19192
J. Phys. Chem. 1996, 100, 19192-19196
Quenching of Chemical Oscillations with Light P. G. Sørensen,* T. Lorenzen, and F. Hynne* Department of Chemistry, H. C. Ørsted Institute, UniVersity of Copenhagen, UniVersitetsparken 5, DK-2100 Copenhagen, Denmark ReceiVed: September 4, 1996X
We show that it is possible to quench oscillations of the ruthenium bipyridyl catalyzed Belousov-Zhabotinsky reaction by a pulse of light when the system is run close to a supercritical Hopf bifurcation. The response to light depends on the phase of oscillations at the exposure and on the total absorbed radiant energy. It is characteristic of a quenching associated with the geometry of a limit cycle and invariant manifolds of a saddle focus created by a Hopf bifurcation. The experimental results impose a constraint on the possible production and consumption of species induced by light. It can help clarify the mechanism of the effect of light on the system. This is important to the use of light for studying chemical waves and for chaos control.
1. Introduction Interaction of light with species participating in chemical reactions has several applications in addition to the obvious one of monitoring concentrations of species. It has been used to induce and maintain dynamical structures in closed systems by photochemical generation of species1,2 or by creating thermal instabilities.3 It has also been used for study and control of reaction systems4,5 to study dynamical structures including unstable states. In distributed systems, light has been used to control locations or movements of spirals and other chemical waves.6,7 The light sensitivity of the ruthenium tris(bipyridyl) catalyzed Belousov-Zhabotinsky (BZ) reaction is particularly important for experiments on chemical waves. It makes it possible to establish predetermined, spatially inhomogeneous initial conditions, provided the effect of light on the reaction mixture is known. The purpose of the paper is to obtain quantitative information that may help elucidate the mechanistic basis of the effect of light on the system.8-14 In essence we compare the photochemical effect of a pulse of light with the effects of additions of the separate species that participate in the reaction. To get a useful response, we compare the results of special “quenching” perturbations (i.e. perturbations that temporarily stop the oscillations). The quenching technique utilizes the special geometry that exists in concentration space near a small limit cycle arising from a supercritical Hopf bifurcation.15 That geometry contains much kinetic information and can be probed by perturbations from the limit cycle. Close to a supercritical Hopf bifurcation on the oscillatory side, it is possible to stop the oscillations (temporarily) by shifting the state of the system in concentration space from the small limit cycle to the stable manifold of the saddle focus from which the limit cycle emerged at the bifurcation. If the effect of an illumination of the oscillating reaction with a short flash of light is to shift the state in concentration space, it will be possible to quench the oscillations by making the exposure with the correct total radiant energy at the right phase of the oscillations. The effect of light can then be analyzed in terms of the quenching data for addition of the separate species that participate in the reaction (basic quenchings). In the following section we show how it is possible experimentally to quench the oscillations with light. In order X
Abstract published in AdVance ACS Abstracts, November 15, 1996.
S0022-3654(96)02741-4 CCC: $12.00
to interpret the result chemically, we need to know the quenching data for each of the essential species of the BZ reaction. (The concept of essential species is discussed in ref 16.) The basic quenching experiments that provide these data are described in section 3 with the experimental details summarized in the Appendix. In section 4 we then present the necessary elements of quenching theory in a form suitable for interpretation of the chemical effect of light, which follows in section 5. The short section 6 summarizes the results of the paper. 2. Perturbations with Light Figure 1 shows the results of two experiments in which the system is exposed to a short, intense pulse of light. It illustrates how the response depends on the phase of the oscillations at which the light pulse is applied. The first experiment, parts a and b, shows that the oscillations are stopped when the exposure is made when the limit cycle oscillations are in a phase of -176°. The second experiment, parts c and d, shows that the same exposure made in the opposite phase, 4°, increases the amplitude of the oscillations temporarily. In all four parts of Figure 1, the instant of a perturbation is marked with vertical line segments. For the oscillations to stop by an exposure to light, the total absorbed radiant energy must have a definite value as well as being made in the correct phase. (The precise definition of phase is given in the Appendix.) In the experiments we keep the intensity and the optical geometry constant and vary the duration of the exposure in search of the quenching dose. For the setup used, the oscillations were stopped if the light exposure at -176° had a duration of 6.6 s, but not if the duration was smaller or larger or if the phase was different from -176°. The quoted quenching phase, -176°, is the mean phase of the exposure relative to that of the maximum of [Ru(bpy)3+ 3 ]. As a simple illustration of the implications of the results for quenching with light, suppose the sole effect of light is to produce some species (such as Br-) in excess of what would have been produced without light. Such a hypothesis can be tested simply by determining experimentally whether it is possible to quench the oscillations by addition of the species considered, at the right phase, viz. -176°. 3. Basic Quenchings In general the effect of light may be an excess production and/or consumption of several species in amounts having © 1996 American Chemical Society
Quenching of Chemical Oscillations with Light
Figure 1. Quenching of oscillations with light. Time series showing (a) the transmission of light at 452 nm and (b) the potential of a bromide electrode during a perfect quenching made by exposing the reaction mixture to a flash of light at phase -176°. The time series in (c) and (d) show the response when a flash of light of the same total energy is made at the phase 4°. The phase dependence of the response in (a), (b) versus (c), (d) is characteristic of the geometry associated with quenchings of simple oscillations.
definite ratios, and the resulting quenching data can then be expressed in terms of basic quenchings, i.e. quenchings by addition of one species at a time. The possibility of such analysis motivates the experimental determination of basic quenching data for as many relevant species as possible. In Figure 2 and Table 1, we report the results of successful quenchings with Ru(bpy)3+ 3 , Br , HBrO2, and HBrO. It was not possible to quench the oscillations with 2+ BrO3 or Ru(bpy)3 . The results reported here differ somewhat from those of a preliminary study17,18 partly because the point of operation is somewhat different. We note that the quenching phases show a pattern broadly similar to that found for the cerium-catalyzed BZ reaction:19 the quenching phase for HBrO2 is completely different from those of the other essential species, which all have comparable quenching phases. Indeed, for the ruthenium system, the quenching phases of Ru(bpy)3+ 3 , Br , and HBrO are all equal to within the experimental uncertainty of approximately 5°. Nevertheless there are significant differences between the quenching data of the ruthenium bipyridyl and cerium catalyzed reactions. In particular, Ru(bpy)3+ 3 is a much more efficient quencher than Ce4+. In the ruthenium system, quenching with hypobromous acid requires the addition of a considerably larger amount than for the other species of Table 1, so it may be less important than these.16 A three-dimensional reconstruction based on
J. Phys. Chem., Vol. 100, No. 49, 1996 19193
Figure 2. Basic quenchings. Time series showing the transmission of light at 452 nm during quenchings by addition of (a) Ru(bpy)3+ 3 , (b) Br-, (c) HBrO2, and (d) HBrO.
TABLE 1: Quenching Concentrations qs in Units of the Amplitude a of the [Ru(bpy)33+] Oscillations and Quenching Phases Os Relative to That of the Maximum of [Ru(bpy)33+] Ru(bpy)3+ 3 Br-
qs/a
φs
0.060 0.172
-95° -97°
HBrO2 HBrO
qs/a
φs
0.485 1.91
115° -96°
Ru(bpy)3+ 3 , Br , and HBrO2 may therefore be meaningful as a test of consistency. We have found the amplitudes and phases of the concentration oscillations of HBrO2 and Br- by the method described in refs 15 and 19. The phase of Br- relative to Ru(bpy)3+ 3 is calculated to 177°, in reasonable agreement with the value of 149° determined from the experiment shown 3+ in Figure 3. (It is assumed that [Ru(bpy)2+ 3 ] + [Ru(bpy)3 ] is constant.) We note that the phase for quenching with light differs from those for addition of the species tested, so the effect of light cannot be explained as production of just one of these species. We shall therefore try to explain the effect in terms of several species, and to discuss the effect quantitatively we must first summarize the relevant theory. This we do in the immediately following section.
4. Theory of Quenching with Light We want to express the quenching data for quenching with light in terms of those for basic quenchings. We must therefore look at a general quenching which is described in terms of the geometry of a limit cycle and an associated saddle focus that exist near a supercritical Hopf bifurcation (on the oscillatory side).
19194 J. Phys. Chem., Vol. 100, No. 49, 1996
Sørensen et al. in concentration space. From a state r on the limit cycle, the perturbation shifts the system to r + q. This results in a quenching of the oscillations if
r+q)t
(5)
where t satisfies eq 3. By multiplying eq 5 from the left with e+ and using eqs 3 and 4, we get
e+‚q ) -a exp(iφ)
Figure 3. Simultaneously recorded time series of transmission of light at 452 nm and potential of a bromide electrode. Their phase difference is -31° corresponding to a phase of 149° for [Br-] relative to [Ru3+ -7 (bpy)3+ 3 ]. The amplitude of the [Ru(bpy)3 ] oscillations is 6.6 × 10 M, and the mean value is 2.5 × 10-6 M. The amplitude of the potential oscillations is 2.1 mV.
To a good approximation, small limit cycle oscillations in concentration space can be described in terms of the pair of complex conjugate eigenvectors of the Jacobi matrix J with eigenvalues (iω at the bifurcation
J‚e( ) (iωe(
(1)
c ) c0 + a Re{e+ exp(iωt)}
(2)
Thus
in which a is the amplitude of the limit cycle oscillations. The real and imaginary parts of e+, denoted e1 and -e2, respectively, span the plane of oscillation. We shall henceforth work with vectors relative to the stationary point c0 and denote a point on the limit cycle relative to c0 as r, thus r ) c - c0 with c given by eq 2. Quenching of oscillations occurs if a perturbation shifts the state from a point on the limit cycle, r, to a point on the stable manifold, which is characterized by the fact that its tangent space is biorthogonal to the left eigenvectors e( of J corresponding to the eigenvalues (iω. Thus, with t a vector in the stable subspace, relative to c0, the stable subspace can be characterized by +
e ‚t ) 0
(3)
Left eigenvectors are automatically biorthogonal to right eigenvectors corresponding to different eigenvalues. The complex eigenvectors e( are normalized implicitly by eq 2. Left eigenvectors are normalized to satisfy biorthonormality relations. For complex eigenvectors these are
e(‚e( ) 2, e(‚e- ) 0
(4a)
For other combinations of real and complex eigenvectors, the relations are
ej‚ek ) δjk
(4b)
in which δjk denotes the Kronecker δ. For quenching from a sufficiently small limit cycle, the stable manifold can be approximated by its tangent space. Consider a perturbation with the effect of shifting the state by a vector q
(6)
in which we have introduced the phase φ of the oscillations as ωt, which we usually refer to as a principal value in the interval -180° < φ e 180°. An important special case of eq 6 occurs if the perturbation arises from an addition of a species (labeled by s) so that the quenching vector is in the direction of the sth concentration axis. If the addition of species s must be made in the phase φs with a change of concentration of species s equal to qs in order to quench the oscillations, then condition 6 shows that the s-component of e+, (e+)s, can be determined as
(e+)s ) -
a exp(iφs) qs
(7)
So if we can quench the oscillations with all of the essential species of the reaction, we can determine the left eigenvector e+ completely. We now consider quenching with light. We shall assume that the perturbation is so short that it can be considered instantaneous and that its effect is a shift in the concentration space of the essential species. In other words, when the flash of light is over, there are no other species present in the reaction mixture than those participating in the reaction when it runs without light. We shall denote the quenching vector for perturbation with light as p. From the onset we do not know which species have changed their concentration as a result of the perturbation and by how much, so we do not know p. But we know the quenching data for quenching with light corresponding to p, in particular the quenching phase φp. If we write eq 6 for quenching with light, i.e. e+‚p ) -a exp(iφp), in terms of concentration coordinates by using eq 7, we get
exp(iφp) ) ∑ s
ps qs
exp(iφs)
(8)
in which the s-component of the vector p is denoted ps. We shall show how constraint 8 on the changes of concentrations ps of species s, caused by an exposure to light, may help determining the effect of light on the chemical reaction. Note in eq 8 that the left-hand side is determined by the quenching phase φp, and on the right-hand side we have a linear combination with unknown real coefficients ps of known complex quantities exp(iφs)/qs. This relation can thus be represented in the complex plane. We now discuss the chemical implications of relation 8. 5. Interpretation of Results To discuss the effect of light in terms of eq 8, we first represent the terms of the equation in the complex plane. Figure 4 shows the quenching phases for the basic quenchings (solid lines) and for quenching with light (dashed line). The quenching phases of Ru(bpy)3+ 3 , Br , and HBrO are equal within the experimental uncertainty. Therefore it is not practical to show the magnitudes of the complex components of e+. For the same reason, it is convenient to rewrite the right-hand side of eq 8 as
Quenching of Chemical Oscillations with Light
J. Phys. Chem., Vol. 100, No. 49, 1996 19195
Figure 4. Polar diagram showing the quenching phases for addition of each of the four species (solid lines) and for exposure to light (dashed line).
a sum of two terms corresponding to the two distinct basic quenching phases, φx ) 115° (for HBrO2) and φw ) -96° (for Ru(bpy)3+ 3 , HBrO, and Br ), thus
exp(iφp) ) bx exp(iφx) + bw exp(iφw)
(9)
We then get py/px ) 0.34 if only Br- and HBrO2 are and HBrO2 are produced, pz/px ) 0.12 if only Ru(bpy)3+ 3 produced, and pu/px ) 3.7 if only HBrO and HBrO2 are produced. The important chemical conclusion from these special cases is that there must be several molecules of HBrO2 produced per molecule of Ru(bpy)3+ 3 or Br produced as the net result of the exposure to light. We now discuss the consequences in terms of possible mechanistic explanations. The exposure to light may result in the emergence of new reactions or in increase of rates of already occurring reactions. We consider only mechanisms involving species of Table 1 and 2+ species like BrO3 or Ru(bpy)3 that cannot quench the oscillations. (If a species cannot quench experimentally, its quenching concentration qs must be very large so that its contribution to the right-hand side of eq 8 is very small because qs appears in a denominator.) Since HBrO2 must be produced and Ru(bpy)2+ 3 undoubtedly is involved in the photochemical reaction, it is natural to consider the following reactions with oxidation of one Ru(bpy)2+ 3 in an excited state13 + Ru(bpy)2+* + Ru(bpy)2+ 3 3 + BrO3 + 3H f
2Ru(bpy)3+ 3 + HBrO2 + H2O (13) 8 or, perhaps less likely, with both Ru(bpy)2+ 3 excited
+ + BrO2Ru(bpy)2+* 3 3 + 3H f
Here the coefficients are defined by
2Ru(bpy)3+ 3 + HBrO2 + H2O (14)
bx ) px/qx
(10)
bw ) ∑ ps/qs
(11)
for HBrO2 and
s
where the sum is taken over the species Ru(bpy)3+ 3 , Br , and HBrO. By multiplying eq 9 by exp(-iφp) and taking the imaginary part, we get the ratio of the two coefficients as
bw sin(φp - φx) ) ) 0.95 bx sin(φw - φp)
(12)
We now discuss the chemical implications of the result 12 on the assumption that only the four species of Table 1 or 2+ species (such as BrO3 or Ru(bpy)3 ) that cannot quench the oscillations are produced or consumed in excess as the result of the perturbation with light. From Figure 4 and eq 9 we note that both coefficients bx and bw must be positive, and it is obviously not possible to explain the quenching with light as the result of a production or consumption of just one of the species from Table 1. Moreover, HBrO2 must be produced by the process. It must be produced together with one or more of the species Ru(bpy)3+ 3 , Br , and HBrO. (It is possible that some are consumed as long as others are produced in sufficient amounts to make the coefficient bw sufficiently large positive.) We may calculate the ratios of the amounts produced if we assume that only one of the species Ru(bpy)3+ 3 , Br , or HBrO is produced together with HBrO2. Let us denote the coefficients p for the four species by px, py, pz, and pu for the changes of concentration of HBrO2, Br-, Ru(bpy)3+ 3 , and HBrO, respectively, generated by the exposure to light. (The notation is similar to one often used in models for the BZ reaction.)
Here an asterisk indicates a species in an excited state. Unfortunately, reaction 13 produces twice as much Ru(bpy)3+ 3 as HBrO2, so it cannot explain the quenching experiments as it stands, and the same applies to reaction 14. It is of no consequence for the interpretation if reactions with nonexcited species occur in parallel with reactions 13 or 14 because these reactions also occur in the basic quenching experiments. It also makes no difference if the species produced subsequently participate in further reactions because the same applies if these species are added to the reaction as in a quenching experiment. As a purely hypothetical explanation of the quenching results, may also be excited by light and become more Ru(bpy)3+ 3 reactive, or it might be produced in an excited state by the produced reactions 13 or 14. If most of the excess Ru(bpy)3+* 3 reacts to produce little or no Br-, constraint 12 might be is satisfied. This would be the case if most of the Ru(bpy)3+* 3 rapidly reduced by malonic acid. The previous discussion is meant to indicate how constraint 12 can be used in analyses of possible mechanisms. We do not claim that the indicated reactions are necessarily the right ones. 6. Conclusion The main result of this paper is the demonstration that chemical oscillations of the ruthenium-catalyzed BelousovZhabotinsky reaction can be quenched with light. It indicates that the effect of a pulse of light is a shift in concentration space and hence can be understood in terms of quenchings by addition of species. The main chemical result is the constraint 12 on species that must be produced and species that can be produced and on the amounts produced, when the reaction is exposed to light. In particular the quenching results show that HBrO2 must be produced and that the amounts of Ru(bpy)3+ 3 and Br produced must be much smaller than that of HBrO2. The constraint 12
19196 J. Phys. Chem., Vol. 100, No. 49, 1996
Figure 5. CSTR used for quenching experiments. Additions of species are made through an inlet. Exposure to light is made through a Plexiglas window using a beam splitter. The concentration of Ru(bpy)2+ is measured by absorption of light at 452 nm; [Br-] is 3 measured by a bromide electrode.
is useful as a guide in a search for the mechanism of the effect of light. It is particularly efficient as a way of excluding impossible mechanisms. The results are strictly applicable at the conditions of the experiments only. Nevertheless we believe the conclusions that Br-, e.g., cannot be the only product,2 that HBrO2 must be produced, and that the production of Ru(bpy)3+ 3 is relatively low should be taken into account when other experiments with light are interpreted. A number of assumptions underlie the conclusions of the paper. We assume that after an illumination, no new species are present in the system. In the mechanistic discussion we have not considered any species not tested in the quenching experiments. (In particular, we have not tested any radicals.) Nevertheless the paper demonstrates that methods based on the theory of nonlinear dynamics can be used for mechanistic analysis of complicated reactions. Acknowledgment. We thank Merete Torpe for experimental assistance and Gabor Marlowitz for participation in preliminary experiments. Appendix: Details of the Quenching Experiments Equipment. The experiments were carried out in a continuous-flow stirred tank reactor (CSTR), shown in Figure 5. The reactor is made of Plexiglas and wrapped in black paper with holes at the windows to minimize the influence of ambient light. is monitored through the The concentration of Ru(bpy)2+ 3 transmission of light at 452 nm (the absorption maximum of Ru(bpy)2+ 3 ), and [Br ] is monitored by the potential of a bromide selective electrode versus a mercurous sulfate reference electrode. The windows also let the reaction mixture be illuminated by a short, intense pulse of light from an incandescent lamp via a beam splitter as indicated in Figure 5. The reactor is fed with three separate flows from piston burets driven by stepper motors with equipment described in ref 19. Each flow enters the reactor through 0.8 mm borings ending close to the stirrer. Surplus reaction mixture is removed through a small hole at the shaft of the stirrer so that the volume of the mixture is kept constant at 33.0 cm3. Perturbations by addition of species were made with a separate stepper motor driven piston buret of 0.55 mm3 step size. Such additions were made through a hypodermic needle with a 0.5 mm bore, ending close to the stirrer. Operating Conditions. The three inflows contain 3.6 × 10-2 M KBrO3, 1.5 M CH2(COOH)2, and 5.5 × 10-5 M Ru(bpy)3-
Sørensen et al. SO4, respectively, each dissolved in 1.0 M H2SO4. In the experiments the three flows were maintained at equal rates with a total specific flow rate of 1.18 × 10-4 s-1. The temperature was fixed at (25.0 ( 0.02) °C, and the stirring rate was 550 rpm. Materials. The chemicals used and procedures of purification and use were the same as those described in the materials section of ref 20 except for the ruthenium complexes. Ru(bpy)3SO4 was prepared from the chloride (Fluka, p.a.) by the method described in ref 21. It was checked that no Cl- was present. The solution of Ru(bpy)3+ 3 used for perturbation was obtained from dissolved Ru(bpy)3SO4 by oxidation with PbO2 as described in ref 21. The solution was kept in 0.05 M BrO3(which does not affect the quenchings) to keep it in the oxidized state. Perturbations. A perturbation by addition of a species was made automatically by a computer at the requested phase with the amount requested. The phase is defined as the phase of the first harmonic of the [Ru(bpy)3+ 3 ] oscillations with the maximum as the zero. The precise phase of a quenching is determined from the recorded signal by Fourier analysis. In perturbations with light, the total radiant energy at 450 nm entering the reactor was 180 µW/nm. The spectral range used for quenching included the absorption bands of Ru3+ (bpy)2+ 3 and Ru(bpy)3 and was limited to wavelengths larger than 350 nm. The path length of the beam in the reactor was 30 mm. The length of the quenching pulse used in these experiments does not correspond to an instantaneous perturbation but extends over a finite part of the period. Ideally, the experiments should be repeated with a high-intensity lamp, reducing the time of exposure for quenching to a fraction of a second. However, the maximum error of the present experiments is sufficiently small to admit qualitative or semiquantitative conclusions about possible mechanisms. References and Notes (1) Ga´spa´r, V.; Bazsa, M. T. Z. Phys. Chem. 1983, 264, 43-48. (2) Kuhnert, L. Nature 1986, 319, 393-394. (3) Nitzan, A.; Ross, J. J. Chem. Phys. 1973, 59, 241-250. (4) Creel, C. L.; Ross, J. J. Chem. Phys. 1976, 65, 3779-3789. (5) Zimmermann, E. C.; Schell, M.; Ross, J. J. Chem. Phys. 1984, 81, 1327-1336. (6) Steinbock, O.; Mu¨ller, S. C. Physica A 1992, 188, 61-67. (7) Ouyang, Q.; Flesselles, J.-M. Nature 1996, 379, 143-146. (8) Srivastava, P. K.; Mori, Y.; Hanazaki, I. Chem. Phys. Lett. 1992, 190, 279-284. (9) Jinguji, M.; Ishihara, M.; Nakazawa, T. J. Phys. Chem. 1992, 96, 4279-4281. (10) Sekiguchi, T.; Mori, Y.; Hanazaki, I. Chem. Lett. 1993, 7, 13091312. (11) Mori, Y.; Nakamichi, Y.; Sekiguchi, T.; Okazaki, N.; Matsumura, T.; Hanazaki, I. Chem. Phys. Lett. 1993, 211, 421-424. (12) Sekiguchi, T.; Mori, Y.; Okazaki, N.; Hanazaki, I. Chem. Phys. Lett. 1994, 219, 81-85. (13) Hanazaki, I.; Mori, Y.; Sekiguchi, T.; Ra´bai, G. Physica D 1995, 84, 228-237. (14) Agladze, K.; Obata, S.; Yoshikawa, K. Physica D 1995, 84, 238245. (15) Hynne, F.; Graae Sørensen, P.; Nielsen, K. J. Chem. Phys. 1990, 92, 1747-1757. (16) Eiswirth, M.; Freund, A.; Ross, J. In AdVances in Chemical Physics; Prigogine, I., Rice, S. A., Eds.; Wiley Interscience: New York, 1991; Vol. LXXX, pp 127-199. (17) Kristiansen, K. R. M.Sc. Thesis, Department of Chemistry, University of Copenhagen, 1994. (18) Kramer, L.; Hynne, F.; Graae Sørensen, P.; Walgraef, D. Chaos 1994, 4, 443-452. (19) Graae Sørensen, P.; Hynne, F. J. Phys. Chem. 1989, 93, 54675474. (20) Kosek, J.; Graae Sørensen, P.; Marek, M.; Hynne, F. J. Phys. Chem. 1994, 98, 6128-6135. (21) Gao, Y.; Fo¨rsterling, H.-D. J. Phys. Chem. 1995, 99, 8638-8644.
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