GENERAL,DISCUSSION Dr. B. F. Gray (Leeds) said We have studied propane oxidation in a stirred flow reactor an apparatus in which questions of fuel consumption etc. do not arise because either perfectly steady states or limit cycles can be achieved exactly and their behaviour with respect to perturbations studied experimentally. Results which can only be guessed at from closed vessel studies (such as the existence of a stable focus) can be studied exactly in this case and in particular limit cycle oscillations can be made to persist indefinitely. Hysteresis and multistability are also shown in this system. In the closely related case of acetaldehyde oxidation the advantages of the flow system are relatively greater since the reaction parameters are such that in a closed system the behaviour is dominated first by the arbitrary initial conditions selected and then almost immediately by the monotonic decay towards equilibrium.The system never gets close to the kinetic stationary state (or sustained oscillation) and thus little information is obtained from experiments in closed vessels. On the other hand in the flow system the kinetic stationary state can be achieved exactly as can conditions of sustained oscillation. The system also exhibits hysteresis and multistability.2 Also as the kinetics of this system are reasonably well understood on a semi-quantitative basis computations have been performed including the energy conservation equation excellent agreement with experiment being obtained con-sidering the uncertainties in some of the experimental values of the rate constants used.This agreement leads one to feel that this thermokinetic oscillation is at least as well understood as the Belousov reaction following the detailed work of Noyes et al. but at the same time it displays a considerably richer phenomenology than the latter (which is not sufficiently nonlinear to show multistability or hysteresis). By analogy it would be interesting to pursue the possibility of obtaining spatial patterns in the acetaldehyde system. Dr. L. J. Kirsch (T/zornton)said At Thornton we have been working for some time on the mathematical modelling of hydrocarbon oxidation phenomena. These models starting with the acetaldehyde oxidation model of Halstead Prothero and Quinn have more recently been developed in a generalised form to describe alkane oxidation phenomena over a wide range of experimental conditions.They employ a degenerate branched chain mechanism with competing reactions of differing activation energies controlling the supply and consumption of branching agent and removal of radicals from the system. This competition leads to oscillation under certain con- ditions and the models are therefore examples of the thermokinetic type described by P. Gray et al. in their paper. In both analytical and numerical treatments of the model we have taken fuel consumption into account. The models give excellent simulations of all the patterns of behaviour illustrated in fig. 2 of the paper by Gray et al. I should like to mention some conclusions resulting from our experience of working with these models which are of relevance to the points raised by these authors.B. F. Gray and P. G. Felton Comb. Flame 1975. B. F. Gray and P. G. Felton Comb. Flame 1975. B. F. Gray P. G. Felton and N. Shank Comb. Flame 1975. M. P. Halstead A. Prothero and C. P. Quinn Proc. Roy. Soc. A 1971 322 377. 150 GENERAL DISCUSSION First with reference to the “ sustained ” oscillation illustrated in fig. 2(d),we have noted in our simulations that cool flame behaviour often terminates when a sizeable proportion of reactant remains unconsumed. There follows a lengthy period of slow combustion. Secondly we have observed during the course of a single simulation transitions froni the damped oscillations of fig.2(a) to the sustained oscillation of fig. 2(d). Both of these observations illustrate a property of the thermokinetic model we have considered-namely that the nature of the stationary point solutions are pro- perties not only of the initial conditions and heat transfer properties but also of reaction co-ordinate in as much as reactant consumption influences the appropriate rate coefficients. It would therefore be of interest to learn whether the statement that cool flame behaviour terminates abruptly when no fuel remains is based upon direct experimental evidence. Also whether Gray et al. have found any experimental evidence for a change in the nature of oscillatory behaviour during the course of a single experiment. This might particularly be anticipated in their experiments where a small quantity of acetaldehyde was added and clearly had a pronounced influence on the pattern of behaviour.Variation of the relative quantities of propane and acetaldehyde present may occur during the course of the reaction due to unequal reaction rates. Some rationalisation of the fact that our models predict changes in the nature of oscillation due to fuel consumption whereas this was not observed in the experiments of Gray et al. may lie in the orientation of the experimentally determined line separ- ating weakly and strongly damped oscillations (fig. 4). If the chemical mechanism does not involve any pressure dependent (termolecular) steps the effect of reactant consumption can to the first approximation be envisaged in terms of motion of the point describing the appropriate time-dependent reactant conditions parallel to the pressure axis as the reaction proceeds.The line separating weakly and strongly damped oscillations lies parallel to this direction of motion over much of its range. Thus transitions from one kind of oscillation to another will be unlikely in this system although an eventual transition into the region of slow combustion seems probable. Prof. P. Gray (Leeds) said The calculations which Kirsch has made attempt the most complete modelling of thermokinetic effects in hydrocarbon oxidations. They are based on the assumption of spatially uniform temperature and concentrations i.e. they represent a perfectly stirred system. The quantitative validation of the model therefore requires comparison with experiments done in well-stirred conditions.Unfortunately nearly all classical investigations of cool flames have not been made under such conditions they are all affected by natural convection and flame pro- pagation usually from a hot regi0n.l We believe that our experiments in a stirred reactor offer Kirsch experimental results with which he can compare his computer analysis ;we can also supply him with experimental values of heat transfer coefficients for this type of reactor. In answer to Kirsch’s point about cessation of reaction we find that temperature excesses and hence rates of heat release by reaction fall to zero almost discontinuously as soon as the oscillations have died away.(The time axis of our fig. 2 is too brief to display this.) The “piq d’arret ” is often exhibited ;Lucquin identifies this with the complete consumption of oxygen and the end of combustion. We have not measured J. F. Griffiths B. F. Gray and P. Gray Thirteenth International Symposium on Combustion (The Combustion Institute 1971) p. 239. L. R. Sochet J. P. Sawersyn and M. Lucquin Advances in Chemistry Series No. 76 Oxidation of Organic Conipounds-ZZ (The American Chemical Society 1968) p. 11 1. GENERAL DISCUSSION reactant concentrations by direct chemical analysis. Kirsch’s calculations that a large proportion of fuel remains and that combustion continues for a lengthy period even after the last undamped oscillation seem to show a real difference between experiment and the model.Alteration of the amount of acetaldehyde added has a pronounced effect on the reaction rate in the earliest stages. If concentrations of acetaldehyde are increased beyond 3 mol % not only are induction times to oscillations reduced but the ampli- tude of the first oscillation is also increased. Kirsch’s second point really has to do with the imperfections of trying to categorize chemical reactions in closed vessels in terms that are only properly appropriate to open systems. True limit cycles in these oxidations are possible only if fresh reactants can be constantly supplied. As experimentalists we can recognize in a closed system behaviour that is like a limit cycle in the phase plane when we see 5 or 6 successive undamped or weakly damped oscillations; the behaviour is more like a stable focus in the phase plane when we see strongly damped oscillations.Even these remarks do not apply to very many closed-vessel oxidations-propane is perhaps the most favourable case giving up to 11 pulses in a closed vessel. Acetal-dehyde is the opposite; no experimentalist seems to have observed more than one or two cool flames in a closed vessel. In an open system of course oscillations can persist for as long as we wish. Recent studies by B. F. Gray and P. G. Felton working in Leeds have revealed oscillatory oxidation supported indefinitely in a variety of conditions including acetaldehyde as well as propane. In our fig. 4 the broken line dividing the regions is correspondingly subjective and it ought perhaps to be drawn in its lower parts with a finite positive slope instead of as a nearly vertical line.(After all a single oscillation can hardly be called un- damped.) Even so we do not encounter either a train of (say 4)undamped oscillations followed by another train of (say 4) damped oscillations or the reverse sequence. It is as though Kirsch’s calculations assume that the development in time of the (real) closed system with a particular initial reactant concentration but suffering continual reactant consumption can be legitimately imitated by putting together a short sequence of open systems each without any reactant consumption at all but differing from one another by successively diminished initial reactant concentrations.So far as fig. 4 is concerned there is a similarity between a vertical traverse of the diagram and the course of events in time but to represent either fuel consumption (or reactant temperature) on it would require a third dimension. Prof. P. Gray (Leeds) (communicated) Among gaseous oxidations often said by reviewers of oscillatory reactions to be accompanied by oscillations is that of hydrogen sulphide. Our work shows it to involve thermal and kinetic feedback by self- heating and chain-branching but neither we nor any authors save Thompson have so far succeeded in producing oscillations. I think multiple ignitions reported may have arisen from their technique of admitting reactants successively to their relatively narrow cylindrical reaction vessel.At the pressures employed mixing could have been so imperfect that the first ignition would have been very incomplete and localized. When it was over time for mixing would be required to elapse before a second partial ignition could occur and so on. If this is correct then successive ignitions in H2S oxidation belong with old observations on phosphorus oxidation in unclosed vessels as originating in slow diffusive processes rather than in the chemistry. P. Gray and M.E. Sherrington,J.C.S. Furuduy I 1974,70,2338. * H. W. Thompson J. Chem. SOC.,1931 1809. GENERAL DISCUSSION Prof. P. Gray (Leeds) said In answer to Shashoua 1would remark that acetal- dehyde is added here principally to shorten the induction period before oscillatory cool flames set in-it is of course one of the many intermediate products of propane oxidation-and it also has some secondary effects on the oscillations observed.Rather than being a sink for radicals as in Shashoua’s polymerisations added acetal- dehyde in these oxidations is a potential radical source. I agree with Shashoua that it (and other added reactants) offer a means of varying conditions controlling be- haviour and investigating the chemical part of the tliermokinetic mechanism. We believe these would be worthwhile investigations though there are other fundamental aspects to clear up first. Dr. R. G. Gilbert and Mr. R. Ellis (University of Sydney) said Concerning Yang’s isothermal mechanism both he and P. Gray have mentioned the possibility of important non-isothermal effects.In a closed adiabatic system inclusion of temper- ature as an additional independent variable in numerical integration of the kinetic equations is fairly straightforward. We have written a general program to do this and can verify that such systems are very sensitive to this temperature variation. This program was written for our study of sound propagation in gas-phase reactions extending our previous work in this fie1d.l As a result of these studies we suggest that acoustic-kinetic interactions in systems which are capable of undergoing homo-geneous oscillations would lead to amplification and interference effects which could provide useful data for deriving kinetic information. Prof. C. H. Yang (Stonybrook) said In general the non-isothermal effects in the oxidation of CO is not as significant as in the case of hydrogen or hydrocarbon 3* oxidation.For sustained oscillations and long-lasting glows (minutes) in the CO system Dove showed that it is essentially isothermal. In the case of explosion or glow (seconds) for wet mixtures (greater than 1 % H20) the slow reaction rate is very low at sub-critical conditions. Limit pressures or temperatures are unlikely to be affected by the non-isothermal effect. Limits for drier mixture may be influenced by “ self-heating ”mildly. Only the duration of explosions will almost certainly be drastically shortened by thermokinetic considerations as the temperature difference between the reacting mixture and the bath will be high in such cases.However the duration of explosion has rarely been used as an important parameter in either theoretical or experimental studies. A unified thermokinetic approach to the problem probably would not add significantly to the understanding of the kinetic mechanism. Dr. L. J. Kirsch (Thornton)said :In the final paragraph of his paper Yangdiscusses the effect on his calculations of the rate constant k2 which describes the recombination rate between atomic oxygen and carbon monoxide O+CO+M -+ C02+M Yang uses the value recommended by Baulch et al. (1968) in his computations. Recently a number of studies of this reaction have been made many employing R. G. Gilbert H.-S. Hahn P. J. Ortoleva and J. Ross J. Chem. Phys. 1972 57 2672; R. G. Gilbert P. J. Ortoleva and J.Ross J. Chem. Phys. 1973 58 3625. K. K Foo and C. H. Yang Comb. Flame 1971,17,223. C. H. Yang and B. F. Gray J. Phys. Chem. 1969,73,3395. C. H. Yang J. Phys. Chem. 1969 73 3407. J. E. Dove Dissertation (Oxford 1956). I 54 GENERAL DISCUSSION direct monitoring of the atomic oxygen decay in a time-resolved This work has indicated that at room temperature the rate coefficient k2 is some orders of magnitude lower than that given by Baulch et al. Measurements over a range of temperature give a value of k = 2.4 x 10’’ exp( -4340(kcal/mol)/RT) cmG mok2 s-I . The reaction therefore appears to exhibit the sizeable positive activation energy that Yang states will improve the fit between calculated results and experimental data. At a temperature of about 700 K the overall value of the rate coefficient is in fact close to the temperature independent value employed by Yang.Perhaps the most striking fact to emerge from these recent studies of reaction (2) is its extreme sensitivity to traces of impurity presumed to be metal carbonyls. Thus the most stringent purification procedures must be followed even with supposedly high purity carbon monoxide in order that the true value of the rate constant be recorded. For this reason many earlier studies of the reaction must be disregarded. It follows that some doubt must be attached to any quantitative interpretation of more complex reactions in which reaction (2) plays a rate determining role unless similar precautions regarding purification of CO have been taken.Prof. C.H. Yang (Storzybrook) said A very interesting point has been raised by Kirsch namely that the rate constant of reaction (2) is extremely sensitive to traces of impurity according to recent experimental evidence. In fact the rate constant for reaction (4) may also be sensitive to impurities. While we have no direct evidence of this at the present time it appears to be true with some other excited species reported in the literature. Cvetanovic cited the data of Slanger and Welge indicating that the rate constants for the de-excitation reaction of the O(ls) atom with third bodies of N and H20 are 3.0 x lo7 and 2.1 x 1014~m-~ mol-’ s-l respectively. Water would clearly be a very sensitive impurity in this case. It is well known that the oscillation phenomenon in the CO system is difficult to reproduce under apparently identical experimental conditions.Often this is mysteriously attributed to the role played by the vessel surface. In view of our calculations,s* it is clear that oscillatory solutions may be completely inhibited by sizeable variations of either of these two rate constants. Perhaps purity of the reactants may be the most essential quality to strive for in kinetic experiments of this kind. Prof. R. M. Noyes (Oregon) said The behaviour of oxygen atoms in Yang’s mechanism is similar to that of the switched intermediate X in the Oregonator. Thus step (1) forms oxygen atoms at a rate independent of their concentration and the sequence of steps (5) and (6) accomplishes the same net reaction at a rate pro- portional to the concentration of OH radicals.Because the concentrations of OH and of 0 are positively coupled through step (7) the sequence of steps (5) and (6)generates oxygen atoms autocatalytically. The sequence (2) + (4) destroys oxygen atoms by a first order process and the sequence (2)+(3) essentially represents second order destruction of oxygen atoms. R. J. Donovan D. Husain and L. J. Kirsch Trans. Faraday SOC.,1971 67,375. F. Stuhl and H. Niki J. Chem. Phys. 1971 55 3943. T. G. Slanger B. J. Wood and G. Black J. Chem. Phys. 1972,57,233. R. Simonaitis and J. Heicklen J. Chem. Phys. 1972 56 2004. W. B. Demore J. Phys. Chem. 1973 76 3527. E. C. Y.Inn,J. Chem. Phys. 1973,59,5431. ’R.J. Cvetanovic Canad.J. Chetn. 1974 52 1452. C. H. Yang Comb. Flame 1974 23 97. C. H. Yang and A. L. Berlad J. C.S. Favaday 1 1974 70,1661. GENERAL DISCUSSION 155 As we have pointed out elsewhere,' systems with unstable steady states can not be modelled by elementary processes involving only two intermediates. Thus the state of the present system can not be described solely by the concentrations of CO; and 0 ; other species like H OH and H202undergo coupled variations with delays that are important for the instabilities exhibited. Although the calculations with this model show impressive similarities to experi- mental observations ; 1 am disturbed about the postulated COY intermediate. This excited species reacts with oxygen atoms very efficiently and without activation energy even though the net reaction involves breaking and forming chemical bonds each much stronger than 100 kcal/mol.However the same excited species must usually undergo over lo8 collisions before it loses its excitation energy by step (4). I am not aware of any precedent for an excited species that can so easily react with rearrangement of strong bonds or that is so inert to loss of excitation energy by collision; it would be particularly remarkable to find these unusual and seemingly contradictory tendencies in the same molecule. I am therefore unconvinced that the mechanism of this reaction is yet demonstrated. Prof. C. H. Yang (Stonybrook) said In reply to Noyes I would remark that Tyson and Light have shown that the type of oscillation prescribed by a limit cycle does not exist in a kinetic model which is constructed with two intermediates and involves only first and second order elementary process.This conclusion is of course invalid for a kinetic system with more than two intermediates. Under appropriate conditions a kinetic model of multiple components may be reduced to a binary system when the concentrations of some intermediates are eliminated by the steady state assumption. Again Tyson and Light's conclusion is inapplicable to such a reduced system. In our earlier work we replaced the concentrations of H OH and OE (where OE represents a vessel wall site which is occupied by an 0atom) by their steady state values. The kinetic model was reduced to a binary system containing only the concentrations of CO; and 0.A stable limit cycle was shown to exist in that system. In our present work on the other hand all intermediates in the proposed scheme have been con- sidered. Clearly the system is not solely prescribed by the intermediates CO,* and 0. It is useful to point out after comparing the present results with our earlier work that the assumption of steady states for some intermediates while greatly simplifying the mathematical complexities has not impaired the ability of the simpler model to predict all important kinetic features qualitatively. It is indeed essential to assume in our current work that the excited Cog molecule is quite stable (actually metastable) as far as the quenching reaction (4) is concerned.As indicated in the paper our sole criterion for selecting the rate constants for reactions (3) and (4) is based on a favourable fitting of the calculations to the oscillation and explosion limit data. There are no known values for these rate constants reported in the literature to the knowledge of the author. They probably remain to be in- dependently determined. However we find it difficult to accept the argument that the stability of the CO; molecule would imply a slow rate for reaction (3). Many bi- molecular reactions which involve a stable molecule and a radical are known to be fast. One example is the well known titration reaction NO+N -+ N 4-0 where a bond of I50 kcal is broken and a bond of 225 kcal is formed for which a rate constant of the value 3 x IOl3 cm3 mol-l s-l with nearly zero activation energy is a~cepted.~ R.M. Noyes and R. J. Field Ann. Rev. Phys. Chem. 1974 25 95. J. J. Tyson and J. C. Light J. Chem. Phys. 1973,59,4164. C. H. Yang Comb. Flame 1974,23,97. 4D. L. Baulch D. D. Drysdale D. G. Hoare and A. C. Lloyd High Temperature Reaction Rate Data (Leeds University 1969) no. 4. GENERAL DISCUSSION This value is almost three orders of magnitude greater than what we proposed for reaction (3). As noted before the rate constants for reactions (3) and (4) may be changed to 5.0 x 1013 and 1.25 x lo8 respectively if only one thousandth of the total excited CO; molecules produced in reaction (2) are effective in reaction (3). All computed results will be invariant for such a change.Reaction (5) is another example. At T = 1000 K the rate constant for reaction (5) is greater than or at least comparable to the one we suggested for reaction (3). Dr. J. R. Bond (Leeds) said Yang's theoretical paper on carbon monoxide oxidation is a carefully fitted complex of elementary reactions and quite narrow restrictions must be placed upon kinetic parameters (e.g. relative third body efficiencies of COz to other gases) to ensure reasonable agreement with experiment. It is also an isothermal model although it is for a highly exothermic reaction. To discover whether temperature changes in real systems are indeed negligible we have studied both " dry " and " wet " oxidations of carbon monoxide using exceedingly finethermo-couples as probes.In all but one case,2 oscillations observed by previous workers have occurred in dry mixtures and we too have observed many oscillations in such systems. However when we deliberately add small amounts of hydrogen to the initially '' dry " system we find that oscillations are still readily obtained over a range of temperatures and pressures. The number of cycles observed in a closed vessel is always far less than for the "dry " case but the reactant consumption in each cycle is much greater. In a mixture containing 0.1 mol % of hydrogen ten or more oscillations can be observed and this mixture is very " wet " indeed when compared with the low hydrogen content of mixtures used in the investigation of " dry " oxidation phenomena. The most striking feature of the oscillations is the size of the temperature pulse accompanying 0 10 20 30 40 50 time/s FIG.1 .-Temperature pulses in a " wet " CO+02 mixture at 480°C and 30 tom each cycle.Temperature peaks exceed 30 C for the first few cycles ; they reduce to 10°Cas reactant consumption nears completion at the end of the sequence. In " dry " mixtures amplitudes are smaller and the train of oscillations in a closed system is longer. Thus although thermal effects may be small or even negligible in " dry " systems it is probably necessary to allow for self-heating in "wet " systems and the isothermal model may be inappropriate. Moreover even the small temperature pulses may be more that the casual effects of reaction pulses-they may interact integrally with the kinetics.C. H. Yang and A. L. Berlad J.C.S. Farnday I 1974,70 1661. J. W. Linnett B. G. Reuben and T. F. Wheatley Comb. Flame 1968 12 325 These oscil-lations were observed during first or "glow " limit determination by the heating method. GENERAL DISCUSSION Prof. C. H. Yang (Stonybrook N. Y.)(communicated) Our calculated trajectories by oscillating 0 atom concentration also compared very well with the recent measure- ments of the successive emission peaks from the CO and O2 system by McCaffrey and Berlad. Their results will be published shortly. The destruction of hydrogen-containing intermediates on the wall will probably produce relatively more stable molecular species H2 and H20. At the present time the detailed mechanisms of these heterogeneous reactions are far from elucidated.We simply assumed that only H,O is produced from these reactions to avoid the complication of introducing many additional reactions into our calculation. Eqn (27) represents a conservation statement of the total oxygen in the vessel. In the early phase of the oxidation process the consumption of water is usually limited to less than a few percent of its initial concentration. Calculated limits of explosion glow and oscillations are not likely affected even if the products of the heterogeneous reactions contain a small fraction of H2. For the calculation of a long sustained oscillation or glow the overall oxidation rate will probably accelerate if H is slowly accumulated to reach a significant level as the rate of the reaction 0+H2 -+ OH +H is undoubtedly faster than the rate of reaction (7).The general kinetic behaviour however will remain unchanged. Dr. A. Perche (Lilfe)said Yang's paper concerning simulation of periodic carbon monoxide explosions would certainly have been much more informative if a direct @ 1 4 I i 4 6 t/min FIG.1.-Pressure variation (Ap) and luminous intensity (I) versus time. (a) Usual evohtiori- reaction vessel tap opened. (6) Tap closed. (c) Tap initially opened closed after the first ex-plosion. (d) Tap initially closed opened after 15 min. R. McCaffrcy personal communication. GENERAL DISCUSSION comparison with experiments had been performed. Thus in the case of high temper- ature oxidation of methane and carbon monoxide mixtures a similar periodicity was observed by me in Lucquin’s laboratory.As is shown in fig. 1 this oscillatory phenomena in our case mainly depends on the diffusion of initial reactants from the external dead volume into the reaction vessel. Another problem is that water for- mation from heterogeneous radical destruction (reactions 8,lO and 14) each producing +H20,does not seem very clear. Prof. P. Gray (Leeds) said Perche makes a valuable experimental point about the carbon monoxide oxidation “ lighthouse ” and speaking as someone with both theoretical and experimental interests in the system I have not the slightest doubt in asserting that however hard the computer calculations may be they are far less difficult in this system than the experiments.The open tap is clearly important in Perche’s study and the repetitive entry of fresh reactant through it may indeed con- tribute to repetitive reactions in his particular system. An extreme case is furnished by old Russian work.2 However in Ashmore’s (1 939) study there was no dead space at all and oscillatioiis were still foundn3 They persisted apparently unchanged when the vessel was reconnected to a ‘‘ dead space ”. Prof. R. M. Noyes (Oregon) (cornrntmicated) After further consideration I have concluded the argument of Section 3 of the paper by B. F. Gray and Aarons does nof invalidate the Brusselator as a model for many successive oscillations in a closed system. Such a model would require that only a small fraction OftheAandBreactants be consumed during a cycle in the concentrations of the intermediates X and Y.A necessary but not sufficient condition for applicability to closed systems is that k2B > k $ k,. (1) An additional restriction imposed by stability analysis of the steady state is that Although I have not yet proved it I am convinced the requirement of minimal fractional depletion of A and B during a cycle imposes a lower limit on k3A2just as eqn (2) above imposes an upper limit. I also believe that for sufficiently large values of k4/kl all the ilecessary restrictions can be satisfied simultaneously. It therefore appears the Brusselator is an acceptable model for oscillations in closed as well as in open systems although such a model requires rate constant ratios somewhat different from those usually used by the Brussels school.If rate constants are properly selected the Oregonator is also a satisfactory model for an oscillator that only very slowly depletes the major reactants. These ideas have been developed further in a manuscript submitted to J. Chenz. Phys. Dr. P. Hanusse (CNRS Talence) said In theoretical chemical models postulating that pool chemicals are held constants is generally a mere definition of pool chemicals rather than a conjecture. On the other hand it is perfectly justified to check the physical implications of such a definition as it is for instance to wonder about the existence of elementary autocatalytic steps. Now B. F. Gray shows that several well-known models may be structurally A.Perche Thtse de 3tme cyck (Lille. 1970). Tokarev and Nekrasov Russ. J. Phys. Chem. 1936 8 504. P. G. Ashmore Nnrure 1951 167 390. I<. J. Field and R. M. Noyes J. Chi. Phys. 1974 60 1877. GENERAL DISCUSSION unstable with respect to the introduction of some new steps. We think that this result depends on the perturbation introduced or the way it is desned. For instance one may replace a step of form A+X + . . . A constant by the two following steps A. + A diffusion at system limits A. constant A+X -+ . . . and A variable but other physically meaningful ways are possible for instance in a steady flow reactor this becomes A. + A influx constant rate A +. .. outflux A+X 4.. . We have already made experiments which illustrate the control of pool species as well as the effect of a small parameter namely temperature.In a steady flow reactor where influxes are controlled we have studied an oscillating reaction derived from Bray’s reaction as proposed by T. S. Briggs and W. C. Rauscher. The evolution of the system is continuously recorded by electrochemical potential spectrophotometric absorption and temperature measurements. Many interesting phenomena occur in this system oscillations are perfectly sustained very stable in magnitude and frequency-better than 0.5 %. Studies may be achieved in iso- thermal or adiabatic conditions. In both conditions temperature oscillation is observed. In some conditions a double frequency oscillation may be observed as those shown by Sarensen ;the high frequency part is due to purely chemical oscillation and the low frequency oscillation seems to be very drastically dependent on temper- ature.Several stationary states have been found and transitions between them have been studied. Hysteresis phenomena occur also when varying the constraints on the system namely mass flows. We think that the investigation of such completely sustained systems is the only way to avoid artefacts and they may lead to interesting new informations on oscillating reactions since the theoretical prerequisites are fulfilled. Dr. B. F. Gray (Leeds) said As Hanusse remarks our results depend on the perturbation of the system which we consider but our main point is that physically realistic improvements in the description of the system (such as trying to say what happens to the “ pool ” chemicals whose concentrations are not quite constant) can alter the behaviour of the intermediates is.remove or produce oscillation. We agree with the comment about completely sustained systems and are ourselves opera- ting such a reactor (see comment on paper by Gray Griffiths and Moule this Symposium). In response to Noyes our section 3 does not invalidate the Brusselator as a model for many successive oscillations in a closed system in general we simply said that it does not oscillate in the region of parameter space chosen by Lefever and Nicolis for their computations. Your condition (1) is stated in the text in Section 3 and its compatibility with the condition for an unstable singularity (your condition 2) has been discussed and shown to be unlikely.Briefly besides (1) one also needs P. Hanusse Compt. Rend. 1975. B. F.Gray Kinetics of Oscillntory Kmrrioiis (Specialist Periodical Reports 30 The Chemical Society London 1974). GENERAL DISCUSSION k3 >> k2. To be a little more precise if we take X = A. = 1 = Yo = Bo for example then the “ low consumption per cycle ” requirements become k3 k,/c k4 N kilt. Substitution of these into the instability conditions gives k < 0 i.e. a contradiction due to the leeway allowed in the -signs. In view of the limited physical interest of the Brusselator we decided not to pursue the matter numerically but the main con- clusion from our paper on this general aspect is that in future the onus is on any proposers of new oscillatory shemes to show that it is stable with respect to small parameter stability of this type.Dr. R. Lefever (Brussels)said 1. The question whether trimolecular steps as in the Brusselator are “physically unrealistic’’ should not be decided in a general a priori way. From the paper of Matsuzaki ef al. at this meeting it can be seen that sometimes they may appear as a good approximation to the behaviour of realistic systems. Often such a step can be regarded as the overall step of several realistic partial steps e.g. (i) of the isomerization process 2x -+ z Y+Z+ 3x (ii) of the enzymatic chain E+X + (EIX) (E,X)+X + (EiXX) (E,XX)+Y -+ (EIXXY) (EIXXY) -+ E” +3X. It can also easily be seen that for some values of the parameters the allosteric model of the glycolytic oscillator which Goldbeter Boiteux and Hess have men- tioned at this meeting would present a third order molecularity or even higher if more than two subunits are considered for the enzyme.2. The criticisms with respect to the effect of small parameters look exaggerated. (a) The Volterra-Lotka model is structurally unstable in any event. Thus we know that the smallest modifications will alter significantly even the qualitative behaviour sometimes in completely opposite ways. For example if instead of considering diffusion of the initial products one considers diffusion of the intermediates and in- vestigates the equations a2x ax- X-XY+& - at dr2 i?Y a2y- -= XY-Y+& at i?r2 then however small E no sustained oscillatory behaviour is possible.If on the other hand one considers the following slight change in the equation then their solution turns out to be a spiral which blows up in the first quadrant. I A. Roiteux A. Goldbeter aid B. Hess Proc. Not. Acnrl. Sci. 1975. GENERAL DISCUSSION (b) For the trimolecular model it is obvious that keeping A B constant is equivalent to infinitely large flows of A and B or perfect instantaneous stirring inside the system. The main test for the consistency of a model is to have it working in a certain limiting case. This is certainly so for the Brusselator. After all in physics the model of a perfect pendulum without frictioii is generally unrealistic but quite useful.Furthermore our own work in Brussels (see for example the recent papers by Nicolis and Auchmuty,l and Kaufman-Herschkowitz 2 has already shown that by modifying the original Brusselator one can obtain new types of behaviour such as localized structures. Dr. B. F. Gray (Leeds) (communicated) I. I agree with Lefever that a third order reaction may be a representation of a more complex system involving perhaps only first and second order steps but in making such an approxiination one is introducing yet another small parameter relating for example [EIXXY]to X and Y besides the original one relating X and Y to A and B. It may be possible to do this correctly but our main point is that this cannot be assumed. 2a. We do not agree that our criticism of the method of neglecting sinall parameters is exaggerated.The Lotka-Volterra model is well known to be structurally unstable (i.e. a characteristic root is zero) but we are not discussing this type of stability here we are discussing stability with respect to a small parameter and consequent increase of the degree of the characteristic equation producing new roots. If one or more of these is positive then it is possible for the system to be unstable even though the unperturbed system was structurally stable i.e. all its characteristic roots were <0. The other examples in our paper treat systems which are not structurally unstable and the structural instability of the Lotka-Volterra system is in this context incidental. 2b.The reference to a perfect pendulum without friction is completely misleading in this discussion since including friction does not increase the degree of the character- istic equation of the number of variables in the problem. This is a simple pertur- bation problem with no possibility of the type of instability we are discussing which is essentially characterised by non-uniform convergence in the neiglibourhood of E 3 0. On the contrary the convergence of a damped pendulum to an undamped one is completely uniform as the damping factor tends to zero i.e. the solution of the damped pendulum equation d2Y dY M-+E-+K~ =O dt2 dt which is y = exp( -Et/M} sin(K/M)ft is well approximated by the undamped solution yo = sin(K/M)*t provided t < M/E,and as E -+0 this interval gets larger and larger.The correct analogy within the realin of damped springs is where the mass of the spring tends to zero and the unperturbed differential equation is only first order dy/dt +K’ = 0; hence a spring with zero mass is not a sensible approximation to a spring with a small mass however small this may be. This is a singular perturbation problem as compared to Lefever’s example which is a secular perturbation problem. Dr. B. L. Clarke (AZberta)said The term “jump point ” used in Section 1 of the paper by Gray and Aarons can have two interpretations. The term is used in the G. Nicolis and A. Auchmuty Proc. Nat. Acad. Sci. 71 1974 2748. M. Kaufman-Herschkowitzand G.Nicolis J. Chem. Phys. 1972 56 5. S 9-6 162 GENERAL DISCUSSION paper to refer to the values of y for which the curve F(x ;y) = 0 has tangents which are perpendicular to the y axis.If eqn (2) moves x to a stable pseudosteady state on this curve x will essentially be determined by y. As y passes through the “jump point ” 5 becomes infinite briefly. However the trajectory (x ;y) through this point may still be a smooth curve. The sudden changes in the dynamical variables shown in the papers by Field and Noyes and by Franck are usually caused by a situation which is the second inter- pretation of &‘jump point ”. Since the stability of the steady state of eqn (2) depends on y the curve F(x;y) = 0 may have regions of stability and instability for pertur- bations Ax ofthe curve. The trajectory (x;y)will follow this curve in the regions of stability and jump off the curve at the points which separate the stable and unstable regions of the curve.The rapid evolution of x at essentially constant y which follows can move x into a limit cycle around the curve. If the solutions of F(x;y) = 0 arc multivalued for a given y the evolution may go to a second branch of the curve which is stable. The “jumping off” points of the curve are found by examining the Hur- witz determinants of the matrix (8F,/8xi). If the general equations of motion are used instead of eqn (I) the variables can often be treated as slowly evolving variables like y over part of their range and rapidly evolving variables like x over much of the remainder of their range. The variable q of the Oregonator behaves in this fashion.The argument of the preceding para- graph can still be used; however the stability of points on the possible curves must be determined for fluctuations in all the variables. Dr. A. Babloyantz (Brussels) said In connection with the paper of Gray and Aarons I would like to mention that structural stability formalism has already been applied to biological problems in the context of prebiotic evolution of informational macromolecules.1 It can be shown that a favourable mutation giving rise to a new macromolecule although in a small quantity may take over and destabilize the origin- ally stable system. Dr. B. L. Clarke (AZberta) said Regarding the paper by Gray and Aarons the model reaction systems which become unstable when the pool chemicals are allowed to fluctuate suggest a question.“ Which network features are necessary and sufficient to guarantee structural stability with respect to the inclusion of the pool chemicals? ” Several theorems given in a paper on the stability of topologically similar chemical networks give conditions which ensure that including the pool chemicals in the dynamics will not alter after the stability. If a pool chemical enters the network through a decomposition reaction of the form A + . . . permitting A to fluctuate is equivalent to adding a reactant X to the network as follows A 3X -+ . . .. I call the reactant X a “ type B flow through reactant ” (FTRB) and from Theorem 5 of my paper it follows that for large steady state concentrations of X the network is stable if and only if the network with X eliminated is stable.The only other way a pool chemical may enter a network is by reacting directly with other reactants as in A+Y + . . .. If A is allowed to fluctuate such a reaction is replaced by the following pair of reactions A -+ X and X + Y -+ . . .. I call the reactant X a ‘‘ type A flow through reactant ” (FTRA) and from Theorem 8 of my paper the following can be said if the original network has deficiency zero and if the reactant Y does not appear by itself on the left or right hand side of any reaction * I. Prigogine G. Nicolis and A. Babloyantz Physics Tuday November and December 1972. B. L. Clarke J. Chem. Phys. 1975 62 3726. GENERAL DISCUSSION of the extended network the extended network also has deficiency zero.This theorem is an extension of work by Horn Feinberg and Jackson and the significance of the deficiency is explained in the immediately following comment by Feinberg. Roughly speaking if the original network has deficiency zero it is stable and because the freeing of the pool species does not change the deficiency the extended network is also stable. From the Hurwitz determinants one can understand in general why pool chemicals which enter the network via reactions of the form A +Y -+ . . . sometimes destabilize the network when they are allowed to fluctuate. The extended network has a new positive feedback loop in which A and Y inhibit one another. Such loops will de- stabilize a network if they pass through reactants which have marginally strong autocatalysis such as Z+Y + 2Y and in addition certain other conditions are met.In all the examples of Gray and Aarons' paper the pool chemicals enter the network in this manner. Dr. M. Feinberg (Roclzester) said In the discussion regarding B. F. Gray's paper Clarke made reference to some theorems proved Horn Jackson and Feinberg which might bear upon the problem of how stability might be affected if a physico-chemical model is structurally perturbed. Horn and I have published a description of the most intriguing of these theorems.' Perhaps I should explain this theorem and some newer results (in terms more schematic than precise) and offer my views regarding their utility in answering questions of the type raised by Gray.According to a rather simple formalism described in the aforementioned article there can be assigned a non-negative integer (called the deficiency 6) to every chemical mechanism according to its algebraic structure. (Some of the reactions of a mech- anism might be " pseudo-reactions " incorporated to reflect special physico-chemical effects for a system under study). Thus mechanisms can be classified according to whether they are of deficiency zero one two etc. Mechanisms of deficiency zero are surprisingly common and it is for these that we have been able to prove what I think is an interesting theorem for open hoino- geneous reactors. For arbitrary positive rate constants (in the context of the usual mass-action kinetics) the existence of an equilibrium for which all species con- centrations are positive suffices for preclusion of pathological statics and dynamics e.g.sustained composition cycles. Moreover there exists such an equilibrium if aid only if the reaction " arrows " in the mechanism are directed such that the mechanism is what we call "weakly reversible ". The weak reversibility constraint is somewhat limiting since most models people have been considering these clays do not fall into this category. However since publication of the article cited above I have been able to prove the following. For mcchanisms of deficiency zero which are izot weakly reversible it is true that for arbitrary kiiretics (subjcct only to very weak constraints) the dynamical equations for open homogeneous systenis cannot give rise to sustained temporal composition cycles for which at some time all species concentrations are positive.(In fact this can be shown to hold for a class of inechanisms somewhat broader than those of deficiency zero.) Moreover for mechanisins of deficiency zero governed by mass-action kinetics with arbitrary positive rate constants sustained temporal composition cycles cannot be obtained at a11 whether or not the meclzarzism is weakly reversible. That is all mechanisms of deficiency zero taken with mass action kinetics are loosely speaking of stable character. ' M. Feinberg and F. Horn C/iet)t.Etiy. Sci. 1974 29 775. GENERAL DISCUSSION It can be shown that if one begins with a mechanism of deficiency 6 the addition of reactions to that mechanism results in a new mechanism of deficiency not less than 6.Similarly if one begins with a mechanism of deficiency S and one removes reactions the resulting mechanism has deficiency not greater than 6. In each case the deficiency may be unchanged. These ideas might in some instances help in deciding the effect of structural perturbations in a model (resulting in addition or removal or reactions) upon stability characteristics. If one begins with a mechanism of deficiency zero and one adds to it new reactions the resulting mechanism may also be of deficiency zero so that it falls within the realm of the theory. Thus the "perturbed " model will be of stable character. On the other hand if one begins with a mechanism of deficiency in excess of zero (perhaps exhibiting static and dynamic "exotica ") and one structurally perturbs the model by removing reactions then the resulting mechanism may have deficiency zero and the modified model will have essentially stable character.I might add that mechanism deficiency has a surprising bearing upon matters quite divorced from stability considerations (e.g. upon the determinability of com-plete sets of rate constants from certain classes of dynamic experiments). I have discussed these matters in a chapter of the forthcoming Wilhelm Memorial Volume on Chemical Reactor Theory several other chapters of which (particularly those by Luss and Bailey) might hold interest for participants in this Symposium. Dr.H. Tributsch (Berlin) said There have been many theoretical investigations on the Lotka Volterra model but there has been up to now a lack of experimental systems which permit a verification of conclusions derived from it. For this reason I would like to present a new and simple oscillating system which in my opinion corresponds to the first example which has been discussed in the paper by B F. Gray namely to that of a Lotka Volterra oscillator which has to be considered as being perturbed by small parasitic parameters. The system consists of a crystal of CuzS or Cu,FeS jn contact with an electrolyte that contains hydrogen peroxide. Oscil-lations appear during the reduction of H202at the sulphide electrode above a certain -0.9 -0.e -0.7 -0,s -0.5 -0.L -O,3 -0.2 -0.1 electrode potential USCE/V FIG.1.-Dynamical reduction curves for HzOz on CuSFeS4 electrode.GENERAL DlSCUSSlON 165 minimum potential. Fig. 1 shows two reduction curves which have been recorded at 2 mV/s in opposite potential-directions. For this system we could not only derive a reasonable kinetic scheme of the Lotka Volterra type but we were also able to find clear experimental evidence for the behaviour which is considered to be characteristic of this kind of oscillator especially an unstable frequency which is dependent on the initial conditions of the system as well as a complementary amplitude-frequency correlation which is a result of its conservative character and of a constant entropy production. (Consider the amplitude decrease which coincides with an accidental increase in the oscillation frequency in the central portion of one of the reduction curves (fig.1)). A closer investigation of the dynamics and shape of oscillations has shown that they are usually very periodic and well formed in the central region of the reduction curve but that they are often composed of compact spike groups of fast rising amplitude in the border regions where now and then oscillations also fail to occur. The available data are all consistent with the concept that the investigated system is basically of the Lotka Volterra type. Its instability is however so pronounced that it will-depending on the predominant chemical perturbation reactions-either slip into a limit cycle type of oscillation (case discussed in Gray’s paper) or result in a gradually rising or damped oscillation (cases discussed by Frank-Kamenetskii and Sal’nikov ’).I D. A Frank-Kamenetskii and I. E. Sal’nikov Zhur. Fiz. Khim. 1943 17 79.