DISCUSSION REMARKS Prof. J. Ross (MIT)said The subject of chemical instabilities was first studied by means of macroscopic equations then statistical theory and is now being approached by means of the method of molecular dynamics. Some interesting results have been obtained by this method by Ortoleva and Yip at M.I.T. So far they have made calculations on a 32-particle system in which the reactions X+Y -+ 2X T(X) X+Y,V Y-,X,P occur with the listed rate coefficients. Autocatalytic character is achieved in the first reaction by taking the transition probability for reactive collisions to be dependent on the number of X particles within a mean field radius or just proportional to the concentration of X. Fig. 1 shows a comparison of the prediction of the calculation (points) with that of the macroscopic equation (solid line) for the steady state mole fraction of X(F,) as a function of v for a value of p = pc such that the analogue of the critical isotherm is obtained (for further discussion on this point see Fluctuations and Transitions at Chemical Instabilities The Analogy to Phase Transitions by A.Nitzan P. Ortoleva J. Deutch and J. Ross J. Chem. Phys. 1974 61 1056). Fig. 2 shows a calculation of the mean square fluctuation in the number of X particles A2 = N-l (N -(Nx)2)as a function of the rate coefficient v. The calculation clearly shows an increase in the magnitude of fluctuations at the critical value of v = v ; this analogue I I I I I I 1 0 0.1 0.2 0.3 0.1 0.5 0.6 0.7 V FIG.1. of critical opalescence was predicted in the quoted reference.Finally in fig. 3 the calculations now made for parameters such that multiple stable steady states are expected show both fluctuations around one stable steady state and transitions from one such state to another. 66 GENERAL DISCUSSION 67 1 a o &a O' am aa a oaaa a a a ma oaa a * am a a* aaa maa ma oa m m m a Fl a m ma am 0 ma .. -* I I I I 0.2 0.4 0.6 V FIG.3. time FIG.2. 11 f f % ..I'0 \ \ 4 2 -2A= F,*-fi 1 \ a 2 01 I I /L / / \ \ \O \ \ \'-- GENERAL DISCUSSION Prof. G. Nicolis and Prof. I. Prigogine (Brussels) (comnzunicated) The interest of undertaking a molecular dynamics study of chemical instabilities became clear after the phase space theory of fluctuations was developed by the authors.' We believe that in this reference as well as in another the abnormal increase of fluctuations at the critical point of instability was predicted for the first time.A molecular dynamics study of fluctuations in nonlinear chemical systems including stable ones has been developed during the last few years by Portnow. Some of his results are reported in a remark following our paper. Further results referring to the trimolecular model the Volterra-Lotka model and an enzymatic model involving hysteresis have been obtained by Portnow Turner and Van Nypelseer. We are glad to see that the in- teresting work of Ortoleva and Yip leads to an independent confirmation of our predictions.Prof. J. Ross (MIT) said; In regard to the issue of the relation of the critical wavelength of an emerging dissipative structure to the size of the system it may be useful to point out a qualitative distinction between two types of symmetry breaking instabilities.3 " As the system subject to instability is driven out of equilibrium k k FIG.1.-Typical behaviour of the real part of the stability eigenvalues Rez as a function of the wave vector k. Case A is that of extrinsic length scaling where increasing the light intensity I beyond a critical value Ic produces an interval 0 < k < km where unstable modes lie. From J. Chem. Pliys. 1974 60 3134. symmetry breaking first occurs on a given length scale. If that length is fixed by the dimensions of a given system of arbitrary length then the symmetry breaking is classified as extrinsic length scaling.In contrast however there are systems for which the pattern length at which symmetry breaking sets in is essentially independent of the dimensions above a minimal length. In such cases the intrinsic length scaling is embedded in the transport and reaction dynamics. From an operational point of view the size of the system chosen to test the character of the length scaling must be greater than the intrinsic length. As the intrinsic length approaches infinity the distinction between the two types of length scaling vanishes. " The different character of extrinsic and intrinsic length scaling is shown in fig. 1 for an example in which G.NicoIis and I. Prigogine Proc. Nat. Acad. Sci. 1971 68 2102. R. Mazo J. Chem. Plzys. 1970 52 3306. A. Nitzan P. Ortoleva and J. Ross J. Chm.Plzys. 1974 60 3134. GENERAL DISCUSSION instability is brought about by increasing light intensity. In the quoted article ex- amples of both types of behaviour are given. Prof. G. Nicolis and Prof. I. Prigogine (Brussels)(communicated) We believe that a qualitative distinction between " extrinsic " and '' intrinsic " length scales may be misleading. In the terminology of our paper the question at issue is whether at the first bifurcation predicted by the characteristic eqn (3.2) the wave number of the bifurcating solution is zero or finite (see comments following eqn (3.2) of our paper). Obviously any finite wavenumber k satisfying the characteristic equation will depend solely on the intrinsic parameters of the system as the size of the latter does not appear in this equation.Nevertheless if one requires that the unstable modes be I I I FIG.1 .-Marginal stability curve as a function of the wave number. km a wave number k predicted by linear stability analysis. ki values of k compatible with the boundary conditions. kc = kz critical wave number corresponding to the first bifurcating solution. compatible with the boundary conditions one will find a set of values { ki}of k which are directly related to the size of the system (see eqn (3.3~)to (3.3~)of our paper). Thus near the critical point the bifurcating solution will actually be dominated by a value k of k which agrees with the boundary conditions (and thus is related to the size of the system) and which is the closest one to the value k predicted by linear stability analysis.Computer simulations confirm this point entirely. The situation is represented schematically on fig. 1. For a more detailed description we refer to Lefever,' and to Nicolis and Auchmuty.2 In that sense therefore the length scale associated with symmetry breaking is always extrinsic as long as the dissipative structure can extend throughout the system. The only intrinsic length scaling that appears to be possible is related to the possi- bility of localization of the dissipative structure as discussed in section 3(ii) of our paper. Dr. P. OrtoIeva and Prof. J.Ross (MIT)said The physical concept of propagation velocity in finite systems is familiar in atomic and nuclear scattering shock waves and acoustic wave packet propagation and waves in the Zaikin-Zhabotinsky reagent. As long as the characteristic structural dimensions of the propagating phenomena R. Lefever Thesis (University of Brussels 1970). (3 Nicolis and J. F. G.Auchmuty Proc. Naf.Acad. Sci. 1974 71 2748. GENERAL DISCUSSION (i.e. the width of a front; (see fig. 4 in the article by Nitzan Ortoleva and Ross this Symposium) is much less than the dimensions of the system then on the basis of causality one expects that boundaries of the system are not important in the concept of a velocity or of propagation (until the disturbance reaches a wall or another disturbance emitted from the wall).Autonomous one-dimensional centres of wave emanation as well as circular and standing waves in two dimensions have been considered by us1 These phenomena are analysed with the aid of bifurcation theory and an alternative scheme " phase diffusion theory " based on the existence of a chemical oscillation in the rate mech- anism. Prof. G. Nicolis and Prof. I. Prigogine (Brussels) (communicated) In our analysis we have been concerned with the asymptotic behaviour in time of reaction-diffusion systems. We have shown (sec. 4 of our paper) that the solutions describing this situation correspond to periodic (or almost-periodic) oscillations which may be space- dependent. These oscillations do not have a well-defined velocity of propagation.Moreover they are often stable and thus independent of the initial conditions. Finally their characteristics depend very strongly on diffusion which triggers the instability leading to these patterns and gives rise to a coherent state compatible with the boundary conditions. In addition to these long-time solutions there may be transient solutions describing the initial stages of propagation of a disturbance in the medium. Such solutions may be Characterized by a velocity of propagation. Moreover they depend on the way the system was excited initially. The comment by Ross as well as the papers by Ortoleva and Ross refer to this latter type of solution or alternatively to an unbounded medium. To our knowledge although bifurcation theory is invoked in these papers the authors have not established the existence or the stability of the various types of wave forms they list.Now the requirement of stability is always a very stringent one a solution whose existence is suggested by linear analysis can be rejected by the system on the basis of stability considerations. Thus it is often necessary to ensure that a certain solution appears at the point of thefirst bifurcation from a reference state in order to be able to guarantee its stability. This question is discussed in some detail in our paper as well as in the paper by Balslev and Degn at this Symposium. Dr. M. Kaufman-Herschkowitz (Brussels) said I would like to add some further remarks concerning the comparison between analytical calculations and computer simulations for the trimolecular reaction scheme.From the linear stability analysis performed for a bounded medium subjected to zero flux or fixed boundary conditions one can infer the evolution of the homogeneous steady state beyond a critical point to new space dependent solutions characterized by a finite wavelength. Mathematically these transitions can be understood as a phenomenon of branching of the solutions of the non linear partial differential equations describing the system. Bifurcation theory enables one to construct analytically the form of the bifurcating solutions. However the analytical expressions one can construct are limited to the neighbour- hood of the marginal stability point. Computer simulations allow one to verify their predictions but also to investigate the behaviour of the system as one enters P.Ortoleva and J. Ross J. Chem. Phys. 1974 60,5090. P. Ortoleva and J. Ross J. Chem. Phys. 1973 58 5673. G. Nicolis and J. F. G. Auchmuty Proc. Nat. Acad. SOC.,1974 71 2748. GENERAL DISCUSSION more deeply into the unstable region. They show that new features can arise in these conditions and I would like to discuss one of these features as an example. Let us consider the case of fixed boundary conditions and use B as bifurcating parameters. Comparison between the analytical calculations (expression (3.4) in the paper by Nicolis and Prigogine) and computer simulations shows (i) that near the critical point (B 2 B,) the two approaches agree very well (fig.1). The new steady state reflects to a good approximation the form of the critical mode. One observes in addition a certain distortion due to the contributions of the non- linear terms which act to enhance the successive maxima (successive minima) from boundary at r = 0 to boundary at Y = L. 1.51 J 0.5 I space (arbitrary units) FIG.1.-Steady state dissipative structure for fixed boundary conditions and B N B,. Dashed line analytical curve ; full line result of the numerical integration on a digital computer. The following numerical values of the parameters have been chosen :A = 2 L = 1 DX = 1.6 x Dy = 6.0 x B = 4.17. The critical wavenumber is n = 8 and Bc = 4.133. The boundary values for X and Yare X = A = 2 Y = BIA = 2.085.(ii) that the agreement between analytical and numerical results becomes poor when the calculations are performed for values of B which do not belong to the direct neighbourhood of the critical point (fig. 2). 0.5 space (arbitrary units) FIG.2.-Steady state dissipative structure for fixed boundary conditions and B =-Bc. Dashed lines analytical curve ;full line :result of the numerical integration on a digital computer. B = 4.5 ;other parameters are as in fig. 1. GENERAL DISCUSSION Moreover one observes from the computer simulations that although the critical mode determines the number of extrema of the final steady state structure the spatial asymmetry is now quite different as for B 21 B near the boundaries the maximum is enhanced and minimum diminished.A great number of calculations performed for various initial conditions show that this new type of asymmetry does not depend on the initial perturbation. These observations suggest the possibility of occurrence beyond a certain distance from the critical point of a secondary bifurcation which could be responsible for the observed change in behaviour of the solution the first bifurcating space dependent solution becomes unstable and one observes the appearance of another space dependent dissipative structure. On the other hand it must be pointed out that for the values of parameters we considered the spacing between the values of B corresponding to the first unstable modes emerging successively from the thermodynamic branch is small.Thus there could exist a relation between the appearance of the new unstable modes and the observed change in behaviour of the first bifurcating solution. The analytical study of secondary bifurcations is presently in progress. The possibility of secondary bifurcations may be of great importance for the under- standing of wave-like solutions which could appear as a result of an instability of a pre-existing dissipative structure even when they are not likely to appear from an instability of the homogeneous steady state. Dr. J. Portnow (Austin Texas) said I would like to report the first results of a comprehensive molecular dynamics study of fluctuations in non-linear chemical systems under far-from-equilibrium conditions. The study is divided in two parts.The first which is described here was aimed at determining the behaviour of fluc- tuations under far-from-equilibrium conditions but far from instability points. The second part of the study which is still in progress is aimed at determining the growth of fluctuations at instability points the mechanism of instability and the critical size of fluctuations necessary to initiate instability. Fluctuations in equilibrium systems linear and nonlinear alike are described by the Einstein relation. Thus for a single intermediate small fluctuations in the par-ticle number density follow a Poisson distribution. For linear systems far from equilibrium the Poisson relation is still maintained. But for nonlinear systems recent theoretical analysis suggests that the Poisson relation is restricted to fluctuations below a certain critical size that above that critical size the Poisson description is no longer valid.The computer experiments reported here were carried out in order to determine " experimentally " the particle number distribution actually obtained in a reaction. In a machine calculation like the one reported here the observed fluctuations are generally small in extent and therefore the calculated distributions are dominated by small fluctuations. The following reaction was investigated* P+Y + Y+X ZX -+ E+D. for this reaction with the assumption of homogeneous fluctuations extending through- out the system a usual birth and death analysis predicts a variance = $ mean a * Several additional reactions were investigated as well and will be reported in a forthcoming paper.See also a report in Php. Letters in press. GENERAL DISCUSSION significant deviation from Poisson behaviour. On the other hand a local phase space description of the reaction (in which the local character of collisions is made explicit) predicts Poisson behaviour for small fluctuations. Initially 75 hard spheres with Ar mass and radius are randomly placed in a cubic box with p = 0.001 783 7 ~m-~. They are given identical speeds p = [3kT/mJf with T = 273 K and random velocity directions. They interact with each other through hard sphere collisions and undergo specular reflections off the walls of the box. Once thermal equilibrium as measured by the H function as attained 25 spheres are given P identity 25 X identity and 25 Y identity.Then whenever two particles collide their identities are changed in accordance with the reaction mech- anism. There are neither activation energy nor configuration requirements for the reaction so that all collisions between reacting particles are reactive and all rate constants are equal. By neglecting all reactive criteria and since the reacting particles have no internal degrees of freedom the Maxwell-Boltzmann velocity distributions are not upset by the reactive collisions. The number of P particles is kept constant at 25 and each time E and D particles are produced they are removed immediately from the reaction box. Whenever a P particle reacts a new one is introduced into the box at a random position and given a speed u = (3kTim>f with random velocity directions.* The reaction was followed for 3.146 x s during which time there were 1 941 reactive and 4 067 unreactive collisions.After 30 observations the mean and variance were calculated mean expected mean 25.2 24.5 calculated variance Poisson variance 23.3 24.5 and in terms of the psquare test for goodness of fit the observed distribution showed no significant deviation from a Poisson distribution. These results are strong evidence that a Poisson distribution is the correct des. cription of the fluctuation behaviour in the present reaction at least. Since the homogeneous birth and death analyses have predicted non-Poisson behaviour the " experiments " dramatize the need for a local description of fluctuations in non linear far from equilibrium systems.Dr. M.-L. Smoes (Dortmund) (communicated) 1. The claim is again made in the paper of Nicolis and Prigogine that steady spatial patterns have been observed in initially homogeneous chemical reactions. Although the authors do not mention any particular reaction I presume that they might have in mind the case of the Zhabo- tinskii reaction. In this oscillatory system nonpropagating waves have been reported by Busse and a spatial dissipative structure is described by Kaufman-Her~chkowitz.~ My own experience with the Zhabotinskii system leads me to think that such steady spatial structures have not really been found. The two observations reported are not supported by quantitative measurements of the position of the wavefronts as * Boundary conditions requiring diffusion in from the walls have been used as well.G. Nicolis and A. Babloyantz J. Chem. Phys. 1969 51 2632. G. Nicolis and I. Prigogine Proc. Nac. Acad. Sci. 1971 68 2102; G. Nicolis M. Malek- Mansour K. Kitahara and A. Van Nypelseer Phys. Letters 1974 48A,217. G. Nicolis J. Stat. Phys. 1972 6 195 ;G. Nicolis P. Allen and A. Van Nypelseer personal communication. H. Busse J. Phys. Chem. 1969 73 750. M. Kaufman-Herschkowitz Compt. Rend. 1970,27OC 1049. GENERAL DISCUSSION a function of time. Is it not possible that those apparently immobile bands of oxidation are in fact slowly moving‘? Indeed the speed of propagaion of the Zhabotinskii waves is known to depend on the period of bulk osci1lations.l One can thus expect quasisteady structures in the Zhabotinskii system for an appropriate choice of the initial concentrations of the reactants.I would like to suggest that steady structures in chemical systems have not yet been demonstrated and that further claims should be based on careful quantitative measure- ments. 2. The mathematical models for oscillatory chemical reactions may lead to high expectations in terms of biological problems. However the experimental facts do not seem to confirm these hopes. The chemical reactions by themselves do not have all the controlling properties postulated and required for biological applications. This is at least what must be concluded in the case of the spatio-temporal phenomena observed in the Zhabotinskii system.The leading centres their period and the speed of propagation of the wavefronts are all independent of the size of the system but depend on the initial concentrations of the reactants and on temperature. The spatial and temporal distribution of the centres appears to be random. These experimental observations have been taken into account in our model of phase waves.2 The model applies not only to the Zhabotinskii system but also to a large class of chemical oscillatory reactions in which the period depends on con- centrations and/or tempreature. Although no structures have yet been observed in glycolysis the dependence of the period of glycolytic oscillation on temperature and concentrations suggests the possibility of formation of phase waves in this case too.Prof. G.Nicolis (Brussels) (communicated) 1. The claim attributed to us by Smoes is nowhere made in our paper. We are well aware of the controversial status of the horizontal bands in the Belousov-Zhabotinski reaction. The main purpose of our paper was to provide a mathematical classification of the various structures that become possible beyond an instability of the thermodynamic branch. In this respect the existence of stable steady state solutions of reaction-diffusion equations is esta- blished rigorously for the first time. It is not unreasonable to expect that such solutions will describe the behaviour of real chemical systems under certain conditions.2. It is important to distinguish between long-time behaviour and transient behaviour. In our analysis we have been concerned with the asymptotic behaviour in time of reaction-diffusion systems. The qualitative properties of dissipative structures like the dependence on size the independence of the initial conditions etc.. . hold only for the long time solutions. Smoes’ remark refers on the other hand to the transient behaviour of an initial disturbance acting locally on a chemical system. Obviously this behaviour will depend strongly on the initial conditions. Moreover it will not be related to the size of the system. Eventually however the disturbance will reach the boundaries and it will evolve to one of the long-time solutions treated in our paper.Thus there is no contradiction between Smoes’ remark and our results. Dr. M-L. Smoes (Dortmuizd) (comnzunicated) :(1) I refer to the claim made in the first paragraph of the Introduction. Since only theoretical steady-state patterns are well established the wording of that paragraph is ambiguous. The main problem concerning the physical realization of stable structures in chemical systems is the following. According to the theory of Nicolis and Prigogine ’ M-L. Smoes unpublished work. ’M-L. Smoes and J. Dreitlein J. Chem. Phys. 1973 59 6277. FIG.1 .-System 1 825 s after start. FIG.2.-System 2 825 s after start. FIG. 3.-After 22 min. FIG.4.-After 34 min. FIG.5.-After 52 min. F~ti. 6.-After 63 min. FIG.3-6.-Progressive development of the waves.Tofucepqqe 751 GENERAL DISCUSSION both constant parameters and appropriate nonlinearities in the rate equations are required. The two conditions are in practice mutually exclusive. Indeed the nonlinearit'ies are met only in very complicated chemical systems for which the con- stancy of the parameters in the rate equations becomes impossible. Even under mixing conditions irregularities can be observed in the successive periods of the Zhabotinskii oscillations. Sudden variations in the period which may be as large as 50 % are observed in homogeneous oscillations with long period. The irregulari- ties are not observed in systems with short periods. Such anomalies are expected as a result of the dependence of the period on concentrations and temperature in the Zhabotinskii system.'* They are also expected in the glycolytic ~ystem.~ This is the important fact which is not taken into account in the paper by Nicolis and Prigogine and which is accounted for in the work of Smoes and Dreitlei~~.~ (2) Transient behaviour is the only behaviour ever observed with the Zhabotinskii structures even when the disturbances have reached the boundaries of the system.In our opinion this is due to the instability of the period of oscillations with respect to the fluctuations in concentrations and/or temperature.' 9 Such an instability precludes the system from evolving toward one of the solutions treated by Nicolis and Prigogine. To demonstrate the point I introduce two sets of pictures. In the first set is shown the influence of the homogeneous bulk period on the general aspect of the waves.System 1 (fig. 1) and System 2 (fig. 2) have a 2 3 ratio for the sulphuric acid concentra- tion resulting in a much shorter period of bulk oscillations for System 2. The size of the Petri dishes the thickness of the layer and the time elapsed since the start of the reaction (825 s) are identical. In the second set fig. 3-6 is shown an example of the renewal of the transients in the Zhabotinskii waves. Between fig. 3 and fig. 6 the waves from one centre progressively fill the whole space available but new centres appear later and new transients develop. No long-time stable structures are observed. Our model of phase waves accounts reasonably for the absence of a long-time stable solution in this chemical system.Dr. J. S. Turner (Austin Texas)said As Nicolis and Prigogine point out a natural mechanism for testing the stability of a macroscopic state exists in every many-body system in the form of fluctuations. Near a transition point therefore a inacroscopic instability may be nucleated spontaneously by a small volume element which suddenly becomes unstable through a fluctuation. The response of the surrounding medium to such an evolving subvolume will be to damp the fluctuation by diffusion. If the unstable region is sufficiently large however diffusion may serve instead to propagate the disturbance. The fluctuation will then grow leading ultimately to a new regime of macroscopic behaviour. In order to make this notion quantitative Nicolis and coworkers have proposed a stochastic model in which the effect of diffusion on a fluctuating volume element is treated in an average way.' If X is the number of molecules in a small volume (for M-L.Smoes Toward a Mathenlatical Descriptioii of the Phase Waves submitted for publica-tion. Oscillatory Processes in Biological and Chemical Systems (Nauka Moscow 1967) p. 181 ff. B. Hess and A. Boiteux Ann. Rev. Biochem. 1971 40 237; A. Betz and B. Chance Arch. Biochem. Biophys. 1965,109,579. M-L. Smoes and J. Dreitlein J. Chem. Phys. 1973 59 6277. G. Nicolis M. Malek-Mansour K. Kitahara and A. Van Nypelseer Phys. Letters 1974 48A 217. GENERAL DISCUSSION convenience consider a single reactive degree of freedom) then the appropriate stochastic master equation has the form in number-of-particles space dP(X;t)/dt = g{X,P(X;t))+g(X)[P(X-1 ;t)-P(X;t)]+ 9[(X+l)P(X+1 ;t)-XP(X;t)].(1) Here the first term on the right-hand side denotes the reactive contribution while the second and third account for diffusion into and out of the subvolume respectively. The presence of the mean value (X) = xXP(X;t)in the second term expresses the X fact that initially the system is globally homogeneous. The diffusion parameter is defined by 9 = D/L2,with D a Fick-type diffusion coefficient and L a characteristic length over which the fluctuation maintains an approximately coherent character. [For more details see ref. (l).] For chemical systems in which a single homogenoeus steady state exists becoming unstable beyond a critical value of a system parameter (usually a measure of affinity) eqn (l) together with the definition of 9,yields a critical coherence length L giving the minimum size necessary to form an unstable nucleus in an initially homogeneous system.' If more than one homogeneous steady state is accessible to the system then several qualitatively new features are possible.Typically the macroscopic kinetic equations yield a region of a system parameter (e.g. A) in which two such states are simultaneously stable and therefore predict a hysteresis in the transition between the two stable branches of the steady state solution (e.g. arrows fig. 1). In analogy to first-order phase transitions however one may ask whether for any value A only one state is actually stable the other a metastable state being unstable with respect to finite fluctuations occurring spontaneously in the To investigate this possibility Lefever Prigogine and I have applied eqn (1) to a simple chemical model which exhibits multiple steady states.I t ble L I 8 r\ I I x X x FIG.1.-Critical coherence length Lc characterizing the stability of states on the metastable portions bl b2 of the macroscopic coexistence region. G. Nicolis M. Malek-Mansour K. Kitahara and A. Van Nypelseer Phys. Letters 1974 48A 217. J. S. Turner Phys. Letters 1973 MA 395 ;Bull. Math. Biol. 1974 36 205. J. S. Turner Adv. Chem. Phys. 1975 22 63. GENERAL DISCUSSION 77 Consider a Langmuir type of exchange process in which X atoms are adsorbed on a surface region of N binding sites.The cooperative nature of the adsorption- desorption process is expressed in the equilibrium constant for this reaction which depends explicitly on the surface density X/N of adsorbed atoms and on a parameter A. The macroscopic kinetic equations give an S-shaped steady-state curve as a function of A the middle branch of which is unstable (fig. 1). Applying now eqn (1) to a small surface element (finite N) in this system (with 9-L-I for surface diffusion) we find for 9= 0 a unique stationary distribution which is bimodal in the multiple steady state region. Hence the macroscopic predictions are recovered as far as small homogeneous fluctuations are concerned.The ratio of the peak heights is large except very near lo,the point at which the heights are equal. Away from lo,there-fore mean values (X)equal roughly Xa(L),the most probable value. For 9 # 0 the master equation becomes nonlinear due to the presence of (X) implying the possibility of more than one stationary distribution. For 9> 0 small (L large) there remains a unique steady solution but for 9 greater than a critical value g,(A) [or L,(A)],two such solutions are found depending on the choice of initial distribution. For all 92 0 an initial (X) near Xa(A) yields a final distribution with (X) -Xa(A). For 9 < gC(A), the same final distribution results from initial mean values near the smaller peak [at X,(A)]as well implying that fluctuations are too large [L > L,(A)] for that branch to be stabilized.If 9> BC(A),however the latter initial condition produces a stable stationary distribution having (X)-X,,(A). This means that initial states on the branch " b " will not be de- stabilized by fluctuations of size L < Lc(A). These results are displayed schemati- cally in fig. 1 where the stability properties of the metastable states " b " are indicated in a plot of the critical coherence length L,. If fluctuations of size L 3 ,Fax = maxL,(A) occur in the medium then these metastable states will not be rl observed any transition between branches 1 and 2 occurring at the point &,. If the largest fluctuations appearing frequently are smaller than L:"" then a corresponding portion of each metastable branch " b " will be stabilized and an apparent hysteresis in the transition point recovered.In this case the transition may still be induced by external perturbations of an appropriate type. In summary by treating explicitly the occurrence of spontaneous localized fluctuations we have verified the existence of metastable states and hence of a kind of first-order phase transition for chemical schemes exhibiting multistationary states. Moreover we have obtained a quantitative characterization of metastability in terms of a critical coherence length of fluctuations necessary for the spontaneous formation of a critical nucleus of one " phase " within another. The evolution of an existing nucleus of one pure phase within another has been treated by Schlog1,l and is dis- cussed in the paper by Nitzan Ortoleva and Ross in this Symposium.The idea of a non-equilibrium analogue to the Maxwell construction of equilibrium first-order phase transitions is also considered by these authors and has been examined from the point of view of nonlinear irreversible thermodynamics by Kobatake and by T~rner.~ Mr. M. Collinsand Dr. R. G. Gilbert (University ofSydney) said " Trigger waves " have been described by Field and Noyes as a concentration gradient moving under diffusion in a system containing effectively only a single autocatalytic reaction. On F. Schogl Z. Phys. 1972,253 147. Y. Kobatake Physica 1970 48 301. J. S. Turner Adv. Chem. Phys. 1975 29,63. R.J. Field and R. M. Noyes J. Amer. Chem.Soc. 1974 96 2001. GENERAL DISCUSSION the other hand in their paper at this Symposium Nitzan Ortoleva and Ross describe this as the moving boundary between two steady states of a multiple steady state system. As Field and Noyes themselves point out the former description is obviously too simple as it allows the product concentration behind the propagating front to grow unbounded resulting in excessively large propagation velocities and a quanti-k tatively inaccurate picture. Consider the single autocatalytic reaction A+ B -+2B far from equilibrium in a closed isothermal system. In one dimension the equations of motion are aA/at = -kAB+ DAa2A/ax2 aB/dt = kAB fDBd2B/aX2. In this system the product coiicentration must remain finite and we avoid any un- realistic effects arising from infinite concentrations.We have shown that approxi- mate analytic solutions in space and time to these equations may be obtained for DA 3 Da. This condition ensures that the pulse or wave in B is significantly sustained by the system. The analytical solutions then show explicitly that the velocity of the pulse maximum is never greater than its value in the absence of reaction and if DA > BB,depletion of the reactants by back-diffusion in front of the pulse results in the velocity falling below that of simple diffusion. Hence it appears unlikely that a single autocatalytic reaction can account for rapid pulse transmission in an unstable excitable system. Thus two questions arise (i) would complete nunierical solution of the complete coupled partial differential equations for e.g.the Oregonator mech- anism account for such effects or (ii) does the Oregonator mechanism allow multiple steady states and if so is the pulse the boundary between these? Prof. R. M. Noyes (Oregon) said Gilbert has examined two equations in A and B associated with the reaction A + B -+ 2B. He asserts they represent autocatalysis and that the velocity of the pulse maximum is never greater than that for simple diffusion in the absence of reaction. I see nothing surprising about this result. The reaction that produces B at a rate proportional to its concentration simultaneously destroys a species needed to produce B; there is both autocatalysis and autoinhibition. The effects are equal at the pulse maximum where A and B have very nearly equal con- centrations.I predict that Gilbert will find a very different result if he looks at the leading edge of B advancing into pure A. I am sure he will find that edge (defined perhaps as B = 0.01 A) is advancing faster than would be expected by simple diffusion without reaction. The equations Gilbert is examining can hardly generate a wave front moving with uniform velocity along its profile. Gilbert’s equations could be modified to something very like ours if his A-equation were left unchanged and his B-equation were modified to read dB/dt = -IcAB+k’B-k“B2 + DBa2B/dx2. Calculations by Dr. J. D. Murray at the Mathematical Institute in Oxford show that these equations do develop a band profile that moves with uniform velocity.The term in k’B provides the autocatalysis the term in kAB prevents the leading edge from running ahead of the main front the way it does in Gilbert’s equation and the term in k”B2ensures a finite concentration of B behind the front. I am afraid that Gilbert confused our full and simplified equations in the paper to which he refers. The full equations ((3) and (4) in that paper) do describe a moving M. A. Collins and R. G. Gilbert Clrern. Phys. 1974 5 49. GENERAL DISCUSSION 79 boundary between two steady states just as is claimed by Nitzan Ortoleva and Ross. The concentration of one variable rises by a factor of lo5during passage between those two states. In the same paper we made a very crude effort to obtain an approximate analytical solution for the full equations just behind the leading edge of the advancing front.Our simplified eqn (7) is essentially equivalent to omitting the kAB and k"B2terms in the equation above. We pointed out that this simplified equation would let B increase indefinitely behind the wave front and we restricted any application to concentrations at least two powers of ten less than that attained in the upper steady state. We still believe that approximation was applicable to the concentration range for which it was iiitended. Our full equations (with only two variables) generate a migration of a boundary between two steady states that are found in different regions of space. The Ore- gonator (with three variables) in a uniform space has a single true steady state which is unstable with respect to a limit cycle trajectory involving repeated switching between two pseudostationary states.Each of these pseudostationary states evolves so as to switch to the other one. I have complete confidence that a coupling of the full Oregonator to diffusion will model repeating trigger waves advancing in space. I am not attempting such computations myself but I understand they are in progress elsewhere. Dr. B. L. Clarke (Alberta) said Whether or not the Belousov-Zhabotinski reaction can be modelled by the Oregonator is an experimental question. On the other hand whether or not the Oregonator models the detailed kinetic mechanism given by Field Koros and Noyes in ref. (5) is a purely mathematical question.My paper Stability of Topologically Similar Chemical Networks J. Chem. Phys. 1975 62 3726 contains theorems proven to answer the latter question. I have de- veloped a model of the Belousov-Zhabotinski system which is related to the detailed kinetics such that these theorems connect the stability of the model with the stability of the detailed mechanism. Surprisingly this model has stable steady states for all rate constants and all concentrations of the pool chemicals! The stability of this model almost proves that the mechanism in ref. (5) is never unstable. Noyes realized that the Oregonator and his detailed mechanism were only con- sistent when f= +. However then the Oregonator is always stable. He has pro- posed that the reaction HOBr +HCOOH + Br-+COz+H20+H+ (A) be added to the original mechanism to increasef.The expanded detailed mechanism can also be represented by the model I used and calculations on this model have mapped the stability domains analogous to fig. 1 in many dimensions. From the nature of these domains it is plausible that the model has limit cycle oscillations even in the case when reaction (A) is not strong enough to make the steady state unstable. Therefore one should be cautious of using steady state stability as an argument for the validity of a model or mechaiiism- especially when there are nearby unstable pseudosteady states. Prof. R. M. Noyes (Oregon) said Clarke is correct that the original FKN mechanism of the Belousov-Zhabotinskii reaction does not generate an unstable steady state.That mechanism was developed from qualitative arguments by experi- mental chemists who did not then know how to do a stability analysis of a steady R.J. Field E. Koros and R.M. Noyes J Atner.. Clietn. Soc. 1972 94 8649. GENERAL DISCUSSION state. We subsequently developed the Oregonator model to incorporate the essential features of our mechanism. Examination of that model showed us that the stoichio- metry of our original mechanism would have generated a stable steady state ;Clarke has independently realized the same fact. At the same time that the theoretical analysis was demonstrating the inadequacy of our original mechanism experimental evidence was requiring a modification to add oxybromine oxidation of the formic acid that was inert to oxidation by cerium (IV).The revisions generate almost precisely the stoichiometry corresponding to maximum sensitivity of the Oregonator model; a paper has been accepted by J. Amer. Chem. SOC. I am not sure of the significance of Clarke’s examination of the revised model when his reaction (A) is “ not strong enough ” to make the steady state unstable. Formic acid will continue to accumulate until its average rates of formation and de-struction during any sufficient period are equal. I believe the argument based on Oregonator stoichiometry will then indicate an unstable steady state. 1 am very interested in his suggestion that limit cycle oscillations might commence even before formic acid had accumulated enough to render the steady state unstable to a con- ventional linear analysis.I am indebted to Clarke for pointing out it is possible to have a locally stable limit cycle trajectory around a locally stable steady state. He assures me this is possible even if all processes are unimolecular or bimolecular provided the concentrations of enough species are varying simultaneously. Much of the previous theoretical work has been concerned with only two simultaneous variables ;we badly need the sorts of theorems Clarke is trying to develop for multi-component systems. Dr. B. L. Clarke (AZberta) (communicated) The mechanism of the Zhabotinski system allows a number of independent overall reactions to be constructed. Those constructed from the mechanism of ref. (5) have formic acid as a product .Reaction A is part of another overall reaction in which formic acid is an intermediate and C02 is a product instead. When both types of overall reactions are occurring together the status of formic acid is ambiguous. The rate of Br- production compared to HBrO production is then not determined by stoichiometry alone but by the relative rates of the various possible overall reactions as well. The stability of pseudosteady states plays an important role in the trajectory calculations shown in fig. 2-5 of the paper by Field and Noyes. By pseudosteady state stability I mean the stability of the matrix M which appears in the linearization of eqn (1 1)-( 13) about arbitrary values of 2 y p. -(sq +s-2qscY) (s -sa) 0 dt W 0 -W When M is stable it is often possible to calculate the trajectory from simplified equations of motion.Otherwise the unstable normal mode of A4 plays a role in the dynamics. First we calculate the equation of the long slow decline in q shown in fig. 2. During this motion a and p are at a pseudosteady state for a given value of q. Solving eqn (1 1) and (1 3) when da/dz = dp/dz = 0 yields a = p = q/(q-i)-qtp/(V-1)3 + . . . for the pseudosteady state values of a and p. These two equations specify a curve R.J. Field and R.M.Noyes J. Gem.Phys. 1974 60 1877. GENERAL DISCUSSION 81 (parametrized by q) which has been termed the slow manifold by Zeeman,' eqn (12) the motion along the slow manifold is determined by From drl -2rl- dt S or log r] = -2t/s+ C.This prediction of the motion agrees with the calculations (fig. 2) because a and p approach pseudosteady state rapidly compared to the motion of q and in addition the pseudosteady state (i.e. M) is stable for all a q p on the trajectory. We can obtain equations for curve A in fig. 5 by this method. a is a fast variable near steady state while q and p are both slow. Tfie slow manifold for this situation is the two dimensional surface a = g(q p) obtained by solving dcr/dz = 0. The motion on this manifold may be linearized about steady state to give the damped sinusoidal curve A. This procedure requires that the trajectory remains within (or almost within) the region of a y p space where M is stable. Curve B shows what happens when the trajectory on the manifold enters a region where M is unstable.The fast variable c1 departs from its pseudosteady state rapidly and exponentially. It forces the slow variables to change rapidly also. There are two additional complications involved in the spike of curve B. First the one dimensional manifold da/dz = dcc/dz = 0 has a separatrix at r] = 1 and as q -+ 1 from below dq/dz -+ 03. Second the rate of departure ,of a is dependent upon the depth of penetration of the trajectory into the unstable region. The faster a is relative to 17 and p the less penetration is needed to leave the manifold. I have duplicated Field and Noyes' calculations and find that the trajectories of both curves A and B enter the unstable region near q = 1.18. Curve A grazes the unstable region but does not penetrate deeply.Curve B penetrates far enough for the instability in o! to decrease q below y = 1 where dq/dr suddenly becomes large and negative. The instability which causes the spike in fig. 2 to 5 appears mathematically as a negative term in the second Hurwitz determinant. When q is depressed about 30% below steady state this term is an order of magnitude larger than the next largest term. Thus the Hurwitz determinant is quite accurately represented by a single term. This term has a physical interpretation as the product of the feedback loops which cooperate to produce the instability. When the stability problem is set up properly approximate equations of the surfaces which divide the manifold of pseudo- steady states into stable and unstable regions may be obtained by equating the dominant terms of two adjacent stability domains.The extension of the stability analysis of networks to pseudosteady states adds one additional parameter to the problem for each dynamical variable but it does not change the number of Hurwitz determinants which need to be examined. I have developed techniques which use a computer to obtain the equations of the boundary of the stability domains in cases where the algebra would be very tedious. It takes only a few seconds to do a complete pseudosteady state stability analysis on a very simple model like the Oregonator. These techniques are currently being used to study models which are closer to the detailed mechanism of the Belousov-Zhabotinski reaction.Prof. R. M. Noyes (Oregon) (communicated) Clarke is not talking.about the same situation that we were. The calculations reported in our paper concerned a system initially in a stable steady state defined approximately by a = p = 4.999 q = 1.250. We then arbitrarily depressed q by a small percentage and let the system evolve E. C. Zeemsn in Towards a Theoretical Siofos;v,4 Essays (Edinburgh University Press 1972) GENERAL DISCUSSION under the dynamic equations that had established the previous steady state. Clarke is discussing the evolution very near a point where a = p and where both are deter- mined by the value of q. As Clarke recognizes almost immediately after the perturbation in our calculations a attained a pseudosteady state determined by the value of q.However p responded more slowly and Clarke is not justified in his assumption that he can set a = p in his pseudosteady state. A comparison of fig. 3 and 4 of our paper shows that the assumption a = p is even less justified for the slow decline in fig. 2 which Clarke discusses in his comment. If that assumption is indeed required I have doubts about the utility of Clarke’s method. Clarke does not seem to have precise criteria for distinguishing curves A and B in our fig. 5. He talks of unstable pseudosteady states when q < 1.18 and admits that curve A penetrates the unstable region but says it does not do so “far enough ” for exponential growth of a to become “ significant ”. We do have a precise criterion not discussed in our paper.If the trajectory is integrated along curve B in fig. 5 the quantity d2(dcc/dt)/dcc2 changes sign at a point approximated by cc = 7.25 p = 5.3 q = 1.158 and a then increases rapidly. The same quantity never changes sign during the decrease of q along the trajectory of curve A. We shall prepare a paper in the near future defining our criteria for the conditions under which one pseudosteady state will switch to a different one. Perhaps our criteria will turn out to be mathematically equivalent to those for leaving the slow manifold in Clarke’s treatment. However neither treatment is yet defined with sufficient precision for a meaningful comparison. Both of our treatments are based on a computation of the stability of a specific point in u,q p space and neither predicts whether a system in a stable pseudosteady state will evolve along a trajectory such that it will subsequently become unstable.Such prediction would be needed to determine whether the trajectory attained after a finite perturbation like that in fig. 5 will later grow to criticality as in curve B or will subside after a limited growth as in curve A. Prof. J. Ross (MZT) said A variety of models have been proposed which exhibit the property of threshold excitation. Here a system far from equilibrium is in one stable stationary state ;upon excitation (change of concentration temperature etc) of the right magnitude and sign there occur much larger changes in concentrations (temperature) prior to the return to the stationary state.We have studied the behaviour of such systems upon imposition of noise (fluctuations) in concentrations. At low noise levels (compared with the threshold excitation) random excitation and return to the steady state takes place and at high noise levels the expected random variations in concentrations occur. At noise levels of the order of the threshold excitations however we find quasi-periodic concentration oscillations. Thus with imposed noise or inherent fluctuations it may be possible to attain quasi-periodic behaviour under less stringent conditions that those necessary for a limit cycle. Prof. R. M. Noyes (Oregon) said The model we used for our calculations was strictly deterministic. However if the control intermediate Y were subjected to random fluctuations of the order of the threshold magnitude for the deterministic model we should also observe quasi-periodic excursions similar to the interesting ones reported by Ross.Because the concentration of Y during an excursion falls to a very small fraction of its steady state value the random fluctuations imposed during such calculations should be by a percentage rather than by an absolute amount. H. Hahn A. Nitzan P. Ortoleva and J. Ross Pruc. Nat. Acad. Sci. 1974 71 4067. GENERAL DISCUSSION 83 Dr. P. Ortoleva and Prof. J. Ross (MIT) said Consider a system subject to threshold excitation with subsequent fast and slow changes of concentration with time as shown in fig. 2-4 in the article by Field and Noyes. We have studied wave propagation in systems with multiple time-scale kinetics ;the propagation occurs upon local (heterogeneous) threshold excitation.In the formulation of the theory we take explicit account of the different time scales and related length scales in a consistent perturbation method. The theory in lowest order yields good estimates of the velocity and concentration profile of propagating pulses and transitions between stable stationary states (in systems with such multiple states). Dr. A. Winfree (Indiana) (partly communicated) :Carbon dioxide bubbles tragically limit experimental enquiry into the stability of various modes of spatial and temporal organization in Belousov-Zhabotinsky reagent. The problem is particularly crippling in studies of 3-dimensional wave propagation :the liquid must be at least 1 mm deep so little C02 escapes through the surface directly and when a bubble rises it tears through a great volume of otherwise motionless liquid.Is there any way to modify the organic acid’s carboxyl groups (e.g. by forming a diester) or otherwise alter it to prevent decarboxylation without upsetting the /3-keto group’s reactivity or forming insoluble byproducts ? As a postscript I am glad to pass on Trahanovsky’s suggestion of replacing malonic acid by ethyl aceto-acetate using acetic acid cosolvent to prevent precipitation of bromo derivatives. It works inarvellously in a recipe similar to that given in A. Winfree Science 1972 175 634. Dr. A. Winfree (hdiana) said Depending on pH the Belousov-Zhabotinsky reagent can either oscillate spontaneously or remain inert until triggered to execute a single oxidative pulse.Such parameter-sensitivity of behaviour is graphically portrayed by Field and Noyes’ 2-component approximation to their own complete kinetic scheme :behaviour is oscillatory or excitable depending on the exact manner of intersection of the nullclines of the two components’ rate equations. If those null- clines were bent a little they would intersect three times giving rise to reaction with two alternative stable steady-states and an intermediate unstable (threshold-like) steady state. Such a reaction diffusion-coupled in space would provide remarkable oppor-tunities for experimental study not only in connection with physical chemistry (see Ross and Ortoleva this Symposium) but also as a model for the differentiation of living embryos into discrete tissues each corresponding to a different stable steady- state of the biochemical-genetic machinery residing identically in each cell.What parameters or reaction rates must be altered slightly to bend the reaction rate equations as required ? Dr. B. L. Clarke (AZberta) said Winfree is asking how the mechanism of the Belousov-Zhabotinski reagent might be modified to give a multistable steady state situation which is called a Riemann-Hugoniot catastrophe by mathematicians. I think there is no simple modification of the FKN mechanism which would do this. Reaction (R7) of Field and Noyes’ ref. (5) may be omitted because it is the back reaction of (R5) and (R6) and therefore it cannot affect the multiplicity of the steady states.If one omits “ flow through reactants ” from the remaining FKN mechanism the multiplicity is also not changed. Next consider all the remaining reactions except (R3) and (R9). This reaction network can be proven to have a unique positive steady state which is always stable. However if one adds Br-from an external source at GENERAL DISCUSSION constant rate there are two steady states and they have a dynamical significance. The steady states of the network with Br- being added at a constant rate correspond to dynamical situations of the actual network when Br- is decreasing at that instan- taneous rate with the other variables at a pseudosteady state for the instantaneous value of the Br- concentration.These two steady states have high and low con- centrations of HBrO and correspond to the steady states involved in the switching of the Oregonator. One cannot have a third steady state for this network because the equation to be solved for the two steady states is quadratic. Next include reactions (R3) and (R9) in the network once again. Since they both consume Br- the second steady state is removed from the network. However the new reaction Noyes has proposed (reaction (A) in my previous comment) adds Br- in such a way that the modified FKN mechanism at steady state can now behave like the original FKN mechanism with a fixed input rate of Br-. Whether or not it has two steady states depends on the relative importance of reactions (R3) (R9) and (A).In order to get three positive steady states the polynomial to be solved must have coefficients which alternate in sign (+ -+ -). I do not see any realistic way to modify the mechanism to achieve this. Prof. R. M. Noyes (Oregon) said Winfree has commented about three different matters that should be discussed separately. These are (a) the possibility of finding a system that does not produce carbon dioxide (b)the transition of a system between oscillatory and inert-excitable conditions and (c) the possibility that nullclines might be made to intersect three times. (a) With regard to carbon dioxide evolution I am sure Winfree is aware that Bowers Caldwell and Prendergast observed oscillations with 2,4-pentanedione CH3COCH2COCH3.I gather this material is objectionable in other ways. The cyclopentanedione and cyclohexanedione ring systems might be more satisfactory. Kasperek and Bruice report a number of other compounds that do or do not oscillate under their conditions but they do not mention extents of carbon dioxide evolution. There is much merit to Winfree’s suggestion to use an ester such as diethyl malonate. Our calculations with the Oregonator indicate that stoichiometry is a very important factor. The desirable compound must brominate readily and the resulting bromo derivative will be most effective if it liberates one bromide ion for each two Ce(1V) or Fe(phen),(III) ions reduced during the time scale of interest. Stoichiometric tests will probably provide the quickest way to screen the bromine derivatives of various organic compounds.The best organic substrate for a particular study of this reaction must ultimately be identified empirically. (6) With regard to the transition between oscillatory and inert-excitable conditions oxygen appears to be very important. As Winfree points out in his manuscript the reagent remains for long periods in inert reducing condition when it is in a film a millimetre or two thick but the same composition oscillates with slow frequency in bulk. Agitation of the bulk solution with a stream of air can also prevent oscillations. Oxygen is most likely to attack radical species and the -BrO intermediates are likely targets. A plausible mechanism is -Br02+O2 S 000Br0 -00Br0 + WBrO -+ 2.BrO3 aBrO +Ce3+-+ BrOj +Ce4+.P. G. Bowers K. E. Caldwell and D. F. Prendergast,J. Phys. Chem. 1972 76 2185. G. J. Kasperek and T. C. Bruice Znorg. Chem. 1971 10 382. R.J. Field and R. M. Noyes J. Chern.Phys. 1974 60,1877. GENERAL DISCUSSION 85 This mechanism is purely conjectural at present but it is consistent with what is known about thermodynamic and kinetic behaviour of related species. The net effect is that some -BrO is oxidised to BrO; instead of being reduced to HBrO ;reduction is essential for the autocatalytic behaviour that generates oscillations. This revised mechanism can be modelled if the third step of the Oregonator is altered to (M3’) with 0 < n < 2. B+X + nX+Z. (M3’) If n < 1 ,this revised model will have a stable steady state for all possible combinations of rate constants.If the system is oscillatory for n = 2 accessibility of oxygen could reduce n sufficiently to make the system inert to oscillation. However a steady state that was barely stable should be excitable just like the state we obtained by man- ipulating the stoichiometric factor f (c) Winfree asks about the possibility that nullclines might intersect three times. The stiffly-coupled Oregonator can model the Belousov-Zhabotinskii reaction in terms of the two intermediates Y and 2. The curve Z = 0 is monotonic in the Y-2 plane but the curve Y = 0 displays a pair of very sharp relative maxima and minima in 2. However even though there are certain values of 2 such that Y = 0 is satisfied for three values of Y there is no combination of rate constants for which the coupled Oregonator model permits more than one solution such that Y = 0 and Z = 0 simultaneously.The kind of situation Winfree is looking for would be obtained if the stoichio- metric factor f could vary so that f < 1 when Y is a little less than kM3B/kM2 and f> 2 when Y is a little more than kMM3/kMZ. Such an effect introduces auto- catalysis to step (M5) as well as that existing in step (M3). Although such a model would permit interesting hysteresis effects of the kind Winfree is looking for I do not see any way to realize it experimentally. Dr. M-L. Smoes (Dortmund) (communicated) 1. I agree with Field and Noyes that the so-called Winfree solution is an oscillatory system with long period.Indeed Winfree has been trying since 1972 to eliminate the bulk oscillations in the distributed Zhabotinskii system.2 To do so he increases the bromide ion concentration and decreases the sulphuric acid content of the reacting mixture. I have shown recently that the period of homogeneous (bulk) oscillations in the ferroin-catalyzed Zhabotin- skii system increases with an increase in bromide ions and a decrease in sulphuric acid. Moreover when the homogeneous period of oscillations is sufficiently long large variations in period are observed. This may explain the fact that oscillations in the Winfree solution are observed only irregularly and that the system appears unstable but non-oscillatory. 2. The period of the Zhabotinskii oscillations depends on the initial reactant concentrations in a highly nonlinear mannere3* This fact suggests another possibility for the amplification of small perturbations in the concentrations.Indeed any period of the form T = 27rn/(a-b) with a and b related to some parametric concentrations can be drastically reduced by very small perturbations in a or b as long as the unperturbed period is sufficiently large. R. J. Field and R. M. Noyes J. Chem. Phys. 1974 60 1877. A. T. Winfree Science 1972 175 634. M-L. Smoes J. Chem. Phys. 1975. Oscillatory Processes in Biological and Chemical Systems (Nauka Moscow 1967) p. 181ff. GENERAL DISCUSSION It is this transformation of the small concentration perturbations into large period perturbations which justifies the phase waves interpretation of the spatio- temporal structures observed in the Zhabotinskii system.The ainplificatioii model of Field and Noyes is limited to nonoscillatory media diile our model can be used for the waves observed in the oscillatory Zhabotinskii sys tern. Prof. R. M. Noyes (Oregon) (communicated) Smoes evidently does not appreciate thc effect of oxygen on Winfree solution. When she talks about bulk oscillations with long period she is considering compositions that are not maintained fully saturated with oxygen. If saturation is maintained the medium can indeed become non-oscillatory. As I point out in my response to the comment by Winfree we now believe the system could be modelled better by changing the stoichiometry of the third step of the Oregonator rather than the fifth step as we did in our paper.How-ever the effect should be very similar with either model. When Smoes talks about the effect of bromide ion on the period of oscillation she must be referring to initial concentration. Bromide ion is one of the species that undergoes limit cycle oscillations and is not a proper parameter by which to characterize a reactant composition. We remain convinced our chemical mechanism can account for virtually all of the essential features of the Belousov-Zhabotinskii reaction. Smoes may attempt an alternative explanation based on phase waves if she wishes but she should then explain why the period of bulk oscillations depends in a very non-linear way on reactant concentrations yet the rate of propagation of waves is closely proportional to [H+]* [Broil* and almost independent of other concentrationse2 Dr.M-L. Smoes (Dortmund) (communicated) (1) I do not exclude the possibility that oxygen too increases the period of bulk oscillation in the Zhabotinskii system. But the changes in oxygen concentration are probably negligible compared to the changes in concentrations of bromide ions and of sulphuric acid which have led to the progressive elongation of the bulk period culminating in the " Winfree solution ". Although I was actually referring to initial concentrations of bromide ions the fact that a concentration oscillates does not prevent it from having an average value. Moreover a distinction between oscillatory compounds and constant parameters is a helpful approximation to the very complex kinetics involved in chemical oscillations.In any case it is an experimental fact that an addition of bromide ions increases the homogeneous period of the Zhabotinskii oscillations. (2) Our phase wave model can account qualitatively for all experimentally well established phenomena in the Zhabotinskii system. Unlike Field 2nd no ye^,^ we do not have to consider as interferences all waves except one. Moreover we have predicted the larger external wavelength which is observed in the waves at the centre of fig. 1 in the paper of Field and Noyes. Finally we have found in a detailed study of the phase waves that the speed of wave propagation is dependent on the homogeneous period of oscillations.This explains an increase in the speed with an increase in the concentrations of bromate and of malonic and sulphuric acid. It explains also the fact that the speed is insensitive to changes in ferroin concentration. M-L. Smoes and J. Dreitlein J. Chem. Phys. 1973 59 6277. * R. J. Field and R. M. Noyes J. Amer. Chem. Suc. 1974 96 2001. M-L. Smoes and J. Dreitlein,J. Chem. Phys. 1973 59 6277. R. J. Field and R.M. Noyes J. Arneu. Chem. Soc. 1974 95 2001. M-L. Smoes Characteristic Properties of the Phase Waues in preparation. GENERAL DISCUSSION 87 The nonlinear dependence of the period of oscillations on concentration is matched in our model by a similarly nonlinear dependence of the speed of propagation on the homogeneous period.As a result the experimental data of Field and Noyes are not in disagreement with the phase wave model. But the interest of our model comes from its generality. We are not limiting ourselves to the interpretation of the Zhabotinskii waves. Indeed our results are expected to apply to a large class of chemical systems with concentration or temperature dependent period of oscillations. Since the glycolytic system belongs to this class glycolytic phase waves can be expected. We are interested in the features that should be shared by the waves in glycolysis and in the Zhabotinskii system. Dr. 0.E. Rossler (Tiibingen) said The Oregonator (eqn (1 1)-( 13)) is as a 3-variable non-linear oscillator not very easy to analyze mathematically.’ It also involves the mathematically crucial but chemically somewhat ad hoc assumption of f> 1 in order to account for monostability.The question therefore arises whether a 2-variable prototype does not also exist especially since the verbal descriptions given (like “ X being switched by Y’s varying between 2 critical values ”) all refer to such a model. Evidence on the inorganic subsystem (sulphuric acid +bromate +cerium 111) suggests that the autocatalytic reaction can be shifted through a hysteresis cycle depending on the value of an exogeneous influx of bromide. The complete Noyes-Field-Koros scheme indeed provides for such a mechanism the fact that bromide is being regenerated (at least partially) from its own products (namely via the reverse reaction of R2 from hypobromous acid and via R8 from bromine) allows a weak influx from x to shift this “catalyst” up and down without requiring the “accumulating” power of an intercalated variable 2.Stipulating that something like a Michaelis-Menten type approximation is valid the following 2-variable model applies where x again refers to the autocatalytic cycle (represented by bromous acid for example) and y is the total concentration of the catalyst (involving bromide hypo- bromous acid and bromine). This equation which is readily analyzed by phase- plane techniques is of Bonhoeffer-van der Pol (BVP4) type the second variable acts as a slowly changing “ parameter variable ” which drives the bistable subsystem (first variable) through a hysteresis cycle and this is an either astable (self-oscillating) or monostable (repeatedly triggerable) way.One adapted set of parameters k = 1 = k5 = 1 K/k (corresponding to the former k2) = 4 x (or less) k = 1 k3 = 4.8 x lo2 k4 = 4 x lo7 k5 = 1 k6 = 0 (astability) or 0.12 (monostability) ; analogue computer results. Prof. R. M. Noyes (Oregon) said There is nothing particularly ad hoc about the assumption that f> 1. The experimental fact is that the excitable Winfree reagent is in the red reduced condition which is the steady state generated by the Oregonator model when kM5is sufficiently large and f> 1. (When f< I the steady state at J. D. Murray J. Chem. Phys. 1974 61 3610. V. A. Vavilin and A. M. Zhabotinski Kinefika i Kataliz 1969,10,83. (10,65 of cover-to-cover translation).See ref. (4) and (5) of preceding paper. R. FitzHugh Biophys. J. 1961 1 445. ’0.E. Rossler Lecture Notes in Biomafhemafics (Springer-Verlag) 1974 4 399 and 546. 88 GENERAL DISCUSSION large k, corresponds to the blue oxidised condition.) Oxygen is apparently involved in maintaining this steady state and a possible mechanism is discussed in my response to the comments by Winfree. Rossler is right about the mathematical advantage of describing the system in terms of only two variables. However Tyson and Light ' have shown that limit cycle behaviour can not be generated by a model based on bimolecular reactions of only two variables. Rossler has introduced a term IC2xy/(x+K) that resembles a Michaelis-Menten situation. However by doing so he has created a third chemical variable.His equation models an intermediate species H2Br202that is present in significant concentration compared to HBr02 and that is more likely to decompose to HBrO,+Br-+H+ than it is to 2HOBr. There is no experimental evidence for such a species and no theoretical justification from what we know about chemical bonding and reactivity. Rossler's equations also omit any effect of metal ion catalyst which is modelled by 2in the Oregonator. It is an experimental fact that such a catalyst must be present if oscillations are to be observed. Fortunately the Oregonator can indeed be modelled with only two variables. As we have shown elsewhere,2 if X at al€ times is approximated by solving X = 0 for existing values of Y and 2,the Oregonator equations generate limit cycle behaviour in Y and 2 differing little from the results of more complicated calculations in three variables.I am confident our excitable medium calculations would be little affected by repeating them in two variables at the same level of approximation. The key pair of variables for approximating the Oregonator is Y and 2 rather than X and Y as suggested by RiJssler. Dr. P. G. Sdrensen(Denmark)said Experimental studies of the Belousov reaction have been carried out in an isothermal stirred tank reactor with constant volume and time-independent external flows. The reactants in the input flows were KBrO, CH,(COOH) and Ce2(S0&. At low flow rate the system shows periodic oscillations n 100 FIG.1 .-Potential variations of a platinum electrode relative to a calomel electrode.Tank volume 8.1 ml. Temperature 25.0"C. Input flow 1 :0.09 M KBrOJ in 3N H2S04. 0.0037 ml/s. Input flow 2 :0.5M CH2(COOH)2 in 3N H2S04 0.0037 ml/s. Input flow 3 0.0008M Ce2(S04)3in 3N H2S040.0037 ml/s. corresponding to an attracting limit cycle but at higher flow rate the oscillations occur in bursts ;fig. 1. The proportion of time spent by the system in the oscillating phase decreases with increasing flow rate and for a sufficiently high flow rate the J. J. Tyson and J. C. Light J. Chem.Phys. 1973 59,4164. R. J. Field and R. M. Noyes J. Chem. Phys. 1974,60 1877. GENERAL DISCUSSION 89 oscillations disappear corresponding to an attracting steady state.Although the number of oscillations in each burst is almost constant for a certain flow rate and the bursts occur with fairly constant time interval there is no fixed proportion between the burst period and the oscillation period. This behaviour can not be explained by the existence of a limit cycle. I suggest that the attractor in this case is a surface in state space and that the system shows almost periodic movements on this surface. The Belousov reaction is thus an example of a chemical system that may have attractors which are neither fixed points nor limit cycles. The kinetic explanation for the pulsed oscillations is probably that the concentration of a compound which is essential for oscillatory behaviour must be higher than a certain limit in order to allow transition from a non-oscillatory to an oscillatory phase and that the compound is removed faster during the oscillatory phase than it is produced.When the con- centration has decreased below a critical concentration. the oscillations stop and regeneration of the compound occurs during the following non-oscillatory phase. I wish to ask if measurements have been made on the rate of production of BrCH (COOH) in the closed system immediately before the oscillations start and imme- diately after. If the production rate is larger in the first case BrCH(COOH) is a probable candidate for the critical compound in the kinetic scheme described above. Prof. R. M. Noyes (Oregon) said Sorensen’s observations are very interesting. Although his description of experimental conditions is not entirely clear I gather that his platinum electrode has a more positive potential when the system is not oscillating than it does on average during the pulses of oscillation.If that is so the non-oscillatory period is behaving just like the oxidising induction period that is observed when malonic acid is the only organic compound initially present. Sarrensen’s observations can then be explained rather easily by the mechanism we have already de~eloped.’-~ The explanation is a detailed amplification of the one he proposes in his comment Oscillations in a closed system involve rapid switching between an oxidising and a reducing condition depending upon the net direction of change in oxidation state of cerium. The reducing condition is characterized by significant but monotonically decreasing concentrations of both Br-and Ce(1V).The oxidising condition is characterised by a very much smaller concentration of Br- and by a monotonically increasing concentration of Ce(1V). When the system switches to the oxidising condition Ce(IV) is produced initially at a very rapid rate. When the system is in the reducing condition the important processes are those of (A-D). BrOJ + 2Br-+ 3Hf -+ 3HOBr (4 HOBr + CH,(COOH) -+ BrCH(COOH) + H,O (B) BrCH(COOH) + 4Ce4++ 2H,O -+ Br-+ 4Ce3+ + HCOOH + 2C02+ 5H+ (C) HOBr + HCOOH -+ Br-+ COz+ H++ H20. (D) The stoichiometry of (A)+2(B)+(C)+(D) yields (R) as the net reaction in the reducing condition. Br0; +2CH2(COOH)2+ 4Ce4+-+ BrCH(COOH) +4Ce3+ + 3C02+ 3Hf + H20.(R) R.J. Field E. Koros and R. M. Noyes J. Amer. Chem. Sac. 1972 94 8649. J. J. Jwo and R.M. Noyes J. Amer. Chem. Suc. 1975. R.M.Noyes and J. J. Jwo,J. Amer. Chem. Soc. 1975 GENERAL DISCUSSION The kinetics in the reducing condition are somewhat complex. If there is in- sufficient bromomalonic acid (BrMA) process (C)is soslow that the depletion of bromide ion by process (A) reduces [Br-] below the critical value necessary to maintain the reducing condition. If there is somewhat more BrMA [Br-] attains a steady state established by equal rates of processes (A) and (C). If there is still more BrMA the cerium(1V) will first be consumed by (C) and the resulting bromide will then disappear by the effect of (A)+2(B)+(D).If there is enough BrMA to generate a reducing condition (R) describes the net stoichiometry for the overall period in that condition. To a good approximation,' the rate during a reducing period is given by eqn (1). -d[BrO;] -d[BrMA] = 2.1[H+]'[Br-][BrO;]. dt dt The numerical value of the rate constant is based on concentrations in mol/l. and time in seconds. Approximate concentrations during such a period are about 1 M for hydrogen ion a few times 0.01 M for bromate and a few times M for bromide ion. When the system is in the oxidising condition the net process (Ox) results from processes (E) and (B). BrO +4Ce3++5H+ -+HOBr+4Ce4++2H,0 (E) HOBr +CH2(COOH)2-+ BrCH(COOH) +H20 (B) BrO; +CH2(COOH)2 +4Ce3++ 5H+ -+ BrCH(COOH) +4Ce4++ 3H20. (0x1 The kinetics in the oxidising condition are also complex.If the intermediate -Br02 radicals irreversibly oxidise cerium(II1) as rapidly as they are formed our previous analysis indicates the rate of (Ox) is given by eqn (2). A d[BrO;] d[BrMA] = 0.6[H+]2[Br0;]2. dt dt If -BrO radicals are reversible oxidants if they disproportionate with each other or if they also reduce cerium(IV) the rate of (Ox)is somewhat less than that of eqn (2). Reaction conditions are such that to a first approximation [BrO;] GS 104[Br-]. Therefore even if eqn (2) somewhat overestimates the rate a comparison of eqn (1) and (2) shows that d[BrMA]/dt during the oxidising condition is many times that during the reducing condition. Therefore it is quite possible to design conditions for an oxidising condition in a stirred tank reactor such that [BrMA] is temporarily too small for process (C) to switch the system to reducing condition but such that d[BrMA]/dt by eqn (2) is greater than the rate at which BrMA is being removed from the reactor by flow.As [BrMA] increases in the reactor the system eventually switches to oscillation just as it does in a homogeneous closed system. However .most of the time during oscillation is spent in the reducing condition and eqn (I) is so much slower than eqn (2) that net d[BrMA]/dt during an oscillatory period is in- sufficient to compensate for loss of BrMA by flow from the reactor ;the system will revert to a non-oscillating oxidising condition. This explanation is entirely consistent with the one proposed by Sarrensen.The above qualitative description can be modelled by the following revised Oregonator and differential equations. R. J. Field E. Koros and R. M. Noyes J. Amer. Chem. SOC.,1972 94 8649. * R. J. Field and R. h?l. Noyes J. Chern. Phys. 1974 60,1877. GENERAL DISCUSSION 91 A+Y -+ X+P (NO X+Y + 2P “2) A+X -+2X+Z (N3) 2X -+ P+A (N4) 2 +fY -fP (N5) P f=-a+P (3) dX/dt = klAY-k2XY+k3AX-2k4X2-UX (4) dY/dt = -k,AY-k,XY+k&-vY (5) dZldt = k,AX-kSZ-uZ (6) dP/dt = kl A Y+ 2k2 XY+ k4X2-1~sfZ-vP. (7) In this model the chemical significance of the letters is intended to be A = BrO; X = HBr02 Y = Br- 2 = 2Ce(IV) P = BrMA. The rate constants for steps (Nl-N4) are determined by those of oxybromine chemistry.’.The stoichio- metry of step (N5) is that of (C) + (D) if P represents either BrMA or its HOBr pre- cursor and if process (C) is first order in Ce(1V). The simplified model developed here assumes that cerium(1V) is consumed at an almost constant rate by an excess of malonic acid (MA) containing a smaller amount of bromomalonic acid (BrMA). Because the radicals from malonic acid may attack BrMA with liberation of bromide, the stoichiometric factor f is calculated by eqn (3). As P increases the resulting increase in fwill cause the system to pass sharply from an oxidising steady state to an oscillating condition as happens in a closed homogeneous system. It appears that a proper selection of u k, and a could reproduce the main features of Sarrensen’s observations.At too great a v the system would remain in an oxidising condition and would not oscillate just as is observed. At too small a v the system would go into continuous oscillation as is also observed. However it is not certain that this simplified model would generate packets of oscillation rather than single pulses of reducing condition followed by return to oxidising condition. Once [BrMA] has risen to the criticalf necessary for oscillations to commence there must be a delay before [BrMA] falls enough to shut them off again. It may be that the thermal effects associated with the different rates of (R) and of (Ox) will need to be coupled in order to reproduce the experimental obser- vations. It may also be that the intermediate tartronic acid HOCH(COOH), changes at a sufficiently different rate to produce the necessary coupling.2* The ideas suggested here can be tested by computations with the revised Oregonator proposed above and additional experimental testing can be provided by adding bromomalonic or tartronic acid to the solutions entering the reactor and by changing the efficiency with which the reactor is therrnostatted.Dr. 0.E.Rossler (Tiibingen)said The same result (periodic bursts of oscillation in a stirred open system version) has also been obtained by J~nkers.~ The stops R. J. Field E. Koros and R. M. Noyes J. Amer. Chem. Soc. 1972 94 8649. J. J. Jwo and R. M. Noyes J. Amer. Chem. SOC.,1975. R. M. Noyes and J. J. Jwo J. Amer. Chem. Soc. 1975. R. J. Field and R. M. Noyes J.Chem. Soc. 1974 60 1877. G. Junkers Uber die periodisch verlaufende Reaktion zwischen Malonsiiure und Bromat in der Gegenwart von Cerionen (Diploma Thesis University of Aachen 1969). GENERAL DISCUSSION occurred in the "high Ce(IV) " state and addition of Br- immediately revived the oscillation. In order to model this type of behaviour at least three variables are required in any case. For example the following third equation may be added to eqn (1) (as indicated in my earlier discussion remark). dzldt = k,x-k,z with y being coupled to z by replacing the term k6 on the right hand side of the second equation by k6z. Z may correspond to a bromide-releasing intermediate compound (even Br2). For example k = 1.7 k7 = k = 3 x k6 = 0.1 ; analogue computer results.Prof. E. Koros (Budapest) said A large number of measurements have been performed by us using the polarographic method on the rate of production of BrCH(COOH) in different Belousov systems during the non-oscillatory (induction) period of the reaction. Especially suitable for this purpose is the bromate-malonic acid-cerium(II1)+nitric acid system where the non-oscillatory period is rather long and the consumption of the reagent during the non-oscillatory period is significant. (Approximately 40-50 %of the initial bromate content is consumed)-a few data are given in table 1. TABLE1.-T = 15°C. MALONIC ACID 0.40 M NiTRiC ACID 5.0 M CBrO,-I/Ml [Ce/llI/] 1O3/M (dCBrMAl/dt)ifttidM min- 0.10 4.00 3.32~10-3 1.oo 1.25~10-3 0.05 4.00 2.47~10-3 1.oo 0.87~10-3 When catalyst was not present the initial rate of production of BrCH(COOH) was 0.16~ 10-3M min-' in the following reaction mixture [BrO;] = 0.10 [MA] = 0.40 and [HNOJ = 5.0 at 15°C.It is my opinion that in a flow system continuous oscillation would be observable if BrCH(COOH) were also added together with the other reagents. Then the chem- ical system would be always in an excitable state. Very recently we measured the rate of formation of bromomalonic acid both during the non-oscillatory and the oscillatory periods of the cerium(II1)- and man- ganese( 11)-catalysed Belousov-Zhabotinsky reactions. A typical curve is given in time FIG.1. See paper by Koros et al. this Symposium. E Koros and M,Burger to be published GENERAL DISCUSSION fig.1. From the curve it is obvious that at the transition from the non-oscillatory to the oscillatory stage a rather sharp decrease occurs in the rate of formation of bromomalonic acid. These results yield experimental support for Snrrensen’s explanation of the phenomena observed in his flow system. Prof. R. M. Noyes (Oregon) said The experiments by Koros and his co-workers contribute important new insights to the function of the metal-ion catalyst in the Belousov-Zhabotinskii reaction. By a somewhat unfortunate coincidence the strong oxidants Ce(IV) and Mn(II1) with potentials of 1.4 V or over are liable to substitution of the oxygen in the inner sphere of coordination while the weaker oxidants Ru (dipy);+ and Fe(phen)i+ with potentials of 1.2 V or less are inert to substitution of the organic species in the inner sphere.Species that were inert and labile to substitution would probably show very different relative kinetic behaviour with oxybromine species and with organic molecules like malonic acid even if they had very similar thermodynamic reduction potentials. It would be very helpful if a catalyst system could be found that would uncouple effects of changing reduction potential from effects of changing substitution lability. In addition to the excellent experiments reported here I can suggest some other types of measurement that would help to sort out the details of metal-ion reactions in this system (1) The oxidation of organic species is very complex.Jwo at the University of Oregon has completed a thorough study of the cerium system. His observations show that malonyl radicals attack bromomalonic acid with liberation of bromide ion and that presence of the intermediate tartronic (hydroxymalonic) acid is important to the detailed behaviour of the Belousov-Zhabotinskii system. It would be useful to know the ways in which ferric phenanthroline differs from cerium(1V) as an oxidant of these organic species. In addition to studies of direct metal ion oxidation of organic species it would be useful to measure d In [Br-]/dt during slow bromide consumption periods with dif- ferent ratios of malonic and bromomalonic acids. Such measurements (at known concentrations of bromate) would establish rates of bromide ion generation by metal- ion oxidation of the mixture of organic species in a Belousov-Zhabotinskii system.(2) The reduction of bromate by metal ion also needs to be understood better for the catalysts discussed here. The studies by Thompson were crucial to elucidating the mechanism 3-5 of the Belousov-Zhabotinskii reaction. However they do not establish the relative reactivities of Br02*radicals with each other and with both the reduced and oxidized forms of the various metal ions. Neither do they establish whether metal-ion oxidation of HBrO can compete with disproportionation of that species. These effects will be difficult to sort out because the very reactive species BrO,. and HBrO exist in very low kinetically established steady state concentrations that can not yet be measured directly.It is suggested that mechanistically useful infor- mation can be obtained by measuring absolute values of [Br-Jcrit which is the bromide concentration at which the system goes into rapid bromide consumption or pro-duction. In the cerium catalyzed system the critical concentration for rapid bromide J. J. Jwo,J. Amer. Chem. Soc. 1975. R. C. Thompson J. Amer. Chem. Soc. 1971,93,7315. R. M.Noyes R. J. Field and R. C. Thompson,J. Amer. Chem. Soc. 1971,93,7315. R. M. Noyes R. J. Field and E. Koros J. Amer. Chem. SOC.,1972,94 1394. R.J. Field E. Koros and R. M.Noyes J. Amer. Chem. Soc. 1972,94 8649. GENERAL DISCUSSION consumption is considerably greater than that for rapid production. It is tentatively predicted that these two critical concentrations will be much closer to each other in the iron phenanthroline system.The above suggestions merely indicate that the study of no chemical system is ever truly complete! Not all the suggested experiments are necessary or even desirable to carry out. However the Belousov-Zhabotinskii reaction is so dramatic it has attracted considerable attention. Further effort is undoubtedly warranted in order to under- stand more about the detailed mechanisms in this remarkable system. It is to be hoped that people undertaking such studies will communicate enough among themselves so that unnecessary duplication is avoided. Prof. E. K6rh (Budapest) said (1) Unfortunately tris(phenanthroline)iron(III) is mstable in dilute sulphuric acid; the comples partly decomposes partly is reduced to tris(phenanthroline)iron(II) and for this reason it is not possible to look at its reactions with organic compounds.(2) The reduction of bromate by tris(phenanthroline)iron(II) was investigated by us and here I refer only to the original paper.' Dr. H. G. Busse (Kiel) and Prof. B. Hess (Dortmund) said The Belousov-type reactions are most suitable systems for investigating oscillatory chemical reaction mechanisms. Indeed the malonic acid +cerium sulphate +bromate reaction in aqueous sulphuric acid is currently being studied in several laboratories. Originally only the range of initial concentrations in which oscillations could occur was deter- mined apart from the shape of the oscillations within this range.2 Later inter- mediate and final products were also analysed.For some of them it was shown that the intermediates oscillate with the same frequency as the system itself. This behaviour would be expected from oscillations of the limit cycle type. We analysed the final product of this reaction system and found besides C02 rnonobromomalonic acid as well as bromoacetic acid. 3-5 Furthermore we investi- gated the reactivity of the oscillating reaction system towards light perturbation.6 From estimates of the quantities of products the overall reactions can tentatively be formulated by the two following equations 3H2C(COOH)2 +2Br0; +2H+ -+ 2BrHC(COOH) +4H20 +3C02 2H,C(COOH) +2BrO; +2H+ -+ Br2HCCOOH+4H20+4C0,. E. K&os M. Burger and A.Kis Reaction Kin. Cat. Letters 1974 1 475. A. M. Zhabotinsky in Oscillatory Processes in Biological and Chemical Systems (Puschino on 0ka-1967). L. Bornmann H. G. Busse and B. Hess Z. Naturforsch. 1973 28b,93. L. Bornmann H. G. Busse B. Hess R. Riepe and C. Hesse 2.Naturjorsch. 1973 28b 824. L. Bornmann H. G. Busse and B. Hess 2. Naturforsch. 1973 28c 514. H. Busse and B. Hess Nature 1973 244,203. FIG.1.-Time course of the optical density changes as recorded in a double-beam spectrophotometer. Initial conditions are :malonic acid (0.1 M) KBr03 (0.1 M) Ce(SO& (2 x M)in 3 N H2S04. (a) Simultaneous record of the optical density changes in time anaiysed at 260 nm and 377 nm. (b) Record of the optical density changes at 377 and 260 nm plotted on a X-Y-recorder as given in fig.16. The experiment starts with the trace at the top side left. The time course is indicated by the stepwise shift of the record of one period towards the right. Each step corresponds to approxi- mately 52s. (c) Time course of the optical density difference analysed at the wavelengths of 270 and 356 nm. The oscillating contribution of the optical density changes of ceric ions is suppressed by recording the difference. The periodicity is indicated in a stepwise increase towards a higher optical density (downward deflection of the trace see arrows) at 270 nm relative to 356 nm. GENERAL DISCUSSION time -i + '3 A260nm GENERAL DISCUSSION It should be mentioned however that the equations give no information about the detailed processes occurring during the oscillatory phase.The only indication of the course of the reaction is given by the results of an experiment in which radioactively labelled malonic acid (labelled in the 2 position with 14C) 10 minutes after the initiation of the oscillatory reaction was added to the reacting mixture. Since in this experiment the radioactivity in the CO slowly increases to the expected value it might be con- cluded that CO may be produced from a compound such as monobromomalonic acid and not from the initial coinpound malonic acid. From this point of view monobromomalonic acid is a final product as well as an intermediate since it is both accumulated during the reaction and also partly decomposed. Furthermore in conjunction with this experiment it should be mentioned that the bromination and the oxidation process might well occur by different reaction pathways.Here the bromination seems to be involved in the conversion of malonic acid to monobromomalonic acid. Probably the subsequent oxidation process decomposes monobromomalonic acid to COz and other products. In our opinion the driving force of this reaction is the chemical affinity rather than the heat of reaction.' In order to get more information on the mechanism of the formation of bromo- malonic acid in the system we analysed the optical density change in the system at two different wavelengths with a double-beam spectrophotometer. The two wave- lengths are chosen to be on both sides of the absorption band of the ceric ions so that the oscillatory contribution of the ions is of the same amplitude at both wavelengths over a given time interval.Typical experiments are shown in the records of fig. 1(a-c). Fig. I (a)demonstrates the periodic optical density change as measured at both wavelengths with the component absorbing at 260 nm slowly increasing in its mean optical density. Fig. 1 (b)shows a simultaneous plot of the optical density changes at 260 nm against 377 nm indicating clearly a stepwise increase of the 260 nm corn-ponent.2 Finally in an extinction difference record obtained in a separate experi- ment at a later time interval as shown in fig. I(c) the oscillatory portion with a lower frequency compared to the experiments given in fig.1(a)and 1(6)is suppressed and the periodic increase is observed relative to the optical density change at 356 nm. Based on the measured optical density increase of bromomalonic acid absorbing in the range between 270 to 250 nm we wish to attribute the stepwise increase in the absorption to the accumulation of bromomalonic acid in the oscillatory phase. This is in agree- ment with observations reported on the basis of polarographic analysis. ' Finally we would like to raise the question as to why it is stated that the nature of the oscillatory reactions implies a periodicity in the rate of heat evolution. It is not clear to us why this is in contradiction to periodic temperature variations. In the two papers referred to,3* it is not claimed that there is no overall heat production "by chemical reactions " but rather that under the experimental conditions oscil- latory variations in temperature are recorded.Prof. E. Khos (Budapest)said Busse and Hess ask why the nature of the oscilla- tory reactions implies a periodicity in the rate of heat evolution. During the oscilla- tory Belousov-Zhabotinsky reaction the rate of formation of both bromomalonic acid and CO is periodic i.e. a " restful " period (during which the rate of reaction E. Khros M. Burger V. Friedrich L. Ladanyi Z. Nagy and M. Orban this Symposium. B. Hess Biochemische Oscillationen in Dechema Monographieri (Verlag Chemie Weinheim/ Bergstrabe) 1973 71 261. H. G. Busse Nature 1971 233 137. U. Frank and W. Geiseler Naturwiss. 1970 58 52. GENERAL DISCUSSION is low) is followed by a “ burst ” period (during which the rate of reaction is high).l Our calorimetric investigations on the different Belousov-Zhabotinsky systems unambiguously proved that the amount of heat evolved has a linear relationship with the amount of bromomalonic acid formed.Thus it can be expected that the rate of heat evolution should exhibit the same periodicity as the rate of formation of bromomalonic acid provided that the heat transfer between the reaction mixture and its environment is slow. Our calorimetric measurements provided the experi- mental proof for that expectation. Dr. J. Re Bond (Leeds) (communicated) Koros correctly disposes of previous suggestions (ref. (10 and (1 1) of his paper) (1) that temperatures fall below ambient as well as rising above it during oscillations in the Belousov reaction and (2) that there is no overall heat production.However temperature peaks are entirely possible ; temperatures only rise monotically if heat transfer is slow and the fact that Koros recorded only monotonically increasing steps is simply due to this. We have measured temperatures in the Belousov reaction in rapidly stirred conditions and have observed peaks and valleys in the temperature-time record. Like Koros we find no evidence for cooling below ambient temperature ; overall the reaction is strongly exothermic. Dr. 0.E. Rossler (Tiibingerz) said Stirred-flow results 2n suggest that an “ upper state ’’ excitable version of the Z-reagent (being blue in the resting state) may also be possible.Even more tricky though perhaps still possible would be the realization of a doubly excitable Z-reagent (switching readily not only from red to blue but also from blue to red). In it two actively propagated moving folds (trigger waves) chasing each other would be possible thus allowing of new nontrivial spatial patterns. Such a reagent would in the simplest case again be described by my eqn (1) (as indicated in the discussion following the paper by Field and Noyes) with k = 2.5 and k6 = 0.3; analogue computer results. In biology a doubly excitable system has been detected recently (an optic nerve propagating impulses of variable length). Prof. R. M. Noyes (Oregon) said Rossler’s suggestion of a blue resting state is stimulating.It is significant he observes such a blue state only in a stirred flow reactor from which bromomalonic acid is constantly being removed. The reasons are discussed in my response to Sarrensen’s comment. A blue resting state will be generated only iff < 1 (and probably alsof < 0.5) in the Oregonator model. It occurs to me that a free radical trap such as the polymer- izable monomer acrylamide might lower fenough to create a blue excitable state. Excitation would require a mechanism that suddenly generated bromide ion instead of depleting it as in my paper. I do not see how a solution could be made to switch from blue excitable to red excitable unless a catalyst could be found for which the equivalent of kM5is several hundred times that for cerium(1V).Dr. P. Ortoleva and Prof. J. Ross (MIT) said The concept that kinematic waves in oscillatory systems depend weakly on diffusion has been analysed by a perturbation series in the effects of diffusion and the weakness of imposed gradients and hetero- E. KQros,Nature 1974,251,703. G. Junkers Uber die periodisch verlaufende Reaktion zwischen Malonsaure und Bromat in der Gegenwart vun Cerionen (Diploma Thesis University of Aachen 1969). P. G. Sarrensen this Symposium (discussion remark following the paper of Kiiros et al.). F. Zettler and M. Jarvilechto J. comp. Physiol. 1973 85 89. s 94 GENERAL DISCUSSION geneities. The developments in ref. (1)-(3) in Winfree’s paper appear as a natural consequence in lowest order of this phase diffusion theory.’ In addition the presence of heterogeneities in an oscillatory reagent has been shown by means of this theory to lead to the emission of localized wave patterns.Expressions were derived for the wavelength of the emitted waves. It was shown that to first order in the strength of the heterogeneity the period is just that of the bulk reagent T. We note that if the heterogeneity tends to slow down the bulk dynamics a pattern of incoming waves emerge although to our knowledge this has not been observed in Z-reagent. Dr. M-L. Smoes (Dortmund) (communicated) 1. It is not correct that the work of Smoes and Dreitlein describes kinematic waves. What we have proposed is an interpretation of the spatio-temporal structures which appear spontaneously in the distributed Zhabotinskii oscillatory system.The model reproduces qualitatively all the well-established features of this system bulk oscillations leading centres with periods shorter than the bulk period from which waves propagate non-uniformity of the wavelengths annihilation of colliding wavefronts formation of simple and double spirals in two dimensions. The fundamental hypothesis is that the period of oscillation depending on temperature and parametric concentrations is subject to perturbations. The leading centre appears as the result of a fluctuation. in parametric concentration for instance. The waves originating from the centre are due to the regression of the initial per- turbation through diffusion. Diffusion is thus important in this model.Although in the work cited above we neglected the diffusion of the oscillating intermediate concentrations this was done only in order to save computer time. We have now evaluated the effect of the diffusion of the intermediates; the results show that the speed of propagation of the wavefronts becomes very constant except near the leading centre and just before annihilation by the bulk oscillations or colliding wavefronts. 2. The distinction between “ pseudo-waves ” and “ trigger waves ” suggested by Winfree on the basis of the behaviour of these waves in the presence of impermeable barriers is unwarranted. As we have shown elsewhere,2* all chemical waves are blocked by truly impermeable barriers. However if two points of the distributed system are in such a state that they will become oxidised in succession due to an established phase difference the introduction of a barrier between the two points will not prevent the successive oxidations to take place as expected.3. I have verified that no special increase in the number of leading centres can be recorded when dust is added directly into the Zhabotinksii oscillatory system. (One must keep in mind that new centres occur spontaneously and randomly during the whole reacting time). Although no positive results are obtained with dust a local small contamination of the solution by concentrated sulphuric acid has been shown to produce propagating waves that cannot be distinguished froin the spontaneous waves which are simultaneously observed.More experiments of this kind should be done in order to determine if a dust theory can really explain the most interesting waves in the Zhabotinksii system. Dr. J. F. G. Auchmuty and G. Nicolis (Brussels) (partly communicated) The classification of waves into “ pseudowaves ” and ‘‘ trigger waves ” based on P. Ortoleva and J. Ross J. Chem. Phys. 1972 58 5673 ;1974 60 5090. M-L. Smoes and J. Dreitlein,J. Cheni. Phys. 1973 59 6277. M-L. Snioes to be submitted. M-L. Smoes Ph.D. Dissertation (University of Colorado 1973). PLATE1 PLATE 3 PLATE 5 PLATE2 PLATE 4 PLATE6 PLATE PLATE 7 8 To face puge 991 GENERAL DISCUSSION the very interesting experiments by Winfree seems to be appropriate for describing the propagation of disturbances in chemical systems.However it seems that there can be other types of oscillations in such systems. In particular chemical system may undergo periodic oscillations which may even be spatially dependent (cf. the paper of Nicolis and Prigogine). These oscillations are different from those appearing in classical electromagnetism and elsewhere in that they are often independent of the initial conditions. Such oscillations appear as the asymptotic behaviour in time of these systems and are very similar to limit cycle behaviour for ordinary differential equations. Their characteristics depend very strongly on diffusion which is responsible for initiating the instability leading to these patterns and for synchronizing the local limit cycles.We believe that it is important to distinguish in chemical systems between transient behaviour and long-time behaviour and to point out that chemical systems can evolve to stable time-periodic and space-dependent states which are not typical propagating waves. Prof. E. Koros (Budapest) said Here I should like to report on our observations with the unstirred tris(dipyridine)ruthenium(II)-[Ru(dipy)~~]-catalysed Belousov system. It is known that Ru(dipy)z+ is a catalyst in the Belousov oscillating system,'. and our investigation on the Ru(dipy)i+-catalysed reacting Belousov system have revealed its close resemblance to the ferroin-catalysed This fact encouraged us to look at the spatial behaviour of the former system. The experiments were performed in a Petri dish of about 10 cm diameter at ambient temperature with the following reagent concentrations in the final mixture 0.3 M sodium bromate 0.065 M inonobromomalonic acid 0.05 M malonic acid 0.3 M sulphuric acid and 0.003 M Ru(dipy),Cl,.To a reaction mixture of about 9 in1 one drop of 0.1 % TritonX-I00 was added and swirled until the mixture was homogeneous. The temporal oscillation started inmediately after mixing the re- agents simultaneously a large number of pacemaker (trigger) centres formed and from some of them oxidation bands started to travel. This phenomenon is visible even in laboratory illumination; the colour contrast however is not marked enough. For this reason the photographs were taken in u.-v. light illuminating the solution with a 360-nni radiation.(Namely Ru(dipy):+ exhibits luminescence when irradiated with a 360-nm radiation ;however Ru(dipy)z+ can not be excited by u.-v. energy.) A series of 8 photographs demonstrates the generation and propagation of the chemical waves. Plate no. 1 was taken first and the following photographs were taken at suitable intervals. Plate no. 8 was taken approximately after 10 minutes. (The small bubbles on the photographs are from carbon dioxide.) With the Ru(dipy);+-catalysed system the development and propagation of trigger waves is accompanied by the occurrence of pseudo waves. During the early period pseudo waves annihilate most of the trigger centres and only a few of them can develop further. Plate 1. shows the system without trigger centres and when the catalyst is in the oxidized form [Ru(dipyi+] ;on plate 2 a few trigger centres (black spots) are already discernible their development can be seen on plate 3 pseudo waves propagating through the medium however annihilate most of the trigger centres (plate 4).Plates 5-8 show the trigger waves in the progressively developed phases. (Especially easily J. N. Deinas and D. Diemente J. Chem. Ed. 1973,50 357 E. KijrGs L. Ladanyi V. Fricdrich Zs. Nagy and A. Kis Reaction Kiiz. Cat. Letters 1974 1 355. E. KBriis h.1. Burger V. Frizdrich L. Ladinyi Zs. Nagy and M. OrbAn paper at this Sym-posium. 100 GENERAL DISCUSSION observable is the propagation of chemical waves from a trigger centre at the upper left part of the plates.) Similar to the ferroin-catalyzed system the leading edge of an oxidizing band is sharp the trailing edge of the band however is diffuse i.e.there is a continuous change from the totally oxidized form of the catalyst to the totally reduced one. Ru(dipy)g+ as a catalyst has a great advantage over ferroin the former being an inert complex and stable even in highly acid medium over a very long period of time. On the other hand ferroin is subjected to protolytic decomposition.l The chemical mechanism controlling the band migration is the same as that proposed by Field and Noyes 2* for the Fe(phen)i+-catalysed system. Dr. A. Winfree (Indiana) said My answers to various questions which have been posed informally are as follows (1) Could the observed scroll waves be transient rather than stable solutions to the reaction/diffusion equation? To which I would reply (a) Numerical solutions to a reaction/diffusion equation not unlike Z reagent include spiral waves rotating without detectable change of shape or angular velocity in a square box for 10 cycles.(b) Fig. 2 shows no change of period in 26 cycles of a scroll ring in Z reagent. Observations were terminated by a committee meeting not by an instability of the reaction. (c) Scroll waves seem to be attracted to interfaces and to counter-rotating scrolls less than 1/2 wavelength away. Tiny scroll rings may therefore be slowly contracting (without change of period according to fig. 2). Twisted and knotted scroll rings may have escaped detection up to now because they are violently unstable ; but possibly only because the required initial conditions have never been arranged.(d) The whole reaction dies out after enough malonate has been decarboxylated so in that sense all the waveforms are transients. (2) Busse asked what are the fuzzy red ripples sometimes seen near the end of an experiment with Z reagent. Well I don’t know what these are. They are not anything like scroll axes though. They seem to be less excitable regions plane-wave trains lying ob- liquely to these ripples are interrupted at crossings. They resemble thermal convection cells and may be regions where oxygen is transported into the medium from the air above. (3) Is the Z reaction really homogeneous or are there suspended particles gas bubbles surface films etc.of importance in determining wave geometry? I would state in answer to this question that (for reasons stated in the manuscript) I believe that the circular-wave pace- makers with diverse periods between T and To are heterogeneous nuclei. I have found no threadlike heterogeneity along the scroll wave’s axis. In reagent filtered at 0.2 pm and covered against dust scroll rings behave normally while microscopic observation reveals no turbidity gas bubbles surface films or suspended precipitate of broininated ferroin. B. Z. Shakhashiri and G. Gordon J. Amer. Chern. SOC.,1969 91 1103. * R. J. Field and R. M. Noyes Nature 1972 237 390. R.J. Field and R. M. Noyes J. Ainer. Chem. Soc. 1974 96 2001. GENERAL DISCUSSION Prof.R. M. Noyes (Oregoiz) said It is unfortunate the dithionite measurements of De Poy and Mason required titration of samples removed by pipette from a solution very sensitive to oxygen-initiated autocatalytic reaction. For the observations re- ported points from no more than two successive titrations deviated in the same direction from a smooth curve representing non-oscillatory autocatalytic dispropor- tionation of dithionite. Hence it is not established with certainty that oscillations even exist although the observations of pH and of turbidity are certainly very suggestive. Let us hope it will be possible to develop an analytical procedure that will continuously monitor dithionite concentration. Even if the reported oscillations in titre are real it is doubtful that they solely reflect changes in concentration of the dithionite starting material.If several percent of starting material is consumed and then regenerated a still greater amount of some intermediate must build up and then break down by disproportionation. The radical intermediates proposed here could not attain the concentrations necessary to produce the observed changes in titre. It therefore seems probable the points in fig. 2 represent composite concentrations of dithionite and of some yet unidentified intermediate species. Furthermore with regard to Matsuzaki’s paper I am disturbed about the pro- posed mechanism because it employs unprecedented species like H21203and H31305 because it regards elementary oxygen as an inert product and because it produces oxygen only by reduction of iodine containing species.It is well established that oxygen is produced most rapidly at times when iodine species are undergoing net oxidation. Sharma has recently shown that visible light can shift the system from a pre- sumably nonradical condition characterized by relatively high iodine ion concentration and slow evolution of oxygen to a radical condition characterized by much less iodide ion and by much faster oxygen evolution. He has also shown that the system becomes grossly supersaturated with oxygen during oxidation of iodine-containing species and that the pressure of oxygen has an important effect on reaction behaviour. These observations indicate that the mechanism involves autocatalytic switching between a nonradical condition during which iodate is reduced and a radical con- dition during which iodine is oxidized.A detailed mechanism of such type will be documented in a longer manuscript. Prof. D. M. Mason (Stanford) (communicated) We concur with Noyes that continuous observatons of the dithionite system are highly desirable and plan to use e.s.r. to follow some of the free radicals with time to determine if their concentrations also oscillate. We feel that the reproducibility of Rinker’s data coupled with the continuous pH measurements does provide strong evidence of the existence of oscillations of the dithionite concentration. We also agree with Noyes that other intermediates may affect the titre but that they could not exist in sufficient concentrations necessary to make the observed oscillations.It is for this reason that we feel that the dithionite is in effect an intermediate being consumed by the thermal decomposition and formed from side products with which it has reached equilibrium at the low temperature at which it is initially prepared. Prof. I. Matsuzaki (Japan) (communicated):In response to Noyes I would like to make the following remarks K. R. Sharma and R. M.Noyes J. Amer. Chem. Soc. 1975,97. GENERAL DISCUSSION 1. On the existence of H21203. This species was postulated by H. A. Liebhafsky.' 2. On the existence of H31305. This species is used in the paper as the key species for the oscillation. However its existence is left to be examined that is I do not at present intend to claim anything concrete about it more than is described in the text.From the standpoint of oxidation number it had better been considered to be a complex H21203 HI02. We merely devised this chemical formula as one possibility in order to embody the theoretical conclusion that the autocatalytic back-activation step to be contained should have an order higher than 1. 3. On the effect of light resulting in the decrease in [I-]. Let me consider the effect of light on the basis of our non-radical mechanism. According to our mechanism a decrease in [I-] results in an increase in [HIO] which in turn causes [HI02] to increase; these increases will favour the functioning of the autocatalytic back-activation step 2HI02+HI0 +H202+ 3HI02 +H20.In con- nection with the mechanism in which light causes [I-] to decrease a few runs of experi-ments with a 150 W tungsten lamp have been made with results for which the well- known reaction light 21-+ +02 +H20 + 12 +20H-might be considered to be responsible. With this finding I have come to think that addition of the above process to our current mechanism would extend the range of application of the mechanism to reaction conditions with irradiated light of varying intensities. 4. On the validity of a mechanism. I have not established the mechanism for the complex oscillation but merely proposed a plausible mechanism. To establish a mechanism for complicated phe- nomena we need to get a number of successful checks on its capability of accounting for a variety of phenomena.Prof. R.M. Noyes (Oregon) (communicated) :Because the hydrolysis of elementary iodine is a rapidly established equilibrium I must disagree with Matsuzaki's claim that a decrease in [I-] will immediately cause a decrease in [HOI] ;the effect will be in the opposite direction. However I agree with Matsuzaki that any decision about alternative mechanisms will require a more detailed analysis of kinetic data than is possible here. H. A. Liebhafsky J. Amer. Chent. SOC.,1931 53 2074.