J . Phys. Chem. 1984,88, 5442-5445
5442
Relaxation Kinetics in a Bistable Chemical System G. Dewel,* P. Borckmans,* and D. Walgraef*t Service de Chimie-Physique 2, Universite Libre de Bruxelles, Campus Plaine, CP 231, Bruxelles, Belgium (Received: February 1 , 1984)
According to the value of the residence time and of the feed concentrations nonlinear chemical reactions occurring in a CSTR may exhibit critical behavior and bistability. Relaxation phenomena are investigated for such a well-stirred system. It is shown that the critical behavior is always of the magnetic field type and that long-lived unstable modes analogous to metastable states may occur near the bistability limits. The possible consequences of a lowering of the stirring rate are also briefly discussed.
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Introduction It has long been predicted that open chemical systems can exhibit two or more stationary states over a range of system constraints.’ A large number of such multiple steady-state systems have thereafter been discovered in continuous flow reactors (CSTR).2 In most of these systems the stirring is sufficiently vigorous to rule out concentration inhomogeneities. In such situations nucleation is absent and the transition between the steady states takes place via the growth of unstable modes at the hysteresis limit. Recently the anomalous behavior of the relaxation time (“critical slowing down”) near the hysteresis limit has been experimentally ~ b s e r v e d . ~In this note we present a theoretical analysis of a well-defined model exhibiting bistability taking into account nonlinear effects and stressing the specificity of the CSTR. Quantitative predictions are made about the relaxation phenomena near the stability limits and the critical behavior which could hopefully be tested experimentally. The conclusions we draw are indeed valid for a wider class of bistable systems. The paper ends with a qualitative discussion of the effects of inhomogeneous fluctuations in systems where the stirring is less e f f i ~ i e n t . ~
X,
=:
( S o b - k1)/3k2
(4)
In order to make contact with the theory of equilibrium phase transitions: it is convenient to define a new variable x by x = X,(1 x) (5)
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In terms of the variables defined by (4) and (9, the stationary condition for X , [eq 31 may equivalently be written as
The System We consider a simple two-variable kinetic model which has been used to describe the autocatalytic oxidation of arsenous acid with i~date:~ X = R , + ko(X0 - X) Y = -R,
+ ko(Y0 - Y)
(1)
where X and Y a r e respectively the iodide and iodate concentrations, ko is the reciprocal fesidence time, X o and Yoare the corresponding feed concentrations, R, is the rate of iodide autocatalysis described by the following empirical law for the Dushman reaction? R , = (k1 k2X)XY (2)
+
where the buffering hydrogen ion concentration has been included in the rate constants k l and k2. We consider first a CSTR where the stirring is sufficiently vigorous to allow a homogeneous description. This situation seems to be well realized in many experiments performed recently on similar bistable systems.
The Stationary States The behavior of the system is studied by varying the residence time ko-’ which is a convenient bifurcation parameter in a CSTR. The steady concentrations of iodide X , as a function of ko are the positive roots of the following equation derived from eq 1: k&:
where So = Xo Yo is the feed concentration of the stoichiometrically significant iodine species. For ko = 0 (batch reactor situation), eq 3 has three real roots, thus three stationary states: X I = So,X 2 = 0 , and X , = - k l / k 2 . When k, grows, X2 becomes negative and finally coalesces with the other spurious state X3,as the product of the roots must remain positive and equal to k d o . Therefore only the solution X I = So remains to be considered. On the other hand, for very high flow rates (ko large) the system is in a state of iodide concentration determined primarily by the input flow (X, = Xo). Furthermore, according to the feed concentrations this transition from thermodynamic to flow branch may be either uniform or two states may coexist over a range of ko (bistability). The crossover between these two regimes corresponds to a critical point related to the triple root of eq 3 :
+ (kl - k2So)X; + (ko - klSo)X, - k&o
= 0 (3)
t Chercheurs Qualifits au Fonds National de la Recherche Scientifique de Belgique.
0022-3654/84/2088-5442$01 .50/0
where K = kO/k2X? K, = 3
+ (klSO/kZX?)
Equation 6 has the standard form of the mean field state equation for the order parameter describing a first-order equilibrium transition; ( K - K,) and (1 - (Xo/X,))(K- Kh)are the temperature and magnetic field terms. But contrary to the equilibrium situation, in the case of CSTR these variables are never independent since the dimensionless residence time K-’ appears in both terms. In order to be able to discuss the behavior around the critical point ( K = Kh = K,), we will restrict the discussion to the case X , > 0. It follows that for Kh > K,, eq 6 will admit only a single (1) Glansdorff, P.; Prigogine, I. “Thermodynamicsof Structure, Stability and Fluctuations”; Wiley-Interscience: New York, 1977. (2) Epstein, I. R.; Dateo, C. E.; De Kepper, P.; Kustin, K.; Orban, M. In “Nonlinear Phenomena in Chemical Dynamics”;Vidal, C., Pacault, A., Eds.; Springer-Verlag: West Berlin, 1981; pp 188-191. (3) Ganapathisubramanian,N.; Showalter, K. J . Phys. Chem. 1983,87, 1098-99. (4) Roux, J. C.; De Kepper, P.; Boissonade, J. Phys. Leu. A 1983, 97A, 168. ( 5 ) Liebhafsky, H. A.; Roe, G. M. Int. J. Chem. Kinet. 1979,Il, 693-703. (6) Stanley, H. E. “Introduction to Phase Transition and Critical Phenomena”; Oxford University Press: Oxford, 197 1.
0 1984 American Chemical Society
The Journal of Physical Chemistry, Vol. 88, No. 22, 1984 5443
Relaxation Kinetics in a Bistable System real root, whereas when K, > Kh, there are three real solutions x 1 > x 2 > x 3 in the range K1 < K < K 2 where K1 and K2 are the corresponding stability limits of the hysteresis (coercive fields in the language of phase transition phenomena). x 1 and x3 lie respectively on the thermodynamic and flow branches. In the experiments in ref 3, the condition K, > Kh is fulfilled and the system therefore exhibits a bistable region for which the simple model (1) provides a near quantitative description. When the dynamics of the system has a potential structure the unstable solution x 2 delineates the boundary between subcritical and supercritical perturbations, Le., those necessary to induce a transition between the stable branches XI and X3.’ However, in the case discussed here this situation only happens when S ( t = 0) = So where S = S = X Y. Otherwise, the boundary between the attracting basins of X I and X 3 also depends on S ( t = 0). Finally, near the critical point K Kh = K, where the three roots and the stability limits coalesce one can define a so-called “critical exponent” characterizing the behavior of the “order parameter” (iodide concentration).
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In the range of concentrations used in the experiments, Kh = K, corresponds to Yo 8(Xo + k1k2-’). This behavior corresponds in equilibrium to a magnetic field type approach to the critical point (6 = 3).6 A temperature-like approach (leading to the well-known square root law p = 1/2) is never possible in a CSTR. All chemical schemes exhibiting bistability in a sufficiently well-stirred CSTR would exhibit the same critical exponent 6 = 3 irrespectively of the underlying chemical mechanisms. It would be interesting to check experimentally this “universality” property. Nonequilibrium critical points have indeed been characterized in recent experiments in a CSTR.8
We consider situations where the system was in a stable stationary state for times t C 0 and at t = 0 the system is driven into a state xo by an appropriate injection of concentration or a sudden change of the flux ko. We then study the subsequent relaxation of the system. This relaxation can be obtained by solving eq 1, which may equivalently be written as a&‘ = (k1 k&(S - X)X ko(X0 - X )
S(t) = X
+
+ Y = So + e-ko‘(S(t=O) - So)
(1’)
However, when approaching the stability limits ( K K1, K2) or the critical point (K Kh = K,) one of the eigenvalues of the linearized problem remains finite and roughly equal to -ko whereas the other tends to zero (“critical slowing d ~ w n ” ) . As ~ is wellknown nonlinear effects then become important and should be taken into account. This is the main purpose of the paper. In these regimes and for times t > k o l the dynamics of the system can be described by a single kinetic equation: -+
+
x = -((I - (xo/Xc))(K- Kh)
+ (K-
K,)X
+ x3)
(9)
where x = allx with t‘= tkJ2. For t > k c ’ , the sum S has indeed relaxed to its stationary value Sodetermined by the input flows. For a given initial condition eq 9 can be integrated exactly (for algebraic details see the Appendix). We here summarize the dynamical behavior of the system in various regimes according to the position of the initial state xo in the (x,K) plane (cf. Figure 1) near the bistability limits: (7) Papsin, G.A.; Hanna, A.; Showalter, K.; J . Phys. Chem. 1981, 85, 2575-82. ( 8 ) Dateo, C . E.; Orban, M.; De Kepper, P.; Epstein, I. R. J . Am. Chem. SOC.1982, 104, 5911-18. (9) (a) Nitzan, A.; Ortoleva, P.; Deutsch, J.; Ross, J. J . Chem. Phys. 1974, 61, 1056. (b) Heinrichs, M.; Schneider, F. W. J . Phys. Chem. 1981, 85, 2112-6 (c) Gray, P.; Scott, S.K. Ber. Bunsenges. Phys. Chem. 1983, 87, 379-82.
Stable states are represented by plain lines and dashed lines correspond to unstable states. The real part of the complex roots of eq 3 for K > K2 is given by the dotted line. The relaxation regimes discussed in the text correspond to initial conditions illustrated by the crosses on the dashed-dotted lines. (1) K 5 Kz. For x,, > x2 the system relaxes to the steady state x 1 (subcritical perturbation) and if the initial state xo is close to the final state x I the relaxation function +(t’) = (x(t’)- x I ) / ( x o - x l ) has a simple exponential behavior:
+(t’) = exp[-(3xI2
Since for K
E
XI
+ (K - Kc))t’l
(10)
K2 we have from eq 6 N
3-‘J2(Kc - Ka)‘/z + c(K2 - K ) ‘ / 2
(11)
we find
+(t?
Relaxation Phenomena
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Figure 1. Schematic phase diagram in the (x, K ) plane (K, > Kh).
N
exp[-c(K2
- K)’/ZtI
(12)
The relaxation time rR diverges when approaching the stability limit (critical slowing down): 7~ =
(Kz - K)-‘l2
(13)
Such an anomalous increase of the relaxation time near the hysteresis limit has been recently observed in the bistable iodate-arsenite ~ y s t e m .If~ the initial state differs appreciably from x I the decay law is more complex but for large times it is always exponential with the same time constant or relaxation time independent of the initial state. For xo C x 2 the system relaxes to the state x 3 (supercritical perturbation); the relaxation function +(t’) = ( x ( t ’ )- x 3 ) / ( x ox 3 ) now becomes for large times + ( t ’ ) exp[-3(KC - K ) t q . In this case the relaxation time remains finite for K K2. (2) K = K2. When xo > xz = x 1 the relaxation function + ( t ’ ) = ( x ( t ’ )- x I ) / ( x o- x l ) now exhibits an algebraic decaygc
-
+(t?
= l / ( x o - x1)3x1t’
(14)
whereas for xo < x2 = x , the relaxation remains exponential for long times with the relaxation time: 3(K, - Kz)-‘
TR
-
(15)
This behavior should not be confused with that occurring in the critical region (K K, = Kh): indeed there the relaxation time has the anomalous behavior: 7~
N
(K,
-
(16)
This critical slowing down characterized by the dynamical critical exponent z = 2/3 is different from the slowing down which develops near the hysteresis limit (cf. eq 13). Again this behavior corresponds to a magnetic field like approach to criticality characteristic of the CSTR. At the critical point one recovers the algebraic decay which now takes the form + ( t ’ ) N t + ’ / 2 . (3) K k K2. Beyond the hysteresis limit K2, the relaxation of
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The Journal of Physical Chemistry, Vol. 88, No. 22, 1984
Dewel et al. X
J
I
Figure 2. Relaxation curves slightly beyond the hysteresis limit for the same initial condition (xo = x1(K2))but for decreasing residence time ko-' ((1) K = 1.01K2; (2) K = 1.05K2; (3) K = 1.1K2; (4) K = 1.5K2) exhibiting the plateau regime.
the system to x3 is different according to the position of the initial state xo with respect to the real part y (see Appendix) of the solutions x i and x2 which become complex for K > K,. The relaxation function 4 ( t ' ) = ( x ( t ' )- ~ 3 ) / ( ~- 0x 3 ) first presents a transient related to the short-time relaxation of S . When xo < y one recovers afterward the usual exponential behavior toward x 3 whereas for xo > y the exponential decay is preceeded by a plateau as shown in Figure 2. The length of this plateau is given near K 2 by TMS
IK2 - K ( - ' / Z
(17)
For instance TMS(K=I.OIKJ
70ko-l
(18)
Hence these unstable states have macroscopic lifetimes. Such long-living unstable states may be classified as metastable states according to a dynamical definition of metastability based on the flatness properties of the relaxation function.1° This definition is particularly constructive in far from equilibrium systems where the conditions for the existence of a generalized potential are rarely satisfied.' The macroscopic nature of the lifetime of these states can lead to an inaccurate experimental determination of the bistability limits. By increasing the flow rate continuously and starting from a stationary state (xl) one can pass through the stability limit if one does not wait long enough to see the decay of the long-living unstable states. (4) K = K 1 . A similar behavior is expected near the stability limit Kl except that when xo > x2 the relaxation to xi is dominated by the residence time kciwhich becomes in this regime the largest time scale and as a result the reduced description in terms of eq 9 is no more valid. Similarly in the central part of the bistable region (see Figure 1 ) the relaxation of the supercritical and subcritical perturbations are again determined by k0-' and in the case of supercritical perturbations overshoots in the relaxation are possible as shown experimentally by Papsin et al.' and illustrated in Figure 3. Discussion and Conclusion Transitions between steady states in CSTR have long been presented as examples of nonequilibrium phase transitions. It is well-known that the mean field theory (Landau) does not give an accurate descritpion of the kinetics of equilibrium first-order transitionsI0 because it applies only to systems with long-range forces. However, the CSTR provides an experimental example where the Landau theory may be applied when the stirring rate is sufficiently high to avoid inhomogeneous concentration fluctuations. When the stirring rate is lowered it has recently been shown by Roux et aL4 that the bistability region is correspondingly reduced. This stirring rate sensitivity has been presented by these authors as an illustration of a nucleation induced transition in nonequilibrium systems. (10) Binder, K. Phys. Rev. B Solid State 1973, 8, 3423-38.
Figure 3. Relaxation curves for different initial conditions in the central part of the bistability region (K = Kh
