J. Phys. Chem. 1996, 100, 19055-19058
19055
Hydrodynamic Modes of the “Beating Mercury Heart” in Varying Geometries S. Smolin and R. Imbihl* Institut fu¨ r Physikalische Chemie und Elektrochemie, UniVersita¨ t HannoVer, Callinstr. 3-3a, D-30167 HannoVer, Germany ReceiVed: June 5, 1996; In Final Form: August 14, 1996X
A mercury drop covered by an aqueous solution which is brought into contact with a metal tip at a fixed potential can be excited to sustained electromechanical oscillations (“beating mercury heart”). The driving force for these pulsations is the electrocapillary effect. We investigated these excitations for the traditional watch glass geometry and for linear and ring-shaped geometries with different lengths and diameters, respectively, varying the potential of the metal tip. We find standing waves in the linear geometry with the number of nodes depending on the potential. In the ring geometry we observe a fast pulsation mode and a slow mode with 2-fold symmetry axis. In addition, we find solitary waves circulating on the ring.
Introduction One of the oldest and also one of the most popular oscillatory reactions in electrochemistry is the so-called “beating mercury heart”. The experiment is known already since the early 19th century, and it was reported by Lippmann in 1873.1 The name originates from the periodic heart-shaped contractions a drop of mercury undergoes when the mercury surface in an aqueous oxidizing solution is brought into contact with the tip of an iron nail. It was soon realized by Lippmann that the basic underlying phenomenon is electrocapillarity, i.e., the dependence of the surface tension of a liquid on its electrochemical potential. Despite its popularity, it was only in 1979 that a detailed mechanistic study of the “beating mercury heart” was conducted by Keizer et al., leading to a mathematical model with three variables.2,3 This model describes the changes in the electrochemical potential and hence in the surface tension of a mercury drop as a consequence of the different electrochemical processes occurring at the surface of the mercury in and out of contact with a metal tip.2 In the traditional experimental setup in which one observes the characteristic heart-shaped excitations the experiment is conducted in a so-called watch glass geometry with the mercury being covered by an oxidizing acid solution and touching an iron nail.4 In subsequent investigations, it was demonstrated that a number of different modes can be observed if one replaces the corroding iron nail by a tungsten tip and applies an adjustable voltage source instead of the fixed potential of the iron electrode.5 These modes which comprise excitations with 2-, 3-, 4-, and 6-fold symmetry axes and also irregular motions have been termed hydrodynamic modes because they appear to be determined by the fluid mechanical properties of the mercury. In this paper we report on the modes we obtain when we replace the watch glass geometry by a linear or ring-shaped geometry and vary the potential of a tungsten tip brought into contact with the mercury. The results show that depending on the given geometry and on the applied potential we can find a number of new modes comprising standing waves and solitary waves. By varying the potential, we can induce transitions between these different modes and we can find mixed-mode oscillations in the corresponding potential variations of the mercury. X
Abstract published in AdVance ACS Abstracts, November 1, 1996.
S0022-3654(96)01642-5 CCC: $12.00
Figure 1. Experimental setups employed for the investigation of the “beating mercury heart” in a ring-shaped geometry. The Teflon (PTFE) disc has a diameter of 120 mm. The disc is covered with 100 mL of a solution with 1 M NaCl and 0.01 M NaOH. (a, top) Setup with a potentiostat. (b, bottom) Setup with an adjustable voltage source.
Experimental Section The experimental setups displayed in Figure 1 are similar to the one introduced by Keizer et al.2 The mercury is covered with a basic solution of 1 M NaCl and 0.02 M NaOH. A tungsten tip in connection with an adjustable voltage source supplies the electrons for the reduction process at the mercury surface. A tungsten wire (0.2 mm diameter) with a sharpened tip whose vertical position can be controlled with a micrometer screw within 10-3 mm is brought into contact with the mercury surface. For keeping the W tip at a fixed potential, ∆φ, a © 1996 American Chemical Society
19056 J. Phys. Chem., Vol. 100, No. 49, 1996 potentiostat is used with a Cl-/AgCl/Ag reference electrode and a 8 mm2 Pt foil as counter electrode as indicated in Figure 1a.6 In some of the experiments an adjustable voltage source instead of the potentiostat is employed with the W tip as cathode and a second W wire as anode. This setup is shown in Figure 1b. The potential difference between the mercury drop and the W tip, U, is digitized with a 12 bit A/D converter and stored onto a computer. Teflon (PTFE) discs in which spherical, linear, and ringshaped grooves are cut serve to investigate the excitations of the mercury in restricted geometries. The linear grooves which are of 40 and 80 mm length have a rectangular profile with 5 mm width and 2 mm depth. The rings which have diameters of 20, 60, and 100 mm are formed by grooves of 10 mm width with a circular profile of 5 mm radius. The volume of the mercury filled into the rings is 1, 4, and 7 mL. For the linear grooves the corresponding volumes are 0.7 and 1.4 mL. Electrical contact to the mercury in these grooves is provided by thin Pt wires fed through holes at the bottom of the channels. The petri dish with the Teflon discs is on a platform whose inclination angle with respect to the horizontal position is adjustable. A regular 50 half-frames/s CCD-video camera serves to record images of the pulsating mercury. It was suggested by Keizer et al.2 that the electrochemical process which takes place when the W tip is in short circuit with the mercury is the reduction of dissolved oxygen according to
Smolin and Imbihl
Figure 2. Pulsations of two galvanically coupled mercury drops of 1 mL (foreground) and 2 mL (background) volume in a watch glass geometry. The W tip visible in the foreground is at a potential of ∆φ ) -1800 mV. The glass tube next to it is the reference electrode. The “watch glass” has a curvature of 30 mm radius and a depth of 8.5 mm.
O2(soln) + 2H2O + 4e- f 4OH-(aq) We find that even without bubbling air through the solution, stable oscillations can be sustained for more than 1 h. When we cycle the potential, we observe, however, pronounced hysteresis effects in the amplitude of the electrochemical oscillations. In the following we neglect these hysteresis effects which are mostly reproducible but complicate the system and focus onto the qualitative behavior of the system. Results Previous investigations of the hydrodynamic modes of a mercury drop conducted by Olson et al. have shown that the mode selection depends on a number of factors such as the applied potential, the tip separation to the mercury surface, the position of the W tip relative to the center of the mercury drop, the mass of the mercury drop, and the curvature of the watch glass.5 In our experiments with a single oscillator in a watch glass geometry, we find that potentials of the W tip with ∆φ ca. -1000 mV favor concentric modes whereas low potentials with ∆φ below -1300 mV select the deformation modes with 2-, 3-, and 4-fold symmetry axis. Irrespective of which mode is selected, the frequency of the electrochemical oscillations is constant within ca. 5% in the investigated potential range from -1000 to -2000 mV. These results are consistent with the view that, at least under the experimental conditions chosen here, the oscillation frequency of the “beating mercury heart” in the watch glass geometry is mainly determined by the fluid mechanical properties of the Hg drop. We illustrate these relations by the experiment displayed in Figure 2. The image shows two galvanically coupled mercury drops with different masses in a watch glass geometry exhibiting different excitation modes. The one in the foreground is the actual oscillator since over this one a W tip is positioned with which the mercury periodically makes contact. The mercury drop in the background oscillates because it is electrically connected to the oscillating potential of the first mercury drop
Figure 3. Pulsations of electrically connected mercury columns in linear grooves of 40 and 80 mm length displaying standing waves. ∆φ ) -1700 mV.
and thus experiences the same periodic changes in the surface tension due to the electrocapillary effect. Both mercury drops oscillate with the same basic frequency, but the mercury drop in the background exhibits concentric pulsations while the one in the foreground displays a mode with a 2-fold symmetry axis. In this mode the long axis of the Hg drop rotates by 90° in each oscillation cycle of the potential so that the period of this mode is twice the period of the potential oscillations. In the experimental setup of Figure 2 the principal factor explaining the occurrence of different vibrational modes is clearly the fact that the two mercury drops have different masses; i.e., the Hg drop in the foreground has 1 mL volume and the other one 2 mL. When we exchange the watch glass geometry against linear grooves we observe the modes displayed in Figure 3. In this linear geometry we obtain standing waves. Similar to the previous experiment, we have connected the mercury in the two grooves by a Pt wire. The standing waves in both grooves exhibit the same wavelength despite the different lengths of the grooves which demonstrates that the potential is mainly responsible for the selection of the wavelength. In the following the vibrational properties of each of the two different grooves are investigated separately for varying voltages. We observe that by changing the potential we can control the vibrational modes, i.e., with decreasing potential the number of nodes, n, increases, starting from n ) 2 at ∆φ ) -1350 mV to n ) 8 at ∆φ ) -1700 mV. This behavior is illustrated by Figure 4 showing waves with two nodes at ∆φ ) -1350 mV in and waves with eight nodes at ∆φ ) -1700 mV in a groove
Hydrodynamic Modes of the “Beating Mercury Heart”
J. Phys. Chem., Vol. 100, No. 49, 1996 19057
Figure 4. Standing waves in the 80 mm groove showing different number of nodes for varying potentials ∆φ of the W tip: (a) ∆φ ) -1350 mV; (b) ∆φ ) -1700 mV. Figure 6. Electrochemical mixed-mode oscillations associated with different excitation modes of mercury in rings of varying diameter. The experiments were conducted with an adjustable voltage source. The voltage UW varies from 1000 mV over 1740 mV and 1200 mV to 1800 mV (top to bottom). The top curve displays the simple oscillations associated with the fast pulsations in a ring of 60 mm diameter; all other curves show the mixed-mode oscillations associated with the superposition of the fast pulsation mode with the slow symmetry breaking mode.
Figure 5. Electrochemical oscillations associated with the standing waves in the 80 mm groove for different potentials ∆φ.
of 80 mm length. In most of the experiments the wave patterns are not pure standing waves, but the nodes periodically move within a fraction of the wavelength. The comparison of modes in grooves of different lengths shows that the selected wavelength is determined by the applied potential. Although the overall behavior of the system is reproducible, the bifurcation voltages where the number of nodes changes varies from experiment to experiment within ca. 200 mV, apparently as a consequence of some experimental parameters that are difficult to control. The main influence in that respect should be the problem to exactly reproduce the position of the W tip relative to the meniscus of the mercury. The electrochemical oscillations that are associated with the hydrodynamic modes are followed by recording the variations of the potential difference between the mercury and the W tip, U. The corresponding time series of the 80 mm groove are displayed in Figure 5 for four different potentials of the W tip, ∆φ. The curves exhibit the characteristic shape reported earlier with a flat (zero) potential part when the W tip touches the mercury surface and a rising part when the W tip has lost contact with the mercury surface.2 The data show that the period of the oscillations increases with decreasing potential of the W tip. Furthermore, one notes that simultaneously the oscillations become increasingly complex, starting from a slight modulation of the maxima to the development of real mixed-mode oscillations and the appearance of irregularities. The oscillation amplitude is characterized by a steep rise below a potential of ∆φ ) -1450 mV. The increase in the period with decreasing potential seems to be in contradiction to the number of nodes increasing in the same direction since for a vibrating string the frequency rises with the number
of nodes. This is also true for the standing waves seen here where the frequency of the up-down motion increases with the number of nodes. However, the frequency of the associated electrochemical oscillations is not determined by the pure standing wave mode but rather by additional slow modes which are superimposed on the fast standing wave mode and which are responsible for the mercury making periodically contact with the W tip. The increasing complexity of the oscillations with decreasing potential can be attributed to the development of these additional modes. These modes become increasingly irregular at very low potentials (∆φ ca. -2000 mV). Alternatively, at very low potentials a second redox process might contribute to the signal. In contrast to the above experiments in which a potentiostat was used, the experiments with a ring geometry are mostly conducted with a voltage source (UW) in between the W tip and a second W electrode. With the ring-shaped grooves, one observes typically rapid oscillations of ca. 20 s-1 at voltages UW in between 1000 and 1500 mV. These rapid pulsations take place in the radial direction, and their amplitude decreases with increasing distance from the position of the W tip. The frequency of this fast mode varies only little with the ring diameter. One obtains 16.9 s-1 for the 20 mm ring, 18.9 s-1 for the 60 mm ring, and 18.2 s-1 for the 100 mm ring. The frequencies are comparable to the frequency of a mercury oscillator in a watch glass geometry which typically varies between 10 and 20 s-1. Further decreasing the potential of the W tip, i.e., increasing the voltage UW, a new mode appears in which part of the mercury mass moves from one half of the ring to the other half and back. This mode with periods on the order of seconds is considerably slower and can easily be followed by eye. In contrast to the fast pulsations, the frequency of this mode varies significantly with the ring diameter. For the largest ring with 100 mm diameter one obtains a period of 1.96 s, for the 60 mm ring the period decreases to 1.51 s, and for the smallest ring with 20 mm diameter the period is only 0.41 s. The corresponding time series displayed in Figure 6 exhibit mixedmode oscillations. The fast oscillations which appear in the flat part are due to the fast pulsations while the slow symmetrybreaking mode is responsible for the slow increase in U. The
19058 J. Phys. Chem., Vol. 100, No. 49, 1996
Figure 7. Standing wave pattern in a ring of 60 mm diameter which is inclined by 1.5°. ∆φ ) -1750 mV.
voltage at which the mode change occurs varies between 1400 and 1700 mV for the different ring diameters. In order to investigate the influence of a deviation from the exact horizontal orientation of the Teflon disc on the mode selection, we conducted experiments in which a small inclination angle was adjusted. These experiments are again carried out with a potentiostat and a Pt foil as counter electrode. Filling the ring only partially with mercury, we observe standing waves of the type displayed in Figure 7. Superimposed to this rapid mode is a second slow mode in which the ends of the mercury ring periodically approach each other. With a closed mercury ring and with an inclination angle of 1.5°, we can induce solitary waves at a potential ∆φ ) -1900 mV. These solitary waves which are initiated at the position of the W tip travel periodically around the ring with a velocity of 140 mm/s. If we adjust the Teflon disc in the horizontal position, we can observe a symmetric slow mode in which the mercury ring undergoes a periodic concentric expansion and contraction in radial direction with periods of the order of tenths of seconds. Discussion Electrochemical oscillations are known since the beginning of the past century, and after being almost forgotten they found renewed interest in recent years.7,8 The beating mercury heart differs, however, from other electrochemical oscillators since its oscillations originate from the coupling between an electrochemical and a mechanical process. With respect to the different modes obtained here, the simplest approach is to assume that these are determined by the property of mercury as a fluid
Smolin and Imbihl whereas the electrochemical processes simply supply the energy necessary to sustain the pulsations. Since the frequency of the Hg oscillator in simple geometries like the watch glass geometry varies little with the applied potential, such an approach at first sight does not appear to be unrealistic. The task to explain the different modes is nevertheless not trivial since one has to solve a complicated hydrodynamic problem. For the watch glass geometry this has been attempted by Caddell, who used the Lagrange formalism and expanded the resulting equations of motion in different modes.9 For the linear and ring-shaped geometries this remains to be done. Since we observe a change in the modes with varying potential, the electrochemical processes cannot be neglected, but again the simplest approach would be to assume that it is simply the amount of dissipated energy which determines which mode is selected. The analysis is also complicated by the fact that it is usually not a simple mode but a superposition of various modes we observe. These modes involve different time scales and can be as fast as 100 s-1 as demonstrated in recent experiments with a high-speed camera (250 frames/s).10 One also has to consider the possibility that the electrochemical processes at the Hg/H2O interface do not take place uniformly but involve propagating potential waves whose existence has been demonstrated recently for Ag electrodes.8 It remains to be shown to what extent the observations made here can be explained by purely fluid mechanical laws and how strongly the electrochemical processes contribute to the selection of modes. In summary, it was demonstrated that the “beating mercury heart” is quite rich in its dynamical behavior and far from being exhausted. By applying different confinement geometries, one can obtain new modes, and by varying the potential, one can select modes and induce transitions between them. Acknowledgment. The authors are indebted to K. Krischer and G. Fla¨tgen for numerous fruitful discussions. References and Notes (1) Lippmann, G. Ann. Phys. 1873, 149, 546. (2) Keizer, J.; Rock, P. A.; Lin, S.-W. J. Am. Chem. Soc. 1979, 101, 5637. (3) (a) Lin, S.-W.; Keizer, J.; Rock, P. A.; Stenschke, H. Proc. Natl. Acad. Sci. U.S.A. 1974, 71, 4477. (b) Keizer, J. In Special Topics in Electrochemistry; Rock, P. A., Ed.; Elsevier: Amsterdam, 1977; p 111. (4) Avnir, D. J. Chem. Educ. 1989, 66, 211. (5) Olson, J.; Ursenbach, C.; Birss, V. I.; Laidlaw, W. G. J. Phys. Chem. 1989, 93, 8258. (6) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, Fundamentals and Applications; Wiley: New York, 1980. (7) Hudson, J. L.; Bassett, M. ReV. Chem. Eng. 1991, 7, 109. (8) Fla¨tgen, G.; Krischer, K.; Pettinger, B.; Doblhofer, K.; Junkes, H.; Ertl, G. Science 1995, 269, 668. (9) Caddell, J. F. Thesis, University of California, Davis, 1994. (10) Krischer, K. Unpublished results.
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