J . Phys. Chem. 1989,93,2731-2737
-
Figure 5. Concentration of C with a center-paint perturbation of 1% in A for the oligomerization reaction 6A Cwith a = IO-’ M, Do= 200 cmys, D, = IO cm’js. k = IO’sC‘, C, = 10-2 M.
the product, the relatively high rate of diffusion can be seen to dampen out smaller fluctuations while allowing the more prominent to grow. This is shown explicitly by comparing the curve of the product to that of the overlaid starting material. Figure 4 shows a sequence in two-dimenstonal space for the reaction 2A C that also started from random seeding. Reaction 14 ( A + B C) showed behavior similar to that seen in Figures 2-4 for 2A C and is not shown here. Figure 5 shows the result of an oligomerization reaction (6A C ) with a seeding perturbation at the center of the grid. This is an aggregation-type behavior.
2731
reactions, to the evolution of patterns from initial fluctuations. The diffusion concentration dependency provides the necessary growth feedback loop through the assumption that the accumulation of product renders slower diffusion rates and provides a smooth transition to the low-concentration,constant D range. As in the well-studied case of spatial and temporal pattern formation through coupling of highly nonlinear reaction schemes with diffusion?O the case we study emphasizes again the apparently paradoxical role that diffusion can play in nonequilibrium situations: Instead of being a mechanism for dissipation of concentration fluctuations, it becomes a source of pattern growth. The consequences of this report to the related areas of polymerizationaggregation” and island formationU phenomena are obvious and will he treated in future reports.
--
Acknowledgment. The Fritz Haber Research Center is supported by the Minerva Gesellschaft fiir die Forschung, mbH, Munich, FRG. Supported by the Israel Space Agency through the National Council for Research and Development. O.C. thanks the Berman Fund for support.
5. Conclusion We have demonstrated the concept that nonlinearities in the diffusion term can lead, when coupled to simple, weakly nonlinear
(20) E.g.: Noyes. R. M. Nature 1988,329,581. For an early review, ye: Noyes, R. M.:Field. R. J. Ace. Chem. Res. 1977, IO, 273. (21) Mcakin, P. In Phare Tmnritionr and Critical Phenomem: Comb, C.. Lebawitz, I. L., Eds.; Academic Pres: New York, 1988. (22) Silverberg, M.;Ben-Shad, A. 1.Chem. Phys. 1987.87, 3178 and references cited therein.
-
-
Quasi-Periodicity and Chaos during an Electrochemical Reaction M. R. Bassett and J. L. Hudson* Department of Chemical Engineering, Thornion Hall, University of Virginia, Charloftesville, Virginia 29901 (Received: June 20. 1988)
The dynamic behavior of the electrodissolution of copper in acidic chloride solution is studied. We show transitions from two- to one-band chaos and from a chaotic attractor to a torus. All the examples were observed during the course of a single experiment, and the transitions take place as the nature of the electrode surface changed slowly relative to the characteristic time of the dynamic behavior.
Introduction Richard M. Noyes, with his co-workers Field and KOr&, published in 1972 an analysis of the oscillatory behavior of the Belousov-Zhabtinskii reaction.’ This seminal paper, along with the numerous other contributions of Professor Noyes, his students, and his colleagues, has led to an explosion in information on oscillating chemical reactions. His work has also led to a greatly improved understanding of the origin of oscillations in chemical reactions and to numerous examples of oscillatory behavior in homogeneous and heterogeneous chemical systems. Oscillations can also occur during electrochemical reactions. Examples include both anodic and cathodic reactions which can electroplating: be found in such diverse areas as electrocataly~is?~~ electropolishing? electromachining? and in the activepassive transitions of many ( I ) Field, R. J.; Kciras, E.: Noycs, R. M. 1.Am. Chem. Soc. 1972, 94, 8649. (2) Wojtowicz, J.; Marincic, N.; Conway, B. E. 1.Chem. Phys. 1968,48, 4333. (3) Tributach, H. Be,. Bunsen-Ges. Phys. Chem. 1975, 79) 570. (4) Clarke, M.; Bernie, J. A. Electroehim. Acta 1967, 12, 205. (5) Tsiuapoulos. L. T.; Tsotsi, T. T.; Webtcr. 1. A. SwJ Sei. 1987.191, 225. (6) Caapcr. I.: Muller. R.; Tobias, C. 1.Electrochem. Soc. 1980. 128, 1733. (7) Franck U. F.; FitrHugh. R. Z. Elrkfrochem. 1961,6S, 156
0022-3654/89/2093-2731$01.50/0
There have been several recent studies of the dynamics of oscillatory electrochemical reactions; these studies have produced numerous experimental examples of different types of dynamic behavior. Electrochemical reactions have several features, in addition to their interesting behavior, which facilitate such studies. First, the measurement of current or voltage is relatively straightforward. Second, the time scale of the oscillations under some conditions is short, both relative to the time scale of changes in the surface conditions so that data are taken under stationary conditions and absolutely so that large amounts of data can be obtained to use in data analysis. Diem and Hudson” have used methods such as the calculation of a correlation dimension12to show the existence of simple and higher order chaotic structures during the electrodissolution of iron in H2S0, at the limiting-current plateau. Copper electrodissolution in H3P04has been studied by Alhahadily and Schell” (8) Pdests, J. J.; Piatti. R. C. V.: Arvia, A. J. Electrochim. A m 1979, 24. 633. (9) Jaeger, N. 1.: Plath, P. J.: Quycn. N . Q. Tempor01 Ordm Springer: Berlin, 1985. (10) McKubre, M. C. H.: Macdonald. D. D. 1.Elccfrochem. Soc. 1981, 128. 524. (11) Diem, C. B.; Hudson, J. L. AIChE 1.1987.33.218, (12) Grasskrgcr, P.; Prmccia, 1. Physico D 1983, 9, 189. (13) Albahadily, F. N.; Schell, M. 1.Chem. Phys. 1988.88, 4312.
0 1989 American Chemical Society
2732 The Journal of Physical Chemistry, Vol. 93, No. 7, 1989 30
Bassett and Hudson
r
it-4
I
s--+
5L
a
TIME
27.8,
25
P
,,P
73 'I
i
cl
F ?.
v 4 H
75
*I5 69
17 17
I(t-2T)
17
25
I (t-4T)
n- 1
C
25 d
Figure 1. Two-band chaos: (a) Time series. (b) Attractor (spline fit); the two projections are made at angles differing by 6 O . The right picture is for the left eye and the left for the right. The attractor was constructed by using the time delay method (7 = s for this and all other figures). The arrow shows the direction of flow (same for other figures). The attractor was spline fit by using IMSL subroutines ICSCCU and ICSEVU (the same holds for the rest of the attractors shown). (c) Poincarb section taken at I ( t ) = 18 mA; increasing I(t-47). This and all other PoincarE sections were calculated by a linear interpolation of two points, one on each side of the cutting plane. (d) Return map constructed from the Poincarb section taken at I(t-27) = 18 mA with increasing I(t-47); using variable I(t-47).
and has been found to undergo Hopf bifurcations to oscillatory behavior and period-doubling bifurcations to simple chaos. This system was also found to give both periodic and chaotic mixedmode oscillations. Lev et al. have studied the galvanostatic electrodissolution of nickel in H2sO4; they have observed Hopf bifurcations, saddle loop bifurcations, period-doubling bifurcations, chaos, and qua~i-periodicity.'~*'~ Apparent subcritical Hopf bifurcations have been found with a cobalt electrode.16 Bassett and Hudson have shown that copper electrodissolution in NaCl and H2S04gives a plethora of interesting behavior, including a variety of periodic oscillations, period doublings, type I11 intermittency, windows of period three oscillations within chaotic regions, and Shil'nikov ~ h a o s . ' ~ J * In this paper we show some additional dynamic behavior not previously seen in electrochemical systems. The experiments are done with copper in a NaCl and H2S04electrolyte. Examples are given of quasi-periodicity, chaos, and chaos with a broken toroidal structure. We show a transition from two-band to (14) Lev, 0.;Wolffberg, A.; Sheintuch, M.; Pismen,
1988. 43. 1339.
L. Chem. Eng. Sci.
(15) Lev, 0.; Wolffberg, A.; Pismen, L.; Sheintuch, M., preprint. (16) Hudson, J. L.; Bell, J.; Jaeger, N. I. Eer. Bunsen-Ges. Phys. Chem. 1988. 92. 1383. ( 1 7 ) Bassett, M. R.; Hudson, J. L. Chem. Eng. Commun. 1987, 60, 145. (18) Bassett, M. R.; Hudson, J. L. J . Phys. Chem. 1988, 92, 6963.
one-band chaos. We also show chaos with a broken toroidal structure that undergoes a transition to quasi-periodic motion on a torus. The dynamic behavior is analyzed with a variety of techniques, including the use of attractor r e c o n s t r u ~ t i o n , ' ~ ~ ~ ~ PoincarB sections,21return maps,21next amplitude maps,22correlation dimension calculations,'2 the calculation of the largest Lyapunov exponent,23and power spectra.21 Experiments
The experiment was carried out using a copper rotating disk electrode 0.54 cm2 in area. The solution, which was 1 N in H2S04 and 0.3 M in NaCI, was held at 25 "C.The disk was rotated at 200 rpm. Experiments were performed potentiostatically at E = 275 mV (versus a saturated calomel electrode). At the beginning of an experiment, the potential was changed in a stepwise manner from its rest potential to 275 mV. The current in mil(19) Packard, N. H.; Crutchfield, J. P.; Farmer, J. D.; Shaw, R. S.Phys. Reu. Lett. 1980. 45. 712. (20) Takens,' F. Dynamical Systems and Turbulence; Springer: Heidelberg, 1981. (21) Eckmann, J. P.; Ruelle, D. Reu. Mod. Phys. 1985, 57, 617. (22) Hudson, J. L.;Rossler, 0. E. Dynamics of Nonlinear Systems; Hlavacek. V. Ed.; Gordon and Breach Science: New York, 1986; Chapter 6. (23) Schuster, H. G. Deterministic Chaos; Physik-Verlag: Wenheim, 1984.
The Journal of Physical Chemistry, Vol. 93, No. 7, 1989 2733
Quasi-Periodicity and Chaos during a Reaction
+5
s--+
17
51
TIME
a
17
25
d
27 h
tT c
3
10
10
2 i o 27
e 3
F
f * .
Figure 2. One-band chaos: (a) Time series. (b) Attractor (spline fit). (c) Poincart section taken at I ( ? )= 17.5 mA; increasing I(?-47). (d) Return map constructed*fromthe Poincart section taken at I(?-27) = 17.5 mA with increasing I(?-47); using variable I(?-47). (e) Series of Poincar.5 sections showing the flow of trajectories around the attractor. Sections 1 and 3 were taken at I(?-27) = 17.5 mA and sections 2 and 4 were taken at I(?-47) = 17.5 mA.
liamps was then measured as a function of time at 60 Hz with the aid of a multimeter and a laboratory computer. More details on the experiments can be found in ref 17.
Results As the experiment begins, a thick, white, CuCl film forms at the perimeter of the electrode while the center of the electrode is covered by a very thin film of CuCl which can only be seen by close inspection. The CuCl film forms via the electrochemical reacti~n~~,~~
+
-
CuCl
+ c1-
+
Cu C1CuCl eand is removed via the r e a c t i o n ~ ~ ~ , ~ ~ and26,27 CuCl
-
-
Cu2+
cuc12-
+ C1- + e~~~
~~
(24) Cooper, R.; Bartlett, J . H. J . Elecfrochem. SOC.1958, 105, 109. (25) Lee, H. P.; Nobe, K. J . Elecfrochem. SOC.1986, 133, 2035.
As the experiment continues the thick, white, CuCl film grows inward, covering more and more of the electrode surface while the center portion correspondingly shrinks. Oscillations begin 0.5 min into the run and die out about 54 min into the experiment, when the electrode is entirely covered by the thick, white, CuCl film. During the 54 min of the experiment, the nature of the oscillations changes. The slowly growing film thus acts as a slowly changing parameter, which in this case is probably the magnitude of a resistance for transport to the copper surface. Several models for the oscillations of copper electrodissolution in chloride solutions have been suggested in which the thickness of the film played a major ro1e.24,27-30 However, the time scale of this parameter change is much greater than of the oscillations, so that data taken over a short interval of time can be considered to be stationary. (26) Flatt, R. K.; Brook, P. A. Corros. Sci. 1971, 11, 185. (27) Pearlstein, A. J.; Lee,H. P.; Nobe, K. J . Electrochem. Soc. 1985, 132, 2159. (28) Bonhoeffer, K. F.; Gerischer, H. Z . Elekfrochem. 1948, 52, 149. (29) Lal, H.; Thirsk, H. R. J . Chem. SOC.1953, 2638. (30) Hedges, E. S. J . Chem. SOC.1929, 1028.
Bassett and Hudson
2734 The Journal oJPhysica1 Chemistry. Vol. 93, No. 7, 1989 19.4 -
25
7 *.
-
-.**
. .. .
..i
.t
. ..f . .. ..t
i= ? . c
-- ... .:fit., .. .. .. .. : f
-
L 1
I..
.f
..
.
1
.
. .. ... .... .“ :2
ft
i t 6 s+
15
TIME
a
22.8 c
4
. . ..
24
-16
16
16 24
Figure 3. Chaos on a broken toroidal S V U ~ U ~ C(a) : Time series. (b) Attractor (spline tit). (c) Poincare section taken at I(f-27) = 20 mA; increasing l(1-27). (d) Series of Poincare sections showing the folding process. All planes used to make the sections are defined by the general equation I ( t ) + l(1-27) - 40 = a(l(f-4r) - 20). For sections I and 5, a = 1: for section 2, a = -I; for section 3, a = -5; and for section 4, a = 7 . These planes are marked by lines an the attractor. Attractor is the same as that shown in Figure 3b.
For example, consider the data presented in Figure 1. One thousand data points are shown which, at a 60-Hz sampling rate, corresponds to 16.7 s, during which there are 89 maxima. Since 16.7 s is short compared to the rate at which the surface changes during the 54 min of the entire experiment, the 16.7 s of data can be used for data analysis. As a check, we varied the length of the section of data used in the analysis, normally by a factor of 2, and obtained essentially identical results. After the oscillations begin 0.5 min into the experiment, the oscillatory behavior changes somewhat rapidly over the next IO min until the stationary behaviodshown in Figure l a is obtained. The attractor (Figure Ib), PoincarE section (Figure IC). and return map (Figure Id) all indicate that this is two-band chaos. The trajectories on the attractor lie in two distinct bands, and the flow on the attractor alternates between these two bands. The PoincarE section appears to be approximately one-dimensional, indicating a simple chaotic struture, but it has two islands into which all crossings fall. The vertical line in Figure I C was drawn so that this two-island structure can be seen more clearly. Points alternate between the islands to the left and right of the line, respectively. In addition, the points on the Poincar6 section are numbered in sequence, and it can be seen that all the odd numbers lie on the left island and the even on the right. The return map (which is also numbered in sequence as the points appear) is also approximately onedimensional and consists of two islands as well. The points here also alternate between each of the two islands; all odd numbers lie on one island of the return map and all even numbers lie on the other island. The two-band chaos changed into the one-band, simple chaos seen in Figure 2a. The attractor (Figure 2b) is similar in structure
to that seen in Figure I b but obviously no longer consists of two bands. The Poincar6 section and return map (Figure 2c,d) are similar to the corresponding results in Figure I, but now the two islands have merged to form one larger island. A sequence of Poincar6 sections is shown in Figure 2e to explain the flow of trajectories around the attractor. The Poincar6 d o n s were made with planes at constant I(r-27) = 17.5 mA and I(f-4r) = 17.5 mA and were constructed by using all the intersections of the attractor of Figure 2b with the above-mentioned planes. In Figure 2e only a few orbits around the attractor are shown so that the location and orientation of the sections can be seen. The arrow on each section is used to mark the points that started on the edge of the attractor in section 1. This sequence shows the stretching, folding, mixing, and contraction processes common to these types of simple chaotic attract or^?'.^^ In going from section 1 to section 2, one observes the stretching of the trajectories. In going from section 2 to 3, the trajectories continue to stretch but the folding process is also observed. In addition, the points that were on the outside of the attractor are now on the inside. In going from section 3 to 4, one sees the contraction of the fold onto itself. This proccss continues as trajectories proceed around the attractor to give back the result seen in section I . From the shapes of the attractor, Poincari sections, and return map, the chaos shown in Figure 2 appears to be a simple type that may be embeddable in three-dimensional space and that has a single positive Lyapunov exponent. For further verification the correlation dimension” and the largest Lyapunov exponent” were (31) R&sslcr, 0.E.2. Nnrurforsch. 1976, 310, 259. (32) Rhsler, 0. E. Bull. Moth. Bio. 1977. 39, 275.
The Journal of Physical Chemistry, Vol. 93, No. 7, 1989 2135
Quasi-Periodicity and Chaos during a Reaction
I -4
J
s--+
19
*
*
‘ f
20.6
23.8
25 d
L” -?
18
0
FREQUENCY (Hz)
b
Figure 4. Quasi-periodicity: (a) Time series. (b) Power spectrum;f, = 0.78 Hz;f2 = 4.14 Hz. (c) Attractor (spline fit). (d) Poincare section taken at I(r-21) = 23 mA, increasing I(t-27). (e) .Return map (next angle) of (d), center point used for the calculation was (24.6,21.3).
calculated. The correlation dimension was found to be 2.2. An approximate value for the Lyapunov exponent was calculated from the return map, Figure 2d. The return map was fit with the equation X,+I = A(x, - C) + B ( x , - Q2 + D(x, - Q3+ C where A = -1.7454, B = -0.24017, C = 21.855, and D = 0.039 885 and the largest Lyapunov characteristic exponent was calculated from 1” X = lim -E log, Idy/dxl pm
n
where n = number of points in the return map and dy/dx 2 slope of the return map. The Lyapunov exponent was found to be 0.6, which indicates the divergence of nearby trajectories in at least one direction and supports the belief that this is chaos. We checked the sensitivity of this approximation by fitting the map with quadratic and quartic equations and recalculated the exponent, obtaining values of 0.41 and 0.47, respectively. We also purposely worsened the fit by choosing arbitrary values for C and recalculating A, B, and D and
the corresponding Lyapunov exponent. For all values of C for which the curve still reasonably fit the data a positive exponent was found. The chaotic behavior shown in Figures 1 and 2 was preceded by a period-doubling bifurcation of periodic orbits. Such period doublings have been seen in a previous study” and are not shown here. Of note, however, is the fact that the period-doubling sequence results in a chaotic structure of more than one band with a Poincare section of more than one island. To within experimental accuracy, we see only a two-band structure. Furthermore, the return map for the resulting fully developed chaos has the familiar quadratic shape (Figure 2d) associated with period-doubling bif~rcations.~~ After some time, the simple, one-band chaos gives way to the behavior seen in Figure 3. This has a strong three-peak tendency, as is seen in Figure 3a, which is slowly modulated to give chaotic behavior that lies on a broken toroidal structure. The attractor is shown in Figure 3b, and a Poincar6 section, a closed curve (33) Feigenbaum, M. Physicu D 1983, 7, 16.
Bassett and Hudson
2736 The Journal of Physical Chemistry, Vol, 93, No. 7, 1989
2Grrl
TIME
0
a
n
18 b
FREQUENCY (Hz)
21 25
e
C
Figure 5. Quasi-periodicity: (a) Time series (the middle half of the data used in the actual analysis of the data. (b) Power spectrum;f, = 0.53 Hz; A = 3.9 Hz. (c) Attractor (spline fit). (d) Next maximum map. (e) Return map (next angle) of (d), center point used for the calculation was (24.5,24.5).
indicating a toroidal substructure, is shown in Figure 3c. A series of Poincar6 sections (Figure 3d) shows the stretching and folding process that indicates the chaotic nature of the flow. The planes used to make the sections are defined by the equation Z ( t ) I(t-27) - 40 = a(Z(t-47) - 20). Changing the value (Y rotates the plane about a line that goes through the center of the attractor and is defined by the intersection of the planes Z(t) I(t-27) = 40 and I(t-47) = 20. The first cut shows a closed cross section. Two areas of the section are marked (A and B) to point out the folding process. The folding becomes noticeable in the second section as A and B are coming together. The folding process is almost complete in the third section as A and B have almost totally come together. By the fourth section the folds have become so close together that the separation can no longer be seen. The fifth section is 180' from section one; in the remaining 180' the shapes of the sections undergo no additional significant changes. Chaotic behavior on a broken toroidal structure has also been observed with the Belousov-Zhabotinskii reaction.34 The chaos changes eventually giving way to the quasi-periodic behavior seen in Figure 4a. The latter has two incommensurate
+
+
(34) Roux, J . C. Physica D 1983, 7, 57.
vi)
frequencies at 0.78 Hz and 4.14 Hz cfi) as is seen in Figure 4b. The frequencyf2 corresponds to the main flow around the center hole of the torus while the frequencyfi represents the progression in the plane of the Poincari section. The power spectrum also shows strong frequencies at linear combinations of the two incommensurate frequencies, a few of which are marked in Figure 4b, as expected for quasi-periodicity. The attractor (Figure 4c) is the torus associated with quasi-periodic behavior, and the Poincari section (Figure 4d) is a one-dimensional closed curve as expected. A return map (next angle) is shown in Figure 4e and appears to be both one-dimensional and invertible, indicating quasi-periodic behavior. This map was made by first translating the coordinate system of Figure 4d to (24.6,21.3) which lies in the hole of the Poincari section. The angle of the n 1st point in Figure 4d with respect to the new origin was plotted against the angle of the nth point for all the points in the map. In time, the hole in the torus shrinks as can be seen in Figure 5ac. The time series shows only the middle 16.67 s of the total 33.33 s of data used in the analyses of Figure 5b-e. The power spectrum (Figure 5b) now has incommensurate frequencies at 0.53 Hz (fJ and 3.9 Hz cf2). Again there are also strong frequencies at linear combinations of these two frequencies. The next max-
+
J. Phys. Chem. 1989, 93, 2737-2740 imum map (Figure 5d) is a one-dimensional closed curve as expected, and its return map (next angle) is also one-dimensional and apparently invertible, consistent with quasi-periodicity. At the end of the experiment the CuCl film covers the surface completely and the oscillations decay to some steady-state value.
Final Comments We have presented examples of quasi-periodicity and chaos during the electrodissolution of copper in NaCl and H2S04. We have seen the transition from two-band chaos to one-band chaos and a transition from chaos lying on a broken toroidal structure to a simple torus. The study thus adds to the compendium of types of dynamic behavior that have been found to occur in electrochemical reactions.
2737
Several phenomena are involved in the overall electrodissolution process: reaction at the copper surface, growth and dissolution of a film, and transport to and from the bulk electrolyte. A complete model of the system would consist of partial differential equations in the three independent variables time, axial position, and radial position. Nevertheless, all the types of dynamic behavior observed in this study appear to have dimension less than three. In addition, it appears that the attractor can be represented in three-dimensional-state space; thus the dynamics may be describable by as few as three ordinary differential equations.
Acknowledgment. This work was supported in part by Grant CBT-87 13070 from the National Science Foundation. Registry No. Cu, 7440-50-8; NaCI, 7647-14-5; H2S04,7664-93-9; CUC1, 7758-89-6.
Measurement of Convection Velocities in “Mosaic” Patterns J. Rodriguez? and C. Vidal* Centre de Recherche Paul Pascal, Domaine Universitaire. 33405 Talence Cgdex, France (Received: July 1 1 , 1988; In Final Form: November 28, 1988)
Experimental evidence is given of the occurrence of convection in a liquid layer of an oscillatory BZ reagent, sandwiched between two glass plates. Quantitative data on the hydrodynamic velocities are reported for three different layer depths. It is shown that convective motions result from inhomogeneities spontaneously occurring in the chemical reaction.
1. Introduction Spatial self-organization phenomena have been known for years to develop spontaneously in certain reacting systems.’ Among them, chemical waves were the first to be discovered at the end of the 1 9 6 0 ~though ,~ their existence was anticipated by Luther3 at the beginning of this century. The occurrence of these waves requires an “active” medium, i.e., one that is metastable, bistable, excitable, or oscillatory. All of their properties (except their birth, which is still a matter of controversy) are well understood in terms of the interaction of chemical reaction and diffusion of matter; neither mass convection nor the release and transport of heat need to be invoked. Stationary patterns were discovered in 1977 by Mocke14 and studied later by Micheau et aL5 and Avnir et aL6 These patterns often occur at a liquid-gas interface, even with a monotonic (e.g., first or second order) reaction. In many cases, they are an outcome of hydrodynamic instability: the Rayleigh-BBnard and/or Marangoni effects set up currents that are revealed by the colored products of the r e a ~ t i o n .Similar ~ patterns are also observed in reacting layers sandwiched between two plates, in spite of this arrangement ruling out the possibility of either evaporative cooling or surface tension inhomogeneities. Provided the (photo)chemical reaction takes place in the upper part of the layer, a density gradient may still appear and give rise to convection beyond the Rayleigh-=nard threshold. This explanation fails in an isothermal medium, but, as was suggested by Dewel et al.’ and shown by Avnir et a1.,8 convective motion may still be generated by a double diffusion process when the reaction products has a diffusion coefficient different from that of the reactant. Finally, active systems, especially the Belousov-Zhabotinsky (BZ) reaction and severaI other bromate oscillators, exhibit stationary patterns named “mosaic structure^"^ from their appearance. Though convection has been thought likely to have Permanent address: Departamento de Fisica de la Materia Condensada, Faculdad de Fisica, Universidad de Santiago, 15703 Santiago de C o m p t e l a , Espafia.
something to do with them, only indirect arguments have hitherto supported this hypothesis. The primary purpose of the work reported in this article is to provide direct experimental evidence, clarifying this issue. To this end, we observed the hydrodynamic currents with a microscope, and we measured their velocities so as to be able to compare them with the velocities that would result from the action of various possible mechanisms. The data collected definitely rule out gas bubble formation, evaporative cooling, and liquid-gas interface inhomogeneities as possible “engines” of these patterns in a layer of BZ reagent sandwiched between two rigid plates. Since the convection velocities are strongly dependent on layer depth, it seems plausible that a buoyancy driven instability may be induced by the spatially inhomogeneous chemical reaction.
2. Experimental Section The BZ reaction was carried out at room temperature (21-22 “C). As was shown to be legitimate in several earlier studies,1° we used trade chemicals without further purification. The initial concentrations (0.08 M malonic acid, 0.22 M sulfuric acid, 0.31 M sodium bromate, and 0.004 M ferroin) were also close to those (1) Vidal, C.; Hanusse, P. Int. Rev. Phys. Chem. 1986, 5 , 1. (2) Oscillations and Traveling Waves in Chemical Systems; Field, R. J., Burger, M. Eds.; Wiley: New York, 1985. (3) Luther, R. Z. Electrochem. 1906,12, 596; translated by Showalter, K.; Tyson, J. J. J . Chem. Educ. 1987, 64, 742. (4) Mockel, P. Natunvissenschaften 1977, 64, 224. (5) Micheau, J. C.; Gimenez, M.; Borckmans, P.; Dewel, G. Nafure 1983, 305, 43. Gimenez, M.; Micheau, J. C. Nafunvissenschaften 1983, 70, 90. (6) Avnir, D.; Kagan, M. Nature 1984, 307, 717. (7) Dewel, G.; Walgraef, D.; Borckmans, P. Proc. Natl. Acad. Sci. USA
1983, 80, 6429. (8) Avnir, D.; Kagan, M.; Ross, W. Chem. Phys. Lett. 1987, 135, 177. (9) Zhabotinsky, A. M.; Zaikin, A. N. J . Theor. Biol. 1973, 40, 45. Showalter, K. J . Chem. Phys. 1980, 73,3735. Orban, M. J. Am. Chem. Soc. 1980, 102, 4311. (10) Vidal, C.; Pagola, A.; Bodet, J. M.; Hanusse, P.; Bastardie, E. J. Phys. 1986, 47, 1999. Pagola, A,; Vidal, C. J . Phys. Chem. 1987, 91, 501. Vidal, C.; Pagola, A,, this issue.
0 1989 American Chemical Society
