfactant. Therefore, it can exist at both interfaces. Second, the turbulent flow is considered. The surfactant monolayer was formed at interface II and the convection flow induced by the Marangoni effect disrupted the layer. Then the formation and destruction of the monolayer repeated themselves periodically. This repetition produces interfacial oscillations. Finally, the source of the electrical potential is considered. On the basis of reports about the electrical potential because of the nonionic surfactant,26-28 we propose that a dipole of nonionic surfactant generates the electrical potential described in the Introduction section. These reports suggest that the presence of monolayer of nonionic surfactant at interface II engenders the difference of electrical potential. Then the monolayer is disrupted; the electrical potential changes. The repetitive cycle of monolayer formation produces the electrical potential oscillations. This viewpoint is almost identical to that for a case described in a previous study with an ionic surfactant.10 For clarification of details, however, further studies must be performed. 4. Conclusions We observed interfacial and electrical oscillation phenomena using an entirely new three-phase system. Electrical oscillations were synchronized with interfacial oscillations between phase II and phase III. The induction period before the electrical oscillations decreased concomitantly with increasing initial concentration of ethanol. The ethanol diffusion affects the occurrence of oscillations. For this occurrence, the threshold of interfacial concentration of ethanol between phase II and phase III was estimated as 0.20-0.25 M. Additionally, the pseudodiffusion coefficient was calculated as several square millimeters per minute, which is several tens of times higher than the free diffusion coefficient. On the basis of the results of this study, the mechanism of oscillations is inferred as the Marangoni effect because of the ethanol adsorption in addition to the generation of dipole of nonionic surfactant at the liquid-liquid interface. Acknowledgment. This work was funded by the Science Frontier project “Nanochemistry for the Creation of Functional Materials” of the Ministry of Education, Culture, Sports, Science, and Technology of Japan. References and Notes (1) Hodgkin, A. L. Brit. Med. Bull. 1950, 6, 322–325. (2) Teorell, T. J. Gen. Physiol. 1959, 42, 831–845.
Spontaneous Oscillation in a Liquid Membrane System (3) Teorell, T. J. Gen. Physiol. 1959, 42, 847–863. (4) Shashoua, V. E. Nature 1967, 215, 846–847. (5) Nicolis, G.; Prigogine, I. Self-Organization in Non-Equilibrium Systems; Wiley Interscience: New York, 1977. (6) Ulrich, F. F. Upsala J. Med. Sci. 1980, 85, 265–282. (7) Langer, P.; Page, K. R.; Wiedner, G. Biophys. J. 1981, 36, 93– 107. (8) Dupeyrat, M.; Nakache, E. Bioelectrochem. Bioener. 1978, 5, 134– 141. (9) Scriven, L. E.; Sternling, C. V. Nature 1960, 187, 186–188. (10) Yoshikawa, K.; Matsubara, Y. Biophys. Chem. 1983, 17, 183–185. (11) Yoshikawa, K.; Matsubara, Y. J. Am. Chem. Soc. 1983, 105, 5967– 5969. (12) Yoshikawa, K.; Matsubara, Y. J. Am. Chem. Soc. 1984, 106, 4423– 4427. (13) Yoshikawa, K.; Shoji, M.; Nakata, S.; Matsubara, Y. Langmuir 1988, 4, 759–762. (14) Miyamura, K.; Morooka, H.; Hirai, K.; Gohshi, Y. Chem. Lett. 1990, 19, 1833–1836. (15) Yoshihisa, H.; Sutou, S.; Miyamura, K.; Gohoshi, Y. Anal. Sci. 1998, 14, 133–136. (16) Yoshidome, T.; Higashi, T.; Mitsushio, M.; Kamata, S. Chem. Lett. 1998, 27, 855–856. (17) Yoshidome, T.; Higashi, T.; Mitsushio, M.; Kamata, S. Chem. Lett. 2000, 29, 550–551. (18) Igawa, M.; Otani, K.; Horiguchi, S.; Okochi, H. Chem. Lett. 1997, 26, 923–924. (19) Szpakowska, M.; Czaplicka, I.; Nagy, O. B. J. Phys. Chem. A 2006, 110, 7286–7292.
J. Phys. Chem. B, Vol. 114, No. 36, 2010 11783 (20) Szpakowska, M.; Plocharska-Jankowska, E.; Nagy, O. B. J. Phys. Chem. B 2009, 113, 15503–15512. (21) Pimienta, V.; Lavabre, D.; Buhse, T.; Micheau, J. C. J. Phys. Chem. B 2004, 108, 7331–7336. (22) Kovalchuk, N. M.; Vollhardt, D. J. Phys. Chem. B 2006, 110, 9774– 9778. (23) Kovalchuk, N. M.; Vollhardt, D. Colloid Surf. A 2007, 309, 231– 239. (24) Kovalchuk, N. M.; Vollhardt, D. J. Phys. Chem. C 2008, 112, 9016– 9022. (25) Ogawa, T.; Shimazaki, H.; Aoyagi, S.; Sakai, K. J. Membr. Sci. 2006, 285, 120–125. (26) Jones, D. L.; Moody, G. J.; Thomas, J. D. R. Analyst 1981, 106, 439–447. (27) Jones, D. L.; Moody, G. J.; Thomas, J. D. R.; Birch, B. J. Analyst 1981, 106, 974–984. (28) Sugawara, M.; Nagasawa, S.; Ohashi, N. J. Electroanal. Chem. 1984, 176, 183–194. (29) Kojima, K.; Sugawara, M.; Umezawa, Y. Anal. Sci. 1985, 1, 477– 478. (30) Dudnik, V.; Lunkenheimer, K. Langmuir 2000, 16, 2802–2807. (31) Yoshidome, T.; Takahama, J.; Mitsushio, M.; Higo, M. Anal. Sci. 2003, 19, 603–606. (32) Crank, J.; Nicolson, P. Proc. Camb. Phil. Soc. 1947, 43, 50–67. (33) Lide, D. R. CRC Handbook of Chemistry and Physics, 84th ed.: CRC Press LLC: Boca Raton, 2003.
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