J . Phys. Chem. 1984,88, 187-198
+
+
transfer rate (reaction 16). If (kWl5 k16)> (kI k2[MV2+]),
-
[*RU(~~~)~~+.-EDTA.-MV’+] [RU(~P~)~~+.*.EDTA.*.MV+.] (16) one would expect to observe a positive deviation of the normally linear Stern-Volmer quenching plot for nonaggregated species with an onset of the deviation at the concentration of MV2+ necessary for aggregation to occur under the [R~(bpy)~’+], [EDTA], and pH conditions of the medium. The concentration of species required for aggregate formation has been termed “critical aggregation concentration” (cac) by analogy with the critical micelle concentration; aggregation of MV2+ has been observed to occur at 10 mM in aqueous solutions (with EDTA absent).I5 Excited aggregates containing EDTA formed above the cac would lead to enhanced static scavenging of R ~ ( b p y ) , ~ + within the solvent cage (reaction 13), manifested by a dependence ~ + ] [EDTA]. of qcr on [ R ~ ( b p y ) ~ and Evidence for the formation of Ru(bpy)?+/MV2+/EDTA triple aggregates is shown in Figure 3. At low [ R ~ ( b p y ) ~ ~the +], Stern-Volmer plot is linear suggesting that triple aggregates do not form in that concentration regime; any formation of MVZ+ aggregates apparently does not affect the dynamics of quenching. At high [ R ~ ( b p y ) ~ ~the + ] ,breaking point occurs at -20 m M MV2+. The data in Figure 2 can be interpreted in the same way; at constant [MV2+](20 mM), [EDTA] (0.10 M), and pH (ll.O), M Ru(bpy);+ and is essentially aggregation begins at -5 X M. The addition of R u ( b ~ y ) beyond ~~+ complete at -5 X
-
(15) Ebbesen, T. W.; Ohgushi, M. Photochem. Photobiol. 1983,38,251-2.
187
that point increases the total number of aggregates but the fraction of the photosensitizer in the aggregated form remains unity. In summary, it appears that R ~ ( b p y ) , ~ +MV2+, , and EDTA form ground-state ion-paired aggregates with photosensitizer, electron relay, and sacrificial electron donor within a single unit. Such aggregates, when directly excited, appear to have significantly higher values of qcr than those exhibited by weakly ion-paired species engaged in bimolecular quenching. It is clear that a(MV+.) and qcr are very sensitive functions of solution medium which must be carefully controlled and specified. Ion pairing of the various cationic species with EDTA increase qcr due to the presence of the sacrificial anion within the solvent cage upon excited-state electron transfer. The dependence of qcr on the concentrations of the various ionic species in solution may be the cause of the wide range and great disparity of the reported values of @(MV+.) from continuous photolysis experiments involving the R U ( ~ ~ ~ ) , ~ + / M V ~ + /~EyDs tTe m A. ’ ~ - ’ ~
Acknowledgment. This research was supported by the Office of Basic Energy Sciences, Division of Chemical Sciences, U S . Department of Energy. Registry No. Ru(bpy)32+,’ 15158-62-0; MV2+,4685-14-7; EDTA, 60-00-4; MV’., 25239-55-8; Na2S04,7757-82-6; NaOH, 1310-73-2; hydrogen, 1333-74-0. (16) Kiwi, J.; Gratzel, M. J . Am. Chem. SOC.1979, 101, 7214-7. (17) Amouyal, E.; Grand, D.; Moradpour, A,; Keller, P. N o w . J . Chim. 1982,6, 241-4. (18) Xu,J.; Porter, G. B. Can. J . Chem. 1982, 60, 2856-8. (19) NenadoviE. M. T.: MiEiE. 0. I.:. Raih, T.: SaviE. D. J . Photochem. 1983, 2 1 , 35-44.
FEATURE ARTICLE Complex Dynamical Behavior in “Simple” Chemical Systems Irving R. Epstein Department of Chemistry, Brandeis University, Waltham, Massachusetts 02254 (Received: August 11, 1983)
Apparently simple chemical systems and reaction mechanisms involving a small number of components may give rise to a remarkable variety of dynamical phenomena if the systems are maintained sufficiently far from equilibrium. These include multiple stationary states, simple and complex periodic oscillation, aperiodic oscillation (chaos), and the growth of traveling waves and spatial structures in initially homogeneous media. Examples of these phenomena are described, both in mathematical models and in experimental chemical systems. The mechanistic treatment of such behavior is progressing rapidly, and several cases are cited. The relationship between studies of exotic dynamical behavior in chemical systems and related investigations in mathematics, physics, and biology is discussed briefly.
I. Introduction Simplicity of the components of a system does not necessarily lead to simplicity of that system’s overall structure or behavior. Cyrstallographers, for example, are well aware that even the lightest elements, such as beryllium, boron, or carbon, exhibit the phenomenon of polymorphism.’ Recent studies on the dynamics of chemical reactions far from equilibrium provide further evidence that apparent simplicity in the makeup of a system is no bar to (1) ~ ~ E. ~ W. f ’The i ~polymorphism ~ , of Elements and Compounds”; Methuen Educational: London, 1973.
0022-3654/84/2088-0187$01.50/0
its possessing a rich variety of complex behavior. In this article, we examine a number of hypothetical reaction schemes and real chemical systems which give rise to complex dynamical behavior. W e focus primarily on “simple systems”, Le., on models with polynomial rate laws, relatively few steps, and only two or three species with variable concentrations, and on reactions involving a small number (usually two or three) of inorganic species in homogeneous (or initially homogeneous) aqueous solution. The types of behavior in which we shall be interested are the existence of multiple stable states, periodic oscillation of varying degrees of complexity, aperiodic or “chaotic” oscillation, and the 0 1984 American Chemical Society
188 The Journal of Physical Chemistry, Vol. 88, No. 2, 1984
Epstein
-j
mmm
Reactant Reservoirs
Il
Q I
L
jacket
Figure 1. Schematic diagram of a CSTR. In the configuration shown, up to three different solutions can be pumped (by the peristaltic pump, PP) into the reactor, R. The detectors shown in the diagram are light absorption (M, monochromator: PM, photomultiplier), platinum (redox), and iodide (or bromide) selective electrodes. Time ( m i d
evolution of spatial structures and/or traveling waves from an unstirred but initially homogeneous solution. 11. The Tools The phenomena to be discussed can occur only in systems which are sufficiently far from equilibrium. Although far from equilibrium conditions exist during an initial transient period for a reaction in a closed system, sustaining such conditions is most easily accomplished in an open sytem. Early s t u d i e P of oscillatory chemical behavior were carried out in closed (batch) configurations. The key experimental development in the study of complex dynamical phenomena, however, was the adoption by the Bordeaux group6 of the continuous flow stirred tank reactor (CSTR). The results of CSTR experiments are often conveniently summarized and analyzed by means of dynamical “phase diagrams” which show the stable dynamical states of the system, much as the more conventional thermodynamic phase diagram represents the stable thermodynamic states of a system. Before proceeding further, we shall find it useful to discuss the CSTR and various types of phase diagrams in more detail. A . The CSTR. The type of CSTR used in our experiments is illustrated in Figure 1. It may be thought of as a well-stirred beaker augmented by a constant temperature bath, potentiometric, optical, and/or thermal probes, and, most importantly, tubes for the input of reactants and for the outflow of reacted material. Ideally, then, the system is open, homogeneous, and at constant volume and external temperature. The internal temperature may vary owing to the exo- or endothermicity of the reactions which occur. In such a configuration, the experimentalist has under his control a number of variables or constraints. These include (a) the temperature of the external bath: (b) the concentrations of the input chemicals X, in the reservoirs (these are normally given as [Xilo,the value that [Xi] would have immediately after mixing (2) Bray, W. C. J. A m . Chem. SOC.1921,43, 1262-7. (3) Belousov, B. P. ‘Sb. Ref. Radiats. Med. 1958”; Medgiz: Moscow, 1959, p 145. (4) Field, R. J.; Koros, E.; Noyes, R. M. J . Am. Chem. SOC.1972, 94, 8649-64. (5) Briggs, T. S.;Rauscher, W. C. J. Chem. Educ. 1973, 50, 496. (6) Pacault, A.; Hanusse, P.: De Kepper. P.: Vidal, C.; Boissonade, J. Acc. Chem. Res. 1976,9,439-45. Independent studies of oscillating reactions in a CSTR were also carried out at about the same time by several groups of chemical engineers, e.g., Graziani, K. R.; Hudson, J. L.; Schmitz, R. A. Chem. Eng. J. 1976,12,9-21. Marek, M.; Stuchl, I. Biophys. Chem. 1975,3,241-8.
Figure 2. Response-time plot for the C102--I- system showing periodic oscillations in absorbance at 460 nm (proportional to [I?]), redox potential, and [I-].
in the reactor but before any reaction takes place); and (c) the flow rate through the pump or, equivalently, the average residence time of material in the reactor, 7 (the flow rate is usually expressed ; flow of as the reciprocal of the residence time ko = 1 / ~ total material through the system is equal to koVwhere Vis the volume of the reactor). In a typical CSTR experiment, a chosen set of constraints is established and the responses (internal temperature, concentrations of species in the reactor) of the system are monitored. After an initial transient period, the responses reach a stable long-time behavior: stationary state, periodic, or aperiodic oscillation. One or more of the constraints is then varied and the new long-time responses monitored. As we shall see, the responses may not be uniquely determined by the constraints, but may depend upon the past history of the system, e.g., the order in which the constraints are varied. While most experiments to date have focused on the long-time responses, some recent work7,*has been devoted to the study of the transient behavior following a perturbation or a change in the constraints. B. Phase Diagrams. There are many ways in which one might represent the data from a CSTR experiment. The most natural plot, since it is obtained on the chart recorder during the course of the experiment, is of a response (e.g., the potential of an ionspecific electrode) as a function of time at a fiied set of constraints. Such response vs. time plots are extremely useful in analyzing perturbation and transient experiments as well as in studying oscillatory systems. They do not, however, allow one to view more than a single point in the multidimensional constraint space. An example of such a plot showing periodic oscillation in the chlorite-iodide systemg appears in Figure 2. An alternative procedure, which gives more of an overview of the system’s long-time behavior as the constraints change (at the cost of less detail for any particular constraint value) is the constraint-response plot. Figure 3 gives an example for the (7) Heinrichs, M.; Schneider, F. W. Ber. Bunsenges. Phys. Chem. 1980, 84, 857-65. J. Phys. Chem., 1981, 85, 2112-216. (8) Ganapathisubramanian, N.; Showalter, K. J. Phys. Chem. 1983,87, 1098-9. (9) Dateo, C. E.; Orbin, M.; De Kepper, P.; Epstein, I. R. J. Am. Chem. SOC.1982, 104, 504-9.
The Journal of Physical Chemistry, Vol. 88, No. 2, 1984 189
Feature Article
I
1
4
6
5
7
8
-
k, x IO-' (P-')
Figure 3. Constraint (ko)-response (redox potential) plot for the Br03--1-system showing steady states (A,V) and oscillatory state as a function of flow rate with [BrO3-Io= 0.005 M, [I-I0 = 0.005 M, [H2SO4lO= 1.5 M, T = 25 OC. Envelopes of vertical segments show upper and lower limits of platinum electrode potential in the oscillatory state. Numbers next to these segments indicate period of oscillation in seconds.
Arrows show spontaneous transitions between states. bromateiodide reaction.1° Stationary states and simple periodic oscillations are easily represented in this way, but chaotic or complex periodic behavior is difficult or impossible to characterize. Perhaps the most useful representation of data from CSTR experiments is the constraint-constraint plot, usually referred to as the phase diagram. In this plot, a two-dimensional section of the phase space is divided into different regions according to the long-time behavior exhibited by the system in response to the constraints. Three-dimensional phase diagrams are occasionally employed" and can be of great utility, particularly for complex systems. A phase diagram showing the many types of behavior encountered in the chlorite-bromate-iodide systemI2 appears in Figure 4. A final and less frequently utilized plot is the phase portrait, or response-response graph, as illustrated in Figure lob. Such a plot (which can be obtained experimentally by connecting two different probes to an x-y recorder) constitutes a two-dimensional projection of the system's trajectory in the higher dimensional response space. Qualitative changes (e.g., the appearance or disappearance of a closed trajectory) in the phase portrait as a constraint is varied are referred to as bifurcations. The study of the nature and location of the bifurcations in a system yields considerable insight into its dynamics. 111. Multiple Stable States
Most chemists tend to think of chemical systems as having a single stable mode of long-time behavior, generally the equilibrium state. Multiple stable states under the same set of external constraints are often thought to be associated with mechanical (e.g., the double-well potential in Figure 5 ) or biological (e.g., asleep-awake, living-dead) rather than chemical systems. The fact is that many chemical reactions under appropriate conditions give rise to two or more different states at a single set of constraint values. In this section we examine some simple models and some real chemical systems which have been found to exhibit multiple stable states. A . Models. We consider several models, all of which are too simple to characterize a real chemical system, but each of which illustrates an important point about the origins of multistability. I . A Simple Differential Equation. Nitzan et al." have studied the equation dx/dt = -(x' - MX
+ A)
(1)
(10) Alamgir, M.; De Kepper, P.; Orban, M.; Epstein, I. R. J. Am. Chem. SOC.1983, 105, 2641-3. (11) De Kepper, P.; Hanusse, P.; Rossi, A. C.R. Acad. Sci. Paris, Ser. C 1975, 251, 215-20. (12) Alamgir, M.; Epstein, I. R. J . A m . Chem. SOC.1983, 105, 2500-2. (13) Nitzan, A.; Ortoleva, P.; Deutch, J.; Ross, J. J . Chem. Phys. 1974, 61, 1056-74.
where x represents a concentration or other state variable, as a model for hard transitions, the analogue of first-order phase transitions. For p positive and Ihl C ( 2 / 3 ) ( ~ ~ / 3 ) 'the / ~ , steadystate equation dx/dt = 0 has three solutions. One solution is unstable; the other two solutions are stable in the sense that small perturbations of x away from the steady-state values tend to increase or decrease in magnitude for unstable or stable steady states, respectively. Equation 1 thus exhibits bistability in a region of the (X,p) constraint space. While eq 1 is unrealistic chemically, it has played a major role, as we shall see below, in the development of a systematic procedure for generating real chemical oscillations. 2. Autocatalysis in a CSTR. Almost all known examples of multistability in real chemical systems involve an autocatalytic reaction studied under flow conditions. Figure 6 presents a schematic picture of the origin of bistability in such systems. The solid line represents the net rate of production of the autocatalytic species X via flow into the reactor and chemical reaction. The dashed lines a-e show the flow of X out of the reactor at various flow rates. Where the dashed and solid line intersect, Le., where the rates of production and outflow are equal, we have a steady state. The outflow rate is simply ko[X], while the production rate rises initially with [XI, but then reaches a maximum and begins to decline at sufficiently high [XI, as the concentration of precursor is then too low to sustain a high reaction rate. For flow rates between those corresponding to lines b and d, there are three steady states. Higher or lower flow rates (e.g., lines a or e) give a unique steady state. If we consider the behavior of the steady states on line c, marked 1,2, and 3, with respect to small perturbations, we note that, at points 1 and 2, a small increase in [XI causes the outflow rate to increase more than the production rate, so that the perturbation will decay. These states are thus stable. At point 3, in contrast, an increase in X will be amplified, since the production line will then lie above the outflow line. The system will thus be propelled away from the unstable state 3 toward the stable state 2. Perhaps the simplest model which exhibits the behavior described above is reaction 2 in a CSTR. A
+ 2X
-
3X
(2)
This system and the simpler reaction A+X-2X
(3)
have been studied in detail by Gray and Scott,I4 who find that the cubic autocatalysis of reaction 2 is necessary to produce the inflection point in the production curve which leads to bistability. By introducing the additional reaction X
-
inert product
(4)
along with input flows of A and X, they are able to generate a remarkable variety of behavior including the constraint-response plots dubbed mushrooms and isolas illustrated in Figure 7. 3. Nonisothermal First-Order Reaction. Gray and ScottI4 point out that strong analogies exist between the isothermal autocatalytic systems discussed in the previous section and nonautocatalytic nonisothermal systems. The simplest and most thoroughly studied such system is the first-order, exothermic irreversible reaction X+P
(5)
in a CSTR. Uppal, Ray, and Poore15have investigated the behavior of this model in considerable detail. The rate equations are dX/dt = -k exp(-E/RT)X dT/dt = (-AH/C,)k exp(-E/RT)X
+ ko(Xo- X)
+ ko(To- T ) + h(T, - 7')
(6) (7)
(14) Gray, P.; Scott, S. K. Chem. Eng. Sci. 1983, 38, 29-43. (15) Uppal, A.; Ray, W. H.; Poore, A. B. Chem. Eng. Sci. 1974, 29, 967-85.
190 The Journal of Physical Chemistry, Vol. 88, No. 2, 1984 I
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3 x i0-4L
I
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I
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4
6
8
10
12
, (small1,amplitu~r) 14 6 18
I
I
I
20
22
24
I
26
IO3 k, (s") Figure 4. Phase diagram for the ClO,-BrO 0, a negative feedback in the constraint parameter X is introduced by means of a second variable y :
- p~ + A) - ky dx/dt = (X - Y)/T
dx/dt = -(x3
- t)”{3k
- 2p-
:}
kl
A-X
(19)
(16)
(17)
The parameter T characterizes the time scale on which the feedback -ky evolves. The corresponding role of the relaxation time of the system to its steady state(s) in the absence of the feedback is played by p-l. If T > p-’, then for [XI < lXzcl where XZc = & { p
the system possesses a stable periodic solution. This solution is a stable limit cycle, i.e., a unique periodic trajectory which is approached in the limit of very long times by all trajectories starting within some neighborhood of that trajectory. Thus, unlike the conservative system which may be found in any of an infinite number of periodic orbits, the limit cycle system after an initial transient will always end up with the same period and amplitude, independent of the initial conditions, so long as the constraints are the same. The system behaves reproducibly! While this model is not of immediate chemical significance, the phase diagram it generates has proved to be a recurrent one in many chemical systems. Consider any chemical reaction in which bistability occurs in the CSTR over a range of some constraint parameter. Equation 1 and X may be taken as the prototype. If we now introduce a second reaction (eq 16 and 17) which modifies that constraint by significantly different amounts on the two branches of steady states and on a time scale slow with respect to the relaxation of the unperturbed system to its bistable steady states (the analog of T > p-l), then the system will oscillate essentially between the unperturbed steady-state response values. A quantitative treatment of this model is given by Boissonade and De K e ~ p e r , while ~ ’ a detailed qualitative rationale may be found in ref 28. The resulting “cross-shaped phase diagram” shown schematically in Figure 13 has been observed experimentally in a wide variety of chemical oscillators and has served as a guide for designing oscillators starting from bistable systems. 3. The Brusselator. Just as eq 1 can be augmented to give the oscillatory model of eq 16 and 17, so the cubic autocatalysis of eq 2 can be developed into a two-variable oscillator. One such model is the B r ~ s s e l a t o r . ~ ~
(18)
and k > (2p + 1 / ~ ) 3all , steady-state solutions are unstable and (28) Epstein, I. R.; Kustin, K.; De Kepper, P.; Orbln, M. Sci. Am. 1983, 248 (3), 112-23.
(29) Nicolis, G.; Prigogine, I. ‘Self-Organization in Nonequilibrium Systems”; Wiley: New York, 1977. (30) Lotka, A. J. J . Am. Chem. SOC.1920, 42, 1595-9. (31) Boissonade, J.; De Kepper, P. J. Phys. Chem. 1980.84, 501-6.
k4
X-E
where [A] and [B] are held constant, and D and E are inert products. Equations 19-22 have received a great deal of attention in the literature. Perhaps the most thorough treatment is that of Nicolis and P r i g ~ g i n ewho , ~ ~ point out that the chemically unlikely termolecular reaction 21 may be thought of as equivalent to an appropriate sequence of enzymatic reactions. The Brusselator exhibits limit cycle oscillations over a sizeable region of the (A,B) constraint plane. 4. The Oregonator. A somewhat more chemically realistic model is the Oregonator, eq 23-27, proposed by Field and NoyeP (32) Prigogine, I.; Lefever, R. J . Chem. Phys. 1968, 48, 1695-730. (33) Field, R. J.; Noyes, R. M. J . Chem. Phys. 1974, 60, 1877-84.
Epstein
194 The Journal of Physical Chemistry, Vol. 88, No. 2, 1984 480
I
I
(0.9935, 450.05)
1
240t I
state
24-4
I
207
195
,
183
164
,
152
1~49(
i IOmn c-
1
Time
0
I O
2.0
Figure 15. Compound oscillation (C) between A and B oscillatory states in the chlorite-bromate-iodide system as flow rate is changed. Fixed constraints: [I-]O = 4.0 X 10“ M, [CIO,], = 1.0 X M, [BrO3-l0 = 2.5 X M, [H2S04]0 = 0.75 M.
3.0
f Figure 14. Phase diagram for the Oregonator withfand k5 (eq 23-27)
a
e
1
as the constraints.
to account for the key features of their mechanism4 of the Belousov-Zhabotinsky (BZ) reaction: kl
A+Y-X+P
(23)
k2
X+Y+2P A + X A 2 X + Z
(24)
b
(25)
ki
2X-A+P
(26)
where X, Y, and Z are identified with HBr02, Br-, and Ce4+, respectively, while A = [BrO 0 and zero otherwise, on a square with zero flux boundary conditions at the edges. A rotating wave solution was found to be stable if the dimensions of the square were sufficiently great. If the square was too small, only the homogeneous steady state was stable. While eq 34 and 35 are far from realistic chemically, they do suggest that the combination of diffusion, an inherently oscillatory system, and a switching term (the Heaviside function) will give rise to spatial waves. 2. The Oregonator. A number of calculations have been undertaken to model the traveling waves observed in the BZ reaction. Most of these5z-54have utilized simplified (one or two independent variable) versions of the Oregonator model eq 23-27 and a single spatial dimension. Reusser and FieldS5have numerically integrated the full Oregonator scheme including onedimensional diffusion using the method of lines, In all cases, wave propagation was found and the dependence of the wave velocity upon reactant concentrations was in qualitative agreement with experiment. 3. The Brusselator. Several treatments of the Brusselator model eq 19-22 in one and two spatial dimensions are summarized by Nicolis and P r i g ~ g i n e . ~Both ~ time-and-space-periodic structures are found, with their nature being quite dependent upon the boundary conditions. 4 . Single Wave Propagation. Probably the most successful model in quantitatively explaining spatial waves in real chemical systems is Showalter’s treatments6 of one-dimensional propagation in the arsenite-iodate system.57 This reaction is not oscillatory and supports only a single wave. The model consists of two overall component processes:
+ 51- + 6H+ = 31, + 3H20 H3As03+ I2 + H 2 0 = 21- + H3As04+ 2H+ 103-
(36) (37)
The rate laws for these two processes have been found experimentally to be
(39) Using the stoichiometric constraint that in the presence of excess (51) Winfree, A. T. SIAM-AMS Proc. 1974, 8, 13-31. (52) Murray, J. D. J . Theor. Biol. 1976, 56, 329-53. (53) Schmidt, S.; Ortoleva, P. J . Chem. Phys. 1980, 72, 2733-6. (54) Rinzel, J.; Ermentrout, G. B. J . Phys. Chem. 1982, 86, 2954-8. (55) Reusser, E. J.; Field, R. J. J . Am. Chem. Soc. 1979, 101, 1063-71. (56) Hanna, A,; Saul, A,; Showalter, K. J . Am. Chem. SOC.1982, 104, 3838-44. (57) Gribschaw, T. A.; Showalter, K.; Banville, D.; Epstein, I. R. J. Phys. Chem. 1981, 85, 2152-5.
The Journal of Physical Chemistry, Vol. 88, No. 2, 1984 197
Feature Article
Figure 18. Spatial wave pattern observed at 5 OC in a thin (2 mm) layer of solution with initial composition [CH,(COOH),] = 0.0033 M, [NaI] = 0.09 M, [NaCIO,] = 0.1 M, [H,SO,] = 0.0056 M, and starch as indicator.
+
H3As03, [IO3-] [I-] = [IO3-lO, Showalter is able to obtain a single partial differential equation for C = [I-]: -ac = - Da2c + at ax2
( h + hC)([I03-10 - C)[H+I2C
(40)
where D is the diffusion constant for I-. Equation 40 has an analytical wave solution of the form C(x,t) =
[IO,-Io
1
+ A exp(k(x - ut))
(41)
The calculated dependence of the wave velocity u on [H+l0and [IO,],, agrees well with that observed experimentally. B. Experimental Results. The number of phenomena which may arise from the coupling of chemical reaction with diffusion, convection, precipitation, etc. is indeed enormous. We shall not discuss periodic precipitations8 here, but shall look briefly at the role of convection, since it may easily be confused with that of diffusion. 1. Periodic Wave Phenomena. Much of the interest in chemical wave phenomena was inspired by the beautiful patterns produced in the BZ system by WinfreeeS9 Bull’s-eyes, spirals, and scroll waves in red and blue make for fascinating viewing and challenging analysis. More recently, De Kepper et aL60 have created similar phenomena in chlorite based oscillators. One such pattern, which arose spontaneously from an initially homogeneous medium, is shown in Figure 18. Thus far, all systems which exhibit periodic waves of this type are multicomponent and contain malonic acid or a similar organic species. Stationary mosaic patterns have also (58) Kai, S.; Muller, S.C.; Ross, J. J. Chem. Phys. 1982, 76, 1392-1406. (59) Winfree, A. T. Sei. Am. 1974, 230 (6), 82-95. (60) De Kepper, P.; Epstein, I. R.; Kustin, K.; Orbln, M. J. Phys. Chem. 1982, 86, 170-1.
been observed in such systems.61*62 2. Single Traveling Wave. A number of systems which show bistability in the CSTR are capable of supporting a single traveling wave in an unstirred closed system. This wave may arise spontaneously, presumably generated by a local concentration fluctuation, or may be induced, e.g., by an electrical pulse. Systems in which such behavior has been observed include arsenite-iodate,56,57ferr0in-bromate,6~ chl~rite-iodide:~ and ferrous-nitric acid.16 3. The Role of Convection. One way of maintaining a system far from equilibrium is to subject it to a constant influx of energy, e.g., in the form of light. Photochemical reactions appear to offer fertile ground for finding new dynamical phenomena. However, recent work shows that one must proceed with extreme caution in any system which may be subject to a temperature gradient. Several report^^^-^^ have appeared in the literature of “photochemical oscillators” in which systems under constant intensity illumination showed periodic and/or aperiodic fluctuations in the intensity of light emitted. The reactions were quite varied, but in all cases the phenomenon was difficult to reproduce quantitatively and disappeared when the solution was stirred. Two independent investigation^^^*^^ have now presented convincing evidence that the observed phenomena resulted not simply from the chemistry but from the convective motion of the fluid being illuminated. A small temperature gradient is induced by the illumination and/or by evaporation from the surface. The liquid at the top becomes cooler and heavier, the system becomes mechanically unstable, and convective motion sets in. The observed variations in emission intensity simply monitor the spatial motion of the fluid through the observation region. The significance of the above experiments is not that photochemical oscillation is impossible; photochemical oscillators will certainly be found. What is important is that, in any experiment in which spatial behavior is of interest, extreme care must be taken to ensure uniformity of temperature. The magnitude of the temperature gradient required to initiate convection in a typical experimental configuration is astonishingly small.
VII. Conclusion This article has presented a survey of a set of phenomena which may have been unfamiliar to many physical chemists. At times we have risked being superficial in order to give a broad overview. More detailed treatments may be found in the references cited and in the published proceedings of three recent meeting^^"^ on nonlinear phenomena and dynamic instabilities in chemical systems. It is appropriate in an article like this to speculate on the future directions that the field may take. In this case such speculation is both enticing and perilous given that nearly all the important developments have occurred within the past 5 years. It seems clear that attention will turn increasingly from simple bistability and periodic oscillation to more complex phenomena such as temporal chaos and spatial periodicity. The deliberate design of new systems which show a desired behavior should soon replace the earlier reliance on accidental discovery or small variations of existing systems. Part of this program will surely be the coupling of systems showing simpler behavior to produce more complex phenomena. (61) Showalter, K. J. Chem. Phys. 1980, 73, 3735-42. (62) Orbln, M. J . Am. Chem. SOC.1980, 102,43 11-4. (63) Showalter, K. J. Phys. Chem. 1981, 85, 440-7. (64) Weitz, D.; Epstein, I. R., J. Phys. Chem., submitted for publication. (65) Yamazaki, I.; Fujita, M.; Baba, M. Phorochem. Phorobiol. 1976, 23, 69-70. (66) Nemzek, T. L.; Guillet, J. E. J . Am. Chem. SOC.1976, 98, 1032-4. (67) Bose, R. J.; Ross,J.; Wrighton, M. S . J . Am. Chem. SOC.1977, 99, 6119-20. (68) Laplante, J. R.; Pottier, R. J. J . Phys. Chem. 1982, 86, 4759-66. (69) Epstein, I. R.; Morgan, M.; Steel, C.; Valdes-Aguilera, 0. J . Phys. Chem. 1983,87, 3955-8. (70),Vidal, C.; Pacault, A., Ed. “Nonlinear Phenomena in Chemical Dynamics”; Springer: New York, 1981 (Bordeaux, 1981). (71) Physica D 1983, 7, 3-362 (Los Alamos, 1982). (72) Nicolis, G., Ed. “NATO Workshop on Chemical Instabilities”; Reidel Dordrecht, Holland, in press. (Austin, 1983).
J. Phys. Chem. 1984,88, 198-202
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A great deal of inspiration and guidance in this field has come who predicted that chemical reactions could oscillate in a chaotic fashion several years before such behavior was experimentally from the other sciences, most notably biology, physics, and established. As the chemical phenomena become better charmathematics. Physicists were dealing with bistable systems and acterized, the mathematics provides a framework which highlights periodic oscillators long before it occurred to chemists that their the fundamental identity of apparently dissimilar phenomena in reactions might behave in similar fashion. Periodic oscillation chemistry, physics, biology, and even geology. is ubiquitous in living systems and the hope that understanding The safest way to predict the future is, of course, to extrapolate simple chemical oscillators will provide insight into biological from the past. On this basis one should expect that new dynamical oscillators motivates many in this area. Similarly, spatial pattern phenomena will be found in chemistry and that these phenomena formation in chemical systems may be of relevance in underwill be describable with simple models and will occur in systems standing cellular differentiation, chemotaxis, and related processes of apparently simple composition. One may further expect that in biology. Certainly the similarities in appearance between the these phenomena will have analogues in systems of interest to chemical waves of the type shown in Figure 18 and the aggregation biologists and physicists and that the mathematical tools for their ~ ~striking. pattern of the slime mold Dictyostelium d i ~ c o i d i u mare analysis already exist. Finally, one may certainly assume that In fact, Turing’s seminal for reaction-diffusion systems complex dynamical phenomena will continue to provide many was inspired by the problem of biological morphogenesis. fascinating problems for physical chemists to ponder. Nearly all of the mathematical apparatus for dealing with new dynamical phenomena in chemistry has been available in the Acknowledgment. My own work in this area has been supmathematical and physical literature before the phenomena were found experimentally. It was a mathematician, David R ~ e l l e , ~ ~ ported by the National Science Foundation through grants CHE7905911, CHE8204085, and INT8217658 and by a NATO Research Grant. I gratefully acknowledge the invaluable help (73) Tomchik, K. J.; Devreotes, P.N. Science 1981, 212, 443-6. of many able co-workers including Kenneth Kustin, Colin Steel, (74) Turing, A. M. Phil. Trans. R. SOC.(London), Ser. B 1952, 237, Gyorgy Bazsa, Patrick De Kepper, Miklds OrbBn, Jerzy Maselko, 37-72. Mohamed Alamgir, Reuben Simoyi, Robert Olsen, Oscar (75) Ruelle, D. Trans. N.Y. Acad. Sci. 1973, 35, 66-71. Valdes-Aguilera, Debra Banville, and Christopher Dateo. (76) Bazsa, G.; Epstein, I. R., unpublished.
ARTICLES Intramolecular Vlbratlonal Relaxation of Benzene E. Riedle, H. J. Nemer,* E. W. Schlag, Institut fur Physikalische und Theoretische Chemie, Technische Universitat Miinchen, D 8046 Garching, West Germany
and S . H. Lin Department of Chemistry, Arizona State University, Tempe, Arizona 85287 (Received: April 19, 1983)
The purpose of this paper is to present a theory for the previously published high-resolution spectra of 14; 1; of C6Hs. The analysis of these high-resolution spectra revealed an intramolecular process which was found to be strongly dependent on the rotation of the molecule. Here, possible intramolecular coupling processes are discussed and a qualitative comparison of theoretical results for the Coriolis coupling effect with the experiment is performed.
1. Introduction Recently there has been considerable interest in the theoretical analysis of systems of coupled anharmonic oscillators.’” As the energy is increased, many classical systems are known to undergo a transition from regular to stochastic motion. Considerable effort in this area has been devoted to finding the phenomenology of the analogous quantum transition. (1) D. W. Noid, M. L. Koszykowski, and R. A. Marcus, Annu. Reu. Phys. Chem., 31, 267 (1981). (2) E. J. Heller and M. J. Davis, J. Phys. Chem., 86, 2118 (1982). (3) I. Hamilton, D. Carter, and P. Brumer, J. Phys. Chem., 86, 2124 (1982). (4) R. Kosloff and S . A. Rice, J . Chem. Phys., 74, 1340 (1981). ( 5 ) R. M. Stratt, N. C. Handy, and W. H. Miller, J. Chem. Phys., 71, 3311 (1980). (6) M. Shapiro and M. S . Child, J . Chem. Phys., 76, 6176 (1982).
0022-3654/84/2088-0198$01.50/0
In a previous paper,’ the density matrix method has been applied to treat coupled anharmonic system. For this purpose, the adiabatic approximation has been employed to find the basis set.* It has been shown that the energy eigenvalues obtained from the adiabatic approximation are in very good agreement with exact and are on the whole better than those obtained from the SCF calc~lation.~ Furthermore, the adiabatic approximation can provide analytical expressions of wave functions and energy eigenvalues of a coupled anharmonic system. Recently it has been shown by using the Schrodinger equation method that both de(7) S. H. Lin, Chem. Phys. Lett., 86, 533 (1982). (8) S . H. Lin, Chem. Phys. Lert., 70,492 (1980); S . H. Lin, X. G. Zhang, 2. D. Qian, X. W. Li, and H. Eyring, Proc. Natl. Acad. Sci. U.S.A., 79, 1356 (1982). -~ ( 9 j H . Kono, X. G. Zhang, 2. D. Qian, X. W. Li, and S. H. Lin, Mol. Phys., 47, 713 (1983). \-
0 1984 American Chemical Society
