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Transduction of electrochemical into mechanical oscillations
Physics Letters A 361 (2007) 488–492 www.elsevier.com/locate/pla
Transduction of electrochemical into mechanical oscillations B. Maiworm, M. Schmick, M. Markus ∗ Max-Planck-Institut für Molekulare Physiologie, Postfach 50247, D-44202 Dortmund, Germany Received 26 January 2006; received in revised form 9 May 2006; accepted 26 September 2006 Available online 5 October 2006 Communicated by A.P. Fordy
Abstract We observe aperiodic mechanical oscillations of a drop of oxidant solution placed around an iron stick on a mercury surface. These oscillations are explained as a visual manifestation of electrochemical oscillations at the iron–solution interface that, via electrocapillarity, lead to changes of the surface tension at the drop’s bottom. © 2006 Elsevier B.V. All rights reserved. PACS: 82.40.Bj; 68.03.Cd; 82.45.Bb Keywords: Electrochemical oscillations; Surface tension; Beating mercury heart
1. Introduction The system investigated here resembles the so-called “beating mercury heart” [1–7] and an analogous system using a gallium–indium–tin alloy [8]. We shall show below that these systems have some properties in common with ours, but that they differ substantially in their mechanism. One of the best known setups of the beating mercury heart consists of a mercury drop separated sideways from an iron needle and covered with a solution of sulphuric acid and potassium dichromate K2 Cr2 O7 . The oxidation of the mercury by the dichromate charges positively the mercury surface and thus reduces its surface tension; the drop thus flattens, touches the iron, recovers electrons, curls up due to the increased surface tension and then the process starts again. In the setup described in this work it is not the mercury drop that beats, but a drop of the dichromate-sulphuric-acid solution placed on a mercury surface. An iron stick passes vertically through the drop and is constantly immersed in the mercury, as shown in Fig. 1. U is the measured voltage due to the elec* Corresponding author.
E-mail addresses: [email protected] (M. Schmick), [email protected] (M. Markus). URL: http://www.mpi-dortmund.mpg.de/markus. 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.09.075
Fig. 1. Scheme of the experimental setup. The angle Θ is given by Eq. (1).
trochemical processes in the system. We also investigated the behavior in the case where U is not endogenous, but imposed by us externally, using a Pt-electrode instead of Fe to eliminate the electrochemical oscillations. Another process that resembles ours is the manipulation of tiny drops (< 1 µl) using electrowetting [9,10]. In this process, an ac or dc voltage is applied between the drop and a metal surface, generally separated from the drop by an insulating layer. If the voltage is applied between two parallel planar electrodes, a drop between them oscillates, alternating between bridging the electrodes and breaking into two drops [11]. Still another system related to ours is a dc-controlled oscillator on nanoscales, consisting of two molten metal particles on a carbon tube [12]. The system with the parallel electrodes and the nanosystem, both mentioned in the preceding paragraph are related to ours in the sense that geometrical properties of the liquid vary due
B. Maiworm et al. / Physics Letters A 361 (2007) 488–492
489
Fig. 2. Experimentally observed, mechanical oscillations of a drop of oxidant solution on a mercury surface. The total time interval for these five pictures is given by T in Fig. 3(c). Times: 1.4 s, 2.4 s, 3.4 s, 4.4 s and 5.4 s. The white spots are H2 -bubbles.
to changes in surface tension, the latter varying due to changes of the voltage. However, these systems differ from ours in that they require an externally applied voltage, while in our system an ac-voltage is generated endogenously by electrochemical oscillations. Such electrochemical oscillations occur on a metal surface (iron in our case) in contact with an electrolyte [13–17]. In particular, such oscillations have been investigated for iron in the presence of a dichromate-sulphuric-acid solution [18,19], as in the present work. In such a system, the iron first goes into solution as Fe2+ ; this is followed by the formation of a nonconducting FeCrO4 -film on the iron, which then dissolves as it reacts with protons so that Fe2+ is released again. The following irreversible processes occur, as oscillations proceed: reduction 3+ of Cr2 O2− 7 to Cr , production of H2 and dissolution of solid 2+ iron into Fe [18]. These are the same overall processes that occur in the beating mercury drop [5]. However, the observations described below indicate substantial differences in the present system, since here the oscillations of the surface tension and of the drop’s radius are driven by the oscillations on the iron surface. In contrast, in the beating mercury drop iron only acts as an electron donor and is not in constant contact with the mercury. 2. Experimental procedures A dish with a diameter of 15 cm was filled with Hg up to a height of 13 mm. The oxidant solution contained H2 SO4 (7 M) and K2 Cr2 O7 (1.16 mM). A drop (volume Vd = 200 µl) of this solution was poured around an iron stick (diameter: 1 mm; length: 4 cm), which had been placed vertically through the Hg at the center of the Hg surface. Its shape self-organized symmetrically as a circle around this stick. In one series of experiments (Figs. 2 and 3), the potential U was measured between the iron and a Pt electrode immersed in the solution, as indicated in Fig. 1. The oscillations of the drop were monitored with a video camera. The radius of the drop was determined (as a function of time) from the image on the monitor. In additional experiments, we drove the system externally, replacing the iron by a second Pt-electrode and applying the potential U between the two electrodes; in these experiments, potential pulses were applied and the temporal response of the drop’s radius was determined (Fig. 4). Also, we measured the drop’s radius r by increasing (quasistatically) the potential U at a rate of 1 V/min (Fig. 5(a)). Note that we increased, instead of decreased U , so that the relaxation times were as small as pos-
Fig. 3. (a, b) Self-oscillations of the potential. (b) Enlarged interval. (c) Oscillations of the radius of the drop, corresponding to (b).
sible (compare the relaxation of r for upward steps of U with that for downward steps in Fig. 4). We want to stress that all potentials appearing in the present work are defined as given in Fig. 1, i.e. between Pt and Pt, or between Pt and Fe. This should be kept in mind when comparing with measurements using standard electrodes, as found in the literature. 3. Measurements The oscillations of the drop are illustrated in Fig. 2 by snapshots after equal time intervals. Fig. 3(a) shows the measured
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B. Maiworm et al. / Physics Letters A 361 (2007) 488–492
Fig. 4. (a) Imposed pulses of potential intended to be an analogon to the self-oscillations shown in Fig. 3(b). (b) Measured responses (drop’s radius r) to the pulses shown in (a).
Fig. 5. Measured drop’s radius r (a) and σ (b) (surface tension at the mercury–air interface minus that at the mercury–solution interface, resulting from measurements and Eq. (6)) vs. the applied potential U .
self-oscillations with the iron stick. These were always aperiodic. Fig. 3(b) displays an enlargement with a few selected periods. Fig. 3(c) shows the changes of the radius r of the drop corresponding to Fig. 3(b). If we replace the iron by Pt and apply short potential pulses (Fig. 4(a)), we obtain a similar response of the radius (Fig. 4(b)) as that for the self-oscillations with iron shown in Fig. 3. The similarity of Figs. 4(a) and 4(b) with Figs. 3(b) and 3(c) indicates that the mechanical oscillations illustrated in Figs. 2 and 3 are driven by electrochemical oscillations at the iron stick. In addition, we determined the drop’s radius r, as a function of U , by quasistatically changing U . The result is shown in Fig. 5(a); the steps in this figure are caused by the limited resolution of our video tool. 4. Determination of the surface tension We shall now describe a handy way to determine the surface tension at the boundary of mercury, solution and air, using our setup. Note, that we cannot adopt the equations used for submicroliter drops [9,10] because gravitation is neglected there. In contrast, to obtain oscillations in our experiments, we require much larger drops, which are considerably deformed by gravitation.
We consider the surface tensions σMA (mercury–air interface), σMS (mercury–solution interface) and σSA (solution–air interface). At the curve shared by the three interfaces, Young’s equation σMA − σMS = σ = σSA cos Θ
(1)
holds. Θ is the angle between the drop’s surface and the mercury surface. Contrary to microdrops [9,10], which have a spherical shape, the shape of the drop in our experiments can be described by a flat lower surface (radius: r) and a flat upper surface (radius: r − ; see Fig. 2 and sketch in Fig. 1). If we approximate the drop’s shape by the frustum of a cone (height: h), the angle between the curved side and the base being Θ, then its volume is Vd = πh/3(3r 2 − 3r + 2 ); its upper surface is Au = π(r − )2 and its curved lateral surface is Ac = πh(2r − )/ sin Θ. From our experiments, we estimate 0.02 cm, while 0.7 cm < r < 1.3 cm (see Fig. 2). Thus, r; we therefore make the approximations Vd = πr 2 h, Au = πr 2 and Ac = 2πrh/ sin Θ, i.e. Ac = 2Vd /(r sin Θ). We shall now consider the equilibrium condition that the change of the total energy must be zero for a virtual displacement dr. We set σSA = 70.5 dyn/cm (measured by us using a capillary tube) and σMA = 471 dyn/cm. We assume that σMA is unaffected by the electrical charge at the drop; thus, σMA
B. Maiworm et al. / Physics Letters A 361 (2007) 488–492
is set constant. The surface energy of the mercury surface is EM = σMS πr 2 + σMA (AM − πr 2 ), where AM is the constant total area of that surface; thus dEM = −2σ πr dr − πr 2 d(σ ).
(2)
The surface energy of the solution–air interface is ES = σSA (Au + Ac ); thus dES = σSA 2πr − 2Vd / r 2 sin Θ dr − 2σSA Vd dΘ/(r sin Θ tan Θ),
(3)
where, by virtue of Eq. (1), dΘ = −d(σ )/(σSA sin Θ).
(4)
Finally, the change in gravitational energy by the change dh = −Vd dr/(πr 3 ) of the center of mass is Vd 2 ρg (5) dr. πr 3 The condition dEM + dES + dEP = 0, along with Eqs. (1) through (5), yields d(σ ) 2Vd 1− dr πr 3 sin2 Θ tan Θ 2Vd 2 Vd 2 ρg 2(σ ) − 4 . = σSA (6) − 2 5 − r r πr sin Θ π r
dEP = −
Considering σ = σSA cos Θ (Eq. (1)), we integrated the differential equation (6), in order to obtain σ (r). Since we observed Θ ≈ 90◦ at r = 0.7 cm, we set σ = 0 for this value of r. σ (r), together with r(U ), as given by Fig. 5(a), yielded the net horizontal surface tension σ (U ), which is shown in Fig. 5(b). 5. Discussion We reported here on a mechanical manifestation of electrochemical oscillations. Besides making the latter visible to the bare eye, this device is—along with the “beating mercury heart”—a novel and easy-to-build chemically driven “motor”. The oscillations that we observed have the shape of sharp recurrent peaks, as those in previous reports [14–16,18–20]. While both periodical and aperiodical oscillations have been reported, we did not obtain the former. One reason may be the variations of the area of the iron–solution interface in our case; another reason may be a feedback to the iron from the mercury– solution double-layer; still another reason may be related to the Pt–solution interface. We showed that the electrochemical oscillations drive mechanical oscillations of the drop. In addition, measurements of the drop’s radius along with an energy analysis allowed to determine the net horizontal surface tension σ at the border of the drop on the mercury surface, as a function of U . We summarize the meaning of Fig. 5 as follows: if U changes in time (either electrochemically or by imposed voltages) there occurs a change in σ , as given by Fig. 5(b), which (according to Eq. (6)) causes a change in the radius of the drop (Fig. 5(a)).
491
6. Outlook An experimental result that remains to be explained is the slow relaxation of r as U decreases, as compared with the fast relaxation of r as U increases. This relaxation asymmetry occurs both in self-oscillations (Figs. 3(b) and 3(c)) and in imposed pulses of U (Fig. 4). The limitations of our setup did not allow to measure the potential UD at the double-layer at the mercury–solution interface. In a preliminary approach [21], we had assumed that UD ≈ U . However, subsequent experiments (using W- or Auinstead of Pt-electrodes) showed that the contribution to U of the potential at the Pt–solution interface cannot be neglected, as compared to UD . A redesigned setup, allowing to measure UD using a standard electrode that eliminates liquid junction potentials (see e.g. [22]), would permit to determine the charge per unit area q = −dσMS /dUD = d(σ )/dUD , as well as the differential capacity CD = −dq/dUD of the double layer. One may then be able to show that the total differential capacity d|Q|/dUD (where Q = qπr 2 ) can become negative; in fact, if we assume that U and UD are positively correlated (as in our approximation U ≈ UD in [21]), then an increase in UD causes a shrinkage of the capacitor’s size (see Fig. 5(a)). Considering q, Q < 0, dQ d(qr 2 ) dr d|Q| =− = −π = πr 2 CD − 2πqr . (7) dUD dUD dUD dUD On the right side of this equation the second, negative term may dominate over the first, entailing the possibility for negative d|Q|/dUD . It is believed that the experiments (using gold in contact with electrolytes) by Hamelin et al. [23,24] may be an indication of negative d.c. differential capacities. However, these are not total capacities, but only the contribution to the capacities by the inner Helmholtz layer at the interface. Thus, finding d|Q|/dUD < 0 in the system presented here would be the first experimental verification of a negative d.c. differential capacity. Acknowledgement We thank the Deutsche Forschungsgemeinschaft (Grants Ma 629/6-) for financial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
A. Volta, vol. XVIII, 1800. F. Runge, Ann. Phys. Chem. 15 (1829) 96. G. Lippmann, Ann. Phys. Chem. 149 (1873) 547. S.-W. Lin, J. Keizer, P.A. Rock, H. Stenschke, Proc. Natl. Acad. Sci. USA 71 (1974) 4477. D. Avnir, J. Chem. Educ. 66 (1989) 211. C.W. Kim, I.-H. Yeo, W.-K. Paik, Electrochim. Acta 41 (1996) 2829. S. Smolin, R. Imbihl, J. Phys. Chem. 100 (1996) 19055. H. Möllencamp, H. Huntermann, W. Jansen, Monatsh. Chem. 130 (1999) 741. C. Quilliet, B. Berge, Curr. Opin. Colloid Interface Sci. 6 (2001) 34. F. Mugele, J.-C. Baret, J. Phys.: Condens. Matter 17 (2005) R705. A. Klingner, S. Heminghaus, F. Mugele, Appl. Phys. Lett. 82 (2003) 4187. B.C. Regan, S. Aloni, K. Jenson, A. Zettl, Appl. Phys. Lett. 86 (2005) 123119.
492
[13] [14] [15] [16] [17] [18] [19]
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U.F. Franck, R. Fitzhugh, Z. Elektrochem. 65 (1961) 156. M.T.M. Koper, J.H. Sluyters, J. Electroanal. Chem. 347 (1993) 31. D. Sazou, M. Pagitsas, Chaos Solitons Fractals 17 (2003) 505. L. Organ, I.Z. Kiss, J.L. Hudson, J. Phys. Chem. B 107 (2003) 6648. J.L. Hudson, T.T. Tsotsis, Chem. Eng. Sci. 49 (1994) 1493. F. D’Alba, C. Lucarini, Bioelectrochem. Bioenerg. 38 (1995) 185. G. Bourceanu, V. Melnig, J. Watamanu, R. Vasiliu, Electrochim. Acta 43 (1998) 1031.
[20] A. Karantonis, Y. Shiomi, S. Nakabayashi, J. Electroanal. Chem. 493 (2000) 57. [21] B. Maiworm, M. Markus, Sci. Asia 31 (2005) 329. [22] D.C. Grahame, Chem. Rev. 41 (1947) 441. [23] A. Hamelin, L. Stoicoviciu, J. Electroanal. Chem. 236 (1987) 267. [24] A. Hamelin, L. Doubova, L. Stoicoviciu, S. Trasatti, J. Electroanal. Chem. 244 (1988) 133.
Transduction of electrochemical into mechanical oscillations B. Maiworm, M. Schmick, M. Markus ∗ Max-Planck-Institut für Molekulare Physiologie, Postfach 50247, D-44202 Dortmund, Germany Received 26 January 2006; received in revised form 9 May 2006; accepted 26 September 2006 Available online 5 October 2006 Communicated by A.P. Fordy
Abstract We observe aperiodic mechanical oscillations of a drop of oxidant solution placed around an iron stick on a mercury surface. These oscillations are explained as a visual manifestation of electrochemical oscillations at the iron–solution interface that, via electrocapillarity, lead to changes of the surface tension at the drop’s bottom. © 2006 Elsevier B.V. All rights reserved. PACS: 82.40.Bj; 68.03.Cd; 82.45.Bb Keywords: Electrochemical oscillations; Surface tension; Beating mercury heart
1. Introduction The system investigated here resembles the so-called “beating mercury heart” [1–7] and an analogous system using a gallium–indium–tin alloy [8]. We shall show below that these systems have some properties in common with ours, but that they differ substantially in their mechanism. One of the best known setups of the beating mercury heart consists of a mercury drop separated sideways from an iron needle and covered with a solution of sulphuric acid and potassium dichromate K2 Cr2 O7 . The oxidation of the mercury by the dichromate charges positively the mercury surface and thus reduces its surface tension; the drop thus flattens, touches the iron, recovers electrons, curls up due to the increased surface tension and then the process starts again. In the setup described in this work it is not the mercury drop that beats, but a drop of the dichromate-sulphuric-acid solution placed on a mercury surface. An iron stick passes vertically through the drop and is constantly immersed in the mercury, as shown in Fig. 1. U is the measured voltage due to the elec* Corresponding author.
E-mail addresses: [email protected] (M. Schmick), [email protected] (M. Markus). URL: http://www.mpi-dortmund.mpg.de/markus. 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.09.075
Fig. 1. Scheme of the experimental setup. The angle Θ is given by Eq. (1).
trochemical processes in the system. We also investigated the behavior in the case where U is not endogenous, but imposed by us externally, using a Pt-electrode instead of Fe to eliminate the electrochemical oscillations. Another process that resembles ours is the manipulation of tiny drops (< 1 µl) using electrowetting [9,10]. In this process, an ac or dc voltage is applied between the drop and a metal surface, generally separated from the drop by an insulating layer. If the voltage is applied between two parallel planar electrodes, a drop between them oscillates, alternating between bridging the electrodes and breaking into two drops [11]. Still another system related to ours is a dc-controlled oscillator on nanoscales, consisting of two molten metal particles on a carbon tube [12]. The system with the parallel electrodes and the nanosystem, both mentioned in the preceding paragraph are related to ours in the sense that geometrical properties of the liquid vary due
B. Maiworm et al. / Physics Letters A 361 (2007) 488–492
489
Fig. 2. Experimentally observed, mechanical oscillations of a drop of oxidant solution on a mercury surface. The total time interval for these five pictures is given by T in Fig. 3(c). Times: 1.4 s, 2.4 s, 3.4 s, 4.4 s and 5.4 s. The white spots are H2 -bubbles.
to changes in surface tension, the latter varying due to changes of the voltage. However, these systems differ from ours in that they require an externally applied voltage, while in our system an ac-voltage is generated endogenously by electrochemical oscillations. Such electrochemical oscillations occur on a metal surface (iron in our case) in contact with an electrolyte [13–17]. In particular, such oscillations have been investigated for iron in the presence of a dichromate-sulphuric-acid solution [18,19], as in the present work. In such a system, the iron first goes into solution as Fe2+ ; this is followed by the formation of a nonconducting FeCrO4 -film on the iron, which then dissolves as it reacts with protons so that Fe2+ is released again. The following irreversible processes occur, as oscillations proceed: reduction 3+ of Cr2 O2− 7 to Cr , production of H2 and dissolution of solid 2+ iron into Fe [18]. These are the same overall processes that occur in the beating mercury drop [5]. However, the observations described below indicate substantial differences in the present system, since here the oscillations of the surface tension and of the drop’s radius are driven by the oscillations on the iron surface. In contrast, in the beating mercury drop iron only acts as an electron donor and is not in constant contact with the mercury. 2. Experimental procedures A dish with a diameter of 15 cm was filled with Hg up to a height of 13 mm. The oxidant solution contained H2 SO4 (7 M) and K2 Cr2 O7 (1.16 mM). A drop (volume Vd = 200 µl) of this solution was poured around an iron stick (diameter: 1 mm; length: 4 cm), which had been placed vertically through the Hg at the center of the Hg surface. Its shape self-organized symmetrically as a circle around this stick. In one series of experiments (Figs. 2 and 3), the potential U was measured between the iron and a Pt electrode immersed in the solution, as indicated in Fig. 1. The oscillations of the drop were monitored with a video camera. The radius of the drop was determined (as a function of time) from the image on the monitor. In additional experiments, we drove the system externally, replacing the iron by a second Pt-electrode and applying the potential U between the two electrodes; in these experiments, potential pulses were applied and the temporal response of the drop’s radius was determined (Fig. 4). Also, we measured the drop’s radius r by increasing (quasistatically) the potential U at a rate of 1 V/min (Fig. 5(a)). Note that we increased, instead of decreased U , so that the relaxation times were as small as pos-
Fig. 3. (a, b) Self-oscillations of the potential. (b) Enlarged interval. (c) Oscillations of the radius of the drop, corresponding to (b).
sible (compare the relaxation of r for upward steps of U with that for downward steps in Fig. 4). We want to stress that all potentials appearing in the present work are defined as given in Fig. 1, i.e. between Pt and Pt, or between Pt and Fe. This should be kept in mind when comparing with measurements using standard electrodes, as found in the literature. 3. Measurements The oscillations of the drop are illustrated in Fig. 2 by snapshots after equal time intervals. Fig. 3(a) shows the measured
490
B. Maiworm et al. / Physics Letters A 361 (2007) 488–492
Fig. 4. (a) Imposed pulses of potential intended to be an analogon to the self-oscillations shown in Fig. 3(b). (b) Measured responses (drop’s radius r) to the pulses shown in (a).
Fig. 5. Measured drop’s radius r (a) and σ (b) (surface tension at the mercury–air interface minus that at the mercury–solution interface, resulting from measurements and Eq. (6)) vs. the applied potential U .
self-oscillations with the iron stick. These were always aperiodic. Fig. 3(b) displays an enlargement with a few selected periods. Fig. 3(c) shows the changes of the radius r of the drop corresponding to Fig. 3(b). If we replace the iron by Pt and apply short potential pulses (Fig. 4(a)), we obtain a similar response of the radius (Fig. 4(b)) as that for the self-oscillations with iron shown in Fig. 3. The similarity of Figs. 4(a) and 4(b) with Figs. 3(b) and 3(c) indicates that the mechanical oscillations illustrated in Figs. 2 and 3 are driven by electrochemical oscillations at the iron stick. In addition, we determined the drop’s radius r, as a function of U , by quasistatically changing U . The result is shown in Fig. 5(a); the steps in this figure are caused by the limited resolution of our video tool. 4. Determination of the surface tension We shall now describe a handy way to determine the surface tension at the boundary of mercury, solution and air, using our setup. Note, that we cannot adopt the equations used for submicroliter drops [9,10] because gravitation is neglected there. In contrast, to obtain oscillations in our experiments, we require much larger drops, which are considerably deformed by gravitation.
We consider the surface tensions σMA (mercury–air interface), σMS (mercury–solution interface) and σSA (solution–air interface). At the curve shared by the three interfaces, Young’s equation σMA − σMS = σ = σSA cos Θ
(1)
holds. Θ is the angle between the drop’s surface and the mercury surface. Contrary to microdrops [9,10], which have a spherical shape, the shape of the drop in our experiments can be described by a flat lower surface (radius: r) and a flat upper surface (radius: r − ; see Fig. 2 and sketch in Fig. 1). If we approximate the drop’s shape by the frustum of a cone (height: h), the angle between the curved side and the base being Θ, then its volume is Vd = πh/3(3r 2 − 3r + 2 ); its upper surface is Au = π(r − )2 and its curved lateral surface is Ac = πh(2r − )/ sin Θ. From our experiments, we estimate 0.02 cm, while 0.7 cm < r < 1.3 cm (see Fig. 2). Thus, r; we therefore make the approximations Vd = πr 2 h, Au = πr 2 and Ac = 2πrh/ sin Θ, i.e. Ac = 2Vd /(r sin Θ). We shall now consider the equilibrium condition that the change of the total energy must be zero for a virtual displacement dr. We set σSA = 70.5 dyn/cm (measured by us using a capillary tube) and σMA = 471 dyn/cm. We assume that σMA is unaffected by the electrical charge at the drop; thus, σMA
B. Maiworm et al. / Physics Letters A 361 (2007) 488–492
is set constant. The surface energy of the mercury surface is EM = σMS πr 2 + σMA (AM − πr 2 ), where AM is the constant total area of that surface; thus dEM = −2σ πr dr − πr 2 d(σ ).
(2)
The surface energy of the solution–air interface is ES = σSA (Au + Ac ); thus dES = σSA 2πr − 2Vd / r 2 sin Θ dr − 2σSA Vd dΘ/(r sin Θ tan Θ),
(3)
where, by virtue of Eq. (1), dΘ = −d(σ )/(σSA sin Θ).
(4)
Finally, the change in gravitational energy by the change dh = −Vd dr/(πr 3 ) of the center of mass is Vd 2 ρg (5) dr. πr 3 The condition dEM + dES + dEP = 0, along with Eqs. (1) through (5), yields d(σ ) 2Vd 1− dr πr 3 sin2 Θ tan Θ 2Vd 2 Vd 2 ρg 2(σ ) − 4 . = σSA (6) − 2 5 − r r πr sin Θ π r
dEP = −
Considering σ = σSA cos Θ (Eq. (1)), we integrated the differential equation (6), in order to obtain σ (r). Since we observed Θ ≈ 90◦ at r = 0.7 cm, we set σ = 0 for this value of r. σ (r), together with r(U ), as given by Fig. 5(a), yielded the net horizontal surface tension σ (U ), which is shown in Fig. 5(b). 5. Discussion We reported here on a mechanical manifestation of electrochemical oscillations. Besides making the latter visible to the bare eye, this device is—along with the “beating mercury heart”—a novel and easy-to-build chemically driven “motor”. The oscillations that we observed have the shape of sharp recurrent peaks, as those in previous reports [14–16,18–20]. While both periodical and aperiodical oscillations have been reported, we did not obtain the former. One reason may be the variations of the area of the iron–solution interface in our case; another reason may be a feedback to the iron from the mercury– solution double-layer; still another reason may be related to the Pt–solution interface. We showed that the electrochemical oscillations drive mechanical oscillations of the drop. In addition, measurements of the drop’s radius along with an energy analysis allowed to determine the net horizontal surface tension σ at the border of the drop on the mercury surface, as a function of U . We summarize the meaning of Fig. 5 as follows: if U changes in time (either electrochemically or by imposed voltages) there occurs a change in σ , as given by Fig. 5(b), which (according to Eq. (6)) causes a change in the radius of the drop (Fig. 5(a)).
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6. Outlook An experimental result that remains to be explained is the slow relaxation of r as U decreases, as compared with the fast relaxation of r as U increases. This relaxation asymmetry occurs both in self-oscillations (Figs. 3(b) and 3(c)) and in imposed pulses of U (Fig. 4). The limitations of our setup did not allow to measure the potential UD at the double-layer at the mercury–solution interface. In a preliminary approach [21], we had assumed that UD ≈ U . However, subsequent experiments (using W- or Auinstead of Pt-electrodes) showed that the contribution to U of the potential at the Pt–solution interface cannot be neglected, as compared to UD . A redesigned setup, allowing to measure UD using a standard electrode that eliminates liquid junction potentials (see e.g. [22]), would permit to determine the charge per unit area q = −dσMS /dUD = d(σ )/dUD , as well as the differential capacity CD = −dq/dUD of the double layer. One may then be able to show that the total differential capacity d|Q|/dUD (where Q = qπr 2 ) can become negative; in fact, if we assume that U and UD are positively correlated (as in our approximation U ≈ UD in [21]), then an increase in UD causes a shrinkage of the capacitor’s size (see Fig. 5(a)). Considering q, Q < 0, dQ d(qr 2 ) dr d|Q| =− = −π = πr 2 CD − 2πqr . (7) dUD dUD dUD dUD On the right side of this equation the second, negative term may dominate over the first, entailing the possibility for negative d|Q|/dUD . It is believed that the experiments (using gold in contact with electrolytes) by Hamelin et al. [23,24] may be an indication of negative d.c. differential capacities. However, these are not total capacities, but only the contribution to the capacities by the inner Helmholtz layer at the interface. Thus, finding d|Q|/dUD < 0 in the system presented here would be the first experimental verification of a negative d.c. differential capacity. Acknowledgement We thank the Deutsche Forschungsgemeinschaft (Grants Ma 629/6-) for financial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
A. Volta, vol. XVIII, 1800. F. Runge, Ann. Phys. Chem. 15 (1829) 96. G. Lippmann, Ann. Phys. Chem. 149 (1873) 547. S.-W. Lin, J. Keizer, P.A. Rock, H. Stenschke, Proc. Natl. Acad. Sci. USA 71 (1974) 4477. D. Avnir, J. Chem. Educ. 66 (1989) 211. C.W. Kim, I.-H. Yeo, W.-K. Paik, Electrochim. Acta 41 (1996) 2829. S. Smolin, R. Imbihl, J. Phys. Chem. 100 (1996) 19055. H. Möllencamp, H. Huntermann, W. Jansen, Monatsh. Chem. 130 (1999) 741. C. Quilliet, B. Berge, Curr. Opin. Colloid Interface Sci. 6 (2001) 34. F. Mugele, J.-C. Baret, J. Phys.: Condens. Matter 17 (2005) R705. A. Klingner, S. Heminghaus, F. Mugele, Appl. Phys. Lett. 82 (2003) 4187. B.C. Regan, S. Aloni, K. Jenson, A. Zettl, Appl. Phys. Lett. 86 (2005) 123119.
492
[13] [14] [15] [16] [17] [18] [19]
B. Maiworm et al. / Physics Letters A 361 (2007) 488–492
U.F. Franck, R. Fitzhugh, Z. Elektrochem. 65 (1961) 156. M.T.M. Koper, J.H. Sluyters, J. Electroanal. Chem. 347 (1993) 31. D. Sazou, M. Pagitsas, Chaos Solitons Fractals 17 (2003) 505. L. Organ, I.Z. Kiss, J.L. Hudson, J. Phys. Chem. B 107 (2003) 6648. J.L. Hudson, T.T. Tsotsis, Chem. Eng. Sci. 49 (1994) 1493. F. D’Alba, C. Lucarini, Bioelectrochem. Bioenerg. 38 (1995) 185. G. Bourceanu, V. Melnig, J. Watamanu, R. Vasiliu, Electrochim. Acta 43 (1998) 1031.
[20] A. Karantonis, Y. Shiomi, S. Nakabayashi, J. Electroanal. Chem. 493 (2000) 57. [21] B. Maiworm, M. Markus, Sci. Asia 31 (2005) 329. [22] D.C. Grahame, Chem. Rev. 41 (1947) 441. [23] A. Hamelin, L. Stoicoviciu, J. Electroanal. Chem. 236 (1987) 267. [24] A. Hamelin, L. Doubova, L. Stoicoviciu, S. Trasatti, J. Electroanal. Chem. 244 (1988) 133.