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(20) Clever, H. L.; Battino, R. In Techniques ofChemistry; Dack, M. R. J., Ed.; Wiley: New York, 1975; Vol. 8, p 386. (21) Schmidt, R.; Afshari, E. J. Phvs. Chem. 1990. 94.4377. (22) Scurlock, R. D.; Ogilby, P. R:J. Phys. Chem. 1987, 91, 4599. (23) Gorman, A. A.; Krasnovsky,A. A.; Rodgers, M. A. J. J. Phys. Chem.
benzene leads to efficient formation of triplet exciplexes.
Acknowledgment. Supported by NSF Grant No. CHE8914366. Registry No. Fluoranil, 527-21-9; oxygen, 7782-44-7; carbon tetrachloride, 56-23-5; methanol, 67-56- 1; chloroform, 67-66-3; 1-propanol, 71-23-8; acetonitrile, 75-05-8; cyclohexane, 110-82-7; benzene, 71-43-2; toluene, 108-88-3; pxylene, 106-42-3; anisole, 100-66-3; chlorobenzene, 108-90-7.
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Reduction and Analysis of Oscillatory Landolt Mechanisms Paul Ibison* The School of Chemistry, The University of Leeds, k e d s LS2 9JT, England (Received: March 4, 1992)
Sensitivity analysis and principal component analysis are applied to the Luo and Epstein (LE) and Edblom, Gyijrgyi, Orbfin, and Epstein (EGOE) mechanisms of the oscillatory Landolt reaction. Once they are reduced to their skeletal structures, these schemes are then compared to an empirical rate law (ERL) model. The routes of positive and negative feedback and the bifurcation structures of the schemes are contrasted. It was found that there was a strong similarity between the EGOE and the ERL models, while the LE model produces nonlinear behavior via a different route.
Introduction Sensitivity analysis and principal component analysis have been recently utilized for the examination of several models of solution-phase systems.14 Once a mechanistic scheme of elementary processes has been proposed, it may be converted to a set of kinetic differential equations. These types of equations may be numerically integrated to produce concentration versus time wave forms which are then compared to experimental results. A favorable comparison of these results is often used as verification of the model’s accuracy, and this situation sometimes occurs in those cases where several mechanistic parameters have been estimated. The use of sensitivity analysis and principal component analysis enables us to examine a set of differential equations and eliminate those reactions and species which do not contribute to the behavior of the reaction scheme. Such a procedure reduces unnecessary complexity, allowing us to discover the most important and sensitivevariables and helping us to achieve a more substantial understanding of the origin of nonlinear behavior in a model. This paper concentrates on the application of these methods to a specific chemical reaction system: the modified Landolt 0022-3654/92/2096-6321$03.00/0
r e a ~ t i o n which ,~ is the reaction between ferrocyanide, sulfite, iodate, and hydrogen (H’)ions. Several attempts have been made to simulate the behavior of this system. Most notably, an “elementary step mechanism” has been proposed by Edblom, Gyiirgyi, OrMn, and Eptein and an “empirical ratelaw mechanism” (ERL) by GBspiZr and Showalter,’ both of which give good agreement with experimental results. More recently, an alternative “elementary step mechanism”, which gives the nearest match to experimental results, has been published by Luo and Epstein8v9(LE). It is important to note that these. three modeling studies are derived from very different analytical approaches and an immediate comparison of them is not easy to make. After introducing these models, sensitivity and principal component analysis is applied to the ‘‘large” EGOE and LE mechanisms. This reduces the mechanistic schemes and helps establish compatibility with and connections to the ERL model.
Background to Sensitivity Methods Full details of sensitivity techniques are given elsewhere:I0 here a brief summary is provided. 0 1992 American Chemical Society
6322 The Journal of Physical Chemistry, Vo1. 96, No. 15, 19
Ibison
The kinetics of a spatially homogeneous reaction system may be modeled by solving the following initial value problem: dc/dr =f(c,k); c(0) = co where c represents a concentration vector and k represents the system's parameters. In general, these parameters may include Arrhenius parameters, temperature, inflow concentrations, residence time, etc., but for our purposes they are simply the rate coefficients. Various types of sensitivity may be distinguished on the basis of the result investigated as a function of parameters, such as concentration, rate, and feature sensitivities (e.g. time period)." Investigation of the rate of production of a species and its sensitivity (usually presented as a matrix, F) is potentially very useful in chemical kinetics. Once obtained, the rate sensitivity coefficients, Xi/akj,supply further mechanistic details about a reaction system which are not inherent in the concentration sensitivity coefficients. Usually, normalized sensitivity matrices are derived, in order to produce directly comparable coefficients. If k denotes the vector of rate coefficients, then the log-normalized algebraic rate sensitivity matrix F' can be computed as a In fi F' = a In kj Where kjis the rate of reaction j and fi is the rate of production of species i. Such an expression represents the percentage change in the rate of production of i, caused by a 1% change in the relevant rate constant. Sensitivity analysis must be carried out at various time points which cover all the important features of the reaction. Typically, time points corresponding to maximum and minimum concentrations and largest and smallest rates of change of concentration of the important species are chosen. Once the concentration data at these time points has been evaluated, the following process is undergone. (i) It is usual first to eliminate unnecessary species. Here, necessary species are defined as those reactants whose concentration value dynamically contributes to the behavior of the most important species. Two alternative methods exist for the identification of redundant species." We used the definition that a species is designated as redundant if the elimination of its consuming reactions does not cause a significant change from the integrated solution of the full model with respect to the concentration of important species. Species which are considered to be "removable" will also cause the elimination of those elementary steps in which they are reactants. This process is easy to perform and simplifies the use of the following methods. (ii) Often combined parameters have considerable influence on the concentrations. These parameter groups cause functional connections between the sensitivity coefficients, and they can be identified by inspection and comparison of the eigenvectors and eigenvalues of matrix F'TF', i.e. a process termed principal component analysis.l* Combined parameters which have consistently low eigenvalues may lead to the elimination of the corresponding reactions. (iii) Simulation of the reduced mechanism can be used as a comparison with the original scheme in order to verify conclusions made. Certain software packages are available to assist in these calculations, and in the following work extensive use has been made of a set of subroutines called KINAL, developed by T. T~r5nyi.l~ Once this process has been camed out, chemical heuristics such as the constant-concentration approximation and stationary-state approaches may be used to further reduce a model. Tbe Modified LMddt Reaction. The reaction between sulfite, iodate, and hydrogen (H+) ions, usually known as the Landolt reaction,I4has been the subject of several thorough investigati~ns.'~ It behaves as a simple clock reaction in a well-stirred, closed system, showing a sharp increase in [I2] and a rapid drop in pH at the end of the induction period. In a stirred tank reactor (CSTR), bistability producing a hysteresis loop has been ~bserved.~ At high flow rates the system exists on a flow branch characterized
I
I
0
18
36 Timdmin Figure 1. Experimental behavior in the oscillatory Landolt reaction. Reproduced from ref 5 by permission. Inverse residence time = 0.0022 s-'. Initial concentrations: [Fe(NC)6C]o= 0.0204 M, [Iop-]~= 0.075 M, = 0.0893M,[H2SO4I0= 0.0045 M, and [I-]o= 1.0 X 10-6 M.
by high pH and low [I2],while the thermodynamic branch, observed at low flow rates, has a low pH and high [I2]. Using the flow rate as a parameter, a transition from the thermodynamic to the flow branch is possible, but once this branch has been reached the reverse transition usually cannot be located, due to limitations on flow rates. Recently, Edblom, OrbBn, and EpsteinSdiscovered that the addition of ferrocyanide to the Landolt reaction (the "modified Landolt", ML, reaction) produces several changes in the observed behavior; the region of bistability is decreased resulting in a closed hysteresis loop, while at temperatures greater than 30 OC, sustained, largeamplitude pH oscillations are otwerved. An example oscillatory time series is shown in Figure 1. The pH oscillations are entirely periodic and are characterized by a long time spent at high pH followed by a fast decline to a minimum. The gradual increase which follows the minimum may be interrupted by a short plateau. This ML reaction has subsequently been shown to belong to a large family of pH oscillators, but in comparison with the other pH systems the ML reaction has received particular attention. The reasons for this are 2-fold. Firstly, the concentrations of an unusually large number of reacting species (H+, I-, Fe2+,Fe3+, 12,I
