J. Phys. Chem. 1992,96, 2953-2961
2953
Nonlinear Dynamics, Multiple Steady States, and Oscillations in Photochemlstry B. Borderie, D. Lavabre, J. C. Micheau, UniversitC Paul Sabatier, UA 470, 31062 Toulouse, France
and J. P . Laplante* Department of Chemistry and Chemical Engineering, Royal Military College of Canada, Kingston, Ontario, Canada K7K 5LO (Received: July 19, 1991; In Final Form: November 14, 1991)
Light is often considered as a type of chemical reagent. As a result, there is a natural tendency to infer that the rate equations describing the dynamics of photochemical systems should not differ significantly from their light-free counterparts. There are, however, two important differences between chemical and photochemical systems. The first one is that photochemical systems are nonisolated closed systems that are maintained away from their thermodynamic equilibrium by the photon flux. The second difference lies in the rather peculiar nature of the interaction between light and the physical medium in which the photoreaction takes place. This interaction is governed by the highly nonlinear Lambert-Beer’s law and has no equivalent in light-free chemistry. In this paper, it is shown that these two differences are of more than academic importance and naturally lead to a variety of situations where instabilities such as multiple steady states and oscillations can be obtained in relatively simple photochemical schemes. Contrary to light-free chemistry, these instabilities can be observed in isothermal systems that are closed to mass flows and do not contain any autocatalytic steps. To support this thesis a number of models based on thermally reversible, consecutive, and cyclic photoreactions are analyzed. By use of these models, it is shown that features such as competitive absorption and excited state quenching can give rise to multiple steady states and sustained oscillations. The nature of the feedback mechanism responsible for the onset of these instabilities is clarified and shown to bear some similarity with substrate inhibition.
Introduction In 1977, Professor Ilya Prigogine was awarded the Nobel Prize for chemistry for his pioneering work on the thermodynamics of far-from-equilibrium processes. Almost 15 years later, it is fair to say that the work of Professor Prigogine and the Brussels School of irreversible thermodynamics has triggered much interest in the applications of nonlinear dynamics in chemistry.’ This field has now matured into a vigorous sector of research with powerful mathematical, numerical, and experimental took2 Nonequilibrium phenomena such as multiple steady states, oscillations, deterministic chaos, and spatial dissipative structures are now well documented for a variety of inorganic and organic systemsa2 So far however, very few examples of instabilities have been reported in photochemistry. In a series of pioneering papers, Nitzan, Ortoleva, and Ross were the first to show that light can be used to induce instabilities in chemical ~ystems.~ In their first papers? they considered cases where the light absorbed by the system is transformed into heat via a radiationless transition. As pointed out by these authors, the rate equations then become nonlinearly coupled to the rate of change in temperature of the system, via the amount of light absorbed. The result of such coupling is that multiple steady states, damped oscillations, and instabilities then become possible in very simple systems, such as isomerization reactions for e ~ a m p l e . ~Such light-induced instabilities were indeed later observed experimentally by Ross and co-workers in a number of systems4 ( I ) The pioneering work of Professors Prigogine, Glansdorff, Nicolis, and other members of the Brussels School is summarized in two excellent monographs: (a) Glansdorff, P.; F’rigogine, I. Thermodynamic Theory of Structure, Stability and Fluctuations; Wiley-Interscience: New York,1971. (b) Nicolis, G.; Prigogine, 1. Self-organization in Nonequilibrium Systems; Wiley-Interscience: New York, 1977. (2) The most recent developments in nonlinear dynamics in chemistry are well summarized in the following books and Conference Proceedings: (a) Field, R. J.; Burger, M. Oscillations and Travelling Waves in Chemical Systems; Wiley-Interscience: New York, 1985. (b) Gray, P., Nicolis, G., Baras, F., Borckmans, P., Scott, S. K.,Eds. Spatial Inhomogeneities and Transienr Behaviour in Chemical Kinerics; Manchester University Press: Manchester, 1990. (c) Proceedings of the International Conference on Dynamics of Exotic Phenomena in Chemistry (Hajduszoboszlo, Hungary, 1989) React. Kinet. Catal. Lett. 1990, 42 (2). (3) (a) Nitzan, A.; Ortoleva, P.; Ross,J. J. Chem. Phys. 1973, 59, 241. (b) Nitzan, A.; Ortoleva, P. J. Chem. Phys. 1974, 60, 3134.
In this paper we focus our attention on multiple steady states and oscillations in isothermal photochemical reactions. More specifically, we seek to understand the mechanisms through which these instabilities can emerge under conditions for which photochemical reactions are usually carried out. In preparative photochemistry for example, most photoreactions are carried out in well-stirred thennostated reactors.s The photoreactions considered in this paper are therefore assumed to take place a t constant temperature. The possibility of light-induced thermokinetic instabilities3p4is therefore ruled out in these cases. Let us note that the only known experimental example of an instability in an isothermal photochemical system is the observation of multiple steady states in the irradiation of a solution of the triphenylimidazyl radical dimer (TPID) in a continuously fed stirred flow reactor (CSTR).6 The mechanism initially proposed for this reaction6 has recently been reexamined and numerical simulations are now in very good agreement with experimental results.’ Let us note that one of the key elements of the mechanism leading to multiple steady states in this system is the competition for light absorption by the various intermediates. This competitive absorption arises as a result of Beer’s law and has no equivalent in light-free chemistry. Its role in the onset of instabilities in isothermal photochemical systems was already pointed out in some of our earlier papers8-’0 and will be explored further in this paper. (4) (a) Creel, C. L.; Ross, J. J. Chem. Phys. 1976, 65, 3779. (b) Zimmerman, E. C.; Ross, J. J. Chem. Phys. 1984,80, 720. (c) Kramer, J.; Ross, J. J. Phys. Chem. 1986, 90, 923. (d) Zimmerman, E. C.; Schell, M.; Ross, J. J . Chem. Phys. 1984,81, 1327. (e) Kramer, J.; Reiter, J.; Ross, J. J . Chem. Phys. 1986,84, 1492. (5) Braun, A. M.; Maurette, M. T.; Oliveros, E. Photochemical Technology; Wiley: New York, 1991. (6) (a) Borderie, B.; Lavabre, D.; Levy, G.; Micheau, J. C.; Laplante, J. P. J. Am. Chem. SOC.1990,112,4105, (b) Lavabre, D.; Levy, G.; Laplante, J. P.; Micheau, J. C. J. Phys. Chem. 1988, 92, 16. (c) Borderie, B.; Lavabre, D.; Levy, G.; Laplante, J. P.; Micheau, J. C. J. Photochem. Photobiol. A: Chem. 1991, 56, 13. (7) Borderie, B.; Lavabre, D.; Levy, G.; Micheau, J. C.; Laplante, J. P. Int. J . Chem. Kinet., in press. ( 8 ) (a) Laplante, J. P.; Lavabre, D.; Micheau, J. C. J. Chem. Phys. 1988, 89, 1435 (note the error in eq 1 Id of this reference, which should read @ = &/q5,). (b) Gregoire, F.; Lavabre, D.; Micheau, J. C.; Ginenez, M.; Laplante, J. P. J. Photochem. 1985, 28, 261. (9) (a) Borderie, B.; Lavabre, D.; Micheau, J. C.; Laplante, J. P. React. Kinet. Caral. Lett. 1990, 42, 401; (b) Ibid. 1990, 42, 407.
0022-365419212096-2953$03.00/0 ~. . , .I 0 1992 American Chemical Society I
,
2954 The Journal of Physical Chemistry, Vol. 96, No. 7, 1992
In order to illustrate the mechanisms through which instabilities can emerge in isothermal systems, we analyze the dynamics associated with three simple model schemes. Our approach focuses on steady states and their stability as a function of molecular and external control parameters. This approach is somewhat different from the one pioneered by Mauser and the German School of Photokinetics,11-12where the attention is focused on absorbancetime traces and their relationship with molecular parameters and kinetic rate constants. Since photochemical reactions are usually carried out in batch reactors, we deliberately focus our attention on systems that are closed to mass flows. Note that since photochemical systems are by definition open to an energy flux, the onset of instabilities such as multiple steady states and oscillations is not restricted to systems open to mass f l 0 ~ s . l As ~ will be shown however, the extension of our analysis to systems open to mass flows presents no special difficulties. Our analysis starts with an overview of the basic rate equation describing the dynamics of the elementary A B photoreaction. Even though no instability is expected for such a scheme, its dynamics is quite revealing of the differences between chemical and photochemical reactions. In the second part of the paper, we present two mechanisms through which multiple steady states can be obtained in photochemical systems closed to mass flows, viz. concentration quenching and competitive absorption (screen effect). The first of these models considers the case of a thermally reversible photoreaction where excited states are subject to concentration quenching. This model allows a one-variable description of the system’s dynamics and is therefore particularly simple. The second model examines the onset of multiple steady states in systems of cyclic and consecutive photoreactions. The mechanism responsible for the destabilization of the photostationary states, although different in each of these models, is shown to be similar to substrate inhibition.14 The third section of the paper focuses on the topic of oscillations in isothermal photochemical systems and presents a photochemical model capable of sustained oscillations in a batch reactor. As a final note, it could be emphasized that none of the photochemical mechanisms considered in this paper make use of autocatalytic steps. As will be shown, the necessary ingredients for the onset of chemical instabilities are natural components of many ordinary photochemical reactions.
-
I. Elementary A 3 B Photoreaction The unimolecular photoprocess by which species A is converted to B via its excited states is usually represented as
Borderie et al. assume that this reaction is taking place in a well-stirred batch reactor (i.e. a system closed to mass flows) under conditions of homogeneous and monochromatic irradiation. In order to describe the dynamics associated with ( R l ) , we must first clarify the underlying mechanism through which this reaction takes place. As well-known, scheme (Rl) is in fact a short notation for a group of photophysical and photochemical processes, the exact nzture of which depends on the system considered. For example, one may consider ( R l ) as standing for the following scheme hu
A-A* A*
-+
A*
+ (hd)
-
A
B
Z,
(R1.1)
kd[A*]
(R1.2)
k,[A*]
(R1.3)
hu
B+B*
B*
-+
B
(R1.4)
+ (hd’)
kd’[B*]
(R1.5)
where processes R1.1 and R1.4 represent the light absorption by A and B, processes R1.2 and R1.5 represent the radiative and radiationless deactivations of A* and B*, and process R1.3 represents the irreversible transformation of A* into product B. (A more detailed description of the photophysical processes depleting A* and B* would not modify the arguments and results presented in this paper.) The usual photostationary state approximation leads to the following expressions for the stationary concentrations of excited species A* and B* [A*] =
(&I
[B*] = I b / k i (2) and as a result, the rate of photoreaction ( R l ) can be written as
rv = -d[A]/dt = d[B]/dt = k,[A*] Defining the reaction quantum yield
(3)
aA2s36J7 (4)
the rate expression (3) becomes
rv =
(5)
@AIa
The absorbed photon flux, Z,, is essentially proportional to the amount of light absorbed by reactant A per unit time. This quantity is in turn proportional both to the total amount of light The dynamics associated with this reaction is of fundamental absorbed by the system and to the fraction of this total absorbed importance in all areas of photochemistry, e.g. a ~ t i n o m e t r y , ’ ~ ~ ~by~reactant A. Using Lambert-Beer’s !aw, one can therefore write organic ph~tochemistry,”-~~ photochromism,20,21 etc. Let us now eq 5 as8.9,11,12
AAB
(R1)
(10) Micheau, J. C.; BouE, S.; Vander Donckt, E. J . Chem. SOC.,Faraday Trans. 1982, 78, 39. (1 1) (a) Mauser, H. Formale Kinetik; Bertelsmann: Diisseldorf, 1974. (b) Polster, J.; Mauser, H . J . Photochem. Photobiol. A : Chem. 1988, 43, 109. (12) (a) Bar, R.; Gauglitz, G. J . Phorochem. Phorobioi., A: Chem. 1989, 46,15. (b) Gauglitz, G. Photophysical,photochemical and kinetic properties in photochromic systems. In ref 20a, pp 15-63. ( c ) Rau, H.; Greiner, G. EPA Newsi. 1991, 41, 40. (13) Although the present paper is concerned solely with isothermal photochemistry, it is to be emphasized that the first example of multiple steady states in a chemical system closed to mass flows was reported by Ross and co-workers for the nonisothermal photochemical system o-cresolphthalein in glycerol (ref 4c). (14) (a) Tyson, J. J. J . Chem. Phys. 1975,62, 1010. (b) See also Luo, Y.; Epstein, I. R. Adu. Chem. Phys. 1990, 79, 269. (15) (a) Adick, H. J.; Schmidt, R.; Brauer, H. D. J . Photochem. Phofobiol. A: Chem. 1988, 45, 89. (b) Brauer, H. D.; Schmidt, R.; Gauglitz, G.; Hubig, S. Photochem. Photobiol. 1982, 37, 595. (16) Calvert, J. G.; Pitts, J. N. Photochemistry; Wiley: New York, 1966. (1 7) Turro, N. J. Modern Molecular Photochemistry; Benjamin-Cummings: Menlo Park, CA, 1978. (18) Cowen, D. 0.; Drisko, R. L. Elements of Organic Photochemistry; Plenum Press: New York, 1976. (19) Rohatgi-Mukherjee, K. K. Fundamenral of Photochemistry; Wiley Eastern, Ltd.: New Delhi, 1978.
where Io is the incident photon flux (einstein L-I s-l), is the molar absorptivity of species A (natural, base e, L mol-I cm-I), [A] is the concentration of reactant A (mol L-I), and d is the optical path length (cm). The function F(E) is sometimes referred to as the photokinetic factor,I1J2and here takes the form F(E) =
{
1 - exp(-E) E
1
(7)
The total absorbance at the irradiation wavelength, E , is defined as22
(20) (a) Durr, H.; Bouas-Laurent, H. Photochromism - Molecules and Systems; Elsevier Science: New York, 1990. (b) Dessauer, R.; Paris, J . P. Adu. Photochem. 1963, 1 , 275. (21) (a) El’tsov, A. V. Organic Photochromes; Consultants Bureau: New York, 1990. (b) Gugglielmetti, R.; Meyer, R.; Dupuy, C. J . Chem. Educ. 1973, 50, 413.
The Journal of Physical Chemistry, Vol. 96, No. 7, 1992 2955
Photochemical Systems
1.0
where [ i ] and ei represent the concentration of species i and its molar absorptivity at the irradiation wavelength. For the photoreaction (Rl), we thus have
E = dlCA[A1+ €BIB])
+ [BI = ([AI + [BIJt=o = AO
I
I
600
BOO
(9)
Note that since the system is closed to mass flows, the following conservation relationship holds [AI
I
X
(10)
In the analysis to follow, it will prove convenient (and much less cumbersome) to use normalized expressions. Defining X = [AI /Ao; c = tg/tA;
y = [BI/AO
(11)
io = Zo/Ao
(12)
allows one to write the total absorbance at time t = 0 as
Eo = tAAod 1 - exp[-Eo(X
+ CY)]
x+Y=
(8)
c = t B / c A , as shown near the corresponding curves. The other parameters a r e @A = 0.1, io = 0.1, and Eo = 5. A corresponding set of quantities is I, = lod einstein L-' SKI,A0 = M, d = 5 cm, tA = l o 5 L mol-I cm-l. Note that curve c = 0 is described analytically by eq 17.
of
-
. .
1
Equation 16 describes the dynamics of the simplest of all photochemical reactions, viz. scheme R1. Even though this result is not entirely new,8-10*23 two important points should be noted: (a) Equation 16 is a nonlinear differential equation for which there exists no simple analytical solution. A solution based on a series expansion was recently suggested by Jackson and L i s h a ~ ~ . * ~ The result of the integration in terms of a such an expansion is unfortunately a rather complicated expression which, although useful in some cases, is of limited use in the analysis of more elaborate photochemical schemes. (b) In a photochemical reaction, the reaction rate depends not only on the reactant concentrations but on the concentration of the photoproducts as well. Although some type of dependency on product concentrations can occasionally be observed in light-free chemistry, it remains the exception rather than the rule. In a photochemical reaction, both reactants and products are in constant competition for the incoming photons, and as a result, the fraction of the incoming light flux absorbed by reactants always decreases as light-absorbing products are formed. This apparent "product inhibition" thus arises as a natural consequence of competitive absorption and is a general feature of the dynamics of many photochemical reactions. As seen in Figures 1 and 2, this screen (or filter1*)effect becomes more important as the molar absorptivity of the photoproducts increases. Competitive absorption is a key element in the description of the dynamics of photochemical systems and, as will be shown in the next sections, can play a most important role in the onset of instabilities in photochemistry. A fairly common assumption with regards to eq 16 is to consider that only reactant A absorbs a t the irradiation wavelength, i.e. eB = 0. (Actinometric measurements for example, are ideally carried out under such c o n d i t i o n ~ ) . ~ , ~Using ~ - ' ~ (14 and 15), eq 16 can in this case be integrated analytically to yield
+ Eo-l In (1 - exp(-E&)J
0.008
I
The photochemical rate, rv (eq 6), can therefore be written as ru = - d X / d t = dY/dt = aAiJg(X,Y) (16)
= a. - aAiOt
(17)
where
a. = 1
400
time
0.012
where
X
200
0
Figure 1. Numerical integration of rate equation 16 for different values
(13)
and the photokinetic factor as
g(X,Y) =
I."
+ Eo-' In (1 - exp(-Eo)J
(18) In the limit where cB is negligible, a plot of the left-hand side of (22) The use of E (or E') as a symbol for the total absorbance has been suggested by Mauser and colleagues (see refs 11 and 12). (23) Jackson, R. L.; Lishan, D. G. J . Phys. Chem. 1984.88, 5986.
3
I
a 4
-
0.004
0.000 0.0
0.2
0.4
0.6
0.8
1.0
X Figure 2. Photochemical rate r, (eq 16) for increasing values of c = as shown near the corresponding curves. Other pzranerers e r e as in Figure 1. Note that the rate of the photoreaction is slowed down significantly when the photoproduct absorbs light a t the irradiation wavelength, i.e. when c # 0.
eq 17 versus time should therefore yield a straight line with a slcpe equal to -aAio(from which io or aAc2n be obtained). E q u d o n 17 can be further simplified in the limit where eA is &her very large or very small. In these cases, the resulting expressions zre
x = 1 - aAiot
large: cA
small:
X
+ Eo-l In (E,&)
= a. - cPAiot
(19)
(20)
with
a. = 1 + Eo-'In (E,)
(21)
Occasionally there has been some confusion in the literature regarding the order of elementary photoreaction^.^^ Let us emphasize that according to rate law 16, the order of a photochemical reaction is in general undefined. Even though :be ccncept of reaction order is of limited use in photochemistry, scme limit cases of rate law 16 do yield well-defined orders and zre iherefore of some interest. The first of these cases is when cA is l2rge 2nd >> tB. In this limit, the integrated rate law (19) is obtained (24) A quite common misconception is that the order of an elemen!ary photoreaction is somewhere between 0 and 1. The authors suggest that the origin of this misconception is related to an approximation of the exponent term in g(X,Y)(eq 14), in the limit where cA is small and cB = 0. If the series expansion of this exponent term is carried out in differential equation 16, before integration, the result yields a first-order-like expression. However, this procedure is mathematically incorrect. The integration must be performed before the series expansion. A numerical comparison of expression 17 with the corresponding first-order expression reveals indeed that first-order behavior is only observed at longer times in the photoreaction and is in serious discrepancy with the exact expression (17) for almost half of the total photoreaction time.
2956 The Journal of Physical Chemistry, Vol. 96, No. 7, 1992
and the photochemical reaction proceeds as a zero-order process. It must however be noted that even though this approximation may often seem to be justified in many experimental situations, its use in the description of the time-dependent behavior of photochemical reactions should be avoided. The second case for which a reaction order can be obtained from the general rate law (16) is when the irradiation is carried out at an isosbestic point, i.e. at a wavelength at which t A = tB. The photokinetic factor g(X,Y) (eq 14) is then a constant and the rate law is first order with respect to A (see Figures 1 and 2, e = 0). 11. Multiple Steady States in Photochemistry A. Thermally Reversible Photoreaction with Concentration Quenching. A quenching process is defined as a process which competes with the spontaneous emission from excited states and thereby shortens their lifetime. Quenching reactions are common in photochemistry and can occur through various mechanisms, such as excimer or exciplex formation, electron transfer, singlet-triplet energy transfer, etc.17-19 In this section, we consider the case of concentration quenching and its effect on the dynamics of a thermally reversible photoreaction. Concentration (or self-) quenching was first observed in pyrene solutions by F o r ~ t e and r~~ was explained as due to the formation of an excimer complex between ground and excited state molecules. The mechanism through which this complex is produced is now well understood and described in detail in most of the recent textbooks on photochemi~try.'~-~~ Let us therefore consider the case of a thermally reversible photoreaction, where reactant A is subject to concentration quenching, viz. hv
A Z B k,
A
+ A*
-
2A
As in the case of section I, scheme R2 is in fact a short notation for the group of photochemical and photophysical processes taking place in solution. Here, we assume that these processes are described by the following elementary steps:26 hv
A-A* A* A
A
+ A*
(AA*)
+ (hd)
- + ---c
(AA*)
2A
A*
-
B-A
B
I,
(R2.1)
k,[A*]
(R2.2)
k,,[A] [A*]
(R2.3)
kq[AA*]
(R2.4)
(hv")
kr[A*]
(R2.5)
k,[B]
(R2.6)
One now considers the case where (a) reactions R2 take place in a well-stirred batch reactor, i.e. a system closed to mass flows, (b) the irradiation is carried out using a monochromatic light source, and (c) only A absorbs at the irradiation wavelength, i.e. CB = 0 (22) By use of rate expression 16 and application of the photostationary state approximation to excited states A* and (AA*), the following rate equation is obtained for reaction scheme R2 dX/dr = -io@(X)Xg(X,Y) (1 - X ) (23) where X = [AI/[AOI; io = I d k A (24) and T = k,t
+
Since only A absorbs a t the irradiation wavelength, the photokinetic factor, g(X,Y), is here a function of X only. It takes the simple form ( 2 5 ) Forster, T.; Kasper, K. Z.Phys. Chem. (Munich) 1954, 1 , 275. (26) A more detailed consideration of the photophysical processes that
could take place in such a scheme (such as excimer fluorescence and dimer formation) only results in a renormalization of the various constants and does not significantly affect the results presented in this section.
Borderie et al.
where
-
The function @ ( X ) in (23) expresses the dependency of the quantum yield of the A B photoreaction on the concentration of self-quencher A, viz. @(X) =
@A 1 +ox
where 0, is the quantum yield of the photoreaction in the absence of concentration quenching, i.e. @A = kr/(kd + kr) (29) The constant /3 which appears in eq 28 is proportional to the Stern-Volmer constant and is here defined as16J7*25
P = kexAO/(kr + kd) (30) The system's steady states can easily be obtained from a graphical solution of eq 23, for dX/dr = 0. Defining = iO@A[1-exp(-Ed?1/[1
+ 6x1
(31)
and r,(X) = 1 - x
(32)
the steady states can be found as solutions of -dX/dr = r,(X) - r,(X) = 0
(33) Figure 3 illustrates the functions 3 1 and 32 as a function of X . According to (33), the steady states of the photoreaction scheme (R2) correspond to the intersections of these curves. As clearly seen, multiple steady states can be obtained for some values of io (see Figure 3b). The function describing the steady states as a function of io is easily obtained from eqs 31-33 as (1 -X)(1 + b X ) (34) @A[1 - exp(-Ed?] Equation 34 is plotted in Figure 4, for different values of Eo.Note that the steady states could also have been plotted as a function of other experimentally convenient parameters, such as tA, which depends on the irradiation wavelength. (Unfortunately in this case, it is not possible to derive an analytical expression such as (34); the steady states must then be obtained from numerical or graphical methods.) Let us add that related schemes involving, for example parallel photoreactions with exciplex formation, can also be shown to lead to multiple steady states. As seen in Figure 3, multiple steady states arise as a result of the presence of a maximum in the photochemical rate, rv(X),as a function of X . This maximum is in fact a consequence of the competition between the quenching process and the A + B photoequilibrium. In order to clarify this point, let us first note that in the absence of quenching, r,(X) is an monotonic function of X (see Figure 2, t = 0): in this case the rate of a photochemical reaction increases with the amount of light absorbed. In the presence of concentration quenching however, the quantum yield of the A B photoreaction decreases as the concentration of A increases, according to (28): it is the competition between these increasing and decreasing functions of X that gives rise to a maximum in the function describing r, as a function of X. From a more physical point of view, this dependency of r,(X) on X is reminescent of what is observed when substrate inhibition is present in light-free chemical schemes (such as the chlorite-iodide reaction in a CSTR).I4 Here we have a somewhat similar situation: a t low concentration of the light absorber A, the rate of absorption, Z,, increases faster than the quenching rate; beyond a certain concentration, the quenching rate predominates and the overall photochemical rate decreases as the concentration of A increases, hence a substrate inhibition-like behavior. A numerical study of the nature and stability of the steady states reveals that out of the three possible steady states, the upper and lower ones are stable nodes while the middle one is a saddle point
io =
-
The Journal of Physical Chemistry, Vol. 96, No. 7, 1992 2957
Photochemical Systems 1.2
,
I
I
,
I
0.6
0.8
0.6
0.8
1.0
I
I
place in a CSTR, fed at a rate ko with reactant A at a concentration A@ It is easy to show that the dynamics of scheme (R2’) is then described by the same rate equation as (R2) (viz. eq 23), where A. and ko simply replace A. and k,: from the dynamics point of view, the two schemes are therefore equivalent. Let us however emphasize that scheme (R2’) is particularly simple and in fact quite common in the literature on concentration quenching.17-19Systems that are described by mechanisms such as R2’ may therefore be the most promising candidates in the experimental search for new examples of multiple steady states in isothermal photochemistry. As a final note, let us add that multiple steady states and oscillations have also been reported recently by Ross and Li in their study of an isothermal model in which one of the steps is photochemi~al.~~ In their model, a photochemical reaction is used to regenerate one of the main reactants. However, this is rather different from R4, as the bistability arises from an autocatalytic step and not from a photochemical nonlinearity. B. Cyclic and Consecutive Photoreactions. Many photochemical reactions are described by cyclic mechanisms involving consecutive photoreactions.’7~18~zo~z~ Molecular photoreaction cycles have been used, for example, as models for proton pumps28and for the storage of information and subsequent regeneration of the information carrier.29 The subject of this section is an examination of the dynamics associated with the following cyclic reaction mechanism
I
...-.-..,.
0.9 -
%.
0.0 I 0.0
%
I
I
0.4
0.2
1.0
A
0.0
1
0.0
-.-..
I
I
0.2
0.4
X 2.5 r
2.0
I
I
h
(4J
A
C-B
hv
A corresponding detailed scheme could be
I
0.0 0.0
A L A *
--. , -- ---.___
I
I
0.2
0.4
0.6
hu
-- ++ --
B-B*
1.0
0.8
X
A*
Figure 3. Graphical solution of the steady-state equation (33) for different values of io: (a) 2.0, (b) 5.0, (c) 8.0. Shown in each of the graphs are the corresponding plots of ru (eq 31), solid line, and r, (eq 32), dashed line. The other parameters are = 0.8,E, = 25, and @ = 20. Note that only one steady state is possible in parts a and c, whereas three are possible in part b.
B*
0.6 0.4
0.2 0.0 0
Zb
(R3.2)
(hv’)
kd[A*]
(R3.3)
B
(hd’)
k,j’[B*]
(R3.4)
A*
B
k,[A*]
(R3.5)
B*
C
k,‘[B*]
(R3.6)
C
A
kJC]
(R3.7)
A
kb[B]
(R3.8)
+
-- --
Note that out of the four reactions taking place in this scheme, two are photochemical (A B and B C) while the other two take place thermally (B A and C A ) . One now considers the case where (a) reactions R3 take place in a well-stirred batch reactor, (b) the reactor is iqradiated with a monochromatic light source, and (c) only A and B absorb significantly at the irradiation wavelength, i.e. e, = 0
0.8
x
(R3.1)
A
B 1 .o
1 ,
2
4
8
6
10
12
io
Figure 4. Steady states of the concentration quenching scheme (R2), as given by eq 34, for three values of E,: (a) Eo = 25, (b) Eo = 10, (c) Eo = 5. Curves a and b correspond to multiple steady states, whereas curve c corresponds to a case where the steady state is unique for all values of io.
(unstable). This can also be shown analytically in the limit where Eo is large, but unfortunately not in the general case. As a final note to our analysis of reaction scheme R2, we comment briefly on the following scheme: hu
A-B A A*
+
-
2A
(R2’) This scheme is similar to (R2), except that the photoreaction is now irreversible. Let us assume that the above reactions take
Using eq 16, the differential equations describing the dynamics of the cycle can be written as dX/dT = -io@AXg(X,Y) + Y + MZ (354 dY/dT = io(@AX- @BeY)g(X,Y)- Y d Z / d r = io@BeYg(X,Y)- MZ where X = [AI /Ao;
Y=
PI /Ao; T
= [C]/Ao
= kbt
(35b) (35c) (36) (37)
and io = Io/k,,Ao; /I = k,/kb;
t
=
t g / e ~
(38)
(27) Li, R. S.; Ross, J. J . Phys. Chem. 1991, 95, 2426. (28) Schulten, K.; Tavan, P. Nature (London) 1978, 272, 85. (29) Abdel-Kader, M. H.; Steiner, U.J . Chem. Educ. 1983,60, 160, and references therein.
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Borderie et al.
The Journal of Physical Chemistry, Vol. 96, No. 7, I992
1
1 .o
1.2
I
z o.6 0.4
0.8
x
0.6
-
r
1 .o
-2C
-
v
4
M
0.0 0.00
0.04
0.4
0.0 0.08
0.12
0.16
A
The expression for the photokinetic factor g(X,Y) is similar to (14), viz. 1 - exp[-Eo(X
+ CY)]
with Eo defined as Note that the conservation relationship now takes the form X+Y+Z=l (41) so that only two out the three differential equations (35) are necessary to describe the system dynamics. The system of eqs 35 can be solved analytically for its steady states in the case where = 1;
a* = aB = 4
(42) In this case, one can show that the steady states are solutions of the following quadratic equation [4g(Z)I2(1 - Z)io2- 2 ~ 4 Z g ( Z io ) - pZ = 0
1 .o
0.8
Figure 6. Plots of the photochemical and thermal rates in scheme R3. The photochemical rate, rv (eq 49), is shown as a solid line while the thermal rate, r, (eq SO), is shown as a dashed line. The parameters used in eqs 49 and 50 are 4 = 0.1, io = 0.1, E , = 2 5 , and p = 8 X lo4.
a quenching process. In order to illustrate this point, let us consider the expression for the reaction rate, dZ/dT, in the case where (as in Figure 5) = aB = 4; t = 1; I.(
