General Mechanism for the Synchronization of Electrochemical

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J. Phys. Chem. C 2007, 111, 1150-1160

General Mechanism for the Synchronization of Electrochemical Oscillations and Self-Organized Dendrite Electrodeposition of Metals with Ordered 2D and 3D Microstructures Kazuhiro Fukami,† Shuji Nakanishi,*,† Haruka Yamasaki,† Toshio Tada,† Kentaro Sonoda,† Naoya Kamikawa,‡ Nobuhiro Tsuji,‡ Hidetsugu Sakaguchi,§ and Yoshihiro Nakato†,⊥ DiVision of Chemistry, Graduate School of Engineering Science, Osaka UniVersity, Toyonaka, Osaka 560-8531, Japan, Department of AdaptiVe Machine Systems, Graduate School of Engineering, Osaka UniVersity, Suita, Osaka 565-0871, Japan, Department of Applied Science for Electronics and Materials, Interdisciplinary Graduate School of Engineering Science, Kyushu UniVersity, Kasuga, Fukuoka 816-8580, Japan, and CREST, Japan Science, and Technology Agency, Osaka 560-8531, Japan ReceiVed: June 4, 2006; In Final Form: September 28, 2006

Mechanisms for potential oscillations and synchronizing self-organized formation of ordered dendrite structures, which is an important example of morphogenesis in Laplacian fields, were studied using Sn, Zn, Cu, and Pb electrodeposition. Electron backscattering diffraction (EBSD) experiments showed that Sn latticework as a typical example of the ordered dendrite structures grew in an epitaxial mode in particular directions from the Sn electrode surface. In situ optical and phase-contrast microscopic inspection showed that the Sn latticework growth as well as the concentration profile for deposited Sn(II) ions near the growing front oscillated in synchronization with the potential oscillation, which led to formation of a highly ordered Sn latticework structure. Similar behavior was observed in other electrodeposition systems. On the basis of these results, a general mechanism for the potential oscillations and synchronized formation of the ordered dendrite structures was proposed, in which autocatalytic crystal growth, passivation in flat surfaces of dendrites, and depletion of deposited metal ions in the electrolyte near and inside the dendrites played the key roles. Numerical calculations by use of a modified coupled map lattice (CML) model confirmed the validity of the mechanism. The clarification of the general mechanism has enabled us to classify all the oscillatory dendrite growth, including reported ones by other research groups, into three types (type-I, type-II, and type-III), which will serve for preparation of designed and controlled micro- and nanostructures at solid surfaces.

1. Introduction There is a variety of shapes in natural crystals, from polyhedral quartz to dendritic snow. Crystals are in general formed under nonequilibrium conditions. It is generally believed1,2 that the morphology of crystals strongly depends on the “distance” of the formation conditions from the thermodynamic equilibrium, as illustrated in Figure 1. Under nearequilibrium conditions, polyhedral crystals surrounded with thermodynamically stable crystal faces are formed so that the surface energy will take a minimum value. With increasing distance from the equilibrium, i.e., with the increasing driving force for crystallization, the growing fronts of crystals with flat surfaces come to be destabilized owing to an increasing contribution of mass or heat diffusion. The increasing surface instability then leads to formation of dendrites. The formation of dendrites due to the surface instability under far-equilibrium conditions has attracted a lot of attention from the point of view of morphogenesis in Laplacian fields3,4 (i.e., fields described by Laplace equations such as a mass diffusion field, a thermal diffusion field, and a field of electrostatic potential for electric-field-induced migration). Common growth * Corresponding author. E-mail: [email protected]. † Division of Chemistry, Osaka University. ‡ Department of Adaptive Machine Systems, Osaka University. § Kyushu University. ⊥ CREST, Japan Science, and Technology Agency.

Figure 1. Schematic illustration of a correlation between the distance of the formation conditions from the equilibrium and the morphologies of formed crystals.

patterns such as diffusion-limited aggregates (DLA) and dense branching morphologies (DBM) (Figure 1) have been observed in such systems. Interesting examples are found not only in crystallization under supersaturation and/or supercooling conditions5-21 but also in growth of bacterial colonies in Petri dishes,22,23 formation of air flow in viscous media,24,25 and formation of two-dimensional monomolecular films26,27 and sputtered thin films28,29 at solid surfaces, though in these examples, the constituent substances, the spatial size, and the time scale for growth are largely different from each other. Electrochemical deposition is a suitable system for study of the morphogenesis under far-equilibrium conditions, because the distance from the equilibrium, which is in general not easy to control,30,31 can be tuned continuously and reversibly by simply changing the electrode potential or current. In fact, the morphology of the electrodeposits can be varied easily from

10.1021/jp063462t CCC: $37.00 © 2007 American Chemical Society Published on Web 12/15/2006

Mechanism of Formation of Ordered Dendrites polyhedrals to dendrites. Interestingly, our recent studies9-12 have revealed that the electrodeposition of some metals under far-equilibrium (diffusion-limited) conditions gives rise to nonlinear electrochemical oscillations, accompanied by formation of dendrites with ordered microstructures. The formation of the ordered microstructures has occurred in synchronization with cycles of the oscillations.9-12 In addition, the spatial period (size) of the ordered microstructures was able to be tuned by changing the oscillation amplitude and/or period through variation in the current density or the concentration of electroactive species.13 In the present work, we have studied in detail the mechanisms for the oscillatory metal electrodeposition accompanied by formation of ordered dendrite microstructures. The advanced modification of a mechanism reported in a previous short communication12 by addition of a “screening effect of dendrites” has finally led to clarification of a general mechanism for the oscillatory electrodeposition and the formation of ordered dendrite structures, which will provide a new possibility to prepare designed and controlled 2D or 3D micro- and nanostructures at solid surfaces. 2. Experimental Section Polycrystalline Sn, Zn, and Cu (99.97% in purity) plates were used as the working electrodes. They were etched in dilute HCl to remove surface native oxide and contaminations just before experiments. Aqueous basic solutions of Sn(II) and Zn(II) ions, as the electrolytes for Sn and Zn deposition, were prepared by dissolving SnO and ZnO powder into 4.0 M NaOH. The electrolyte for Cu deposition was 0.6 M CuSO4 + 3.0 M lactic acid, including a small amount of NaOH to adjust the pH to 9.0. In some cases, 1.0 g/L gelatin was added to the electrolyte. Special-grade chemicals, together with pure water obtained with a Milli-Q water purification system, were used for preparation of the electrolytes. Organic solutions, such as N,N-dimethylformamide (DMF) containing 0.1 M SnCl2 + 0.1 M (C4H9)4NClO4, were also used as the electrolytes in some experiments. The solutions were not deaerated in the present study. Electrochemical measurements were carried out using a normal three-electrode system, together with a beaker-type cell. A Pt plate (1 cm2) was used as the counter electrode. The reference electrode was a “Ag|AgCl|saturated KCI” electrode for the aqueous electrolytes and a “Ag|AgNO3|0.1 M (C4H9)4NClO4 in CH3CN” electrode for the organic electrolytes. Current (i) versus (external) electrode potential (U) and U versus time (t) curves were measured with a commercial potentiostatgalvanostat (Nikko-Keisoku, NPGS-301) and recorded with a data-storing system (Keyence, NR-2000) with a sampling frequency of 100 Hz. The electrolytes were kept stagnant in all measurements. The ohmic drop in the electrolyte was not corrected in the present work. Inspection of electrodeposited surfaces was carried out with a high-resolution scanning electron microscope (SEM, Hitachi S-5000) and an optical digital microscope (OM, VH-5000, VHZ450, and VH-Z25, Keyence). The concentration profiles for deposited metal ions in the electrolyte were observed by a phasecontrast microscope (POM, BX51, Olympus). Electron backscattering diffraction (EBSD) measurements were carried out with a Phillips XL30S SEM operated at 20 kV, using a program called TSL OIM Data Collection ver.3.5. The EBSD analysis was done using a program named TSL OIM Analysis ver.3.0. 3. Experimental Results The main purpose of the present paper is to clarify a general mechanism for the potential oscillations and synchronized

J. Phys. Chem. C, Vol. 111, No. 3, 2007 1151

Figure 2. Dendrites of (a) tin (Sn), (b) zinc (Zn), and (c) copper (Cu), formed in the electrodeposition under the diffusion-limited conditions, where (a) is an OM image and (b) and (c) are SEM images. The electrolyte: (a) 0.2 M Sn(II) + 4.0 M NaOH, (b) 0.2 M Zn(II) + 4.0 M NaOH, and (c) 0.6 M CuSO4 + 3.0 M lactic acid + a small amount of NaOH to adjust the pH to 9.0. The applied potential: (a) -1.7 V, (b) -1.9 V, and (c) -1.5 V vs Ag|AgCl.

formation of ordered dendrite structures. Therefore, we include in this section not only new results but also reported ones in a brief form for better understanding. 3.1. Growth of Dendrite Crystals in Electrodeposition under Potentiostatic Conditions. Figure 2 shows, as representative examples, dendrites of tin (Sn), zinc (Zn), and copper (Cu), formed in the electrodeposition at sufficiently negative (fixed) potentials (U) at which the diffusion-limited condition is achieved. The Sn dendrites (a) are composed of needle-like crystals, whereas the Zn dendrites (b) are composed of waferlike crystals. The Cu dendrites (c), on the other hand, are composed of round particles, whose size is much smaller than those of the Sn needles and Zn wafers. We can call the needlelike (1D) dendrites as type-I, the wafer-like (2D) dendrites as type-II, and the round-particle-like (3D) dendrites as type-III. This classification will be discussed later again. Note that we define the term “dendrite”, in the present paper, as the deposits with markedly rough surfaces formed under diffusion-limited conditions. Thus, we call even the round-particle-like deposits classified into type-III as “dendrite”, although the deposits in type-III may not look like dendrite. The formation of dendrites in electrodeposition under the diffusion-limited conditions can be explained in terms of the surface instability,32 in the same way as for other systems of the morphogenesis in Laplacian fields.3,4 The key is that spherical diffusion layers for deposited metal ions (such as Sn(II) and Zn(II)) are formed in the electrolyte near the peaked parts on the substrate, with the concentration gradient steeper than in the electrolyte near flat parts (Figure 3a). This leads to much faster diffusion of the metal ions to the peaked parts, resulting in much faster growth of the peaked parts (Figure 3b). Thus, the peaked parts grow autocatalytically more and more, resulting in the sharp crystal growth, hence in the roughening of the initial substrate surface. In other words, the thermodynamically stable flat surface is destabilized (or the surface instability arises) under far-equilibrium (diffusion-limited) conditions. The sharp crystal growth is generally accompanied by frequent bifurcations in crystallographically equivalent directions (Figure 3c), which leads to the formation of dendrites. Whenever the electrodeposition proceeds under the diffusionlimited condition, the above mechanism works, and dendrites grow as far as other factors such as surface diffusion of deposited adatoms are not important. An interesting point in the above mechanism is that the electrode surface area, i.e., the total surface area of growing dendrite, increases continuously and endlessly with the dendrite growth. This implies that the total electrodeposition current (total amount of deposits per unit time), which is proportional to the

1152 J. Phys. Chem. C, Vol. 111, No. 3, 2007

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Figure 3. Schematic illustration of autocatalytic crystal growth producing dendrites under diffusion-limited conditions.

total surface area, also increases continuously and endlessly with time, even though the applied potential is fixed at a constant value. 3.2. Oscillatory Growth of Ordered Structure Dendrites under Galvanostatic Conditions. The dendrite crystals can also be formed in electrodeposition at a sufficiently high (fixed) current, which exceeds the diffusion-limited one (idif). Interestingly, in electrodeposition of dendrites under the constant-current (galvanostatic) conditions, oscillations of the electrode potential (U) are often observed.5-16 Moreover, the dendrites growing under the oscillations usually have ordered periodic structures, indicating that the dendrite growth proceeds in synchronization with the potential oscillations.9-12 A typical example of the dendrites with ordered periodic structures is Sn latticeworks12 (Figure 4a, right), which are produced during the potential oscillation (Figure 4a, left) appearing in the electrodeposition of Sn. Microscopic inspection has confirmed that one stage of the lattice is produced by one cycle of the oscillation, clearly indicating that the dendrite grows in synchronization with the oscillation. Figure 4b-e shows other examples of the dendrites with ordered microstructures, formed during the oscillations under the galvanostatic conditions. The formation of stacked Zn hexagonal wafers in Figure 4b was already reported,9-11 but other examples are first reported here. Figure 4c shows a potential oscillation and a formed dendrite in the case of the Sn deposition from an acidic aqueous solution containing a small amount of gelatin. A difference between parts a and c of Figure 4 is mainly due to a difference in the solution pH. The appearance of an oscillation in the Sn deposition in acidic solutions was previously reported by Wen and Szpunar,33 though the formation of dendrites with ordered structures was not reported. Figure 4d shows the potential oscillation and formed dendrites for lead (Pb) deposition from an aqueous basic solution, whereas Figure 4e shows those for the Sn deposition from an organic electrolyte, with DMF used as the solvent. In the latter two examples, unique 3D microstructures are formed.

Figure 4. Examples of (left) potential oscillations under the constantcurrent conditions and (right) dendrites formed during the oscillations. The electrolyte and the applied current density: (a) 0.2 M Sn(II) + 4.0 M NaOH, -36 mA cm-2; (b) 0.2 M Zn(II) + 4.0 M NaOH, -21 mA cm-2; (c) 0.3 M SnSO4 + 0.3 M H2SO4 + 1.0 g/L gelatin, scan rate, 1.67 mA/s; (d) 0.15 M PbSO4 + 4.0 M NaOH, -14 mA cm-2; (e) 0.1 M SnCl2 + 0.1 M NaClO4, -7.2 mA cm-2. The solvent for the electrolytes was water, except that dimethylformamide is used in (e).

In all the examples, we confirmed that the dendrite growth occurred in synchronization with the oscillations. 3.3. Mechanistic Studies on the Oscillatory Growth of the Sn Latticeworks. The results of Figure 4 show that the electrodeposition of metals under the diffusion-limited conditions at a constant current in general shows spontaneous potential oscillations and synchronized formation of dendrites with ordered microstructures. This fact strongly suggests that a general mechanism exists in these phenomena. Thus, with an aim to clarify the mechanism, we have made mechanistic studies by taking the Sn latticework formation (Figure 4a) as a representative example.

Mechanism of Formation of Ordered Dendrites


Figure 6. (a) SEM images in a high magnification for (upper) the cuboid crystals formed at the positive end of the potential oscillation and (lower) the needle-like crystals growing from the cuboids. (b) Expected structures and orientations for the cuboid and needles in the Sn latticework.

Figure 5. (a) Potential oscillation observed for the Sn deposition under the same experimental conditions as in Figure 4a. (b) OM images of the Sn latticework taken at stages, 1-6, of the oscillation of (a) under in situ conditions. The Sn latticework is seen in the upper part of the images, the growing front being located roughly in the middle of the figures. Grayish white lines stand for the Sn needles, whereas darkgray small rectangles refer to the cuboid Sn crystals.

Figure 5 shows the results of Figure 4a in a detailed form. In particular, Figure 5b shows how the Sn latticework grows in synchronization with the potential oscillation. Namely, OM images, 1-6, of Figure 5b represent the structures of the Sn latticework at stages, 1-6, of the oscillation of Figure 5a, respectively. A similar result was already reported.12 We can see that needle-like crystals (grayish white lines in Figure 5b) grow at the tip (growing front) of the latticework during stages 2 and 3. This growth of the needle crystals causes a potential shift from the negative to the positive, as explained later. On the other hand, at stages 4 and 5, at which the potential stays at the positive end of the potential oscillation, cuboid crystals (dark-gray small rectangles in Figure 5b) are formed at the tips of the needles produced just in the previous stage. Stage 6 is equivalent to stage 1 except that one cell of the lattice is added. The excellent synchronization of the oscillation and the latticework growth can be seen more clearly by making use of dynamic video images. Figure 6a shows SEM images for (upper) the cuboid crystals formed at the positive end of the potential oscillation and (lower) the needle-like crystals growing from the cuboids, which were obtained by pulling out the electrode from the electrolyte at stages 1 and 3 of Figure 5a, respectively. The lower image

clearly indicates that the needle-like crystals grow from the corners of the cuboid crystals. More precisely, the needle-like crystals grow in four crystallographically equivalent directions,12 as schematically illustrated in Figure 6b, though one of them is used as the support of the cuboid crystal itself in the course of growth of the latticework. The orientations of the facets in the cuboid and the growth directions for the needles will be discussed later again on the basis of the results obtained by an EBSD technique. In order to investigate why the needle-like crystals grow at stages 2 and 3 of Figure 5a, we have made an in situ phasecontrast optical microscopic (POM) inspection of the electrolyte near the growing front of the Sn latticework. The results are shown in Figure 7a, where the POM images, 1-6, are taken at stages, 1-6, of the potential oscillation of Figure 5a. The black part in the images stands for the Sn latticework, whereas the bright reddish part refers to the electrolyte. In the bright reddish part, a belt-shaped region near the growing front of the Sn latticework (or near the latticework/electrolyte interface) is brighter than the region of the bulk electrolyte, indicating that the concentration of Sn(II) ions in the belt-shaped region (or in the electrolyte near the growing front) is lower than that in the bulk electrolyte. Figure 7b shows more quantitative data for the profile of the Sn(II) concentration near the growing front, obtained by digital analyses of the POM images along the white lines (line A-B) of Figure 7a. At stages 2 and 3, the color of the electrolyte adjacent to the growing front is much brighter than in the electrolyte bulk, in contrast to the other stages. This indicates that at stages 2 and 3, the Sn(II) concentration at the growing front becomes much lower than the bulk concentration, or in other words, the diffusion-limited condition for Sn(II) is achieved at these stages. This result is in harmony with the aforementioned growth of the needle-like crystals at stages 2 and 3 (Figure 5) because the autocatalytic dendrite growth occurs under the diffusion-limited condition (cf., Figures 2 and 3). This result is also in harmony with the fact that the needle-like crystals grow from the corner of the cuboid crystals (Figure 6) because the autocatalytic dendrite growth starts from peaked parts of the electrode surface (cf., Figure 3). On the other hand, the concentration gradient for Sn (II) in the electrolyte near the growing front at stages 1, 4, and 5 of Figure 7b is much less pronounced or nearly absent, indicating that the diffusion-limited condition disappears (or the reaction-limited deposition proceeds) at the stages. For the purpose of obtaining more detailed information on the orientation of the needle growth as well as that of the Sn latticework with respect to the electrode surface, we have made the electron backscattering diffraction (EBSD) analyses in


Figure 7. (a) POM images for the electrolyte near the tip (growing front) of the Sn latticework. The black part stands for the Sn latticework, whereas the bright reddish part refers to the electrolyte. (b) Profiles of the Sn(II) concentration near the growing front, obtained by digital analyses of the POM images along the white lines (line A-B) in (a). The numbers 1-6 correspond to stages, 1-6, of the potential oscillation of Figure 5a. The experimental conditions were the same as those in Figure 4a.

combination with the SEM inspection. Figure 8a is an SEM image of a polycrystalline Sn electrode surface, obtained after the Sn latticeworks were deposited on it and then washed out with flowing water. We can see that many needles grew from the electrode surface, in the same direction within a domain of one Sn single crystal. The same direction of the needle growth within a domain of one Sn single crystal implies that the Sn deposition (Sn crystallization) occurs in an epitaxial manner. Figure 8b is an expansion of a part of the electrode surface of Figure 8a, in which the needles grow in two directions. The angle between the two needles was 76°, the same as that for the latticework in Figure 5 or 6, suggesting that the direction (or angle) for the needle growth is essentially determined by the crystal structure of Sn. The accurate direction of the needle growth was determined using the EBSD technique as explained below. Figure 8c displays the normal-direction map (ND-map) and the rollingdirection map (RD-map) for a polycrystalline Sn electrode substrate surface, which was prepared in the same way as in Figure 8a. A combination of the ND- and RD-maps allows us

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Figure 8. (a) SEM image of a polycrystalline Sn electrode surface, taken after the Sn latticeworks were deposited on it and then washed out with flowing water. (b) An expansion of a part of the electrode surface of (a). (c) EBSD pattern maps (left, ND; right, RD) for a polycrystalline Sn electrode surface. (d) Pictures for explaining the procedure to determine the direction of the needle growth against the Sn crystal in the electrode by the EBSD technique. Pictures (R), (β), and (γ) are pole figures, SEM images, and representations of the direction of a sample (Sn crystal) in an XYZ coordinate system. See text for details.

to determine the orientations of randomly oriented Sn crystals in the polycrystalline electrode, namely, it allows us to yield the pole figure for each constituent crystal. As an example, pictures (R), (β), and (γ) in column 1 of Figure 8d show the pole figure for a certain constituent crystal, an SEM image for its surface, and schematic drawing for defining the orientation of the crystal in the XYZ coordinate system, respectively. For simplicity, in (γ), the X axis is placed vertical to the crystal surface, though the Y and Z axes are defined appropriately. The needle-like crystals seen in (β) grow with a certain angle with respect to the crystal surface. To determine this angle, the crystal is first rotated by a θ degree around the X axis in the counterclockwise direction so that the needles in the top view (or the needles projected on the Y-Z plane) will be in parallel to the Y axis (see column 2 of Figure 8d). The crystal is then rotated by a φ degree around the Z axis until the needles are directed just upward or in the direction parallel to the X axis


Figure 9. Schematic drawings for explaining the mechanism for the oscillatory growth of the Sn latticework.

(column 3 of Figure 8d). Since the needles growth occurs in an epitaxial manner as mentioned above, we can determine the growth direction of the needles by analyzing the pole figure of column 3 obtained by EBSD pattern from substrate, in which the direction of the needles is located at the center. By this way, we actually determined the growth direction of the needles in six crystals, numbered 1-6 in Figure 8c, which were chosen in an arbitrary manner. All the crystals showed that the growth direction of the needles was 〈113〉. As the direction of the needle growth at the electrode surface is the same as that in the latticework, as mentioned earlier, we can conclude that the needles in the Sn latticework grow in the 〈113〉 direction or equivalents, as illustrated in Figure 6b. On the basis of this result, the facets of the cuboid crystals formed at the positive end of the potential oscillation are estimated to be the thermodynamically stable (100) and (010) faces. This result is quite reasonable because at the positive end of the potential oscillation, it is expected that the diffusion-limited condition disappears and the reaction-limited deposition proceeds under near-equilibrium conditions, as already mentioned (cf., Figure 7). 4. Numerical Simulation The experimental results described thus far have revealed the mechanism of the potential oscillation and the synchronized Sn latticework formation to a considerable extent. However, one important problem still remains unresolved. In order to explain what problem remains, let us first summarize the main features of the mechanism. The results of Figures 5, 6, and 7 show that the diffusionlimited condition is achieved at stages 2 and 3 of Figure 5a and the autocatalytic needle growth proceeds at these stages (top and second top pictures of Figure 9). This result is conceivable because the Sn deposition rate at the negative potentials of stages 2 and 3 is expected to be high enough to achieve the diffusion-limited condition. On the other hand, the needle growth increases the effective area of the electrode (Sn deposit) surface, which in turn causes a positive potential shift,

J. Phys. Chem. C, Vol. 111, No. 3, 2007 1155 as observed, to maintain the constant (total) current. The positive potential shift will continue until the diffusion-limited condition disappears and the autocatalytic needle growth stops. At the positive end of the potential oscillation, i.e., at stages 4 and 5 of Figure 5a, the diffusion-limited condition disappears and Sn electrodeposition proceeds under reaction-limited condition, resulting in the formation of the Sn cuboid crystals surrounded with the thermodynamically stable (100) and (010) faces (lowest picture of Figure 9). This argument is also reasonable in view of the consideration of Figure 1. Now, a problem arises at the next stage. The experiments show that when the cuboid crystals grow to a certain size, the potential is shifted toward the negative; thus, the system is moving back from stage 5 to stage 6 (essentially equivalent to stage 1). What is the reason for such a negative potential shift? We have discussed in a previous communication12 that the cuboid surface is gradually oxidized at such a positive potential as at stages 4 and 5 and comes to be covered with an inactive (passivating) thin oxide or hydroxide layer, which decreases the effective electrode surface area for the Sn deposition and hence causes the negative potential shift to maintain the constant (total) current. The possibility of the oxidation of the Sn surface in a strongly alkaline solution of 4 M NaOH at a potential of stages 4 and 5 was supported by a Pourbaix diagram.34 Besides, we were able to assume an autocatalytic mechanism for the oxidation at the Sn cuboid surface as well.12 A problem in the above-reported mechanism lies in that it is difficult to explain other examples of the potential oscillations and synchronized formation of ordered dendrites shown in Figure 4. For instance, it will be difficult to assume the formation of the inactive oxide or hydroxide layer at the Sn surface in an aqueous acidic electrolyte, though a potential oscillation and formation of ordered dendrites are observed (Figure 4c). The formation of the oxide or hydroxide layer will also be difficult to assume in the case of an organic electrolyte (Figure 4e). These results seem to imply that there is another general mechanism for the negative potential shift in the potential oscillation. Careful inspection of video movies for the Sn latticework growth indicates that the latticework grows only at the tip (growing front) of it. The inside part of the latticework, which was produced at the previous cycles of the oscillation, shows almost no change when the rapid needle growth proceeds at the tip. This suggests that the Sn(II) ions are exhausted inside the latticework. Thus, we can assume, as a possible general mechanism for the negative potential shift, that the Sn(II) ions in the electrolyte near (or a little inside) the tip of the latticework come to be exhausted while the cuboid Sn crystals are growing at the tip. An important consequence of the exhaustion is that it causes a decrease in the current inside the latticework. In other words, the inside parts of the latticework cannot work as reaction (electrodeposition) sites any more. Thus, the effective surface area comes to be decreased, thus leading to the negative potential shift to maintain the constant (total) current. In order to investigate the validity of the above mechanism, we did numerical calculations for the potential oscillation and synchronized Sn latticework formation, using a coupled map lattice (CML) model, first proposed by Sakaguchi for simulating crystal growth patterns in a diffusion field.35 In this model, a diffusion field is expressed as a lattice map (see, e.g., Figure 10). Namely, a (normally 2D) diffusion field is divided into a large (but finite) number of small square cells, and differential equations are converted into discrete-difference equations, using discrete time steps together with the discrete cells. In the present


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Figure 10. Coupled map lattice (CML) model used for numerical calculations for the oscillatory Sn latticework growth.

work, the original CML model was slightly modified to suit the electrochemical crystal growth, as explained below. The Sn deposition and the hydrogen evolution reactions were considered as the electrochemical reactions in the present system. For simplicity, these reactions were expressed as follows:

k1(E) Sn2+ + 2e- f Sn

(1)

k2(E) 2H+ + 2e- f H2

(2)

where k1(E) and k2(E) are the rate constants, depending on the (true) electrode potential (E). The E is sometimes called the double layer potential and is connected with the externally applied (or observed) electrode potential (U) by an equation, E ) U - iR, where iR is the ohmic drop in the electrolyte between the working and the reference electrodes. The backward reactions for reactions 1 and 2 were in the present work ignored because all experiments were carried out at sufficiently negative potentials. The diffusion of Sn2+ in the electrolyte was expressed by using a discrete-difference equation as follows:

[cn+1(i, j) - cn(i, j)]/∆t ) D[cn(i, j + 1) + cn(i, j - 1) + cn(i + 1, j) + cn(i - 1, j) - 4cn(i, j)]/δ2 (3) where cn(i, j) is the amount (mass) of Sn2+ in a cell of the position (i, j) in the electrolyte at the nth time step, D is the diffusion constant for Sn2+, ∆t is the interval of the time step, and δ is the step size, i.e., size of the each cell. This equation is a forward time centered space (FTCS) expression of diffusion process and regarded also as the discretization of the Laplacian equation. The consumption of Sn2+ at the electrode surface by the electrodeposition was expressed as follows:

[cn+1(i, j) - cn(i, j)]/∆t ) -k1(En)cn(i, j)

[xn+1(i, j) - xn(i, j)]/∆t ) k1(En)cn(i, j)

p ) 1 and 2 (5)

where kp0 is the rate constant at E ) Ep0, Ep0 is the equilibrium redox potential for reaction p with p ) 1 and 2, Rp is the transfer coefficient, ne is the number of transferred electrons, F is the

(6)

This equation is also only applied to cells in the electrolyte, adjacent to the interface, similar to eq 4. The equation means that the decrease in the c at a cell (i, j) by the electrodeposition, expressed by eq 4, leads to the increase in the x at the same cell (i, j) by the same amount as the decrease in the c, thus the total amount (mass), (c + x), being conserved. In the present calculation, we judged that when the x(i, j) increased and reached a critical value (xcr), the Sn crystal was formed at the cell (i, j), i.e., the Sn deposition was completed. Thus, the x(i, j) may be regarded as expressing the amount of a precursor of the Sn crystal. The total electrochemical circuit can be expressed by using an equivalent circuit in the same way as in previous work.37 From the conservation of the current, we obtain38,39

I ) IC + IF ) (En+1 - En)/∆t

∑surf CDL + IF

(7)

where I is the total current, IC is the charging current, and IF is the faradic current. The IC is expressed as IC ) (En+1 - En)/ ∆t∑surf CDL, where CDL is the double layer capacitance and assumed to be same for all the cells located at the interface. The summation (∑surf) is taken for all cells in the electrolyte adjacent to the interface. In case where the I is gradually scanned under the current-controlled conditions, eq 7 is replaced by eq 8:

I(1 + nu) ) IC + IF ) (En+1 - En)/∆t

∑surf CDL + IF

(8)

where u is a proportional constant which determines the scan rate. In the present system, IF is composed of the currents of reaction 1 and reaction 2, denoted as I1and I2, respectively:

IF ) I1 + I2

(4)

This equation is only applied to cells in the electrolyte, adjacent to the electrode (Sn crystal)/electrolyte interface (see Figure 10). The rate constants k1(E), and k2(E) discussed later, are expressed in a common form as

kp ) γpkp0 exp[-RpneF(En - Ep0)/RT]

Faraday constant, R is the gas constant, and T is the temperature. γp is a parameter introduced to express a difference in the rate constant between a flat (or terrace) part and a corner (or step or kink) part of the electrode (Sn crystal) surface. This parameter was used to consider a change in kp, for instance, by formation of an inactive thin surface oxide or hydroxide layer, because the flat surface may be more easily oxidized by an autocatalytic oxidation mechanism.12,36 In the present work, the γ1 for the Sn deposition (reaction 1) at the flat surface was varied in a range from unity to 0.2, with the γ1 at the corner part being kept at unity. The γ2 for the hydrogen evolution was taken to be unity in all cases. We have assumed also that the maximum value for k1(E) is 1/dt, because otherwise the c(i, j) value happens to take a negative value. The γp is the only parameter that depends on space in this model. The growth of the electrode (Sn crystal) by the Sn deposition was expressed by using the order parameter, x(i, j), which has the same dimension as c, as follows:

I1 ) neF

(9)

∑surf (k1(En)cn(i, j))

(10)

∑surf (k2(En)cH)

(11)

I2 ) n e F

where cH is the amount (mass) of H+ ions in a cell, which is in the present work assumed to be constant in all cells in the electrolyte. The summation (∑surf) is taken for all cells in the electrolyte adjacent to the interface, in the same way as mentioned above.


Figure 11. Results of numerical calculations: (a) and (b) represent E vs t, (c) and (d) do ∑surf c vs t, and (e) and (f) do the morphology of the deposited Sn crystal, obtained through calculating (a) and (b), respectively. Drawings a, c, and e are for the case where the total current I was fixed at a constant value of 3.1 × 10-8 A, whereas drawings b, d, and f are for the case where I was scanned (increased in the absolute value) at a rate (u) of 7.0 × 10-5 A. The other parameters were taken as follows: c ) 2.0 × 10-17 mol µm-3, CDL ) 1.25 × 10-8 F µm-2, ∆t ) 0.0002 s, D ) 2.0 × 102 µm2 s-1, F ) 96500 C mol-1, R ) 8.31 × 10 J K-1 mol-1, T ) 300 K, δ ) 1 µm, k10 ) 10 µm s-1 for the corner-shared cells and 2.8 µm s-1 for the side-shared cells, k20 ) 1.0 s-1, E10 ) -1.3189 V, E20 ) -2.1061 V, R ) 0.5, ne ) 2, xcr ) 4.0 × 10-17. The initial potential (En at n ) 1) was set at -1.3 V.

Figure 11 shows results of numerical calculations, with the γ1 value at the flat surface being taken to be 0.28 and the xcr value to be 4.0 × 10-17. The lattice used was composed of 400 × 400 cells with periodic boundary condition along the horizontal direction, and the unit-cell size was 1 µm2. Seed crystals were prepared on every 50 cells as the initial substrate. In this simple simulation, we ignored how the current flows from the dendrites (WE) to the CE. Therefore, the positions of CE and RE were not defined. In Figure 11, parts a and b represent the potential (E) versus time (t), parts c and d do the sum of c (∑surf c) versus t, and parts e and f do the morphology of the deposited Sn crystal, obtained through calculating Figure 11, parts a and b (i.e., for 6 s after initiating the calculation), respectively. Drawings a, c, and e are for the case where the total current I was kept constant, whereas drawings b, d, and f are for the case where I was scanned (increased in the absolute value) at a constant rate. We can see that the calculations reproduce the appearance of the potential oscillation and the formation of dendrites with ordered structures. In addition, for the constant I, a damped oscillation appeared (Figure 11a), whereas for the scanned I, a stable oscillation was obtained (Figure 11b), which were also in good agreement with the experiments, as discussed later (Figures 13a and 14a). It should be noted that the density of the seed crystals strongly affects the dynamic behavior of the potential and the morphology of deposits. For example, the life span of the potential oscillation became half compared with those in Figure 11a when the seed


Figure 12. Snap shots, in an expanded form, of the calculated morphology of the growing front of the dendrites, together with a calculated profile of the Sn2+ concentration, in which the black and white colors represent the high and low Sn2+ concentrations, respectively. The number added to each picture is marked such that it corresponds to the number added to Figure 5a. The mark “X” in stages 4-6 is added for explaining the mechanism of the oscillatory growth of dendrites (see text).

Figure 13. Graphs of i vs U for the type-I (Sn), type-II (Zn), and type-III (Cu) systems, observed when the |i| is gradually scanned from zero to a high value. The electrolytes were the same as those in Figure 2.

crystals were put on every 100 cells, and the oscillation amplitude became unstable when the seed crystals were prepared on every 25 cells. More calculations are necessary to obtain a definite conclusion. The detailed analyses of the simulation will be discussed elsewhere. Figure 12 shows, in an expanded form, the calculated morphology of the growth front of the dendrites, together with


Fukami et al. ably far from the experimentally reasonable values. For example, a value of 2.0 M for the bulk concentration of the Sn(II) is largely deviated from the experimental value of 0.2 M. This is mainly because the present calculations are done not in a 3D system but a 2D system. Thus, the quantitative argument is impossible from the present calculations.

Figure 14. Graphs of U vs t for the same systems as in Figure 13, observed after the |i| was stepped from zero to (a) -21.5 mA cm-2 (Sn), (b) -21.5 mA cm-2 (Zn), and (c) -17.9 mA cm-2 (Cu), which were, in the absolute value, twice as high as the diffusion-limited current (idl).

a calculated profile of the Sn2+ concentration in the electrolyte in which the black and white colors represent the high and low Sn2+ concentrations, respectively. The number added to each picture is marked such that it corresponds to the number added to Figure 5a. In stage 1, the Sn2+ ions are depleted in the electrolyte near the tips of the dendrites, suggesting that the diffusion-limited condition is achieved, thus resulting in the needle growth in stages 2 and 3. In stage 4, where the needles grow long enough, the Sn2+ ions are distributed to a considerable amount in the electrolyte near the needles, suggesting that the diffusion-limited condition almost disappears, and thus the cuboid crystals grow at the tips of the needles. It is to be noted that with the growth of the cuboids in stages 5 and 6, the Sn2+ ions become gradually depleted in the electrolyte near the cuboids (for example, the point marked by X in stages 4-6). This depletion causes again the needle growth at stage 6 (essentially equivalent to stage 1). Thus, the calculated result reproduces well the experimental result of Figure 7. Importantly, the result indicates clearly that the negative potential shift between stages 5 and 6 is mainly due to the depletion (exhaustion) of the Sn2+ ions near (and inside) the dendrites, supporting the expectation mentioned in the beginning of this section. As mentioned earlier, the above-calculated results were obtained when the γ1 value (for the Sn deposition) at the flat surface was taken to be 0.28, nearly one-third of that at the corner part. At present, no potential oscillation has appeared in repeated calculations if the γ1 value at the flat surface is taken to be unity, i.e., it is taken to be equal at all surfaces of the Sn crystal. For this result, two possibilities can be pointed out. (1) In the two-dimensional (2D) CML model used in the present work, the accelerated diffusion in the actually three-dimensional (3D) spherical diffusion layer at the peaked parts cannot be expressed sufficiently. (2) Some passivation of the flat surface at the positive end of the potential oscillation plays a role in the appearance of the oscillation, as suggested in a previous communication.12 The latter factor might affect the morphology of the dendrites as well, together with the surface diffusion of deposited adatoms (not included in the present calculation). Further details are now under investigation. It is finally noted that some of the parameters used in the present calculations (see the caption of Figure 11) are consider-

5. Discussion The experimental and calculated results in the present work have clarified a general mechanism for the potential oscillation and synchronized ordered dendrite formation. The main features of the general mechanism were mentioned in the previous section, especially in the beginning part of it, and are not repeated here. An important fruit of the clarification of the general mechanism is that it has enabled us to classify the oscillation-induced formation of ordered dendrites into three types, type-I, type-II, and type-III. All examples for the oscillatory dendrite formation, including reported ones by other research groups, can be classified into either of these types. The main characteristics for each type are summarized in Table 1. These types are different in the growth dimension in the autocatalytic crystal growth, namely, the growth dimension is 1D (needle) for typeI, 2D (wafer) for type-II, and 3D (round particle) for type-III (see also Figure 2). Interestingly, even when the electrodeposits are constituted of a metal, they can be classified into a variety of classes depending on their growth dimensions. For example, Zn electrodeposition from acidic solution gives rise to needlelike dendrites and is categorized to type-I, whereas that from basic solution produces wafer-like dendrites and is classified into type-II (Table 1). These types are also different in the rate of the increase in the effective electrode surface area (Seff) per a unit amount of the deposit in the autocatalytic crystal growth, i.e., the rate of the increase is high for type-I, moderately high for type-II, and low for type-III. Such differences in the growth mode lead to various differences in oscillatory behavior. Figure 13 compares the current (i) versus U for type-I (Sn), type-II (Zn), and type-III (Cu) systems, experimentally observed when the |i| was gradually scanned from zero to a high value, whereas Figure 14 compares the U versus time (t) for the same systems, observed after the |i| was stepped from zero to a constant current twice as high as the diffusion-limited current (|idl|). In all types, potential oscillations appear spontaneously when the |i| exceeds the diffusion-limited current (|idl|). (The Cu deposition shows another potential oscillation in a region of |i| < |idl|, as reported by Switzer et al.40,42 This oscillation does not accompany the dendrite formation and will not be considered here.) The dotted and dashed lines in Figure 14a-c indicate the potential at which the reaction-limited metal deposition occurs and that at which the hydrogen evolution occurs, respectively. When the |i| is stepped from zero to a high value exceeding the |idl|, the potential for type-I initially remains at the value of the reaction-limited condition (Figure 14a) probably because the surface concentration of metal ions (cs) is high enough at the beginning. With the decrease and depletion in the cs by the deposition reaction, the potential shifts toward the negative down to the hydrogen evolution potential. Then, after an induction period of about 20 s, the potential shifts toward the positive again owing to the needle growth and the oscillation follows. For type-II (Figure 14b) and type-III (Figure 14c), essentially the same behavior is observed, apart from the differences in the stability of the oscillation. A notable point in the above results is that the potential oscillation for type-I is sustained stably and continues for a long



TABLE 1: Classification and Main Characteristics of the Oscillatory Dendrite Growth type-I growth dimension in autocatalytic crystal growth rate of the increase in the Seff oscillation under a constant i potential after the oscillation decayed refsc

type-II

type-III

1D

2D

3D

high rapidly appear but soon decay reaction-limited potentialb Sn in base (refs 12, 14, 40) Sn in acid (refs 40, 41) Sn in org solv (ref 40) Zn in acid (ref 43) Pb in case (ref 40)

middle gradually appear but fairly stable intermediate Zn in base (refs 5, 6, 11, 40) Au in acid (ref 16)

low difficult to appear and very unstablea hydrogen evolution potentiala Cu in base (ref 40) Cu in acid (ref 43)

a Potential can oscillate only a few times as shown in Figure 14c. b The oscillation of type-I can appear stably with a long lifespan when the applied current is scanned to the negative direction at an appropriate scan rate (see text). c Attached numerals refer to the reference numbers.

time when the i is scanned at an appropriate rate (see Figure 13a), though this oscillation decays in 30 s when the |i| is stepped from zero to a constant value exceeding the |idl| (Figure 14a). For type-III, the potential oscillation decays not only when the |i| is stepped (Figure 14c) but also when the i is scanned (Figure 13c). The behavior of the oscillation for type-II appears to be between type-I and type-III. Another notable point is that the potential after the oscillation decayed out for type-I is located at the positive end of the oscillation (or the potential of the reaction-limited condition) (Figure 14a), whereas that for typeIII is located at the negative end of the oscillation (or the potential of the hydrogen evolution reaction) (Figure 14c). The behavior of the oscillation for type-II also appears to be between type-I and type-III. The above results can be understood as follows. As already mentioned, the autocatalytic needle (1D) growth (under the diffusion-limited condition at the negative end of the potential oscillation) for type-I causes the large increase in the effective electrode surface area (Seff), whereas the round-particle (3D) growth for type-III causes the small increase in the Seff. The wafer (2D) growth for type-II gets between type-I and type-III. Thus, the round-particle (3D) growth for type-III is difficult to cause the stable oscillation because the increase in the Seff is not large enough to cause the positive U shift from the negative end (the hydrogen evolution potential) of the potential oscillation (see Figure 14c). Actually, the potential after the oscillation decayed for type-III is at the hydrogen evolution potential (Figure 14c), as mentioned above. On the other hand, the autocatalytic needle (1D) growth for type-I easily causes the oscillation because the increase in the Seff is large enough to cause the positive U shift from the negative end (the hydrogen evolution potential). However, the dendrites of this type cannot cause the sustained oscillation under a constant i (Figure 14a) because the increase in the Seff is so large that the Seff is apt to increase to a large value enough to keep the reaction-limited deposition, in particular when the dendrites grow to a large size. In accordance with this argument, the potential after the oscillation decayed for type-I is at the value of the reaction-limited condition (Figure 14a). The oscillation of type-I can be stable only when the |i| is increased at an appropriate rate (Figures 13a and 11b). 6. Conclusion The present work has clarified a general mechanism for the electrochemical oscillations and the synchronized growth of dendrites with ordered microstructures. Detailed experiments and numerical calculations have revealed that the autocatalytic crystal growth, some surface passivation at the flat surface of the deposits, and the depletion of deposited metal ions near and inside the dendrites (or the screening effect of the dendrites on

the metal-ion diffusion) act as the key roles in the appearance of the oscillations as well as the formation of ordered dendrite structures. An important point is that spatio-temporal synchronization of all these processes leads to self-organized formation of dendrites with ordered microstructures. In addition, the clarification of the general mechanism has enabled us to classify the oscillatory dendrite formation into three types (type-I, typeII, and type-III) according to the differences in the growth dimension (1D, 2D, and 3D). The general mechanism and the classification in the present work will provide a promising way to prepare designed and controlled ordered microstructures at solid surfaces. Acknowledgment. The present work was supported partly by the Grant-in-Aid for Scientific Research in Priority Area “Molecular Nano Dynamics” from the Ministry of Education, Culture, Sports, Science and Technology and The Kurata Memorial Hitachi Science and Technology Foundation. References and Notes (1) Kuroda, T.; Irisawa, T.; Ookawa, A. J. Cryst. Growth 1977, 42, 41-46. (2) Oaki, Y.; Imai, H. Cryst. Growth Des. 2003, 3, 711-716. (3) Ihle, T.; Mullerkrumbhaar, H. Phys. ReV. E 1994, 49, 2972-2991. (4) Arneodo, A.; Argoul, F.; Couder, Y.; Rabaud, M. Phys. ReV. Lett. 1991, 66, 2332-2335. (5) Pierre, J. St.; Piron, D. L. J. Electrochem. Soc. 1990, 137, 24911990. (6) Shinohara, N.; Kaneko, N.; Nezu, H. Electrochemistry (in Japanese) 1993, 61, 1211-1213. (7) Wang, S.; Zhang, K. Q.; Xu, W. Y.; Wang, M.; Peng, R. W.; Zhang, Z.; Ming, N. B. J. Phys. Soc. Jpn. 2003, 72, 1574-1580. (8) Suter, R. M.; Wong, P. Z. Phys. ReV. B 1989, 39, 4536-4540. (9) Nakanishi, S.; Fukami, K.; Sakai, S.-I.; Nakato, Y. Chem. Lett. 2002, 31, 636-637. (10) Fukami, K.; Nakanishi, S.; Sakai, S.-I.; Nakato, Y. Chem. Lett. 2003, 32, 532-533. (11) Fukami, K.; Nakanishi, S.; Tada, T.; Yamasaki, H.; Fukushima, S.; Sakai, S.-I.; Nakato, Y. J. Electrochem. Soc. 2005, 152, C493-C497. (12) Nakanishi, S.; Fukami, K.; Tada, T.; Nakato, Y. J. Am. Chem. Soc. 2004, 126, 9556-9557. (13) Tada, T.; Fukami, K.; Nakanishi, S.; Yamasaki, H.; Fukushima, S.; Nagai, T.; Sakai, S.-I.; Nakato, Y. Electrochim. Acta 2005, 50, 50505055. (14) Piron, E. L.; Nagatsugawa, I.; Fan, C. L. J. Electrochem. Soc. 1991, 138, 3296-3299. (15) Kaneko, N.; Nezu, H.; Shinohara, N. J. Electroanal. Chem. 1988, 252, 371. (16) Saliba, R.; Mingotaud, C.; Argoul, F.; Ravaine, S. Electrochem. Commun. 2002, 4, 629-632. (17) Argoul, F.; Arneodo, S.; Grasseau, G. Phys. ReV. Lett. 1988, 61, 2558-2561. (18) Fleury, V. Nature 1997, 390, 145-148. (19) Fleury, V.; Watters, W. A.; Allam, L.; Devers, T. Nature 2002, 416, 716-719. (20) Wang, M.; Vanenckevort, W. J. P.; Ming, N. B.; Bennema, P. Nature 1994, 367, 438-441.

1160 J. Phys. Chem. C, Vol. 111, No. 3, 2007 (21) Epassaka, D. B.; Ohshio, S.; Saitoh, H. J. Mater. Sci. 2003, 38, 3239-3244. (22) Matsuyama, T.; Matsushita, M. Crit. ReV. Microbiol. 1993, 19, 117-135. (23) Ben-Jacob, E.; Cohen, I.; Gutnick, D. L. Annu. ReV. Microbiol. 1998, 52, 779-806. (24) Ben-Jacob, E.; Garik, P. Nature 1990, 343, 523-530. (25) Ben-Jacob, E.; Godbey, R.; Goldenfeld, N. D.; Koplik, J.; Levine, H.; Mueller, T.; Sander, L. M. Phys. ReV. Lett. 1985, 55, 1315-1318. (26) Brune, H.; Romainczyk, C.; Ro¨der, H.; Kern, K. Nature 1994, 369, 469-471. (27) Sibert, E.; Ozanam, F.; Maroun, F.; Behm, R. J.; Magnussen, O. M. Surf. Sci. 2004, 572, 115-125. (28) Cuenat, A.; George, H. B.; Chang, K-C.; Blakely, J. M.; Aziz, M. J. AdV. Mater. 2005, 17, 2845-2849. (29) Valbusa, U.; Boragno, C.; de Mongeot, F. B. J. Phys.: Condens. Matter 2002, 14, 8153-8175. (30) Hoyt, J. J.; Asta, M.; Karma, A. Mater. Sci. Eng., R 2003, 41, 121163. (31) Herlach, D. M. Mater. Sci. Eng., R 1994, 12, 177-272.

Fukami et al. (32) Mullins, W. W.; Sekerka, R. F. J. Appl. Phys. 1963, 34, 323-324. (33) Wen, S.; Szpunar, J. A. Electrochim. Acta 2005, 50, 2393-2399. (34) Pourbaix, M. Atlas of Electrochemical Equilibria in Aqueous Solutions; National Association of Corrosion Engineers: Houston, TX, 1974. (35) Sakaguchi, H. J. Phys. Soc. Jpn. 1998, 67, 96-98. (36) Nakanishi, S.; Mukouyama, Y.; Karasumi, K.; Imanishi, A.; Furuya, N.; Nakato, Y. J. Phys. Chem. B 2000, 104, 4181-4188. (37) Mukouyama, Y.; Hommura, H.; Nakanishi, S.; Nishimura, T.; Konishi, H.; Nakato, Y. Bull. Chem. Soc. Jpn. 1999, 72, 1247-1254. (38) Koper, M. T. M.; Sluyters, J. H. J. Electroanal. Chem. 1991, 303, 73-94. (39) Koper, M. T. M.; Sluyters, J. H. J. Electroanal. Chem. 1993, 347, 31-48. (40) Switzer, J. A.; Hung, C. J.; Huang, L. Y.; Switzer, E. R.; Kammler, D. R.; Golden, T. D.; Bohannan, E. W. J. Am. Chem. Soc. 1998, 120, 35303351. (41) Reported in the present paper. (42) Aragon, A.; Figueroa, M. G.; Gana, R. E.; Zagal, J. H. J. Appl. Electrochem. 1992, 22, 558-562. (43) Unpublished results.

General Mechanism for the Synchronization of Electrochemical

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