6408
J . Phys. Chem. 1991, 95, 6408-6411
Pimsemnds
Flyc 4. Singlawavelength kinetic trace at 615 nm for CO-ligated CcO. The fit shown is the convolution of the instrument response with a 3 - p exponential rise occurring simultaneously with a 6-pexponential decay. See text for assignments.
in Fe,’-CO (uFe = 520 cm-l) is 64 fs. Subsequent thermal chemistry may be observed following photodissociation of CcO-CO at wavelengths near the isosbcstic point found for unligated CcO. For example, kinetic measurements at 615 nm, a wavelength where relaxation of transients due to cytochrome a would lead to absorbance increases on a 3-59 time scale, show absorbance decreases following the initial absorbance change (Figure 4). This decrease in absorbance can be described by a 6-ps exponential decay which occurs simultaneously with the expected 3-ps exponential increase in absorbance. Preliminary results in the Soret band a t 450 nm are consistent with this. We assign the 3-ps relaxation to cytochrome a and the 6-ps relaxation to the cytochrome a3site in CCOCO. The origin of this relaxation is under investigation but may be due to the transfer of a ligand to the cytochrome a3site from the nearby Cue, as has been previously suggested, or to a local conformational change of the protein.I5
Furthermore, the difference spectrum that exists before and after this relaxation differs significantly from that generated by subtracting a conventional static spectrum of unligated CcO from C C ( X ~ O .For ’ ~ example, while the positive transient in the Soret band is at the expected 443 nm, the positive transient in the a band appears at 620 nm, 15 nm redder than observed in the static difference spectrum. Previously, small wavelength shifts were seen in the Soret region for HbCO, but not for MbCO or protoheme-CO model complexes.” Our results suggest, however, that the a band may be a much more sensitive indicator of the environment surrounding the heme center. In this case in particular, the shifts may be caused by structural changes at the cytochrome u3 center induced by the transfer of CO to Cue which is known to occur in 2000 rpm)
leads to classical’ bistability hysteresis; (b) isola formation at intermediate stirring (1250 < S < 1750 rpm). Feedstrcam concentrations: [ClOc]f = 2.35 X IO-’ M and [I-]f 4.49 X 10-4 M. Buffered to p~ = 1.60; T = 21.0 0.2 OC.
*
some detail, restricting the description to the bistable regime and to nonpremixed feedstreams. A more comprehensive account that includes oscillatory behavior, the role of premixed feedstreams, and batch relaxation will be published elsewhere.” The experiment was conducted in the CSTR shown in Figure 1. Together with its standard ancillary equipment it has been described elsewhere.12 During most of the experiments, the reactants were fed through two ports A, B into the weakly stirred region at the bottom of the reactor. Results from injecting alternatively through the tubes A’, B’ into the highly turbulent zone near the stirrer blade will be briefly mentioned.” The reactant feedstreams were prepared as usual,I2 and their concentrations are given in the caption of Figure 2. The iodide feedstream was prepared by premixing two equal flows of KI and a Na2S04/ H2S04buffer. Hence, it entered at twice the flow rate of the NaCIOz stream. The response t of the Pt electrode was monitored as a function of flow rate (inverse residence time ko) at constant stirring rate S. Results The results obtained under intense stirring (S > 2000 rpm) are shown in Figure 2a. They are in essential agreement with earlier measurements:’ the HEP state is stable from the lowest finite
6410 The Journal of Physical Chemistry, Vol. 95, No. 17, 1991
Letters
I-
(0) GF model
I
P
3000,
(b) A+0
-0001
0
IO
IO-^ sec-1
Figure 3. Bifurcation diagram in (k,,S) space: locus of instabilities of
the HEP branch.
Model Calculations and Discussion To reproduce the data and gain insight into the operating mechanisms, two coupled-reactor (i.e., macromixing) models, differing in their coupling topology, were employed together with the kinetic mechanism of Citri and Epstein.” The Gyorgyi and Field (GF) model,3aillustrated by Figure 4a, places a prereactor in series with one of the reactant flows. It represents the localized reaction near injection port A, inside the feed tube and/or in the surrounding plume, between the injected species and bulk material diffusing into this zone at the coupling flow rate Q,. In agreement with the experiments the flow rate through port B (of KI + buffer) was taken twice that through port A (of NaC10J. The second model, due to Kumpinsky and Epstein (KE),Io mimics a stagnant
1
”I-
T‘
20
flow rate up to a critical flow. This hysteresis limit is a sensitive function of S,low stirring destabilizing this state.l Under less intense stirring (S C 1750 rpm), however, Figure 2b shows that the HEP branch may also be destabilized by decreasing the flow. Under these circumstances, the HEP branch can be accessed only by quickly filling the reactor, setting the flow to zero, and waiting until the autocatalytic reaction ignites,’ indicated by a rapid rise of the electrode potential, and by suddenly starting the reactant flow, Le., by jumping across the gap onto the isola. As the stirring rate is further decreased, the high-potential branch continues to shrink in width, at about equal rates at the upper and lower limits, until it disappears altogether near S = 1250 rpm. The resulting bistable structure could be a mushroom or an isola. The former was disproved experimentally for the great majority of experiments by the absence of up transitions from the flow branch at several flow values within the kola limits and computationally as described below. Only once could such an up transition be observed. Together with the calculations, this confirms that a mushroom must exist as a transitional structure between bistability hysteresis and isola, within some small volume of parameter space. The bifurcation diagram, i.e., the locus of the isola’s stability limits in the parameter space (ko,S), is given in Figure 3 . Bistability is found above the line and monostability below. At high stirring rates (S > 1750 rpm), a single instability of the HEP branch at high flow rates marks the well-known’ stirringdependent hysteresis limit, while at intermediate stirring (1200 C S C 1750 rpm) the second destabilization mechanism comes into play at low flow rates, giving rise to a mushroom or isola. At low stirring (S C 1250 rpm) the latter mechanism dominates over the entire range of flow rates, eliminating the HEP state altogether. The critical point a t which the isola disappears9* corresponds to the minimum in Figure 3 . I d a formation could be suppressed by injecting the feedstreams through ports A’, B’ into the turbulent zone in the vicinity of the stirrer. Then the response resembles that of Figure 2a over the full range of S,and the bifurcation diagram consists of a monotonically rising curve. Obviously, the mushroom/isola structure develops only when the reactants are fed into a poorly mixed zone of the reactor.
00
K E model
Figure 4. Coupled reactor models: (a) Gyorgyi-Field, (b) Kumpinsky-
Epstein. dead zone coupled to the main reactor through the cross flow Q, but outside the main flow Qo of the single stream of premixed reagents. For both mixing models, the parameter vector is partitioned into the vector of rate constants k, the feedstream concentrations [ If, the total reactant flow 3Q, and finally the quantities reflecting the mixing intensity, the inverse cross flow QCI and the volume ratio V/V’. These mixing parameters are believed to be some monotonically increasing functions of S. They were effectively reduced to the single parameter V’by assuming that the small reactor represents an approximately spherical plume surrounding the inflow port and that the cross flow = c( Vq2I3is proportional to its surface area, where c is a constant. It is further assumed that the volume V’of this plume decreases with increasing stirring due to the contraction of the surface of constant cross flow per unit area. The C E mechanism contains six dynamical variables. The structure of the 12 rate equations describing the coupled reactors follows directly from the coupling diagrams (Figure 4). We used the set of rate constants due to Boukal~uch*~ and the feedstream concentrations [If given in the caption of Figure 5. Starting from an initial steady state, obtained by numerical integration with L S O D E , ~the ~ response diagram, i.e., the set of steady-state values of -In [I-], was calculated with the bifurcation following package BIFPACK” as a function of Qo with (V’,c) held constant. The results for the G F and KE models are presented in Figure 5, a and b, respectively. The solid lines represent stable and the dotted lines unstable steady states. The intensity of stirring, expressed by ( V9-2/3, decreases from curve a (homogeneous case) to curve f. The G F results were obtained with C 1 0 fed ~ through channel A into the small reactor and with I- (at twice the rate, see above) into channel B (Figure 4a). Reversing the feedstreams made the kola disappear, and the results became qualitatively similar to those of the KE model (Figure 5b). The G F model reproduces quite well the experimentally observed (Figure 2) erosion of the HEP branch both at high and a t low flow rates, as a function of increasing V’ (and Q,, S decreasing). These results confirm that the observed structure is indeed an isola and that is arises through a localized reaction of C10,- reactant with material from the bulk. Already at very low coupling (high S), -log [I-] decreases sharply, and at V’ = 1.75 mL (curve d) the steady-state manifold is deformed into a mushroom. At slightly higher V’the mushroom is pinched off, and the resulting isola remains over a wide range of V’ until it finally disappears. Calculations with uncoupled mixing parameters yield qualitatively similar results. This agreement with experiments is another piece of evidence supporting the Citri-Epstein mechanism.’” It should be noted, however, that the computations were done for somewhat
a
(14) Hindmarsh, A. C. G E A R Ordinary Differential Equation System Solver; Lawrence Livermore Laboratory, University of California.
The Journal of Physical Chemistry, Vol. 95, NO. 17, 1991 6411
Letters
6.5
/VAL ,
3.5
6.5
V."
0
,
,
,
,
,
,
0.01
0.02
0.03
0.04
0.01
0.02
0.03
0.04
L
0
Qo
Figure 5. Computed steady states (full lines, stable; dotted lines, unstable). (a) Gyorgyi-Field model: curves a-f represent V'= 0.0, 0.001, 1 .O, 1.75, 3.0, and 4.0 cm3, respectively. (b) Kumpinsky-Epstein model: curves a - e represent Y'= 0.0, 1.0, 3.0, 6.0, and 10.0 cm3, respectively. Feedstream concentrations: [C102-]f = 2.91 X l p M and [I-], = 4.49 X lo-' M. Buffered t o p H = 2.6; c = 0.001; V + Y'= 31 mL.
different reactant concentrations than those used in the experiments, in particular at a higher pH. When the experimental feedstream concentrations were used, the pronounced stirring dependence' (here, V'dependence) of the upper isola limit (Figure 2) virtually disappeared. It is not clear to which extent this discrepancy is due to a flaw in the rate constantsI3or to the mixing model. At the two lowest values of V', V' = 0.0 and 0.001 cm3, the ~lassical'~~ bistability hysteresis is obtained, and the HEP branch loses its stability through a Hopf bifurcation before the turning point is r e a ~ h e d . ' ~At b V' = 1.0 cm3there occurs an abrupt drop and an equally sudden rise of the PI= -log [I-] value at flows between Qo = 0 and 0.005 cm3/s, still preserving however the topology of a bistability hysteresis. At V'= 1.75 cm3, this hole in the HEP branch develops into a left hysteresis and the entire manifold into a mushroom. Indeed, a mushroom with a very narrow left hysteresis loop was once observed in an experiment. Finally, at still higher values of V' = 3.0 and 4.0 cm3, the head of the mushroom becomes detached and is transformed into an isola. The KE model, on the other hand, with its stagnant dead zone and premixed feedstream, fails to account for the destabilization at low Qoand for the formation of mushroom and isola, while it qualitatively reproduces the stirring dependence of the upper hysteresis limit,' as has already been shown by its authors.1° This model specificity of the stirring effect with respect to the two isola limits demonstratesk that the isola-forming destabilization at low flow rate is of a fundamentally different nature than that at high flow. Previously, the formation of mushroom and isola has been established and analyzed in the model of Gray and ScottIS and
in the reaction between iodate and arsenous acid.I6 Both system are characterized by quadratic autocatalysis and slow decay of the autocatalyst. According to the classification of oscillators by Eiswirth et al.,I7 these systems are destabilized by so-called 'strong current cycles" and thus belong to a universality class that is distinct from that of the chlorite iodide reaction which is characterized by a "critical current cycle". In the present case, the mechanism of the negative feedback responsible for isola formation may be understood on hand of the computations" as follows. First, they show that separate feed streams are required for isola formation, in an arrangement that allows the bulk material to react selectively with incoming C10; rather than with the I- stream. Second, the simulations show" that as the lower and the upper isola limits are approached from within the isola, the a ~ t o c a t a l y s t ~concentrations .~ [HIO] and [HIO?] in the main reactor are minimal while the reactant concentrations [C102-] and [I-] reach their peaks. This confirms that autocatalysis is extinguished when the autocatalyst concentration drops below critical values. Third, it is found that in the small reactor the reactant concentration [C102-] decreases rapidly as the lower isola limit is approached from within, due to its reaction with HOI and I2 entering from the bulk, facilitated by the increasing residence time. Thus, due to this early consumption in the prereactor, less C l o y enters the main reactor than it does under well-stirred conditions. Consequently, the reactant ratio [CIO~]o/[I-]o(the subscripts in [lo refer to absence of reaction), which is known7J8to be the principal bifurcation parameter effecting a switch from HEP state via the bistable domain to the flow state, decreases with flow rate Qo.This has the effect of destabilizing the HEP branch, eventually giving way to the flow state as the only stable state at low reactant flow rate.I8 This interpretation will be documented in greater detail elsewhere." Although the present simulations successfully reproduce the experiments qualitatively, quantitative agreement is not called for since it is not clear how to relate the mixing parameters to the physical details of the actual mixing process (circulation pattern, points of injection, etc.). The above noted discrepancies between calculation and experiment call for an improvement of the model. The most promising avenue probably lies in circumventing the complexities of quantitative modeling of the mixing process by performing high-quality experiments as close to homogeneity as possible and simulating their dynamical signature. Until now little effort has been made to minimize inhomogeneities in CSTR experiments.l9 In conclusion, experimental evidence was presented for the topological change of the bifurcation structure through reactor inhomogeneities. One out of two macromixing models that have been tried is in good accord with the observed effect. The nonequivalence of mixing models and the computed responses of the dynamical variables provide insight into its origin. Acknowledgment. This work is supported by the NSERC of Canada. (IS) Gray, P.; Scott, S. K. Chem. Eng. Sci. 1983, 38, 29-43. (16) Ganapathisubramanian, N.; Showalter. K. J. Chem. Phys. 1984,80, 4111. (1'7) Eiswirth, M.; Freund, A,; Ross,J. Adu. Chcm. Phys., in press. (18) The bifurcation diagram in log [CIO;], log [I-] space (see Figure 4
in ref 7b) is cross-shaped, with the narrow bistable domain along the main diagonal separating the monostable regimes of the HEP branch at high [ClO,]/[I-] from that of the flow branch at low [ClO;]/[I-]. Decreasing the [ClO,-]/[I-] ratio while on the HEP branch within the bistable domain therefore destabilizes the former. (19) One exception is the experiment by Schneider and Muenster (ref 3b) in which the chemical chaos of the BZ reaction was suppressed by intense stirring in a compact CSTR.
