ISSN:0301-5696    

DOI:10.1039/FS97409FX001

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

A. Introductory and Inorganic Oscillations Thermodynamic Aspects and Bifurcation Analysis of Spatio-temporal Dissipative Structures BY G. NICOLIS*’ AND I. PRIGOGINEJ~ FacultC des Sciences UniversitC Libre de Bruxelles 1050 Brussels Belgium Keceiryed 6th December I 974 The thermodynamic prerequisites for the emergence of spatio-temporal patterns of organization in nonlinear chemical systems are reviewed. Analytic expressions for steady-state time-periodic and travelling wave like solution of reaction-diffusion equations are reported. A comparison with the results based on computer simulations is outlined. 1. INTRODUCTION-THE THERMODYNAMIC BACKGROUND It is well-established that certain types of chemical reactions subject to appropriate conditions organize themselves spontaneously in space and time to give rise to regular steady state spatial patterns or to time-periodic flashes of chemical activity.Obviously self-organization phenomena of this kind are cooperative in the sense that they require a strong coupling between the different subunits constituting the system. However in contrast to phase transitions and other familiar examples of cooperative processes which come from equilibrium thermodynamics self-organization in the context of chemical reactions seems always to involve large scale macroscopic elements. Thus the patterns which are eventually established in a reacting mixture reflect in many respects the global properties of the system such as the size the symmetry of the spatial domain and most importantly the nature of the constraints acting from the external world.Because of this thermodynamics of irreversible processes constitutes the natural framework for studying these phenomena. The principal goal of this discipline is to describe the properties of macroscopic systems in terms of the properties of certain state functionals like entropy or entropy production. Now in the neighbourhood of the state of thermodynamic equilibrium the behaviour of these functionals is deter-mined by the theorem of minimum entropy production ’:In a system subject to time- independent constraints evolution leads to a steady state where the entropy production per unit time P takes a minimum value compatible with the constraints acting on the system dP -< 0.dt At the same time by the second law P 2 0. (1-2) Inequalities (1.1) and (1.2) imply the asymptotic stability of the branch of states t Also Center for Statistical Mechanics and Thermodynamics The University of Texas at Austin Austin Texas 78712 U.S.A. 7 SPATIO-TEMPORAL DISSIPATIVE STRUCTURES constituting the continuation of the equilibrium states hereafter referred to as the thermodynamic branch. Thus for a single phase system where the equilibrium state itself is stable the emergence of new types of cooperative behaviour is ruled out. The extension of thermodynamics away from equilibrium to the so-called nonlinear range of irreversible processes has proved that inequality (1.1) cannot be extended to this range. Instead we have derived a stability condition for the thermo- dynamic branch of states.It is found that stability will be ensured whenever for all t > to. Here S2P is the second order excess entropy production and SJ and SX are respec- tively the excess flows (e.g. reaction rates) and forces (e.g. chemical affinities) due to the deviation of the system from the reference state i.e. from the thermodynamic branch. A close analysis shows that inequality (1.3) can be violated in open systems obeying nonlinear kinetics and driven beyond a critical distance from equilibrium In such systems an instability of the thermodynamics branch will develop and evolve subsequently to a new regime. The threshold for this instability will be determined by the equation PP((l,}) = 0 (1*4) where (Af} is a set of critical values of the parameters-such as the intensities of the external constraints the chemical rate constants or the diffusion coefficients-which influence the evolution of the system.In this case we can see that cooperative behaviour is allowed as the system can conceivably leave the thermodynamic branch and thus exhibit markedly non-equilibrium behaviour. We arrive therefore at the conclusion that spatio-temporal organization in chemical systems is a supercritical phenomenon accompanied by an instability of the thermodynamic branch. This phenomenon is created and sustained by the constraints that is by the dissipative processes inside the system. Dissipation becomes here an organizing factor contrary to what common intuition would suggest.To stress this point we have called these supercritical organized states dissipative structures. Ironically the first chemical model showing some form of ordering-the Volterra-Lotka oscillator-turns out to be a non-typical example of cooperative behaviour.2* It belongs to the class of conservative systems which do not exhibit an instability of the thermodynamic branch. As a result the oscillatory behaviour is not asymptoti-cally stable and can therefore be destroyed by the least external perturbation. As we shall see in detail in the subsequent sections the crossing of a critical point of instability (see eqn (1.4)) is a sufficient prerequisite to guarantee the stability of the subsequently emerging pattern. We close this section by a short remark concerning the origin of the deviations from the reference state causing the appearance of an excess entropy production in relations (1.3) and (1.4).A macroscopic system involving many degrees of freedom is always endowed with an internal mechanism permitting such deviations namely the fluctuations. Thus a system near the state determined by relation (1.4) will always have a non-vanishing probability of reaching the unstable region through fluctuations. An interesting and most unexpected result of our recent work on fluctuations in nonlinear systems shows that only those fluctuations whose range exceeds some critical value will be able to amplify and induce cooperative behaviour. This critical size is the result of a competition between the natural growth rate of a local fluctuation and the influence of the surroundings tending to damp this fluctuation.We expect therefore to have an interesting phenomenon of nucleation of a gew kind whic4 G. NICOLIS AND I. PRIGOGINE suggests surprising analogies between chemical instabilities and phase transitiom6 Molecular dynamics calculations developed by Portnow and aiming to substantiate these ideas are presently in progress. These methods have been applied recently to questions such as metastable and unstable states as described in a discussion remark by R. Lefever 1. Prigogine and J. Turner in this Symposium. 2. MATHEMATICAL FORMULATION Chemical reactions are the most important elements responsible for evolution beyond the thermodynamic branch.Because of their nonlinear character they are capable of generating instabilities by amplifying small effects like small inhonio- geneities induced by external disturbances or by internal fluctuations. The latter can then propagate into the medium through the long range coupling between spatial regions provided by transport processes. Thus we expect that diflusion and heat conduction as well as convection should be important factors in the understanding of cooperative behaviour in chemical systems. In this paper we focus attention on the phenomena originated by the chemical reactions themselves and by the spatial inhomogeneities. Hence we shall assume that the system is isothermal and at mechanical equilibrium. Moreover we shall ignore the influence of electric or of any other types of fields.Let N1,.. .,N be the concentrations of the n chemical substances involved. These variables which by our assumptions provide a complete macroscopic description of the system evolve according to the reaction-diffusion equations dNi (i = I,. . . n) __ = v~((N~])+D~v~N~ a’t where D1 are the diffusion coefficients of species i (assumed to be constant) and ui the rates of production of ifrom the chemical reactions. In general the presence of feedback processes of various kinds will cause 2ii to be nonlinear functions of the (N,)’s. Thus eqn (2.1) constitute a nonlinear partial differential system of the parabolic type. In order to have a well-posed problem we will have to supplement these equations with appropriate boundary conditions on the surface I:of the spatial domain of volume V.Two types of conditions will be considered (i) Dirichlet conditions (Nf) = {const.) (2.2a) (ii) Neumann conditions (n-ViVf> = {const.). (2.2b) The latter type of condition with (const.) = (0)applies in most of the experimental investigations on the Belousov-Zhabotinski reaction although quite recently Marek reported experiments on this reaction under open system conditions. Conditions (2.2a) and (2.2b)represent the constraints acting on the system which in the most general case will maintain it away from the state of thermodynamic equilibrium. Note that the entire formulation adopted in this section assumes bounded media. One reason for this is that in chemistry and to an even greater extent in biology the size and the surface of the system play a very important role owing to the long range nature of diffusion which establishes a means for communication inside the system as well as between system and environment.The other reason for dealing with bounded media is that the mathematics becomes much more trans- parent. In particular one can construct explicit expressions for the various types of SPATIO-TEMPORAL DISSIPATIVE STRUCTURES solution of system (2.1) and study their stability. The requirement of stability is of course extremely important as it will determine among the solutions available those which will actually be realized. Owing to the nonlinearity the mathematical theory of systems of equations of the type (2.1) is still at a rather primitive stage.* At best one has some qualitative information about the behaviour of the solutions based on bgurcation theory lo supplemented with stability considerations.The purpose of this theory is to study the possible branchings of solutions that may arise under certain conditions. This is linked of course to the point of view of thermodynamics developed in the previous section according to which the emergence of dissipative structures implies the instability of the thermodynamic branch. Under appropriate boundary conditions eqn (2.1) admit a ungorm steady sfate solution on the thermodynamic branch. In this paper we shall analyze the transitions from this state to new types of solutions such as (a) space-dependent steady states ; (b)time-periodic (and space-dependent) solutions and (c) travelling waves and more general types of time-dependent solutions.Part of our analysis will be performed on the general form of the eqn (2.1). However quite often we will find it useful to illustrate the ideas on the simple trimolecular model 2* A+X 2X+Y + 3x (2.3) B+X-*Y+D X+E which was analyzed recently in detail from the standpoint of bifurcation theory.’ The astonishing element of most of the transitions leading to patterns (a) to (c) is their symmetry-breaking character. Beyond a certain critical set of values of the parameters defining a bifurcation point of the differential system (2.1) the most symmetrical solution of this system ceases to be stable.One then obtains a macro-scopic quantization? of the various new solutions whose properties are determined by a set of a few “ quantum numbers ” expressing the influence of the rate constants and the diffusion coefficients the symmetry of the spatial domain and the boundary conditions. In the subsequent three sections we shall review briefly the most characteristic phenomena arising in self-organizing chemical systems. 3. STEADY-STATE DISSIPATIVE STRUCTURES 3(i). D E L o cA L I z ED sT R ucT uREs Consider first the case for which the solutions on the thermodynamic branch are spatially uniform. Let -k, 4mbe respectively the eigenvalues and eigenfunctions of the Laplace operator within the spatial domain of interest. Their specific properties will of course depend on the details of the geometry.We shall come back to this point later in this subsection. The stability properties of the thermodynamic branch will be determined by the characteristic equation detI(aui/aNj)o-6:J(co +Dikm)I= 0 (3.1) * In this respect we may note that Thorn’s theory of catastrophes refers to systems described by ordinary differential equations. A brief comparison between dissipative structures and catastrophe theory has been attempted in ref. (9). t This term is due to M. P. Hanson J. Chern. Pliys. 1975. G. NICOLIS AND I. PRIGOGINE where (i,j) = 1 . . . ti and the spectrum of k is determined by the Laplace operator and the spatial domain. Suppose eqn (3.1) predicts a critical set of bifurcation parameter values such that Re Wa = 0 Im 001= 0 Re On*a < 0.Then one will have bifurcation of a new steady state solution which will depend on space. The condition for this is A = det I(au,/aNj)o-dk,’D,k,l = 0. (3.2) The important point is that for n 2 2 eqn (3.2) may be fulfilled for norz-trivial values of k,, provided the bifurcation parameters (A) are within certain limits. Let (Ac] be the first values of (3.) compatible with these conditions k the corresponding values of k,. Then in the neighbourhood of {Ac) the emerging steady state structure will have a non-trivial space-dependence determined to first approximation by this ki and by the corresponding eigenfunction 4;. For any given model one can compute this as well as higher order approximations using bifurcation theory.The method is described in detail in ref. (9) and (10) and will not be reproduced here. As an illustration we consider a one-dimensional system of length L. Then : @r k = (m= 0 1,. . .) (3.3a) and 4mK sin rnrir 0 6 r < L (3.3b) L for the boundary condition (2.2a),whereas mnr ~,occos-L O<r<L (3.3c) for zero flux boundary condition. Thus k describes the wavelength of the emerging dissipative structure. The critical value of this wavelength is determined by eqn (3.2). I 0 1 space/arbitrary units FIG.1.-One-dimensional steady state dissipative structure for X in the model reaction (2.3). NA = 2, NB= 4.6 Dx = 0.0016 Dy = 0.0080 and L = 1. The boundary values for Nx,Ny are & = NA = 2 Ry =NB/NA= 2.3.The critical wave number mC= 8. SPATIO-TEMPORAL DISSIPATIVE STRUCTURES The important point is first that for two or more chemical substances m will be finite ; and second that the critical wavelength will depend on the size of the system. We find here a striking manifestation of the global character of dissipative structures. An additional important element is that the symmetry properties of m influence profoundly the form of the solutions. When m,is even the system exhibits a symmetry- breaking transition to two possible new states both of which are stable. Fig. 1 represents the corresponding spatial pattern for the model reaction (2.3). For the same model the analytic calculation based on bifurcation theory yields m nr sin -(3.4) X(r) = A & a(B-Bc)* sin + b(B-B,) C m 1 mnr L ,, odd (m2-m:)2 m2-4m where a b are well-defined functions of the chemical parameters and the diffusion coefficients and the concentration B of the initial substance B has been used as bifurcation parameter.When m is odd one observes more complex phenomena like hysteresis. The corresponding spatial patterns for model (2.3) are shown in fig. 2. In both cases the emerging structures are shown to be stable. U I spacelarbitrary units FIG.2.-Steady state dissipative structure in the case of an odd critical wave number m = 7. Dx = 0.0018 and all other parameters have the same values as in fig. 1. Consider now a two-dimensional medium for instance a circle of radius R. Then (3.5a) where (r 0) are polar coordinates and J, is the Bessel function of order n.The eigenvalue k of the Laplacian is determined by the boundary conditions. For instance for zero flux boundary conditions J;(k,R) = 0. (3.5b) Again for two or more chemicals both n and k can be different from zero at the point of the first bifurcation. Fig. 3 represents the spatial patterns resulting beyond bifurcation for the model system (2.3) in the case where n = 0 k; # 0. This pattern G. NICOLIS AND I. PRIGOGINE is .cylindrically symmetric. If on the other hand both k; and n are different from zero the resulting pattern exhibits apolarity in the sense that a macroscopic gradient sets up spontaneously across the system.ll Fig. 4 describes the spatial form of the solutions for the model reaction (2.3).FIG.3.-Cylindrically symmetric steady state dissipative structure on a circle for the trimolecular model (2.3). The radius of the circle is R = 0.20 and all other values are the same as in fig. 1. FIG.4.-Polar dissipative structure on a circle for model (2.3) in the case of zero flux boundary conditions. R = 0.10 Dx = 0.00325,Dy = 0.0162 A = 2 and B = 4.6. In the simplest cases of bifurcation the resulting pattern reflects to a good approximation the properties of the critical modes i.e. of the eigenfunctions of the Laplacian at the bifurcation point. In situations involving high symmetries like for example a square domain it is not uncommon to have as first bifurcation a state corresponding to a degelzerate eigenvalue of the Laplacian.The spatial structures are then a complex mixture of the degenerate eigenfunctions and the multiplicity of solutions can become very high. The situation is described in fig. 5. A closer study reveals that as the size of the system increases different dissipative structures can be realized successively. Thus below a critical size R,,,the only SPATIO-TEMPORAL DISSIPATIVE STRUCTURES possible solution is the uniform one on the thermodynamic branch. Between R, and R,,the system may exhibit a non-polar pattern like the one in fig. 3. Then between A, and R, the latter could be succeeded by a polar pattern such as that in fig. 4. Beyond R,,all steady state solutions may become unstable and a propagating wave can emerge (see section 4).Finally for still larger sizes the system may undergo homogeneous bulk oscillations with the same phase everywhere within the reaction space.12 Now growth is a very general phenomenon occurring in all living organisms. The fascinating point is that according to the previously developed ideas growth will engender a whole succession of different forms or patterns. FIG.5.-Steady state dissipative structure on a square for model (2.3) in the case of double degeneracy. The side L = 0.132 and the boundary conditions are zero flux ones. A = 2 B = 4.6 Dx = 0.0016 and Dy = 0.0080. To arrive at this structure initial perturbations along a diagonal of the square had to be considered. The intrinsic dependence on length described above is a particular case of a more general phenomenon occurring quite frequently in nonlinear chemical systems.Namely when a bifurcation parameter takes different values a whole succession of instabilities can take place leading to more and more complex spatial forms. These can be either bifurcations from the thermodynamic branch leading to a multiplicity of dissipative structures each one having its own domain of attraction of a set of initial conditions; or they can be the result of a secondary bifurcation from a previously established dissipative structure. Unfortunately in both cases bifurcation theory very rarely permits a rigorous study of the stability of the various states. An interesting thermodynamic property of these multiple transitions is that they may lead to an increase in dissipation as the wavelength of the structure becomes shorter.3(ii). LO cA L I zED STR uCTuREs In many multicomponent systems the diffusion of some of the chemicals may create an inhomogeneous environment for other chemical intermediates. In this G. NICOLIS AND 1. PRIGOGINE case the solutions on the thermodynamic branch are no longer space-independent. Computer simulations l4 have revealed the existence of spatial patterns which remain localized within a part of the reaction space. Recent theoretical analysis of these states 13? l5 has shown that these patterns are again of the dissipative structure type as they arise beyond the instability of the thermodynamic branch. Fig. 6 describes the spatial distribution of concentrations in a one-dimensional system for the model reaction (2.3).Xt 0 0.5 I space/arbitrary units FIG.6.-One-dimensional localized steady state dissipative structure for mode1-(2.3). DA = 0.197, Dx = 0.001 05 Dy = 0.005 26 B = 30 X = k = 14 = B/A = 2.14. It is important to realize that this localization phenomenon may be spontaneous. No spatial gradient needs to be preimposed. On the other hand localization may result in a weaker dependence of the properties of the dissipative structure on the system as a whole e.g. on the size. Still complete independence of the size is difficult to imagine or to justify mathematically. It appears tempting to suggest that localized structures are good candidates to explain the formation of leading centres in some systems like the Belousov-Zhabotinski reaction.In this respect however it is necessary to point out that the complete independence of those leading centres of the size of the system postulated by Zaikin and Zhabotinski l6 is never fulfilled rigorously in localized structures. 4. TIME-PERIODIC SOLUTIONS According to linear stability analysis a time-periodic solution will bifurcate in the neighbourhood of the thermodynamic branch provided the characteristic eqn (3.1) admits a pair of purely imaginary roots +ioG1 at the bifurcation point. In chemical systems involving only two variables the first bifurcation leads necessarily to a state where ki = 0.l’ If this state is allowed by the boundary conditions then the system will presumably exhibit this most symmetrical solution whose space dependence will be trivial.This actually happens in the case of zero flux boundary conditions where the system exhibits limit cycle oscillations throughout the reaction space. In contrast constant boundary conditions will in general rule out the state with ki = 0. The first allowed bifurcating solution will then exhibit both space structure as well as temporal periodicity. Fig. 7 describes this solution for the model reaction (2.3). Using bifurcation theory one can calculate explicitly these solutions in the neighbourhood of the bifurcation point. For contrast boundary conditions and for a SPATIO-TEMPORAL DISSIPATIVE STRUCTURES one-dimensional system involving two chemical intermediates one obtains the following form for the deviation X(Y t) from the thermodynamic branch sin -cos pt + x(r) = a(~-~,)f (3 00 cos 2pt) sin (Y) b(B-B,) (a i-d -+O((B-Bc)).(44 m=l odd The important point in this expression is the presence of a fundamental mode of period 2n/p and of its first harmonic. This is a general property of all systems in which the first nonlinearity around the thermodynamic branch is quadratic.” t = 7.97 spacelarbitrary units FIG.7.-Characteristic stages of evolution of the spatial distribution of X during one period of the wave for model (2.3) for a length L = 0.80and for constant boundary conditions. A = 2 B = 12.6, Dx = 0.0080 Dy = 0.0040. Relation (4.1) describes a superposition of standing waves. However the numerical evaluation of this formula for model (2.3) reveals a pattern which is strikingly similar to fig.7. It appears therefore that in a bounded medium propagating concentration fronts can appear during certain time intervals but not during the entire period of the phenomenon. Waves of this kind are capable of transporting matter during macro-scopic time intervals as the (Vx) at a certain point retains a given sign during these intervals. However they differ markedly from the waves encountered commonly G. NICOLIS AND I. PRIGOGINE in mechanics or in electromagnetic theory in the sense that it is meaningless to define for them a propagation velocity depending on x through the usual relationship In chemical systems involving more than two coupled variables an additional important element appears in that the first bifurcation can lead to states with k; # 0.’ Thus even for zero flux boundary conditions these systems will exhibit symmetry- breaking transitions leading to states with non-trivial space dependence. For instance in a circular domain the first bifurcating solution will in some cases be approximated by the function (4.3a) with (4.3b) At each instant of time the equal concentration curves +ml = const. will be symmetrical with respect to a diameter of the circle. As time proceeds this diameter will rotate at constant speed. Thus the time-periodic solutions of this form represent rotating waves although their form is not spiral in the approximation provided by relations (4.3). It is quite possible however that a bifurcation theoretical calculation leading to an expression such as eqn (4.1) will produce spiral waveforms arising from the influence of higher order terms.The latter contain a series having J,J(kmr)as coefficients which could break the symmetry of the equal concentration curves with respect to the diameter of the circle. The numerical evaluation of the bifurcation theoretical expression for a circle for the model reaction (2.3) is in progress. Let us recall that the appearance of angle-dependent solutions beyond the first bifurcation from the thermodynamic branch occurs typically in systems with three or more variables. In this respect we may note that the Noyes-Field model for the Belousov-Zhabotinski reaction involves three intermediates and therefore fulfils these conditions.Additional variables which could play an important role in the Belousov-Zhabotinski reaction as well as in other chemical systems are the tempera- ture and the electrical field. The latter could be especially crucial in the generation of electrical waves capable of interfering with or even triggering the chemical waves. Other possibilities of wave forms having a symmetry less than the symmetry of the spatial domain include secondary bifurcations or multiple stable solutions as described at the end of subsection 3(i). At present it appears to be very difficult to investigate the stability of these solutions although one can construct them explicitly using bifurcation theory. 5. TRAVELLING WAVES The requirement of temporal periodicity imposed on the solutions calculated in the previous section turned out to be incompatible with the existence of a well-defined speed of wave propagation although as we saw expressions such as eqn (4.1) could approximate propagating fronts as closely as may be desired.In this section we review briefly the problem of existence of travelling waves by requiring solutions of eqn (2.1) and (2.2) of the formf(v-uvt). For details we refer the reader to a recent paper by Auchmuty and Nico1is.l’ Work in this direction in the case of unbounded media has been reported recently by Kopell and Howard l8 and by Ortoleva and Ross. SPATIO-TEMPORAL DlSSIPATIVE STRUCTURES 5(i). WAVES IN RINGS Let 2n be the length of the ring. The Laplace operator reduces to a single derivative along the ring.We want to find solutions of the form &(r t) =L(r-of)=A(<> (5.la) (5.1 b) The reaction-diffusion equations take the form . n). (5-2) -l;f,/(<) = Difl(t)+vi((fj>) (i = 1 We wish to determine those values of u for which there are non-trivial periodic solutions of (5.2) Ni = N;+ui exp(im<) 112 = 0 1,. . . (5.3) Note that both (5.16) and (5.2) are compatible with the existence of a constant speed of propagation. A linear analysis of eqn (5.2) confirms this point and provides expressions for the critical values of the bifurcation parameter as well as for the speed of propagation. The solutions can then be constructed using bifurcation the0ry.l’ As in section 4 for systems with two variables the first bifurcation occurs at rn = 0.Thereafter the system undergoes a uniform limit cycle type oscillation. For three or more intermediates the first bifurcation can lead to a non-trivial spatial dependence in the form of a propagating pulse. 5(ii). MORE GENERAL ONE-DIMENSIONAL SYSTEMS We now consider a system with open ends subject to the boundary conditions Ni(O t) = Ni(L t) = NP (5.44 or dNi dlvi -dr (0,t) = -(L 1) = 0 (5.46) a). where NP are the values of the (uniform) chemical concentrations on the thermo- dynamic branch. It can easily be seen l7 that such boundary conditions rule out automatically the existence of solutions of the form (5.14 with v constant. We relax these conditions and seek solutions of the form A detailed analysis shows l7 that a solution of the form (5.5) can emerge beyond asecondary bifurcation of a steady-state dissipative structure.The solvability condition of the bifurcation equations will provide us with a set of partial differential equations which determine the propagation velocity v. The ability of these waves to arise in a non-uniform background provides a mecha- nism of localization of these waves as well as a possible formation of leading centres. 5(iii). TRAVELLING WAVES IN SPACES OF HIGHER DIMENSION By an argument similar to that developed in the previous subsection one can show that plane waucs of the form f(k Y-cot) are ruled out in two or higher spatial G. NICOLIS AND I. PRIGOGINE dimensions.The possibility remains however of having superpositions of such waves resulting in a wave packet of the form Ni(r t) = S dt d~ dCf(S V C) exPti(k. r-u(k)t)l (5.6) where k = (c q 5). The group velocity drldk of this wave packet will play the role of the propagation velocity. Again when the geometry of the problem is specified one can look for specific wave forms like cylindrically or spherically symmetric waves spirals and so on. 6. CONCLUDING REMARKS From this analysis of nonlinear chemical systems we have seen that a wealth of structures presenting a wide variety of sizes and shapes becomes possible beyond an instability of the thermodynamic branch. It is remarkable that in spite of the extreme diversity of these far-from-equilibrium phenomena it has been possible to reach general conclusions on two fundamental aspects of dissipative structures.On the one side nonlinear thermodynamics and bifurcation theory have enabled us to provide a rough classijication as well as to construct explicit expressions for the various possible types of organized states. On the other side nonlinear thermodynamics supplemented with fluctuation theory has enabled us to get a better understanding of the mechanisms prevailing in the transitions between states. Finally it is important to stress the analogies between these and similar phenomena encountered in various branches of natural sciences such as fluid dynamics shock waves and detonations ecology and even social behaviour. One can hope that the progress accomplished during the last few years in the field of nonlinear chemical kinetics will provide the tools necessary to tackle complex problems in these fields which until now have eluded quantitative analysis.It is a pleasure to acknowledge that a substantial part of the mathematical results of this paper have been obtained in collaboration with Prof. J. F. G. Auchmuty. We are also indebted to Dr. Herschkowitz-Kaufman Dr. Lefever Mr. Malek-Mansour and Mr. Erneux for their important contributions in the research reported in this paper and for many stimulating discussions. The financial support of the “ Ministhe de I’Education Nationale et de la Culture fraqaise et le Ministke de I’Education Nationale et de la Culture NCerlandaise ” from Belgium and of the Welch Foundation Houston Texas is gratefully acknowledged.’I. Prigogine Bull. had. Roy. Belg. 1945 31 600. P. Glansdorff and I. Prigogine Thermodynamicsof Structure Stability arid Fluctuations (Wiley-Interscience New York 1971). For a historical survey of the concept of dissipative structure and of its precursors we refer to a recent review paper by R. Glansdorff and I. Prigogine Bull. Acad. Roy. Belg. 1973,59 672. G. Nicolis and J. Portnow Chem. Rev. 1973,73 365. G. Nicolis M. Malek-Mansour K. Kitahara and A. Van Nypelseer Phys. Letters 1974 48A 217. ti G. Nicolis J. Stat. Phys. 1972 6 195 ;G. Nicolis and I. Prigogine Proc. Natl. Acad. Sci. 1971 68 2102 ;A. Nitzan P. Ortoleva and J. Ross this Symposium. ’J. Portnow and Svobodova Phys. Letters 1975 51A 370.* M. Marek Biophys. Chem. 1975 3 263. G. Nicolis and J. F. G. Auchmuty Proc. Natl. Acad. Sci.,1974,71 2748. lo D. Sattinger Lecture Notes iit Mathematics vol. 309 (Springer Verlag Berlin 1973). A. Babloyantz and J. Hiernaux Bull. Math. Biol. 1975. SPATIO-TEMPORAL DISSIPATIVE STRUCTURES l2 A. Goldbeter Proc. Nutl. Acud. Sci. 1973 70 3255 M. P. Hanson J. Chem Phys. 1974 60,3210. l3 J. F. G. Auchmuty and G. Nicolis Bull. Math. Biol. 1975 37,323. l4 M. Herschkowitz-Kaufman,Bull. Math. Biol. 1975. J. Boa Ph. D. Dissertation (Calif. Inst. of Technology 1974). l6 A. M. Zhabotinski and A. N. Zaikin J. Theor.Biol. 1973 40 45. J. F. G. Auchmuty and G. Nicolis Bull. Math. Bid 1975. l8 N. Kopell and L. Howard Studies in Appl. Math. 1973 52 291. l9 P. Ortoleva and J. Ross,J. Chem. Phys. 1974,60 5090.

ISSN:0301-5696    

DOI:10.1039/FS9740900007

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

A Model Illustrating Amplification of Perturbations in an Excitable Medium BY RICHARD AND RICHARD J. FIELD M. NOYES Department of Chemistry University of Oregon Eugene Oregon 97403 U.S.A. Received 6th July 1974 The oscillatory Belousov-Zhabotinskii reaction can be modelled approximately by five irreversible steps A+Y-+X (M1) X+Y+P (Ma B+ X-+2X+ Z 043) 2X+Q (M4) Z+fY. (M5) These equations are based on the chemical equalities X = HBr02 Y = Br- 2 = 2Ce(IV) and A = B = BrO;. If the rate constants kMl to kMs are assigned by experimental estimates from oxybromine chemistry the kinetic behaviour of the model depends critically upon the remaining parameters kMs and$ Whenfdoes not differ too greatly from unity and when kMsis not too large the steady state is unstable to perturbation and the system oscillates by describing a limit cycle traject- ory.Whenfand kMs lie outside the range of instability the steady state is stable to very small perturb- ations. However the steady state may still be excitable so that perturbation of the control inter- mediate Y by a few percent will instigate a single excursion during which concentrations of X Y and Z change by factors of about lo5before the system returns to the original steady state. This ability of a small perturbation of the steady state to trigger a major response by the system is just the type of behaviour necessary to explain the initiation of a trigger-wave by a heterogeneous"pacemaker " as has been observed by Winfree. The same type of excitability of a steady state has important implica- tions for the understanding of biochemical control mechanisms.The so-called Belousov-Zhabotinskii reaction occurs in one of the most versatile of chemical systems. If a sulphuric acid solution of bromate ion and malonic acid contains a catalytic amount of a one-equivalent redox couple (such as Ce(II1)-Ce(IV) or Mn(1I)-Mn(II1)) with a reduction potential between about 1.0 and 1.5 V several remarkable types of behaviour are possible. In a stirred homogeneous solution the degree of oxidation of the redox couple may oscillate repeatedly with time. In an unstirred solution containing a gradient in temperature or in the concentration of some reactant variations in the phase of local oscillation can cause alternating bands of oxidation and reduction to propagate through the system ; these phase-related oscillations have been called pseudo-waves by Winfree.' In an unstirred but initially homogeneous solution local disturbances such as dust particles may generate regions of oxidation that propagate more rapidly than individual molecules could diffuse and leave refractory reduced regions behind them ; these disturbances are called trigger-waves by Winfree.' Finally if developing trigger-waves are disturbed in various ways they may develop very complicated spiral structures called scroll-waves by Winfree.2 References to these various types of behaviour may be found in a recent re vie^.^ 21 AMPLIFICATION OF PERTURBATIONS The detailed chemical mechanism of the temporal oscillations has now been elucidatedy4# and a simplified mathematical model developed to exhibit the same type of limit cycle behaviour.(j The same mechanism has been used both quali- tatively ’and quantitatively to describe the chemical processes taking place in the sharp leading edge of a trigger-wave.Although the previous discussion *demonstrated how a region of oxidation could propagate through the medium as a trigger-wave it did not concern itself with the initiation of such oxidation. In the present paper we show that the same model suggests how a modest perturbation in the concentration of bromide ion could initiate a transient region of oxidation that would then propagate as a trigger-wave. CHEMICAL MECHANISM The significant features of the chemical mechanism can be summarized by the following processes BrO +Br-+2H++HBr02 +HOBr HBrO +Br-+H++2HOBr BrO; +HBrO +H+-+2Br02 +H,O Ce3++BrOz+H+-,Ce4++HBrO 2HBrO,-+BrO +HOBr +H+ (C4) nCe4++BrCH(COOH),+nCe3+ +Br-+oxidized products.(C5) Steps (Cl) (C2) and (C4) are assumed to be bimolecular elementary processes involving oxygen atom transfer and accompanied by rapid proton transfers ; the HOBr so produced is rapidly consumed directly or indirectly with bromination of malonic acid. Step (C3a) is rate-determining for the overall process of (C3a) +2(C3b). The Ce4+ produced in step (C3b) is consumed in step ((25) by oxidation of bromo- malonic acid and other organic species with production of bromide ion. The complete chemical mechanism is considerably more compli~ated,~ but the simplified version presented here is sufficient to explain the oscillatory behaviour of the system.COMPUTATIONAL MODEL The significant kinetic features of the chemical mechanism can be simulated by the model we have called the Oregonator? A+Y+X X+Y+P B+X+2X+Z 2x-Q z+.. This computational model can be related to the chemical mechanism by the identities A =B =BrO; X =HBrO, Y =Br- and Z =2Ce(lV). The Oregonator functions because the switched intermediate X is generated by steps (Ml) and (M3) that are zero and first order in X and is destroyed by steps (M2) and (M4)that are first and second order in X. When the concentration of the control R. J. FIELD AND R. M. NOYES intermediate Y is sufficiently large the concentration of X attains a steady state approximated by Xsmall.When the concentration of Y is sufficiently small X attains a different steady state approximated by X,,,,,. kM3B-kM2Y Xlnrgc = 2kM4 ' The concentration of X is switched between the two steady states whenever the concentration of Y attains the critical value Ycrit. ycrit = kM3B/kM2. (3) The regeneration intermediate is produced in significant amount only when X is in the Xlargc steady state and it regenerates Y until the critical concentration is attained and X is switched to Xsma,,.The Concentration of Zthen decreases and Yis destroyed until it again reaches Ycritand permits the concentration of X to be switched again. Thedifferential equations describing temporal behaviour can be cast in terms of the dimensionless variables u 9 p and z and the dimensionless parameters q s and w.= JkM3B/kMlA kM5 w= Jk,,khf3AB' These differential equations then become da/dz = s(q -qu +a -qa2) dvld7 = (1 Is)( -9 -9a+.fP) (12) dp/dr = w(u-/I). (13) We have examined the Oregonator under conditions such that the rate constants for the first four steps correspond to our best estimates for the situation in a solution with 0.06M KBrO and 0.8 M H2S04. Then q = 8.375 x s = 77.27 and w = 0.1610 kM5. When these values are selected the kinetic behaviour depends upon the two parametersf and kM5. APPLICATIONS TO UNSTABLE AND EXCITABLE SYSTEMS It is possible to determine steady state values for the concentrations of the three intermediates such that daldz = dq/dz = dp/dr = 0.Then a secular equation analysis AMPLIFICATION OF PERTURBATIONS of the linearized kinetic equations will reveal whether small perturbations from the steady state will decay or whether they will grow and lead to a limit cycle trajectory around the steady state. I I 1 (a9935,4so.os) 0.0 1.0 2.0 F FIG.1.-Regions of stable and unstable Oregonator steady states for q = 8.375~ s = 77.27 and IV = 0.1610kM,. The steady state is stable to oscillation except for sufficiently small values of kM5 with fin the neighbourhood of unity. The results of such an analysis are shown in fig. 1. This figure differs slightly from one presented previously which was based only on a sufficient condition for instability of the steady state.It is impossible to have a steady state if k, = 0 and fig. 1 demonstrates that the steady state is always stable for sufficiently large values of kM5. Unstable steady states are observed for some finite positive values of kM5provided the stoichiometric parameter fdoes not differ too much from unity. Stable steady states with fappreciably less than unity correspond to Y < Ycritand X = Xlarge. Stable steady states with f appreciably greater than unity correspond to Y > Ycrit and x = L3ll. Trigger-waves of the type studied by Winfree involve regions of oxidation that advance into regions of excess bromide ion (Y > Ycrit)that are marginally stable to oscillation. Such a region can be modelled by a stable point near to the right bound- ary of the region of instability defined by fig.1. As Winfree has pointed out trigger-waves initiate at a " pacemaker " such as a speck of dust that locally perturbs the marginally stable solution and generates a region of oxidation that then propagates as a trigger-wave. If our model is appro6 priate a small perturbation of the steady state should generate a single pulse of oxidation followed by return to the steady state. Moreover the system should be particularly sensitive to perturbation of the control intermediate Y. Exactly these predictions are demonstrated by fig. 2 to 5 illustrating the behaviour of a system initially at the stable but excitable steady state for f= 1.5 and kM5= 2 s-I. Fig. 2 to 4 are logarithmic plots showing the behaviour of the various inter- mediates following a discontinuous reduction of Y (or q) by 6.5 % at time z = 153.R. 3. FIELD AND R. M. NOYES I -__ 1----r--I-si 1 I I 50 150.0 300.0 450.0 7 FIG. 2.-Logarithmic plot of v(Y or Br-) following a discontinuous perturbation of -6.5 :< from the steady state. Perturbation occurred at T = 153. 7 Fro. 3.-Logarithmic plot of a(X or HBrOJ following a 6.5 % decrease of 7from the steady state situation. Perturbation occurred at 7 = 153. 7 FIQ,4.-Logarithmic plot of p(Z or Ce(1V)) following a 6.5 % decrease of 7 from the steady state situation. Perturbation occutced at 7 = 153. AMPLIFICATION OF PERTURBATIONS -lr----- 7 II I 1 I500 I600 1700 180.0 7 FIG.5.-Linear plot of q(Y or Br-) showing subsequent effects of 6.0 and of 6.5 % discontinuous decreases from the steady state at T = 153.The sensitivity of the system clearly is very dependent upon the magnitude of the perturbation. A similar reduction of 6 % generated a ripple barely observable on this logarithmic scale. Fig. 5is a linear plot of the region near Ycritshowing that the 6.5 % perturbation is sufficient to cause the system to attain the critical switching condition while the 6 % perturbation is not sufficient to do so. Even much larger perturbations of the switched (Xor a) and regeneration (2or p) intermediates generated only very minor excursions from the steady state situation. DISCUSSION The above calculations offer a convenient model for the very large (non-linear!) amplification of a finite but minor chemical perturbation of a steady state system.Just such amplification must be involved in whatever process initiates a trigger-wave in a marginally stable Belousov-Zhabotinskii solution. Similar amplification is presumably involved whenever a minor change in conditions triggers a complicated sequence of events in a living organism or biological system. The transmission of a nerve impulse is an example of just such a triggered response. Although these calculations were initiated to model the excitable medium into which the trigger-waves of Winfree would advance the analogy may not be exact. Trigger-waves are observed in a thin sheet of solution very few millimetres thick.The solution in such a sheet is indeed marginally stable but excitable. However we find that the same solution in bulk is unstable to temporal oscillation although with a very long time period. We suspect that the thin sheet is maintained almost saturated with atmospheric oxygen and that the reaction is thereby influenced. We do not know enough detailed chemistry to model this suspected oxygen effect with confidence. Presumably the free radical processes of M3 or M5 are affected. We find that even a factor of two change in kM3changes by only 0.2 % the critical kM5 at which the steady state becomes unstable. It therefore appears that oxygen exerts its influence on the stoichiometry of the system. If oxygen inhibits the attack of BrO radicals on cerium(III) less than two X species would be produced in step M3.If oxygen promotes radical attack on bromomalonic acid with resulting bromide ion production theffactor in step M5 would increase; fig. 1 shows that such increase could stabilize a previously unstable steady state. Therefore the fact that Winfree solution is unstable to bulk oscillations does not affect the basic validity of the model we develop here for chemical amplification of a marginally stable system. R. J. FIELD AND R. M. NOYES ' A. T. Winfree Lecture Notcs 011Biomatlrenratics ed. P. van den Driessche (Springer-Verlag Berlin) in press. A. T. Winfree Science 1973,181 937. R. M. Noyes and R. J. Field Ann. Reu. Phys. Chem. 1974,25,95. R. M. Noyes R. J. Field and E. Koros J. Arner. Chern. Soc. 1972,94 1394. R. J. Field E. Korljs and R. M. Noyes J. Amer. Chem. SOC.,1972 94 8649. R. J. Field and R. M. Noyes J. Chem. Phys. 1974,60,1877. 'R. J. Field and R. M. Noyes Nature 1972 237 390. R. J. Field and R. M. Noyes J. Arner. Chem. Soc. 1974 96 2001. A. T. Winfree Science 1972 175,634.

ISSN:0301-5696    

DOI:10.1039/FS9740900021

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

Chemistry of Belousov-type Oscillating Reactions BYE. K~ROS,M.BURGER L. LADANYI, V. FRIEDRICH Zs. NAGY AND M.ORBAN Institute of Inorganic and Analytical Chemistry L. Eotvos University H-1443 Budapest P.O.B. 123 Receified 30th Jirly 1974 The heat output and the accumulation of monobromomalonic acid during chemical oscillation in the bromate malonic acid sulphuric acid and catalyst [Ce(III) Mn(II) Fe(phen)$+ Ru(dipy);+] systems were measured and the results interpreted. The behaviour of the reacting oscillating systems depends to a certain extent on the catalyst applied. Both the rate of heat evolution and the rate of accumulation of monobromomalonic acid are periodic. The heat of the cerium-catalysed Belousov oscillating reaction is 130+ 5 kJ mol-l bromate.Many teams are involved in studying either experimentally or theoretically the Belousov oscillating systems. Most of the experimental investigations have been performed on the cerium(III) bromate malonic acid and sulphuric acid system and now much is known both about its chemistry and mechanism. Recently Field and Noyes have generalized the chemical mechanism of Field K6ros and Noyes for the Belousov reaction by a model composed of five steps involving three independent intermediates (Br- Ce(1V) and HBrO,). The model exhibits limit cycle behaviour. Koros has treated chemical oscillation as a " mono-molecular " reaction regarded the oscillatory Belousov reaction a series of auto- catalytic reaction bursts (bromate bursts) occurring with a certain frequency and calculated the " activation parameters "for reacting oscillatory systems with different catalysts.Also the preliminary studies on the Ru(dipy)i+-catalysed Belousov system have been rep~rted.~.~ K6rOs et aZ.6 have shown periodicity in the rate of heat evolution and Degn 'in the rate of carbon dioxide evolution during temporal chemical oscillation. Despite the large body of information available on the Belousov systems there are still points to be cleared and revealed. Our researches have been directed (a) to perform calorimetric measurements on reacting Belousov systems (b) to reveal the mechanism of the important composite reactions and (c) to determine the products of the reaction early during oscillation. Here we report on our calorimetric investigations on the Ce(II1)- Mn(I1)- Fe(phen)$ and Ru(dipy)$+-catalysed bromate malonic acid and sulphuric acid +-systems and on our polarographic measurements aimed to follow the accumulation of monobromomalonic acid (BrMA).EXPERIMENTAL MATERIALS All reagents used were of analytical grade. 28 BOROS BURGER FRlEbRlCH LAbANY'I NAGY ANb ORBAN 29 METHODS POTENTIOMETRY Chemical oscillation was followed either by monitoring the M("+l)+/Mn+ratio or by following the bromide ion concentration.2 (M stands for the catalyst.) POLAROGRAPHY The concentrations of BrMA and brornate in the oscillation mixture were measured polarographically. The polarographic determination of BrMA has been described by US.^ Bromate ion gives a well-defined wave on dropping mercury electrode its E+ value varies widely with the hydrogen-ion activity of the solution.* In order to separate the two waves an appropriate pH value was chosen.BrMA concentration was measured by withdrawing aliquots from the reaction mixture and adding to an acetate buffer of about pH 3.5 of the same volume. Then the BrMA wave was recorded. As a maximum suppressor 0.002 % Triton X-100was used. In order to determine bromate concentrations the buffer samples were diluted 50 times with 0.5 M sulphuric acid and the bromate wave recorded. The actual concentrations were calculated from calibration curves. CALORIMETRY The calorimetric experiments were performed in a calorimeter of a constant temperature environment type. The measuring vessel was doubly jacketted and thermostatted by an ultrathermostat.A 6 WZ thermistor forming one arm of a d.c. Wheatstone bridge was used as a sensor; the change in its resistance was continuously recorded during the reaction. This system allowed 0.2 J to be detected equivalent to about 1x lop4"C temperature change in the calorimeter vessel. The initial temperature of reactants and environment were kept at 25°C. The oscillating reactions were started by adding the catalyst to the reaction mixture. The total volume of the reaction mixtures was 200mI. The solutions were stirred. The calorimeter was calibrated both chemically and electrically. PROCEDURE The chemical oscillation was initiated by adding an appropriate amount of a catalyst solution to a mixture of malonic acid potassium bromate and sulphuric acid thermostatted to 15.0 25.0 and 35.0+O.l0C,respectively.From the reacting oscillating reaction mixture aliquots were withdrawn at certain intervals and analysed for BrMA and bromate contents. With Fe(phen)$+ as a catalyst nitrogen was bubbled through the reaction mixture to remove dissolved atmospheric oxygen which is an inhibitor for the rea~tion.~ RESULTS CALORlMETRIC MEASUREMENTS ON REACTING OSCILLATING SYSTEMS The heat outputs of the different reacting Belousov systems are compiled in tables 1-4. TABLE 1.-ce(III)+ MALONIC ACID+ KBr03+ H2SO4 SYSTEM MA = 0.40 M ;HzS04 = 0.5 M heat output/J induct ion i 2 3 5 10 war)x 1041~ period 0.100 5.74 602 21.3 20.1 19.7 18.9 17.2 0.100 11.5 582 39.0 38.1 37.2 36.1 31.5 0.190 46.0 569 89.5 83.3 79.1 72.4 64.0 0.050 11.5 259 22.4 21.6 20.9 18.9 14.2 P 163 11.5 837 53.1 51.0 49.4 47.3 41.s 30 BELOUSOV-TYPE OSCILLATING REACTIONS ACCUMULATION OF BrMA IN REACTING OSCILLATING SYSTEMS One of the main products of the oscillating Belousov reactions is BrMA.We regarded as of importance to follow the accumulation of BrMA in different reacting TABLE 2.-Mn(lI)+ malonic acid+ KBrO,+ H2S04 system MA = 0.40 M ;HzS04 = 0.5 M heat output/J induction 1 2 3 5 10 KbrO3/M Mn(1I)x lO4/M period 0.100 1.15 305 3.4 3.3 3.2 3.1 2.7 0.100 4.60 3 10 23.5 22.6 21.5 29.8 17.7 0.100 11.5 305 49.8 46.4 43.5 41.4 35.1 0.100 46.0 308 111 105 97.9 89.5 78.2 0.025 11.5 17.6 17.2 14.9 13.6 6.4 1.9 0.050 11.5 97.1 34.5 30.9 29.5 26.2 20.5 0.163 11.5 458 62.3 57.7 54.8 49.0 45.6 TABLE 3.-Fe(phen)$++ MALONIC ACID+ KBr03+H2S04 SYSTEM MA = 0.4 M ;H2S04 = 0.25 M heat output/J KbrO3/M Fe(phen);+ X 104 /M intrqductoryperiod/min induction period 1 2 3 5 10 0.100 1.44 1.5 11.9 8.4 8.5 8.6 0.100 5.74 2.2 37.1 27.8 28.2 30.0 30.4 34.9 0.100 11.5 4.2 90.8 64.0 64.5 65.0 69.0 0.050 5.74 >60 34,3 no oscillation even after 1h 0.060 5.74 18 37.2 34.8 35.6 0.070 5.74 6.2 36.8 30.0 32.4 34.1 38.1 0.130 5.74 1.6 38.1 28.8 29.7 30.8 32.0 37.7 0.175 5.74 1.1 38.2 28.0 28.5 29.0 29.3 31.8 TABLE 4.-Ru(dipy);++ MALoNlc ACID+ Dr03f &So4 SYSTEM MA = 0.40 M ;HZS04 = 1.0 M heat output/J Ru(dipy)$+xlo* induction 1 2 25 KBrOa/M IM period 0.100 0.92 14.9 2.9 3.1 5.4 0.100 2.76 23.9 7.8 8.O 21.2 0.100 4.60 31.8 13.0 13.6 37.4 0.125 4.60 30.6 13.7 14.4 30.6 0.150 4.60 30.6 14.6 15.6 37.2 Belousov systems aiid at different temperatures early during oscillation when other brominated products (dibromomalonic acid dibromoacetic acid) are not likely to be formed.To meet this goal a polarographic method has been developed to determine even M BrMA with a few percent accuracy in the presence of 100to 500times bromate. Also the concentration of bromate has been measured polarographically. The results on the different Belousov systems are compiled in fig. 1-9. KOROS BURGER FRIEDRICH LADANYI NAGY AND ORBAN 1 5 I0 IS 20 25 time/min FIG. 1.-The accumulation of BrMA as a function of time at 15 25 and 35"C respectively.The chemical system containing Ce(NO& (0.005 M) malonic acid (0.30 M) KBrO (0.10 M) and HzS04 (0.50 M). 25 I 51I 5 10 15 20 25 D period number FIG.2 .-The accumulationof BrMA as a function of period number at 15,25 and 35"C respectively.The chemical system containing Ce(N03) (0.005 M),malonic acid (0.30 M) KBrO (0.10 M) and HZS04 (0.50 M). BELOUSOV-TYPE OSCILLATING REACTIONS 18 rr) 2 16 x iL 7 12 3 10 a8 6 4 2 5 I0 15 20 25 30 35 40 ' ti melmin FIG.3.-The accumulation of BrMA as a function of time at 15 25 and 35"C respectively. The chemical containing MnS04 (0.005 M) malonic acid (0.30 M) KBr03 (0.10 M) and H2S04 (0.50 MI.i 5. period number FIG.4.-The accumulation of BrMA as a function of period number at 15,25 and 35"C respectively. The chemical system containing MnS04 (0.005 M) malonic acid (0.30 M) KBrO (0.10 M) and HzS04 (0.50 M). KOROS BURGER FRIEDRICH LADANYI NAGY AND ORBAN 25OC b 10 20 30 LO 50 60 70 timelmin Fx. 5.-The accumulation of BrMA as a function of time at 15 25 and 35"C,respectively. The chemical system containing Fe(phen);' (0.0005 M),malonic acid (0.40 M) KBr03 (0.10M)and HZSO (0.25 M). 1 c 5 I0 15 20 25 30 35 period number FIG.6.The accumulation of BrMA as a function of period number at 25 and 35"C respectively. The chemical system containing Fe(phen)i+ (0.0005 M) malonic acid (0.40 M) KBrOJ (0.10 M) and H2S04 (0.25 M).1 5 10 15 20 25 30 35 time/min FIG.7.-The accumulation of BrMA as a function of time at 15 25 and 35"C,respectively. The chemical system containing Ru(dipy)$+ (0.0005 M) malonic acid (0.40 M) KBr03 (0.10 M) and HzS04 (1 .O M). S 9-2 BELOUSOV-TYPE OSCILLATING REACTIONS I8 I 5 10 I5 20 25 30 35 40 period number FIG.&-The accumulationof BrMA as a function of period number at 15,25 and 35"C,respectively. The chemical system containing Ru(dipy);+ (0.0005 M) malonic acid (0.40 M) KBr03 (0.10 M) and H2S04 (1.0 M). t I I I 5 I0 15 20 25 30 35 period number FIG.9.-The accumulation of BrMA as a function of period number at 25°C. The chemical systems containing malonic acid (0.40 M) KBr03 (0.10 M),H2S04(1.0 M) and Ce(N03)3 (0.0005 M) or MnS04 (0.0005 M) or RuCdipy):' (0.0005 M).DISCUSSION In the Belousov oscillating systems heat starts to evolve immediately after addition of the catalyst. With the Fe(phen)$'- and Ru(dipy):+-catalysed systems there is no induction period; thus the first period will be called the introductory period. For the Fe(phen)g +-catalysed system it is characteristic that independent of the bromate concentration approximately the same amount of heat evolves during the introductory period the rate of heat evolution however differs considerably (table 3). At the same time the heat output during the introductory period increases with increasing catalyst concentration. The same is observable with the Ru(dipy)$+- catalysed system; there are however not enough results on this system so far (table 4).On the other hand the Ce(II1)- and Mn(1I)-catalysed systems exhibit an induction period. The heat evolved during the induction period is practically independent of the catalyst concentration but increases however closely linearly with the bromate KOROS BURGER FRIEDRICH LADANYI NAGY AND ORBAN concentration (tables 1 and 2). This indicates that irrespective of the catalyst concentration in a particular system a certain amount of BrMA should accumulate (vide infra) the system to be converted to an oscillatory one. The BrMA concentra- tion required for this depends on the initial bromate concentration. In other words only after the necessary reactant concentrations are adjusted are conditions set for the initiation of chemical oscillation.With the same initial conditions higher BrMA concentration is required for the onset of oscillation in the Ce(II1)-catalysed system as in the Mn(I1)-catalysed one. After the induction (or introductory) period periodicity in the rate of heat evolution is observable. This is expected from the nature of oscillatory reactions ; thus reports on temperature fluctuations lo*l1 or the claim that there is no overall heat production in the reaction are err0neous.l During the oscillatory phase of the reaction the Fe(phen)g+- and Ru(dipy)i+- catalysed systems behave differently from the Ce(II1)- and Mn(I1)-catalysed ones. With the former systems early during oscillation the heat output in one period increases as the reaction proceeds then it reaches a maximum and starts to decrease.(Similar features can be observed with the potential oscillations.) This phenomenon is more pronounced with the Ru(dipy):+-catalysed system and a detailed investigation of it is in progress. When Fe(phen)S+ is used as a catalyst the decomposition of the complex should be considered and thus the concentration of the catalyst decreases in time. Namely iron(I1) formed by decomposition is not a catalyst for the Belousov system.12 In spite of a decrease in catalyst concentration potential oscillation could be recorded throughout a period of 36 h. With the Ce(II1)- and Mn(I1)-catalysed systems the heat evolved in one period decreases as the reaction proceeds the damping factor being higher in the case of the Mn(I1)-catalysed reaction.With the Fe(phen)$+- and Ru(dipy)$ +-catalysed systems the heat output of one period increases closely linearly with the concentration of the catalyst-which means that during the autocatalytic reaction the total amount of the catalyst is oxidized- and only slightly or not at all depends on the bromate concentration. With the Ce(II1)- and Mn(I1)-catalysed systems the above linearity does not hold i.e. only a certain proportion of the catalysts is oxidized in the autocatalytic reaction; there is however a considerable dependence on the bromate concentration. All these consequences follow from the redox potential data of the catalysts. The results on the accumulation of BrMA provide us with valuable information concerning the reacting Belousov oscillating systems.It has been established that during the induction (or introductory) period a considerable amount of BrMA accumulates and this increases during the oscillatory phase. The ([BrMA] time) plots with the Ce(II1)- and Mn(I1)-catalysed systems show an excessive increase in the rate of accumulation of BrMA with increase of temperature (fig. 1 and 3) ;on the ([BrMA] period number) plots however the curves of 35°C lie below those of 25°C (fig. 2 and 4). The same temperature dependences regarding the rate of BrMA accumulation are observable with the Fe(phen);+- and Ru(dipy)g+-catalysed systems (fig. 5 and 7). The ([BrMA] period number) curves however have nearly the same slopes (see fig.6 and 8).This indicates that although the period time decreases with increase of temperature the amount of BrMA formed during one period is nearly the same. With the Fe(phen)g+-catalysed system at 15°C the oscillatory conditions have not been met even after 100 min. The BrMA concentration reached a value of 6 x M after 5 min and 1.6 x M after 100 min. The results of the experiments performed on systems of the same initial concentra- tions (fig. 9) show that the rate of accumulation of BrMA depends on the nature of BELOUSOV-TYPE OSCILLATING REACTIONS the catalyst however the " activation energy " (Em)and ''entropy of activation " (AS;) of chemical oscillation are the same in the case of the three system^.^ (The Fe(phen)z+-catalysed system does not oscillate in 1 M sulphuric acid.) Polarographic measurements of BrMA and bromate in the same mixtures have revealed that early during the oscillating reaction the BrO -+BrMA transformation holds.The changes in bromate concentration however could not have been followed accurately enough since very little differences were to be measured and thus only an accuracy of 10 % has been achieved.* The method developed for the direct measurements of BrMA in low concentrations enabled us to gain more detailed information on the chemistry of the oscillatory Belousov reactions as obtained by Hess eb ~l.'~-'~ They identified the reaction products (BrMA and dibromoacetic acid in a ratio of about 6 :1) extracted by ether from the reaction mixture after two hours reaction time by gas chromatography and mass spectrometry.After a short reaction time (7 min) only BrMA was identifiable by thin-layer chromatography. BrMA is formed during the autocatalytic step in the reacting Belousov system and thus periodicity in its rate of formation is exceptable. To prove this a run has been performed with Ce(II1)-catalysed Belousov system thermostatted to 15.0 & 0.1"Cwhere the period time is long enough-about 2 min. Samples were withdrawn from the reaction mixture after the autocatalytic step before the autocatalytic step (at the end of the " restful" phase) and again after the autocatalytic step and analysed for BrMA content. The results are given in fig. 10. * 14 16 18 20 22 t irnelmin FIG.10.-Bromide ion and BrMA concentration against time curves during the 6th and 7th periods.The initial chemical system containing Ce(NO& (0.005 M) malonic acid (0.30M) KBrO (0.10 M) and H2S04 (0.50 M). Temperature 15°C. Thus we were able to provide experimental evidence for the periodic rate of formation of BrMA. By making use of the calorimetric and BrMA-concentration measurements we were able to calculate the heats of Belousov oscillating reactions referred to 1 mol bromate. For the cerium-catalysed Belousov system the heat of the oscillating reaction in 130+ 5kJ mol-l bromate. * Recently an accuracy of +3 % has been achieved by iodometric titration. KOROS BURGER FRIEDRICH LADANYI NAGY AND ORBAN With our aim in mind of a deeper insight into the heat effects we have started to measure the heats of reaction of some of the composite reactions of the Belousov systems.The results obtained so far are summarized briefly below (a) The catalyst- bromate reactions have rather low heats of reaction. Considering the BrO; +4Mn++5H+ -+ HOBr+4M"+')++2H2O reaction,15 and this has been proved also for the Fe(phen)$+-BrO; reaction,16 the heat of reaction in 0.25-1 M sulphuric and referred to 1 mol bromate is 10.1 kJ with Ce(1II) and 6.70 kJ with Fe(phen)i+ as reductant. (b)The heat of reaction of the bromination of malonic acid (MA + 3 Br * BrMA) in 0.5 M sulphuric acid referred to 1 g atom of bromine is 138 kJ. CONCLUSION The driving force of the Belousov oscillatory processes is the high heat of reaction of the bromate-malonic acid reaction.This reaction however proceeds in steps following a rather sophisticated mechanism. The amount of bromate present in the system can deliver its stored energy only at that stage (phase) of the reaction (and only in small portions) when the concentration of bromide ion present in the system drops below a critical value. This provides a precondition for the autocatalytic reaction to be switched on. Thus bromide ion acts as a trigger and its concentration changes periodically in time. This happens also with the oxidized form of the catalyst and with the bromous acid (HBr02).2 However this latter can not be followed analytically. Besides these intermediates exhibiting temporal concentration oscillation the end products of the reaction (e.g.C02 BrMA) are the non-recycling species. All non-recycling species show periodicity in their rates of formation. With carbon dioxide and BrMA this has been verified experimentally. It is however obvious that the rate of consumption of bromate and malonic acid should also be periodic in nature. Finally the periodicity in the rate of reaction manifests itself in the periodicity of heat evolution. R. J. Field and R. M. Noyes J. Chem. Phys. 1974,60,1877. R. J. Field E. Karos and R. M. Noyes J. Amer. Chem. SOC. 1972 94 8649. E. KGros Nature 1974. E. K6ros L. Laddnyi V. Friedrich Zs. Nagy and A. Kis Reaction Kin. Car. Letters 1974,l. J. E. Demas and D. Diemente J. Chem. Ed.,1973 50 357. E. Kdros M. Orbhn and Zs. Nagy Nature (Phys. Sci.) 1973 242 30.H. Degn Nature 1967 213 589. * J. Heyrovsky and J. Kuta The Principles of Polarography (Publishing House of the Czecho- slovak Academy of Sciences Prague 1965). M. T. Beck and Z. Vhradi Chem. Comm.,1973,30. lo U. Franck and W. Geiseler Naturwiss. 1970 58 52. l1 H. G. Busse Nature (Phys. Sci.) 1971 233 137. l2 L. Bornmann H. Busse and B. Hess 2.Narurforsch. 1973 28b 93. l3 L. Bornmann H. Busse B. Hess R. Riepe and C. Hesse 2.Naturforsch. 1973,28b 824. l4 L. Bornmann H. Busse and B. Hess. 2. Naturforsch. 1973 28b 514. l5 R. C. Thompson J. Amer. Chem. SOC. 1971,93,7315. l6 E. Khras M. Burger and A. Kis Reaction Kin. Cat. Letters 1974 1.

ISSN:0301-5696    

DOI:10.1039/FS9740900028

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

Two Kinds of Wave in an Oscillating Chemical Solution BYA. T. WINFREE Department of Biological Sciences Purdue University West Lafayette Indiana 47907 USA. Received 6th September 1974 Wavelike phenomena in a chemical solution oscillating at period T fall into two classes (1) diffusion-independent but oscillation-dependent structures repeating at intervals T and (2) diffusion-dependent but oscillation-independent structures derived from threadlike filaments of “ scroll axis ” repeating at intervals TOmuch less than T. This paper experimentally examines a typical example of each. In unfiltered solution one usually observed a third class of wave source forming “ target patterns ” at diverse periods intermediate between T and To. KINEMATIC WAVES AND TRIGGER WAVES Distributed throughout a large enough space any unstirred chemical oscillation generally exhibits local parameter variations affecting the local period.Thus phase gradients develop and steepen even from initial synchrony of the bulk oscillation. Unless reactants and products have identical optical properties these changing phase gradients can be seen as moving bands of colour passing every point at regular intervals of time equal to the bulk oscillation period T. Called “ kinematic waves ” by Kopell and Howard,’ such waves have also been studied theoretically by Smoes and Dreitlein,2 by Th~enes,~ and by Ortoleva and Ross.~’ Kopell and Howard,l and Thoenes have additionally exhibited such waves in the oscillating reagent of Belousov and of Zhabotinsky and Zaikin.’ Provided that the oscillating reagent is initially well stirred and left in a homo- geneous environment no chemical gradients remain.The oscillation period being therefore everywhere the same any phase gradients subsequently introduced by a local disturbance will persist without change until molecular diffusion becomes sig- nificant on a scale embracing the affected area. In the diffusion-free uniform- period approximation the corresponding kinematic waves have been called “ pseudo-waves ”. With or without spatial variations of period such waves merely expose a shallow timing gradient in the spatially distributed but locally autonomous oscillation. Wave velocity is inversely proportional to the steepness of the phase gradient and so has no upper bound.Such waves do not involve diffusion are not conducted through the medium and are not impeded by impermeable barriers. However in sufficiently steep phase gradients diffusion cannot be ignored. In the chemical reagent of Belousov6 and of Zhabotinsky and Zaikin7-called 2 reagent henceforth-a new phenomenon emerges from steep phase gradients which assigns a lower bound to pseudowave velocity a pulse of chemical activity is trig- gered and propagates at a velocity v characteristic of the chemical medium. These have been called “trigger waves ”.9* They are arrested at impermeable barriers and have spatially uniform velocity rather than spatially uniform repeat time. Both 38 A. T. WINFREE solitary trigger waves and regularly spaced wave trains appear in both oscillating and non-oscillating versions of Z reagent.THE SLOWEST PSEUDOWAVES Though pseudowaves slower than the conduction velocity are not observable (because they trigger the faster wave) pseudowaves at all velocities down to this limit are readily demonstrable in oscillating Z reagent. In fact pseudowaves travelling at almost exactly the conduction velocity are the commonest sort. This is because every volume element turns blue spontaneously at intervals Tafter passage of a trigger wave. 35 END OF BULK OSCILLATION 30 25 c -3 2 8 20 M (d 2 cw 0 p, 2 15 10 5 0 0 10 20 40 60 period/s FIG.1.-The periods of three kinds of wave in filtered Z reagent are plotted against age of the reagent.Open circles represent high-frequency waves deriving from scroll rotation all have the same period To,which gradually increases. Solid circles represent bulk oscillation ; its period T approaches infinity after half an hour. Letters represent concentric rings filling the holes left by extinct scroll sources the period Tidentifies them as pseudowaves. Raw data underlying the dotted circle and the open circle connected to D by dashes are given in fig.2. For example when a source of trigger waves abruptly ceases to radiate (see Appendix) the last wave emitted (at t =0) sets up a radial phase gradient as it propa- gates away leaving an initially wave-free hole. But within that hole volume elements at distance r from the centre spontaneously turn blue at times nT+r/u.Thus behind the packet of trigger waves all spaced apart by distance Too,there appear concentric pseudowaves spaced apart by distance Tu > Tov all following the last trigger wave TWO KINDS OF WAVE IN AN OSCILLATING CHEMICAL SOLUTION outward at approximately* the conduction velocity u. This abruptchangeof period is shown in fig. 1 and 2. In fig. 1 are recorded the periods of various phenomena in a 2 mm deep layer of oscillating Z reagent at short intervals during a 40 min period after mixing of the ingredients (see Appendix for details). The open circles record a slow increase in the SCROLL RING @ # PERIOD 13 SECONDS PSEUDOWAVE D SCROLL WAVE PERIOD 29 SECONDS PERIOD II SECONDS event number FIG.2.-Successive closures of the inward wave from a tiny scroll ring are recorded by plotting event time against event number in the upper trail of data points.The lower trail shows a scroll source discontinuing its periodic wave emission and the same space being filled with pseudowaves ; their times of appearance at the centre are recorded. period of diverse trigger-wave sources this is the period of scroll rotation. The solid circles record an increase in the period of bulk oscillations in wave-free regions and their eventual termination after half an hour (note that scroll sources continue). The letters A-F record the intervals between eruption of circular waves at the centre of the “hole ” left by a receding packet of nearly circular trigger waves after its source vanishes each letter represents the mean period of all the waves within one expanding hole.* Since u does increase somewhat with wave spacing it is possible that the conduction slightly outraces the pseudowave. If so then if an impermeable barrier could be implanted without deform- ing the liquid (I have not found a way!) the wave striking one side would continue from the other side after a slight delay proportional to the barrier’s distance from the centre or from the next wave approaching from the centre whichever is less. A. T. WINFREE The observations underlying three of these data points are shown in fig. 2 where the time of each event (appearance of a wave wave passing a marker or collision of waves) is plotted in serial order.The series of 12 observations plotted in the lower right depicts a scroll source emitting progressively less eccentric oval waves at an 11 s period until the last one is nearly circular. Thereafter circular waves appear at 29 s intervals close to the simultaneously-measured period of bulk oscillations. This transition is indicated by the dashed line in fig. 1. The other 5 lettered cases show that the period of these hole-filling target patterns approximates to the bulk oscillation period. After the end of bulk oscillations no such waves appear in new holes. The upper series of observations in fig. 2 shows the 13 s period of a horizontal scroll ring approximately ;t mm in diameter shedding waves to the inside and outside all of which appear circular in projection.This observation is entered in fig. 1 as the dotted open circle. In unfiltered reagent the space between curves T and Toin fig. 1 is typically filled with many trails of data points depicting the diverse and increasing periods of indi-vidual pacemaker centres each generating a “ target pattern ” of concentric trigger- wave circles at its own period intermediate between Tand To. ROTATING PSEUDOWAVES? The phase gradients constituting pseudowaves A-F were radial spanning many cycles of bulk oscillation from centre to receding periphery. But unlike most thermodynamic variables the phase of an oscillation is defined mathematically on the unit circle rather than on the real line. Thus it seems possible in principle for phase to continually increase through any number of complete cycles around a closed path in space.The pseudowave seen on such a circular gradient would rotate about a fixed centre. Such pseudowaves have been invoked to explain the rotating spiral waves seen in 2 reagent but these turn out to be trigger waves by the criteria that their period To is much less than T that they are totally blocked by impermeable barriers and that they appear in non-oscillating reagent as well as in oscillating reagent. Rotating pseudowaves remain to be observed in chemically oscillating media though there may be biological examples.lo ROTATING TRIGGER WAVE SOURCES However the principal mode of diffusion-dependent organization in Z reagent (at the ubiquitous period To)seems to be rotation of appropriately crossed concen- tration gradients.A two-dimensional region (a thin film of liquid) organized in this way emits a rotating trigger wave shaped like an Archimedes’ spiral. Its colour pattern rotates about a slightly wobbly pivot. In three dimensions the pivot becomes a 1-dimensional filament threading through the liquid like a vortex line in classical hydrodynamics. However here there is no fluid motion. From this filament there emerges a scroll-shaped wave of excitation. Its perpendicular cross-section resembles an Archimedes’ spiral. The geometry of this pivot-filament the scroll axis deter- mines the spatial and temporal organization of the oscillating reaction. The scroll axis tends to close in rings except where a nearby interface prevents it.Rings several centimetres in circumference are common. The smallest rings yet seen are about one wavelength in circumference leaving just room enough for organization of the scroll core (itself of circumference equal to one wavelength) around the circular scroll axis. TWO KINDS OF WAVE IN AN OSCILLATING CHEMICAL SOLUTION DlSSECTlON OF A SCROLL RING In order to examine the anatomy of a scroll ring in more detail it is convenient to use a version of the oscillating reagent which conducts waves well while absorbed in the pores of a Millipore filter (see Appendix for details). Impregnated filters can be stacked to make a horizontally-laminated three-dimensional medium with roughly isotropic conduction properties. By assembling such stacks from filters already bearing waves it is possible to induce three-dimensional concentration patterns of peculiar geometry.The waveforms which eventually develop from such contrived initial conditions can be examined in detail by tossing the Millipore stack into a preservative bath then reassembling an image of the wave from its fixed horizontal sections. From computer siniulations of two-dimensional media resembling a thin layer of Z reagent one learns that crossed concentration gradients of the sort needed to create a scroll axis are formed when a solitary wave is suddenly brought into contact with inert medium for example as in fig. 3. The pivot of a spiral wave appears near the FIG.3.-A spiral wave can be created by abutting a block of inert reagent against the endpoint of a plane wave in another block.The spiral's centre appears near the initial point of contact. initial point of contact. Imagine fig. 3 spun about its right edge to form a pair of cylinders the bottom one now containing an outward propagating cylindrical wave. This suggests a trick suitable for implementation in a Millipore stack. Two stacks each about 0.7 mm deep are prepared by stacking 5 filters permeated with Z reagent as described in the Appendix. A hemispherical wave is started in one stack by touching the centre of its upper surface with an electrically heated filament. After a minute the wave becomes a cylinder several mm in diameter extending vertically through the stack and propagating outward. When the second stack still inert is set on top the situation of fig.3 is created in every vertical plane through the axis of the cylinder. The locus of pivots for the spiral wave emerging in each such plane is a horizontal circle roughly coincident with the upper edge of the cylindrical wave at the moment of contact. From this ring-shaped filament a scroll wave emerges propagating in all directions throughout the Millipore stack (fig 4.) Every horizontal cross-section of this wave consists of concentric circles. Every FIG.5.-A scroll ring in a stack of 10 Millipore filters is shown in serial section parallel to the circular scroll axis perpendicular to the axis of circular symmetry. The top side of each filter is shown. They are about 0.15 mm thick. Number 10 was on top.Each panel is a 1cm square. The black slash across each panel shows the section line along which the view in fig. 6 was taken. To face page 431 A. T. WINFREE vertical cross-section through the symmetry axis shows segments of two spirals one on the left and a mirror image on the right . Fig. 4 attempts a schematic of 6 stages during one rotation of the spiral showing only the left side spiral; the right edge of the box is the vertical symmetry axis of the Millipore stack. It will be noted that the wave erupts alternately at equal intervals of one half period through the top and bottom Millipores above and below the initial ring of contact. The inside half of this wave continues inward to annihilation on the symmetry axis while the outer half continues outward.rj 4-r2 FIG.4.-A spiral wave in a rectangular piece of medium is shown at 6 equally spaced times during the cycle. The successive panels were drawn by rotating an involute spiral 60" each time. In the actual experiment circular waves emerged through the opaque upper surface (filter No. lo) approximately above the initial cylinder after 25 s then after 50 s more then after another 50 s then after 40 s. The 40-50 s period of the 3-dimensional wave is close to the 45 s rotation period of a well-developed spiral in a parallel 2-dimensional control experiment in a single thickness of Millipore. (The bulk oscillation period is several minutes). 25 s later the stack was dispersed into cold fixative while the trailing edge of the fourth wave was still passing vertically through filter No.1. Fixed in this way wave thickness is about 3mm. Successive waves follow one another by 13 mm at a velocity of 2 mm/min. Fig. 5 shows the upper surfaces of corresponding 1 cm squares cut out of the ten filters in order from 1 (the bottom) to 10 (the top). Waves propagate with the sharp edge foremost the inner circle is moving inward the outer circles outward. In filters 1 and in 4-7 we also see vertically propagating waves only the trailing edge of one in filter 1 but the full thickness of 3 mm in filters 4-7. At the left edge a wave is TWO KINDS OF WAVE IN AN OSCILLATING CHEMICAL SOLUTION entering from another source. A gas bubble seems to have inhibited propagation between filters 3 and 2.Filters 5 and 8 show air bubbles caught under the glass during photography. Each of the 10 prints is marked with a section line fig. 6 assembles a vertical cross-section through the stack along this line shown as though seen from the top the upper half of each print discarded. The position of each wavefront in each layer is marked by a wedge pointing in the direction of propagation. The imaginary spirals connecting these data include arcs about 3wave spacing above the top filter J 1 CM. x FIG.6.-Wave fronts along the section lines in fig. 5 ale marked by wedges pointing in the direction of propagation the apex at the wave front. The successive sections are stacked 0.15 mm apart in this scale reconstruction of a vertical section through the Millipore stack.An imaginary spiral and scroll core boundary are superimposed. (fixed 25 s = 3cycle after passing a vertical wave) and less than 3 mm below the bottom filter (which shows that wave’s trailing edge as a broad swath). The assumed position of the scroll axis is indicated by two circles bounding the scroll core. Core circumference being 14 mm,9 its diameter is not distinguishable from the + mm thickness of waves fixed in this way. Fig. 6 may be compared with stage 2 of fig. 4. CONCLUSION Wave phenomena of two kinds distinguished by their periods arise in Z reagent Those with period T derive from phase gradients in the spontaneous bulk oscilla- tion. They have been analyzed extensively in the theoretical literature. They have been exhibited in Z reagent in the one-dimensional long-wavelength case.l* Their independence of diffusion has been shown experimentally by their passage through impermeable barriers1 Fig.1 and 2 describe such waves in the two- dimensional case in the short-wavelength limit. Those with period To < T occur only in two- and three-dimensional situations as spirals and scroll waves respectively. Some theoretical discussion of such waves has appeared. * By obstructing their passage with impermeable barriers they have been shown to be “ trigger waves ” critically dependent on diffusion. Fig. 5 and 6 show a ring-shaped scroll wave in Z reagent. Wave phenomena at intermediate periods are much more commonly observed in unfiltered Z reagent.7 Since most of these trigger wave ‘‘ target patterns ” are eliminated by careful filtration and can be restored by deliberate contamination with dust they are viewed as consequemxs of local shorter-period oscillation near a hetero-geneous nucleus.A. T. WINFREE These experiments were made possible by NSF Grant 37947 and an NIH Research Career Development Award. APPENDIX The pseudowave experiment uses an oscillating cerium+ malonate reagent similar to that of Kopell and Howard,l but with excess ferroin to enhance visibility 2.15 g cerous nitrate hexahydrate in 250 ml water 1 volume 36.0 g malonic acid in 250 ml water 2 volumes 13.5 g potassium bromate in 250 ml water 2 volumes 55.0 ml sulphuric acid plus 200 ml water 4 volumes 1.0 ml Triton X-100 surfactant in 15 1.water 1 volume 25 millimolar ferrous 1,lO-phenanthroline sulphate 2 volumes It is important to avoid the E. Merck ferroin widely available in Europe. This is made with a chloride salt ; the chloride is not removed and poisons the reaction. The solution is filtered through a Millipore GSWP 0.2 micron " Millex "filter into a fresh Falcolnware petri dish. Lining the dish with Sylgard silicone resin helps in eliminating pacemakers and the "target patterns " of trigger waves they emit. The dish is placed over a blue-green fluorescent light-box baffled to avoid heating. The lid is coated with a 61mof 0.1 % Triton X-100 to prevent fogging. A magnifying glass is helpful. The few remaining pacemakers produce circular waves (or a hot needle is used if there are none) which must be sheared by gently tilting the dish before the bulk oscillation annihilates them.From their wreckage a diversity of scroll sources emerge generating diversely con- voluted patterns (" intestines ") all characterized by the ubiquitous period and wavelength of the involute spiral wave. Some form tiny scroll rings or fragments of rings interrupted by a single airlliquid interface. Some of these contract until they abruptly vanish leaving a series of closely-packed waves to propagate away. The empty hole expanding around the annihilated source then fills from centre outward with concentric pseudowaves. The times of their appearances at the centre are noted to the nearest second with a stopwatch. Some-times instead their passage by a fixed scratch or bubble on the floor of the dish is noted.The results are similar and are plotted in fig. 1(A-F). The bulk oscillation period is measured by recording the times at which the reagent suddenly turns more transparent (in blue-green light) at a fixed wave-free place in the dish as well as in a separate wave-free dish of the same reagent. The interval between this event and the next appearance of a blue spot in the centre of one of the above holes remains the same within a few seconds attesting the synchrony of both bulk-oscillations. Scroll rotation period is measured by recording the times when the inward circular wave in a scroll ring contracts to a point and vanishes ; or when waves from two or more separate scroll sources collide and annihilate each other the intervals are the same within several percent.Usually several of these diverse sources were watched simultaneously noting event times in as many columns of a table. The times were later plotted and periods were read from the slopes of the resulting line segments as in fig. 2. The results from two independent experi- ments were indistinguishable ; fig. I combines them. Herman Gordon pointed out that these curves become much smoother when observations are restricted to one place in the dish. This is because the oscillation period varies several percent from place to place in the dish and because intervals between wave collisions or wave passages reflect the (generally shorter) period of wave appearances at their source some time earlier.The Millipore stack experiments use a published recipe * but with 1 ml instead of 2 nil of sulphuric acid. Consistent wave propagation depends on the purity of the sodium brom- ate ; even reagent grade samples sometimes need to be recrystallized. This reagent does not oscillate when poured in thin layers exposed to the air presumably due to oxygen interfering with free-radical processes. However it does oscillate in bulk with a period of several minutes when confined in a sealed tube or in a thin layer under oil or in Millipores under a glass coverslip. (Sometimes the reagent poured in a petri dish will oscillate in the deeper meniscus though not in the shallower centre ; then a blue wave starts around the rim of the dish and TWO KINDS OF WAVE IN AN OSCILLATING CHEMICAL SOLUTION propagates into the centre once every several minutes).The more acid reagent * behaves similarly in air but must be spread in a much thinner layer or divided into fine droplets to inhi bit oscillation completely. The filters are 23 mm diameter Millipore GSWP numbered with tiny Magic Marker dots and filled with 2 reagent by floating glossy-side-up on the liquid surface. With nylon forceps they are lifted free stacked on Plexiglas (Perspex) covered with a microscope cover- slip and drained of excess liquid into the corner of a Kleenex tissue. The wave is started with a 3 V penlight its bulb removed and the tungsten filament carefully stretched out to a point which becomes hot but does not glow.The two stacks are joined before a bulk oscillation erases this single cylindrical wave. Because carbon dioxide placques quickly accumulate between the filters the whole process must be terminated within several minutes (at 20°C) the stack is impaled with a needle to assist in orienting the numbered filters later then dispersed in cold 3 % perchloric acid. After several minutes each filter is sealed under a coverslip and photographed on high-contrast film in blue light. About 20 such experiments were run. Scroll rings were caught at several of the stages of rotation depicted in fig. 4. Several stacks produced very complicated structures possibly due to invisible gas bubbles separating the filters. In some of the very regular ones like fig. 5-6 the period of wave eruption on the visible surface of the stack was as long as 70 s.This sometimes happens in a single-Millipore spiraI wave too. The reason is not known but chemical contamination perhaps from sodium chloride or perchloric acid fingerprints is suspected. N. Kopell and L. Howard Science 1973 180 1171. M. Smoes and J. Dreitlein J. Chem. Phys. 1973 59 6277. D. Thoenes Nature (Phys. Sci.) 1973 243 18. P. Ortoleva and J. Ross J. Chem. Phys. 1973,58 5673. P. Ortoleva and J. Ross J. Chem. Phys. 1974 60 5090. B. Belousov Sb. Ref.Rod. Med. 1958 145. A. Zhabotinsky and A. Zaikin Nature 1970 225 535. A. Winfree Science 1972 175 634. E. Zeeinan Towards n Theoretical Biology ed. C. Waddington (Aldine New York 1972) vol. 4 pp. 8-67. A. Winfree Mathematical Probleiiis in Biology Lecture Notes in Biomatlzematics (Victoria Conference 1973) Vol.2. ed. P. van den Driessche (Springer-Verlag Berlin 1974) p. 241. A. Winfree Sci. Amer. 1974 230 82. A. Winfree Muihenmtical Aspects of Chemical aiid Biological Problems and Quaitturn Chemistry ed. D. Cohen (Amer. Math. SOC. Providence 1974) vol. 8 in press.

ISSN:0301-5696    

DOI:10.1039/FS9740900038

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

Periodicity in Chemically Reacting Systems A Model for the Kinetics of the Decomposition of Na2S204 BY PHIL E. DEPOYAND DAVID M. MASON Stanford University Stanford California U.S.A. Received 23rd July 1974 Sodium dithionite undergoes thermal decomposition in aqueous solution to form sodium bi- sulphite and sodium thiosulphate. It has been observed previously that in the decomposition under some conditions the concentration of the reactant oscillates in time. Although various theoretical mechanisms such as the Lotka system have been proposed which would produce oscillations in the rate of reaction and in the concentrations of intermediate species in a reacting system none of the theoretical mechanisms can explain the oscillations including periodic increases of the reactant concentration observed in the dithionite system.In this paper the observed behaviour of the dithionite system is described. Three mechanisms are discussed which have been proposed by others to explain various characteristics other than the oscillations of the decomposition. It is then shown how these mechanisms could with certain modifications produce oscillations such as are observed with the dithionite system. Sodium dithionite (Na2S20,) is a powerful reducing agent used in the manu- facture of various organic chemicals and in dyeing and bleaching processes. It under-goes thermal decomposition in aqueous solution to form sodium bisulphite (NaHS03) and sodium thiosulphate (Na2S,03) according to the stoichiometry 2Na2S204+H20-+2NaHS03 +Na2S203.(1.1) Although relatively little has been published regarding the kinetics of the decomposi- tion those findings available are contradictory. The reaction has two distinct regimes as shown by the data in fig. 1 taken from the work of Lem and Wayman.' There is an induction period in which the dithionite concentration decreases slowly with time followed by a rapid autocatalytic reaction. Most of the published work I I I I 1 2 3 4 5 6 tirne/s x lo-* FIG.1.-Sodium dithionite concentration against time. 47 DECOMPOSITION OF SODIUM DITHIONITE deals with behaviour during the initial induction period. Different investigators have reported the order of the reaction in this initial period to be first three-halves and second with respect to the dithionite concentration and one-half and first with respect to the hydrogen ion concentration.Rinker Lynn Mason and Corcoran first reported that at some temperatures marked oscillations in the dithionite concentration were observable even in a system closed with respect to mass. Fig. 2 shows some of their results obtained at 60°C where reproducible oscillations including periodic increases in the dithionite con- centration as high as j 15 % are evident. At 70°C the magnitudes of the oscillations were reported to be considerably less and at 80°C practically no oscillations were observed. time/s x lo-* FIG.2.-OscilIations with time in sodium dithionite concentration.2 Rinker and associates reported as have other investigators that the addition of the products of the reaction to fresh dithionite greatly increased the initial rate of decomposition.If products were added in sufficient concentration the induction period was completely eliminated. It was also observed that the dithionite solution is turbid probably due to the presence of colloidal sulphur. It was reported that in an unbuffered solution the concentration of the hydrogen ion is oscillatory. Fig. 3 shows the time-dependence of the concentration of dithionite and hydrogen ion measured in situ with a glass electrode. As can be seen the hydrogen ion behaviour appears to be nearly a mirror image of the dithionite concentration. PROPOSED MECHANISMS Rinker et aL2 proposed a mechanism which is consistent with the observed rate for the induction period except that it does not account for the oscillatory behaviour.The mechanism that was suggested is as follows S20; +H+fztHS20; HS2Oq +H++2HSO,* fast HSO2*+HS2O,-+HSO,*+HS20 controlling HSO,*+HSO2*+H,O+2HSO3 +2H+. fast P. E. DEPOY AND D. M. MASON This reaction scheme leads to a rate expression of the form r = k[H+]f[HS,Oi]*. To be in accord with their observed kinetics i.e. a rate first-order with respect to [HSOJ variable fractional order in [H+] and zero order in [SO$-],Spencer and Burlamacchi Guarini and Tiezzi have concluded that the rate-determining step must be HSO,.+ SO; +HSO; +intermediate products. (2.6) Wayman and Lem proposed the following sequence of reactions S20 +H++HS,O fast S2O,'+H++HSO +SO (2.8) HS204 +H,O+HSO +HSO; +H+ (2.9) HSO +HS,O,-+HSO +S205 +H+.(2.10) They assumed that the reaction step eqn (2.10) is slow in the absence of a catalyst such as H2S or colloidal sulphur. They have proposed three reactions of sulphoxy-late (HSO;) which might produce H2S or S H++3HSOj -+H$ +2HSO, H++2HSO; +S +HSO3 +H20 (2.1 1) (2.12) and H++HSO +H2S-+2S +2H20. (2.13) They suggest that the catalytic action of hydrogen sulphide may be due to its ability to accept electrons and that its presence may improve the ability of the dithionite and sulphoxylate to react by facilitating the formation of the sulphoxylate free radical to produce a more rapid reaction (2.14) The free radical product becomes the ion by reacquiring an electron from the H2S HSO,-+e-+HSO,.(2.15) Polysulphides formed by the reaction of hydrogen sulphide and sulphur would be even more effective radical stabilizers and explain the observed catalytic effect of colloidal sulphur. Assuming that reaction (2.9) (the reaction of hydrogen dithionite with water) is rate-limiting during the induction period and that reaction (2.10) (the reaction of hydrogen dithionite ion with sulphoxylate) is limiting during the rapid decomposition the overall rate equation is r = kl [H+][S,OT] +k,[H+][S20 T][HSO,]. (2.16) Assuming that the catalytic effect of products of the reaction controls the rate during the fast decomposition Wayman and Lem obtain an overall rate expression of the form where C represents the concentration of dithionite.The Wayman-Lem expression (eqn (2.17)) does appear to give reasonably good agreement with the observed behaviour as shown in fig.4 and 5. Fig. 4 compares their best fit of the rate equation with data obtained by them for a solution of 10 mM DECOMPOSITION OF SODIUM DLTHIONITE dithionite buffered at a pH of 4 and at a temperature of23"C. Fig. 5 is their com- parison of the best fit of the rate equation to data obtained by Rinker et aL2 for a solution of 11.5 mM dithionite at a pH of 6 and a temperature of 60°C. 4-2-0 A R 12 16 20 24 time/s x FIG.3.-Concentration of dithionite and hydrogen ion against time ';0,dithionite ;A,hydrogen ion. -0 I 2 3 4 5 6 timels x FIG.4.-Comparison of Wayman-Len1 data with mechanism of eqn (2.16); -experimental --best fit of eqn (2.16).3. DISCUSSION It can be easily shown that no oscillations can occur with the Wayman-Lem mechanism or the Rinker mechanism since the rate-controlling steps are not auto- catalytic. It has been demonstrated theoretically by several investigators e.g. Higgins,6 that at least one autocatalytic step is required to produce damped oscilla- tions of intermediates and two or more autocatalytic steps are required to give undamped oscillations in open systems. It has also been shown that damped oscil- lations can be produced by autocatalytic reactions in systems which are closed with P. E. DEPOY~ANDD. M. MASON respect to mass.7 The rate-controlling step proposed by Spencer and by Burla- macchi Guarini and Tiezzi is autocatalytic but with respect to one of the products and this also would not produce oscillations.I time/s x FIG.5.-Comparison of the Rinker et af.’ data with mechanism of eqn (2.16). As mentioned earlier the most unusual characteristic of the dithionite system is the periodic increases which are observed in the dithionite concentration itself (fig. 2). Although closed-loop “ feedback ” mechanisms i.e. mechanisms in which one or more of the intermediates or products react to form the reactant can be postulated which could produce such increases it can be shown that these require that the con- centrations oscillate through an equilibrium point and this has been demonstrated in accordance with the Wegscheider constraint * to be thermodynamically unfeasible.Thus no closed-loop system can produce such oscillations. The most appealing explanation of the periodic increases in the dithionite con- centration is that the dithionite does in fact behave as an intermediate during the decomposition process. This behaviour can be accomplished if the dithionite reacts to form some side-products (other than the products or intermediates of the principal decomposition reaction) and reaches equilibrium with them possibly at a low temp- erature at which the concentrated dithionite solution is initially prepared. As the dithionite is consumed by the decomposition the equilibrium with these side-products would be shifted and they would react to form more dithionite thus producing a singular or quasi-singular point in the dithionite concentration.Inview of the observed appearance of what is believed to be colloidal sulphur in the solution the formation of various complexes such as polythionic acids (H,S,06) observed in the decomposi- tion of sodium thiosulphate in acid is a suspected side-reaction. Several mechanisms for the decomposition can be postulated which produce oscillations. One which appears to satisfy nearly all the observed characteristics of the reaction is as follows. As postulated the dithionite reaches equilibrium in a side reaction with various sulphur-compound complexes and possibly with colloidal sulphur S,OT+complexes+[S]+ . . . .. (3.1) As proposed in the other mechanisms the dithionite is in rapid equilibrium with DECOMPOSITION OF SODIUM DITHIONITE HS204 in a manner such as proposed by Rinker et aL2 and the HS204 is in equi-librium with sulphoxylate ions and SO similar to the stoichiometry of eqn (2.8) H++S2O%+HS2O fast HS204%HSO; + SO2 (3.3) fast Sulphur dioxide forms sulphurous acid SO +H20+H+ +HSO;.(3.4) Some sulphoxylate ions give up electrons to become the radical HSO,.. As assumed by Rinker et al. the sulphoxylate radical reacts slowly with HS,O, as follows HSO2*+HS2O,+HSO,*+HS20,. (3.5) Modifying their assumption about the reaction between sulphoxylate and bisulphite radicals eqn (2.4) we postulate that the radicals might react to form disulphurous acid which reacts rapidly with HS20i HSO2*+ HS03*+H2Sz05 (3.6) H2S205 + HS204 +2HS03*+ HS203.(3.7) As assumed in the Weyman-Lem mechanism the bisulphite radical absorbs an electron to give the ion HSO,-+e-+HSO;. (3.8) Overall this reaction is very similar to the theoretical oscillating mechanism first proposed by Lotka in 1910 i.e. XcI4Xl XI +xp2xz (3.9) (3.10) X +products (3.1 1) where the precursor Xo corresponds to various sulphur-compound complexes the first intermediate XI corresponds to the dithionite ion in equilibrium with the sulphoxylate radical and the second intermediate Xz corresponds to the bisulphite radical. time/s x FIG.6.-Behaviour of proposed mechanism. P. E. DEPOY AND I>. M. MASON Assuming a first-ordtr decay of the precursor the rate equations for our mech-anism are io = -klxo (3.12) 11 = klxo-2k2xl/e-2k3xlx2/e (3.13) i2= k2x2/e+k3xlx2/e-k4ex2 (3.14) and i3= k2x1/e+k3x1x2/e+k,ex2 (3.15) where xo is the concentration of the precursor x1 is the concentration of dithionite x2 is the Concentration of the bisulphite radical x3 is the concentration of the bi- sulphite ion and e is an electron availability factor.If the electron availability factor is constant based on the results derived for the closed Lotka system,' it would be expected that this system would exhibit multiple damped oscillations when the factor (k3x;/kle)*is of the order of 5 or greater. If the electron availability factor is related to the precursor and product concentrations by e = xo/(xo+cx3) (3.16) (which can be explained by assuming that the precursor or a substance in equilibrium with it is an electron donor and that the bisulphite and another substance in equi- librium with the precursor are electron sinks) the autocatalytic behaviour of the dithionite is produced.This relationship was selected because it produces the observed autocatalytic-type behaviour (since the precursor concentration xo falls off ex-ponentially with time) and it does not dampen the oscillations as is the case if the electron availability is a strong function of the dithionite concentration xl. As an example of the behaviour of this system of reactions fig. 6 shows the di- thionite concentration when parameters are chosen to approximate the observed behaviour shown in fig. 2 i.e. a period of the oscillations of approximately 150-200 s and an induction period of about 600 s.Parameters on which the figure is based are kl = 2 x k2 = k3 = 290 k4 = 1.6 c = 0.07and x6 =4 x (all based on concentrations in mol/l-l and time in seconds). As can be seen the mechanism compares reasonably well with the observed behaviour. This new mechanism is similar enought to the other three mechanisms presented by others that nearly all the observed characteristics of these mechanisms are preserved ; 1. the reaction appears autocatalytic i.e. the decrease in dithionite concentration is slow at first then rapidly accelerates 2. the reaction appears to be catalyzed by the addition of products (in the new mechanism the addition of bisulphite ion would produce more of an electron sink thereby more rapidly reducing the dithionite concentration) and 3.the reaction is catalyzed by the addition of sulphur (in the new mechanism sulphur would also act as an electron sink). In addition other results observed by Rinker et aZ.,2 e.g. the oscillation of the hydrogen ion concentration are also consistent with our mechanism. 4. SUMMARY In summary we have briefly discussed the behaviour of an oscillating chemical system the thermal decomposition of sodium dithionite. We have discussed three mechanisms which have been proposed by others to explain the general behaviour not including oscillations of the dithionite system. Finally we have suggested modi- fications to those mechanisms which could produce the type of oscillations which have been observed.DECOMPOSITION OF SODIUM DITHIONITE Obviously the mechanism of the dithionite decomposition is complex and a great deal more work would be required to gain a thorough understanding of it. Because of its uniqueness in the domain of oscillating chemical systems further investigation is warranted. W. J. Lern and M. Wayman Canad. J. Chem. 1970,48,776. R. G. Rinker S. LYM,D. M. Mason and W. H. Corcoran Ind. and Eng. Chem. Fund. 1965 4 282 ;S. Lynn Ph.D. Thesis (California Institute of Technology 1952). M. S. Spencer Trans. Faraday Suc. 1967 63,2510. L. Burlamacchi G. Guarini and E. Tiezzi Trans. Faraday Suc. 1969 65,496. M. Wayman and W. J. Lem Canad. J. Chem. 1970,48,782. J. Higgins Ind. and Eng. Chem. 1967 59 19. 'P. E. DePoy and D. M. Mason Combustion and Flame 1973,20,127. R. Wegscheider 2.phys. Chem. 1902 39,257. A. J. Lotka J. Phys. Chem. 1910,14,271.

ISSN:0301-5696    

DOI:10.1039/FS9740900047

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

Mechanism of the Oscillatory Decomposition of Hydrogen Peroxide in the Presence of Iodate Ion Iodine etc. BY ISAO MATSUZAKI NAKAJIMA AND TSUYOSHI Dept. of Synthetic Chemistry Faculty of Engineering Shinshu University Nagano Japan AND HERMAN A. LIEBHAFSKY Dept. of Chemistry Texas A & M University College Station Texas U.S.A. Received 25th July 1974 Various mechanisms are constructed with the aid of information on associated reactions and ~ tested in the light of the theory of two-variable oscillating reactions as well as by means of computer simulations. Reactions were run mainly at [HC1OJo= 0.035 to 0.080 [KIO3l0 = 0.40 and [H,O& = 0.30mol I.-' and compared with computer results for a promising mechanism. Agreement is so good with respect to the induction period the pulses of [I-] and the rate of O2evolution the abrupt decrease in [Iz]etc.for one to conclude the mechanism to be plausible. The plausible mechanism contains HI02 HIO I- H21203 and H31305 as intermediatesand the second-order back-activation step 2HI02+HIO+ H20z -f 3HIo2+H20as the key step for the oscillation which results from the sequence of HIOz+HI0 +HzIzO~,H2Iz03+HI02 + H31305 and H31305+H202-+ 3HI02+ HzO. The oscillation source of the mechanism has been found to be in conformity with the Brusselator. 1. INTRODUCTION Since Bray discovered the oscillatory decomposition of H202 by the 103-12 couple not a few investigations 2-9 have been carried out without giving any concrete mechanisms. Recently we have succeeded in finding a plausible mechanism which enables us to understand the capability of the system to oscillate.Success was attained in the following way. First information was collected on possible intermediates and elementary reactions. Second mechanisms were con- structed and examined for the capability of oscillation both theoretically and by means of a computer. Finally experimental data were compared with computer results. This paper is concerned with the process leading to the plausible mechanism its nature and experimental evidence for it. 2. THE PROCESS LEADING TO THE PLAUSIBLE MECHANISM A survey of the literature 2-9 has shown that aqueous solutions capable of oscil-latory decomposition contain IO, H+ 12,I- and H20z as stable species. In such solutions the following reactions also can take place.210; +2H+ +5H202+ I2+502 +6H20 lo (2.1) I2+5H2O2+ 210;+2H++4H20 11* l2 (2.2) 10 +51-+6H+ + 312+3H,0 13-15 (2.3) 21-+2H++H202 4 12+2H20l6 (2.4) 12+H20+ HIO+I-+H+ l7 (2.5) H202 3H2O+O.5 02. '* (2.6) 55 OSCILLATORY H202 DECOMPOSITION Here we adopt a view that if we find a mechanism which can explain all the above reactions mechanistically the mechanism has a good chance of accounting for the oscillatory decomposition. Previous investigations on the six reactions have suggested a number of inter-mediates such as HIO, HIO H,IO+,l9* 2o and H21203.12 Using such intermediates and supplementary intermediates H210 * and H31305,ta comprehensive mechanism of an oscillatory nature was constructed by the trial and error method as shown in fig.1 in which $ denotes so rapid a reversible step as to be usually in equilibrium 16 FIG. 1 .-The comprehensive mechanism of oscillatory nature for the HzOz-IO;-12 system. The numerals denote the step number. and each step is specified by an Arabic figure; HI03 a medium acid is used as a generic symbol for HI03 and 10 also in what follows ; species IO, 10- and 1; are neglected since HIO and HI0 are weak acids and no I; is formed under the oscil- lation conditions. The detail of each step is listed below. Step 1 HI03+H202+ HI02+H20+O2 (2.7) Step 2 H2IOi +HZ02 + HIOj+H++HzO (2.8) Step 3 H210; + H++HI02 (2.9) Step 4 HTOz+H202 + HIO+H,O+02 (2.10) Step 5 H210++H202-+ HT02+H++H20 (2.1 1) Step 6 HlIOf + HI0 +H+ (2.12) Step 7 HIO+H202 -+ H++I-+H,O+O (2.13) Step 8 I-+H++H,O2 -+ HIO+H20 (2.14) Step 9 I,+H20 f HIO+I-+H+ (2.15) * This species was devised by analogy to H210f but its presence is probable from the Occurrence of X0;+2H+ + H2XO (X = CI Br or t This species was devised without experimental support but its formation is probable since large molecules 24s 25 such as H2130:;and H4130:4 and those 26 such as [I03(HI03),]-(n > 1) are suggested.I. MATSUZAKI T. NAKAJIMA AND H. A. LIEBHAFSKY Step 10 HI02+H++I-+2HIO (2.16) Step 11 HZ1203+H++I-~t 3HI0 (2.17) Step 12 HI02 +HI0 +€I21203 (2.18) Step 13 HIO,+H++I-+ H21203 (2.19) Step 14 H2T203+H202-+ 2HI02+H20 (2.20) Step 15 H21203+HT02+ H31305 (2.21) Step 16 H31305+H202-+ 3HI0 +H,O.(2.22) According to the comprehensive mechanism reactions (2.1) to (2.6) take place via the sequences of steps shown in fig. 2. Reactioii @,I) Reaction (2,4) Rcaition<:,5) ;i ~ HIO -HIO -HI0 -I-A HI0 -I-HI0 1-Reaction (2,2) n HIO HIOz HI0 -I-\# \J HJO,' H,IO+ HIO -HIO -,,HI0 s I-\J HJO FIG.2.-The sequences of steps for reactions (2.1) to (2.6) according to the comprehensive mechanism. According to the theory 27 of oscillating reactions a feedback step is indispensable for oscillation so in the trial-and-error search a mechanism equal to the compre- hensive mechanism minus steps 15 and 16 was first subjected to computer simulations with 92 sets of values of rate constants for each step since it contains a feedback step HIO +HI0 +H,02 -+ 2HI02+H20 (first-order back-activation) as a consequence of the sequence of steps 12 and 14 but without giving any oscillations.Here we examined the sequence theoretically (see section 3 (ii)) HIO 7+ HI02 * HI0 (2.23) U (act)" and found that the feedback step HI0 -+ HIOz should have an order nhigher than 1. Based on this finding we chose 2 the smallest favourable integer for n,and added steps OSCILLATORY H202 DECOMPOSITION 15 and 16 so that the second-order back-activation step 2HI0,+HIO+H202 + 3HIO2+H20might participate. Of course it is possible to get a favourable sequence of steps using such intermediates as 100, HO- and HOz- but at present we would like to retain the use of radical species as a possibility.3. NATURE OF THE COMPREHENSIVE MECHANISM (i) STEPS DOMINANT IN THE OSCILLATION The failure of the associated mechanism to yield oscillation mentioned in section 2 suggests that any steps but steps 15 and 16 are not responsible for the oscillation. On this basis we picked up such steps as would be dominant under oscillation condi- tions arriving at the practical mechanism shown in fig. 3 in which step 2' is the combi- I HIQZ H102 __* HI0 7 I-2' 1 FIG.3.-The practical mechanism derived from the comprehensive mechanism as applicable for oscillation conditions. nation of steps 2 and 3 with k2t =k2/K3 where k2 is the rate constant for step 2 and K3the equilibrium constant for step 3 ;steps 11 and 13 were eliminated on an idea that' step 10 should outweigh them in the interaction with I-; steps 5 6 and 14 were also eliminated since step 16 should be the major step for supplying the key species HI02.The practical mechanism was fed into computer simulation. The simulation is based on the Euler method.28 Steps 12 and 15 were regarded as being in equilibrium while steps 9 and 10 were each decomposed into two reversible steps 9' and 9" and 10' and lo" respectively. The rate law for each step is determined according to the mass action law as exemplified by 08 = k,[I-][H,O,][H+] for step 8 and v16 =k16K12K15[H102]2[H10][H202] for step 16; that for step 1is given by the experimental one lo for reaction (2.1) ul/moll.-l min-l = 2.6 x 10-4[H103][H202]+ 129 x 10-4[H+][HI0,][H202]. (3.1) Based on previous experimental results the simplification was adopted that [HI03] [H202] and [H+] might be regarded as constant for a run as long as the switch- over of the system to oscillation is concerned leading to condensed rate constants E as exemplified by u1 =eqn (3.1) =El tlg =k8[I-][H202][H+]=E8[1-] 016 = kl6K12K15[H102]2[HIO][H202J =k,,[HI02]2[HIO].The sets of k and R values employed for the simulation are summarized in table 1 and the time courses obtained I. MATSUZAKI T. NAKAJIMA AND H. A. LIEBHAFSKY are shown in fig. 4. In the sets of ki and Ei values employed only ki was changed systematically. In choosing the sets we did not adhere to experimentally determined ki and Ki values because in this computer simulation we aimed at making the oscil- latory nature of the mechanism visualizable.It is clear from fig. 4 that the practical mechanism is so versatile as to reproduce almost all the aspects of the system such as the oscillatory decomposition initial stage of reaction catalytic decomposition and one-way I2 formation. TABLESUMMARY OF THE COMPUTER RUNS CONDUCTED BASED ON THE PRACTICAL MECHANISM runno. Ela kztb k4C Elad E7’ xgl Eq,g kg, klo klo, nitofconc.h (mol L-1) 1 2 3 3 4 5 2 2 2 10 10 10 4 4 4 5 5 5 5 5 5 2 2 2 0.05 0.05 0.05 4 4 4 4 4 4 u,/E~u,/kl VJEl 4 6 2 - 10 4 5 5 - 2 0.05 4- 4 vl/kl -Qkl = eqn (3.1). -bk2f= (kz/K3)[H+][H+l. ck4 = k4[H2021. dk16 = (k16K12K1s)[HzOzl. e k = k7[H202]. fka = k8[H+][H2O2]. kgt = kg[H+]. h For example run 1 gives results for which u1/3 corresponds to one unit of concentration where v1 is the actual rate of step 1 given by eqn (3.1) (cf.the [I2] calculation in section 4 (ii)). OSCILLATORY HZ02 DECOMPOSITION (ii) OSCILLATION SOURCE The failure of the associated mechanism mentioned in section 2 coupled with the success of the practical mechanism indicates that the first-order back-activation step consisting of steps 12 and 14 is not effective whereas the second-order one composed of steps 12 15 and 16 is the oscillation source. It is now necessary to make a theoretical check on the above indication. For this purpose the following sequence is the simplest including a nth order back-activation = step. Since this sequence is a two-variable system (because [HI03]const.under consideration) its necessary condition for oscillation can be derived along the line by Higgins 27 as follows. K1 E4 HIO HIO s HI0 E2' E (act)" The rates of increases in A and B are given by dA/dt = -(k2t+ EJA + EAnB (3.3) dB/dt = k4A-EAnB (3.4) where A and B stand for [HIO,] and [HIO] respectively. The singularity is obtained as A = E1/K2? and B = (K4/k)(k2,/E,)"-' (3.5) by setting both the above derivatives equal to zero. The self- and cross-coupling terms at the singularity are obtained as by differentiating the derivatives of eqn (3.3) and (3.4) with respect to A or B and substituting eqn (3.5). From eqn (3.6) to (3.9) we obtain under the condition of n > 1 and (n-l)E4 > k2#at singularity IAb x BaJ-IAa x Bbl = kA"(n-l)kAn-'B -EA"{(n-l)kA"-'B-E2') = JE2fEA"> 0 (3.10) and AaxBb < 0 and BaxAb < 0 (3.11) the other necessary condition being Aa+ Bb > 0 at singularity i.e.(n-1)E4-k2e-E(El/?i2#)n > 0. (3.12) From the condition for (3.10) and (3.11) or eqn (3.12) it is evident that the first-order (n = 1) back-activation step is ineffective whereas the second-order (n = 2) can confer possibility of oscillation in favow of the above-mentioned indication. I. MATSUZAKI T. NAKAJIMA AND H. A. LIEBHAFSKY It should be pointed out that the part of the comprehensive mechanism corres- ponding to sequence (3.2) with n = 2 is equivalent to the Bru~selator,~~’~~ which according to Tyson 35 is the only known two-variable chemical scheme which admits limit cycle oscillations.This equivalence can readily be understood if we apply a replacement of A E = HI03 X = HI02 B = H202 Y = HIO and D = O2on the Brusselator A-+X B+X + Y+D 2X+Y -b 3x X -+ E. On the other hand the unfavourable sequence (3.2) with n = 1 corresponds to the Brusselator with its step 2X+Y + 3X replaced by X+Y + 2X ;with respect to such a replacement Tyson 35 has stated in support of our unsuccess in getting oscillations that the steady state of such a replaced scheme is always stable for positive values of the parameters. 4. EXPERIMENTAL EVIDENCE FOR THE PRACTICAL MECHANISM (i) EXPERIMENTAL The present experiment was designed to get time courses comparable with those of fig. 4 on the basis that the change in k1 corresponds to a change in [H+] when [KIO& and [H20210are kept constant.All the chemicals used were of guaranteed grade and used without further purifica- tion. The hydrogen peroxide contained no inhibitor. Two kinds of reaction vessels open-type and closed-type both made of Pyrex were used. Both the vessels were used in previous experiments 6-9 and are to be described in detail elsewhere. The open-type was designed especially for the spectrophotometric [I,] measurement. The [I-] measurement was made by means of an Orion I-selective electrode and the d02/dt (rate of O2evolution) measurement by means of a Matheson Company LF-20 mass flowmeter. A solution containing KIOJ and HC104 was kept at 323 K and then an amount of 30 % H202was added to initiate the reaction.In table 2 are summarized the runs of reaction conducted and the time courses TABLE 2.-sUMMARY OF THE RUNS OF REACTION CONDUCTED AT 323 K initial conc./mol1.-1 volume type of quantities run no. WC104lo WOdo [HzOdo (V/ml) vessel used followed 1’ 0.035 0.40 0.30 150 closed 2‘ 0.051 6 0.40 0.30 150 closed 3’ 0.060 0.40 0.30 150 closed 4’ 0.070 0.40 0.30 150 closed 5‘ 0.080 0.40 0.30 150 closed 6‘ 0.070 0.40 0.30 140 open 7’ 0.0576 0.282 0.494 140 open 8’ 0.0576 0.282 0.494 150 closed obtained are shown in fig. 5to 7. A number of preliminary runs have concluded that the series of runs 1‘ to 5’ has shown all kinds of time courses for [I-] and d02/dt. Run 6’ is a separate run of run 4’ for the time course of [I2]. In fig.4 and 5 to 7 lower case letters are used to make clear the synchronization among [iJ [I-] and d02/dt. The initial period a-b where [I-3 [I2] and d02/dt increase will be called the OSCILLATORY H2Oz DECOMPOSlTION I I I I 1. 0 10 20 30 40 d " h time /min FIG.5.-The experimentally obtained time courses of dO,/dt and [I-] for runs 1' to 5' of table 2. Thc numerals indicated denote the run number timelmin FIG.6.-The experimentally obtained time courses of [I,] and [I-] for run 6' of table 2. induction period and during the flat part after point gthe smooth catalysis H202+ H20+0.502proceeds almost exclusively. Fig. 5 to 7 coupled with observations in preliminary experiments characterize each time course as follows (all concentrations in mol I.-') I.MATSUZAKI T. NAKAJIMA AND H. A. LIEBHAFSKY 4 Run 1’ ([HClO4Io= 0.035) ... . [I-] and [I,] increase monotonously to reach a constant where I2 begins to precipifate. d02/dt increases similarly but decreases as H202is consumed. Run 2’ ([HCIO,] = 0.051 6) . . . . The induction period is followed by pulse-like oscillations where [I-] first decreases inducing [I2] to decrease and dO,/dt to increase. Run 3’ ([HCIO,] = 0.060) and runs 7’ and 8’ ([HC10410 = 0.057 6). . . . The same as in run 2’ with oscillations appearing earlier with smaller ampli- tudes and shorter periods. Run 4‘ ([HC10410 = 0.070) . . . . The induction period is followed by smooth catalysis via a half-pulse and oscillations appear later via several wiggles.Run 5’ ([HClO,] = 0.080). . . . The induction period is followed by smooth catalysis. Runs 1’ to 5’ . . . . The changes in the position and size of [I-] and d02/dt pulses in runs 2’ and 3’ and the descents of the line for the smooth catalysis in runs 4‘ and 5’ result from the consumption of H202 ;the later oscillation in run 4’ will be absent if H202is supplied constantly. time/min FIG.7.-The experimentally obtained time courses of [I-] and [I2] for run 7’ and that of dOzldr for run 8’ of table 2. OSCILLATORY H202 DECOMPOSITION The nearly linear increases in [I2] in the induction period are in accord with the rate law (3.1) which indicates that the formation of I2 is due to reaction (2.1). One quantity of interest is the ratio (d02/d12)ina.of the amount of 0 evolved to that of I2 formed in the induction period. Values of this ratio estimated from the combined data of runs 4' and 6' and of runs 7' and 8' increase from 1 to about 15 near the end of the induction period indicating that the catalytic decomposition resulting from elementary reactions involving intermediates becomes predominant before the start of a pulse or smooth catalysis. Another quantity of interest is the ratio (d02/d12)pulse of the amount of O2evolved to that of I2 disappearing during a pulse. For the second pulse in fig. 7 the amount of I disappearing is 1.6 x moll.-' and that of O2evolved from 150 ml of reaction mixture is 25.08 ml (stp) hence the value of the ratio is as large as 46.7. (ii) COMPARISON BETWEEN THE COMPUTER AND EXPERIMENTAL RESULTS It is readily seen from a comparison between fig.4 and fig. 5 to 7 that the change in the time courses of runs 1 to 4with Elis qualitatively similar to that of runs 1' to 5' with [H+l0. For quantitative comparison with the experimental results we have to carry out computer simulations with actual kiand Ei values and with consideration paid on the effect of [H+] upon all the kiand ki values. Let us make a quantitative check on some significant quantities. The value of [I2]at point b for run 4 is obtained from fig. 4(c) as 30(2.6 x 10-4[H~02][HI'0,]+ 129 x lO-"[H+][HIO~][Hz02])/6 = 30(2.6 x x 0.30 x 0.40-t-129 x x 0.07 x 0.30 x 0.40)/6 = 7x (mol l.-l) which agrees with the experimental value of 7.6 x (run 6').The value of (d02/d12)pulse for run 3 is calculated as 0.5(55 +35) x 25/(59-27) = 37 in approxi- mate agreement with the experimental value of 46.7. Values of (d02/d12)ind. at the end are about 17 (run 2) 16 (run 3) and 15 (run 4) in agreement with the experi- mental value of about 15. From the good qualitative agreement pointed out above it may be concluded that the practical mechanism of fig. 3 is a plausible mechanism actually in operation. We thank the Robert A. Welch Foundation for supporting the experiment. W. C. Bray J. Ainer. Chem. SOC.,1921 43 1262. W. C. Bray and H. A. Liebhafsky J. Amer. Chem. SOC., 1931 53 38. M. G. Peard and C. F. Cullis Trans. Faraday SOC.,1951 47 616. H. Degn Acta Chem. Scand. 1967 21 1057. P. Lindblad and H.Degn Acta Chem. Scand. 1967,21,791. J. H. Woodson and H. A. Liebhafsky Nature 1969 224 690. J. H. Woodson and H. A. Liebhafsky Anal. Chem. 1969 41,1894. * I. Matsuzaki R. B. Alexander and H. A. Liebhafsky Anal. Chem. 1970,42,1690. I. Matsuzaki J. H. Woodson and H. A. Liebhafsky Bull. Chern. SOC.Japan 1970,43,3317 loH. A. Liebhafsky J. Amer. Chem. SOC., 1931,53 896. l1 W. C. Bray and A. L. Caulkins J. Amer. Chem. SOC.,1931 53 44. l2 H. A. Liebhafsky J. Amer. Chem. SOC.,1931 53 2074. l3 E. Abel and K. Hilferding,2.phys. Chem. 1928 136 186. l4 R. Furuichi I. Matsuzaki R. Simic and H. A. Liebhafsky Inorg. Chem. 1972 11,952. R. Furuichi and H. A. Liebhafsky Bull. Chem. SOC.Japan 1973 46,2008. l6 H. A. Liebhafsky and A. Mohammad J.Amer. Chem. SOC.,1933,55,3977. E. Eigen and K. Kustin J. Amer. Chem. SOC.,1962 84 1355. Is H. A. Liebhafsky J. Amer. Chem. SOC.,1932,54 3504. I. MATSUZAKI T. NAKAJIMA AND H. A. LIEBHAFSKY I9 R. P. Bell and E. Gelles J. Chem. SOC., 1951 2734. 2o I. Matsuzaki R. Simic and H. A. Liebhafsky Bull. Chem. SOC. Japan 1972,45 3367 21 H. Taube and H. Dodgen J. Amer. Chem. SOC. 1949,71 3330. 22 A. F. M. Barton and G. A. Wright J. Chem. Sac. A 1968 1747. 23 M. Anbar and S. Guttmann J. Amer. Chem. SOC.,1961 83 4741. 24 H. Siebert Forts. Chem. Forsch. 1967 8 470. 25 M. DrBtovsk9 and L. PaEesovB Russ. Chem. Rev. 1968 37 243. 26 J. G. Dawber J. Chem. SOC.,1965 4111. ”J. Higgins Ind. Eng. Chem. 1967 59 (9,19. 28 H. Levy and E. A. Baggott Numerical Solutions of Diflerential Quations (Dover New York 1950) p.92. 29 G. Nicolis Adv. Chem. Phys. 1971 19 209. 30 R. Lefever and G. Nicolis J. Theor. Biol.,1971 30 267. 31 I. Prigogine and R. Lefever J. Chem. Phys. 1968 48 1695. 32 R. Lefever J. Chem. Phys. 1968 49,4977. 33 M. Herschkowitz-Kaufman and G. Nicolis J. Clzem. Phys. 1972 56 1890. 34 B. Levanda G. Nicolis and M. Herschkowitz-Kaufman,J. Theor. Bid 1971 32 283. 35 J. J. Tyson J. Chem. Phys. 1973,58 3919. s 9-3

ISSN:0301-5696    

DOI:10.1039/FS9740900055

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

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.

ISSN:0301-5696    

DOI:10.1039/FS9740900066

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

B. Thermokinetic Oscillations Thermokinetic Oscillations accompanying Propane Oxidation BY PETER AND R. J. MOULE GRAY,J. F. GRIFFITHS Department of Physical Chemistry The University Leeds LS2 9JT Received 7th August 1974 Gaseous hydrocarbon oxidations are often accompanied by remarkable non-iso thermal pheno- mena such as oscillatory cool flames and complex ignitions. These owe their existence to an inter- play between kinetics and self-heating via thermal feedback in a system involving chain branching. The present work illustrates these phenomena and provides a quantitative assessment of thermo- kinetic concepts applied to them. Propane is oxidised in a stirred closed reactor. Characteristically time-dependent oscillations occur in the pressure range 40-100kNm-2at vessel temperatures between 570 and 660K.Sub-stantial temperature changes accompany the oscillating reaction rate and these changes are measured by a fine (25 pm) thermocouple and rapid recording equipment. Oscillating temperatures may vary in form from roughly sinusoidal with damping to steep cusp-like maxima of hardly diminished amplitude multiple events being terminated abruptly by the consumption of fuel. Thermokinetic oscillations depend vitally on thermal feedback and hence on heat transfer :by the deliberate variation of heat transfer properties such as heat capacities and thermal conductivities (with diluting gases) and intensities of stirring we are able to test this dependence. The spontaneous oxidation of hydrocarbons in the gaseous phase not only shows " slow " and explosive modes of reaction but also may be accompanied by periodic pulses of light emitted during repetitive bursts of enhanced rea~tivity.~'~ The light emission is very feeble and an associated pulse in gas temperature is generally less than 200 K; these phenomena are called " cool flames " to distinguish them from normal explosive burning5 Cool flames are now recognised as oscillatory reactions in closed reactors up to 11 successive events are observed,6 and in open systems they may be sustained indefinitel~.~The temperature attained in a cool flame pulse varies between about 5 and 200 K according to the initial reactant temperature and concentration,8 though in closed reactors successive flames become progressively weaker since moderate proportions of reactants are consumed in each of them (up to 20 %).9 The earliest recognition of oscillatory reaction attracted contrasting interpretations.At first isothermal explanations were offered based on superficial analogy to the Lotka-Volterra autocatalytic scheme already applied in some electrochemical and biological contexts. However this simple isothermal explanation fails to offer a satisfactory interpretation of hydrocarbon cool flames. Amongst the reasons why the Lotka-Volterra mechanism does not survive detailed scrutiny are (i) its failure to predict an alternate growth and decay in concentrations of intermediates that is not determined by the initial composition (it predicts conservative oscillations not limit cycles),l and (ii) the impossibility of making a satisfactory chemical identification with the participants of the isothermal kinetic model.* Recent more elaborate iso- thermal models are hardly more acceptable since schemes that predict limit cycles have to invoke chemically unlikely elementary reactions.' 3*l4 (The best success so far with 103 THERMOKINETIC OSCILLATIONS an isothermal scheme seems to be that of Field and Noyes Is who have fitted the rigorous criteria for isothermal oscillatory models to the chemical framework of the Belousov l6 reaction.Autocatalysis in their kinetic mechanism occurs via chain branching.) In hydrocarbon oxidation there is an obvious alternative feedback mechanism to isothermal autocatalysis. This is thermal feedback which takes account of the strongly exothermic properties of reaction.The earliest thermal feed back model was proposed by Salnikoff l7 who showed that two consecutive first order reactions (viz. A 3 B -+ C) will generate sustained oscillations in the concentration of the inter- mediate (B) provided that certain (plausible) criteria apply to the exothermicities and Arrhenius parameters of each step. Present day thermokinetic models such as that of Gray and Yang or of Halstead Prothero and Quinn l9 remain simple though now they feature chain branched autocatalysis. Through it these schemes predict not only oscillatory regimes but also other striking and unusual features of hydrocarbon oxidation such as the negative temperature-dependent heat-release rate the single and multiple-stage ignitions and the " lobes " associated with the ignition limit in a pressure-temperature ignition diagram.Implicit in each of these treatments is that simply to devise a plausible model chemical scheme is not enough proper analytical methods must be applied to prove that stable oscillatory behaviour is possible. Moreover extension beyond a qualita- tive test requires not only correct information on stoichiometries kinetic constants and thermochemistry but also correct thermal parameters of the system. We need to supplement conventional kinetic investigations with measurements of heat release and loss rates and of how they depend on temperature. Where theoretical treatments have been simplified such as by assuming spatially uniform temperatures and concen- trations we need to devise experimental techniques in which nature is made to imitate art.Accordingly the present paper describes new investigations of the thermal effects accompanying propane oxidation. A closed reactor is used in which the temperature excess of reactants is made uniform by mechanical stirring. Heat-loss rates are measured and by deliberately changing them the dependence of cool flame oscillations on thermal feed back is tested. THEORETICAL FOUNDATION Most order has been brought into the classification and analysis of oscillatory reactions by phase-plane analysis. Even in systems that are chemically as complex as hydrocarbon oxidations many of the features can be rationalised in terms of a single key intermediate participating in branching and termination processes.l8* The starting point for analysis of a thermokinetic model is the pair of expressions for the generation and loss of heat and for the generation and loss of the chain branching species (X). These conservation equations may take the general forms for energy ;CF= %(T,P,X)-Z(S/ Y)(T-T~) (1) for mass; X = -f(~, x> (2) where B(T,P,X) is the rate of heat release per unit volume andf(T X) is a function involving the rates of reactions from which X is formed or in which X is removed. Each of these parameters depends upon the concentration of X. I is the heat transfer coefficient between gas and reactor walls (T-To)is the spatially averaged temperature P.GRAY J. F. GRIFFITHS AND R. J. MOULE I05 excess of the reactants c is their heat capacity per unit volume and S/Yis the surface to volume ratio for the reactor. Solutions of these equations for Tand X begin by location of singular points in the (T,X)plane and identifying the behaviour close to them. Closed-curve trajectories in the (T,X)plane correspond to oscillatory behaviour in time illustrated by either T(t)or X(t). This connection causes us often to describe the experimental time- dependent observations of undiminished sustained oscillations by phase-plane terminology i.e.,as " limit cycle behaviour ". Experimentally T(t)may be observed via direct measurements of temperature (as in our experiments ** 2o and in those of others 4* l) or X(t)by following the changing concentration of one or more appropriate intermediate species.So far experimental evidence demonstrating the alternate growth and decay of chemical intermediates in cool flames is scant 21* 22 ;this is because the important branching intermediates are extremely difficult to identify and measure continuously. EXPERIMENTAL MATERIALS Propane (instrument grade B.O.C. Ltd,) and acetaldehyde (Analar B.D.H. Chemicals Ltd.) were distilled in vacuu before use. Oxygen and diluent gases (B.O.C.Ltd.) were taken directly from cylinders. APPARATUS AND PROCEDURE The apparatus and general procedure have been described previously.2o* 23 Good mixing of reactants was achieved by a double-vaned rotor spinning round a vertical axis within the reactor (Pyrex 0.5 dm3 spherical).The rotor magnetically driven was made of stainless steel previously coated with a ceramic layer to minimise surface reactions. The reactor was thermostatted in a re-circulating air furnace to 1 K over its surface in the range 550-650 K. Propane oxygen and reactive additives or inert diluents were pre-mixed and stored in a conventional Pyrex vacuum line. Reactants were admitted to the reactor via an electro-magnetic valve opened reproducibly for 0.1 s. Initial pressures in the reactor (typically 30-100 kN m-2) were measured from the output e.m.f. of a pressure transducer (Ether Ltd. UP4). To detect and follow accurately the temperature changes in gaseous reactions the measur- ing device must have a fast response a very small thermal capacity low thermal conductivity adequate sensitivity and it must be of robust construction.Our thermocouples gas welded from very fine Pt-Pt/l3 % Rh wire (25 pm diam.) come close to satisfying these criteria. Their response time is less than 20 ms in moving gas 24 and so they are able to give a faithful record of all behaviour except hot ignitions. A very fine junction coated with a thin layer of silica from a methanol+silicon oil flame was situated within the reactor and the reference junction (100pm diam. wire) was placed on the outside of the vessel wall. The e.m.f. generated from these junctions was amplified and displayed on a light sensitive chart by an ultra-violet recorder (Southern Instruments Ltd.). The interior probe was moveable across a horizontal diameter of the reactor so that temperature-time histories could be obtained at any position across the reactor.Temperature-time records for the same initial conditions but for different positions of the probe were combined to allow temperature-position profiles to be mapped at successive instants of reaction. Records for the same propane+ oxygen+ additive mixtures at different initial temperatures and pressures were combined to map the various types of non-isothermal behaviour on pressure-temperature ignition diagrams. Heat transfer coefficients (I) for all of the reactant mixtures were measured at various temperatures and pressures in additional experiments. With knowledge of vessel dimensions and of reactant heat capacities I may be evaluated from the measured temperature excess of THERMOKINETIC OSCILLATIONS reactants (eqn (1)).Two routes are open; either via a steady slate ti-= 0) if energy is supplied at a known rate,". 23 so that Z(S/V)(T'-T") = &!is (3) or by a dynamic method from the rate of exchange of heat between the gas and reactor walls. If 9= 0 as is the case when inert gas is heated or cooled adiabati~ally,~~ T = (ZS/C Y)(T-To) = (1 /z)( T-7-01 AT = ATo exp (-t/z). The heat transfer coefficient has a natural identity with the characteristic relaxation time (2)for heat dissipation between the gas and reactor walls.25 We have chosen this dynamic method to measure 1. Gas was expanded adiabatically from the reaction vessel and the temperature-time history was folIowed as the residua1gas warmed back to ambient tempera-ture.The characteristic relaxation time (2) was derived from the gradient of the graph of log AT against time. RESULTS TEMPERATURE UNIFORMITY HEAT LOSS RATES AND HEAT TRANSFER COEFFICIENTS We have measured 23 the spatial variation of temperature in the reactor. Except near the walls the effect of stirring is to make temperature excesses very nearly uniform across the reactor. It is this which is the experimentalists' justification for expressing heat loss rates in a Newtonian form (eqn (I) (3) and (4)) and is consistent with some theoretical models for cool flame l7-I9 and other 26* 27 non-isothermal gaseous reactions. Similar spatial temperature distributions prevail for stirring speeds in the range 1200-2400r.p.m.Heat loss coefficients are determined for different gas mixtures in a variety of conditions by a dynamic method. A typical adiabatic quenching and relaxation curve is shown in fig. 1 and with it a semi-logarithmic plot of log AT against time -2.0 -A & RELAXATION 1 e -1.5 I h W I h -8 W -1.0 0 0.I 0.2 0.3 0.4 0.5 0.6 0.7 time/s FIG.1.-Curve (a) An adiabatic cooling followed by thermal relaxation back to ambient temperature (620 K) for nitrogen at 50 kN m-' in a stirred reactor (0.5 dm3). Temperaturechanges are measured directly by a Pt-PtlRh (13 %) thermocouple (25 pm diam. wire). Curve (b) A plot of log (T-To) against time for thermal relaxation. The relaxation time from which I is evaluated (eqn (4)) is determined from the gradient of this line.Values for z and I at various conditions are given in table 1. The actual values of I may be expected to depend on transport properties especially ;L (here altered by dilution) on density (here varied by varying pressure) and on P. GRAY J. F. GRIFFI’krib AND R. J. MOULE stirring efficiency (here varied by the speed of the rotor). The lowest values of / prevail at the lowest pressures and stirring rates (table 1). High concentrations of propane yield high values of 2 thus the simple expedient of dilution with inert gases of low thermal conductivity and heat capacity gives scope to reduce the heat transfer coefficient. T is hardly dependent on pressure the variation of 2 arises mainly from the pressure dependence of c.TABLE 1.-HEAT TRANSFER COEFFICIENTSFOR GASEOUS MIXTURES IN A SPHERICAL VESSEL (0.5 dm3) AT VARIOUS CONDITIONS composition/mol % pressure/kN m-2 temp./K rotor speed/r.p.m. T Is l/W K-1 m-2 N2 = 100 40.0 623 2400 0.16 19.9 53.3 26.2 66.6 32.5 C3Hs = 8 40.0 623 2400 0.18 27.9 N = 92 53.3 34.4 66.6 45.2 CSHs = 33 40.0 623 2400 0.26 28.8 Nz = 67 53.3 38.6 66.6 50.2 623 1 200 0.32 23.0 28.8 36.0 623 2400 0.27 36.4 44.8 54.0 C3Hs = 50 40.0 623 1200 0.34 26.8 N2 = 50 53.3 35.6 66.6 44.4 TEMPERATURE-TIME RECORDS FOR EXOTHERMIC OXIDATION The four main types of non-isothermal behaviour during propane oxidation are depicted in our (temperature time) records. These are (i) a damped oscillatory approach to a quasi-steady state (fig.2a b and c) (ii) nearly undamped oscillations ending abruptly (fig. 24 (iii) a monotonic approach to a quasi-steady state (fig. 2e) and (iv) two-stage ignition (fig. 2f). In damped oscillations successive amplitudes diminished appreciably even after the first temperature peak. For pure propane + oxygen mixtures all multiple cool flames show damped characteristics their damping factors depending upon the initial conditions of temperature and pressure. Fig. 2a shows cool flame oscillations typical of those observed at low temperatures (roughly in the range 580-62OK). Although amplitudes clearly diminish the damping factor is low ; oscillations occur up to the end of reaction. Initial amplitudes are large ( > 100 K) and the peaks have a cusp-like shape interspersed by periods longer than 1 s.At higher temperatures (beyond about 620 K) damping is sufficiently high for oscillations to have died away before the fuel is completely consumed (fig. 2c). These temperature histories are roughly sinusoidal starting with a maximum amplitude that is usually less than 100 K and sometimes as low as 5 K. Periods are generally very short (<2 s). THERMOKINETIC OSCILLATIONS Cool flame oscillations with barely diminished amplitudes occur at low tempera- tures in the oxidation of propane to which acetaldehyde is added (even small amounts say <0.5 mol %). They have steep cusp-like maxima interspersed by shallow minima ;::I50-1 1) -2004 0246 0 24 6810 200-I 50i I 001 I 0 2 4 6 8 1012 02 4 time/s FIG.2.-(Temperature time) histories for propane oxidation at different conditions in a stirred reactor (0.5 dm3).(a)to (d)multiple oscillations ;(a)C3H8(50 mol %) To = 610 K P = 75 kN m-2 ; (6) C3Hs (47mol %)+CH3CH0 (3 rnol %) To = 635 K,P = 65 kN m-2 ; (c) C3H8 (50 rnol %), To = 640 K P = 84 kN m-’ ; (d)C,H (47 rnol %)+CH3CH0 (3 rnol %) To = 600K P = 60kN rn-’ ; (e) slow reaction ; C3H8 (50 mol %) To = 640 K P = 60 kN m-2 ; (f)two-stage ignition ;C3H8 (47 rnol %)+ CH,CHO (3 rnol %) To = 590 K P = 85 kN m-2.and they give the impression of sustained oscillations. That is to say if fuel were to be fed continuously to the system they would propagate indefinitely; in our closed reactor they terminate abruptly when no fuel remains (fig.24. Amplitudes vary between 50 and 150 K and periods from 3 to 15 s the highest temperature excess and largest times being associated with the lowest reactor temperatures (c590 K). These “ sustained ” oscillations occur below about 630 K above which there is a marked change to damped sinusoidstl oscillations (fig.2b). P. GRAY J. F. GRIFFITHS AND R. J. MOULE Slow reaction may be sufficiently exothermic to increase the reactant temperature monotonically (by up to 35 K) to a quasi-steady state (fig. 2e). In two-stage ignition the temperature rises by about 200 K characteristic of a cool flame. This is followed instantly however by a violent hot ignition sometimes after the temperature has begun to fall from its first maximum (fig.2f). Induction times to the occurrence of these non-isothermal phenomena may extend from several seconds to several hours. For this reason fig. 2a d and e are drawn with an arbitrary time zero. Generally the longest induction times are associated with the lowest initial temperatures and pressures but they are reduced dramatically by the addition of acetaldehyde (or other reactive compounds 28). Induction times in hydrocarbon oxidations are rarely exactly reproducible ;they are susceptible to changes of surface such as the deposition of carbon during a hot ignition. PRESSURE-TEMPERATURE IGNITION DIAGRAMS It is most convenient to display these varieties of non-isothermal behaviour according to the initial reactor temperature and the initial reactant pressure for each reactant composition i.e.as a (P,To)ignition diagram (fig. 3 and 4). Because the 90 \ I 80 I N I E I I / E 70 ;ti 60 50 40‘ I I I I I I I J 570 S@CJ ‘330 600 610 620 630 640 650 ambient temperat ure/K FIG.3.-Ignition and cool flame boundaries in a stirred reactor (0.5 dm3) for propane (50 mol %) and oxygen. precise location of boundaries is affected by the reactor dimensions and its surface treatment each experimentalist has to map his own diagram :experimental ignition diagrams usually agree qualitatively but rarely quantitatively. The ignition diagram is divided into the main regions of slow reaction ignition and THERMOKINETIC OSCILLATIONS oscillatory reaction (cool flames) ;it is this last region that is of particular interest to us.Between 1 and 7 consecutive cool flames occur during propane oxidation and fig. 3 and 4 indicate the approximate pressure and temperature locations of their boundaries. However as the temperature histories show (fig. 2aand d) at least for low temperatures in closed conditions the number of successive oscillations appears to be limited by complete consumption of fuel :the exact number is of less significance than their existence. At high temperatures oscillations characteristically die away before reaction is complete (fig. 2b and c). The ignition diagram for pure propane +oxygen mixtures (fig. 3) differs from that when acetaldehyde is added to the reactants (fig. 4). When CH3CH0 is present the cool flame zone extends to lower temperatures (570 K compared with 590 K for pure C3H8/O mixtures) moving with it the two-stage ignition boundary.Moreover there is sufficient distinction between weakly damped and strongly damped oscillations to justify a boundary between them in the (P,To)ignition diagram. For C3H8/02 mixtures damping increases progressively through the oscillatory region. 1 IGNlTlO \ ‘-A’ \ I \WEAKLY DAM? E0 ‘SSCI LLATIONS/ ‘\ / i NS I.. . /I * / \ / \ -\ 3 50 SLOW ---L I GEACTIONS qgfL I I I I I 1 I -1 570 580 50C 530 610 620 630 640 650 700 ambient temperat ure,K FIG.4.-Ignition and cool flamc boundaries in a stirred reactor (0.5 dm3) for propane (47 mol %) and oxygen with acetaldehyde (3 mol %) added.When diIuting inert gases (N2and Ar) are added the cool flame boundary moves to higher pressures. As far as we can tell the new position is determined exactly by the partial pressure of inert diluent. This restricts our studies because the region in which more than 3 cool flames occurs is moved beyond the pressure limit of our system \vhen sufficient diluent is added to affect I markedly. P. GRAY J. F. GRIFFITHS AND R. J. MOULE THE EFFECT ON TEMPERATURE HISTORIES OF VARIATIONS IN HEAT TRANSFER RATES The dependence of the non-isothermal behaviour on the magnitude of 2 is tested most effectively in the closed reactor by altering the stirring rate. As table 1 shows a decrease of 50 % in the speed of the rotor causes an approximately 30 % decrease of 1.Fig. 5 exemplifies how multiple oscillations in the combustion of a 1 1 propane+oxygen mixture are changed when I decreases from 45 W K-’ m-2 (at 2400 r.p.m.) to 35 W K-l m-2 (at 1200 r.p.in.). In particular as I decreases so the damping factor decreases and amplitudes of temperature oscillations are increased. 1501 0 4 8 12 04812 tiinels FIG.5.-(Teniperature time) histories in a stirred reactor (0.5dm3)at 625 K for propane (50 mol %) and oxygen at 60 kN nr2. Curve (a)stirring rate = 2400 r.p.m. Curve (6) stirring rate = 1200 r.p.m. DISCUSSION HEAT TRANSFER RATES Over our ranges of temperature pressure reactant compositions and stirring rates I varies from 20-55 W K-’ ni-2 for the corresponding relaxation times 0.16-0.34s.These values for 1 agree satisfactorily with those that we have determined in other ways using stationary 29 and quasi-stationary 23 methods. A value for I = 25 W K-l m-2 implies that when a reaction is producing 1.5 W dr3 (e0.15 kJ mol-I s-l at 0.5 atm and 620 K) a 1 K temperature excess will be maintained in a spherical reactor. Altering the stirring speed changes heat losses; but even in an unstirred system at appreciably less than 0.5 atm. convection is ~ignificant,~~ so that further forced convection enhances the heat loss coefficient less than linearly. Stirring of the reactants not only produces temperature uniformity and enhanced heat loss rates but also destroys any flame structure. (Temperature time) histories thus become a better mirror of a uniform system.8 PRESSURE-TEMPERATURE IGNITION DIAGRAMS So far as (P,To)diagrams are concerned the areas on them are identified by different temporal behaviour the properties of which correspond to those of the singularities in the (T,X) phase-plane that are experimentally accessible from fixed initial temperatures in closed reactors i.e.limit cycles (“sustained ” oscillations) stable foci (damped oscillations) stable nodes (slow reaction) and unstable saddle points (two-stage ignition). The boundaries between each of these in the (P,To) THERMOKINETIC OSCILLATIONS diagram which represent criticality in the experimentalists’ sense (viz. marginal achievement of ignition or marginal achievement of oscillation) in the phase-plane correspond to the merging of singularities.THE DEPENDENCE OF THERMOKINETIC OSCILLATIONS ON THERMAL PARAMETERS Multiple cool fiames are sensitive to changes of reactor temperature and heat transfer coefficients :increases of each cause decreases in amplitudes and increases in frequency and damping factors (fig. 2 and 5). A logical connection between the cause and effect which can be tested by our results is provided by phase-plane analysis.l8 For example the damping factor depends on (d%/dT-I); if d%/dT becomes negative as is thecase in propane oxidation 2o when reactor temperatures extend through the range 610-650 K or if I is increased then damping of oscillations is enhanced. CONCLUSIONS Although theoreticians have stressed the link between the occurrence of spatial and temporal oscillations and dissipative systems far from equilibrium combustion systems have rarely been invoked as illustrations.They are excellent examples reaction starts far from equilibrium and proceeds through a complex path network to multiple steady states ;it is strongly exothermic and is accompanied by large changes of free energy. Commonly the concept of feedback is interpreted in chemical terms taking the form of either an autocatalytic or an inhibitory kinetic me~hanism.~~ In oscillatory combustion systems chemical autocatalysis even with the unusually responsive changes of rate induced by chain branching only partly describes what is happening. There is also a thermal contribution to a feedback mechanism which because of the Arrhenius temperature dependence of elementary reaction rates is very strongly non-linear.Isothermal oscillatory systems are apt to depend on artificial reaction schemes with elementary steps of order greater than two in active intermediates; the corres- ponding degree of non-linearity may be more easily and naturally attained via the temperature dependence of velocity constants. G. J. Minkoff and C. F. H. Tipper Chemistry of Combustion Reactions (Butterworth London 1962) p. 200. ’R. Ben-Aim and M.Lucquin Oxidation and Combustion Reviews Vol. 1 ed. C. F. H. Tipper (Elsevier Amsterdam 1966) p. 1. B. Lewis and G. von Elbe Combustion Flames and Explosions of Gases (Academic Press New York 1962). 4R.Hughes and R.F. Simmons Twelfth Symposium (International) on Combustion (The Combustion Institute 1969) p. 449. M. Prettre Bull. Sac. Chim.France 1932 51 1132. R. E. Ferguson and C. R. Yokley Seventh Symposium (International) on Combustion (Butterworth London 1959) p. 113. ’P. G. Felton and B. F. Gray Combustion and Flame 1974. J. F. Griffiths B. F. Gray and P. Gray Thirteenth Symposium (International) on Combustion (The Combustion Institute 1971) p. 239. V. Ya Shtern The Gas Phase Oxidation of Hydrocarbons English Translation B. P. Mullins (Pergamon London 1962). lo D. A. Frank-Kamenetskii,Zhur. Fiz. Khinz. 1940 14 30. ’’D. A. Frank-Kamenetskii Difusion and Heat Transfer in Chemical Kinetics English Transla- tion J. P. Appleton (Plenum New York 1969) p.508. l2 A. D. Walsh Trans. Faraday SOC.,1947 43 305. J. J. Tyson and J. C. Light J. Chern. Phys. 1973,59,4164. P. GRAY J. F. GRIFFITHS AND R. J. MOULE l4 J. J. Tyson J. Chem. Phys. 1973 58 3919. R. J. Field and R. M. Noyes J. Chem. Phys. 1974 60 1877. l6 B. P. Belousov Sb. Ref. Radiuts. Med. 1959 1958 145. J. E. Salnikoff,Zhur. Fiz.Khim. 1949 23 258. B. F. Gray and C. H. Yang (a) Trans. Faraduy Soc. 1969 65 1614. (6)J. Phys. Chem. 1969 73 3395. l9 M P. Halstead A. Prothero and C. P. Quinn (a) Proc. Roy. Soc. A 1971,322,377 ;(b) Com-bustion and Flame 1973 20 21 1. 2o J. F. Griffiths P. G. Felton and P. Gray Fourteenth Symposium (International) on Com-bustion (The Combustion Institute 1973) p. 453. 21 J. A. Barnard and A. Watts Twelfth Symposium (International) on Combustion (The Combus-tion Institute 1969) p.365. 22 J. Bardwell and C. N. Hinshelwood,Proc. Roy. Soc. A 1951,205,375. 23 J. F. Griffiths P. Gray and K. Kishore Combustion and Flume 1974 22 197. 24 D. Thompson unpublished results. 25 A. M. Grishin and 0. M. Todes Doklady Akad. Nauk SSSR 1963,151 365. 26 N. N. Semenov Some Problems in Chemical Kinetics and Reactivity Vol. 2 trans. M. Boudart (Princeton University Press 1958) p. 87. 27 P. Gray and P. R. Lee Oxidation and Combustion Reviews. Vol. 2 ed. C. F. H. Tipper (Elsevier Amsterdam 1967) p. 1. 28 M. D. Scheer and H. A. Taylor J. Chem. Phys. 1952 20 653. z9 D. Thompson and P. Gray Combustion and Flame 1974. 30 B. J. Tyler and A. F. Tuck Int. J. Heat Mass Trans. 1967 10,251. ''G. Nicolis and J. Portnow Chem. Reu. 1973 73 365.

ISSN:0301-5696    

DOI:10.1039/FS9740900103

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

Oscillatory and Explosive Oxidation of Carbon Monoxide BY CHINGH. YANG Department of Mechanics State University of New York at Stony Brook Stony Brook New York 11794 U.S.A. Received 30th July 1974 The glow and explosion limits of the CO and O2system are calculated in wide ranges of composi- tion water content surface efficiency temperature and pressure. The boundary of the oscillatory region in the P-Tplane is theoretically predicted. Results are compared with the existing experimental data. The well-known experimental observation that traces of water sensitively affect the oxidation process of carbon monoxide is quantitatively examined and verified. The kinetic oscillation in the low temperature oxidation of carbon monoxide reported by Dickens Dove Harold and Linnett is perhaps one of the most striking observations in gas kinetics.Obviously the autocatalytic chain mechanism in its classical form does not explain this intriguing phenomenon. In an attempt to construct the kinetic oscillation mechanism mathematically a scheme which focused on the interaction between an autocatalytic chain carrier (0atoms) and an inhibitive intermediate (CO molecules) was postulated.2 A binary non-linear model was derived from the scheme. Analysis of this simple model not only yielded solutions for sustained oscillation it also predicted the glow and explosion phenomena that are associated with the oxidation of CO. In order to examine the detailed kinetic mechanism with realistic rate constants a general model which conserves all reactants catalytic impurities (H,O) and ntermediate species was formulated to contain all proposed reactions without invoking the assumption of steady states.Machine-integrated solutions again predicted the various phenomena. Particular effort was directed to the study of the characteristics of kinetic oscillation. The wealth of experimental data on CO oxidation on the other hand is con- centrated on the measurements of glow and explosion limits. It is evident that an extensive calculation programme is needed to map out the limits and oscillatory regions in the P-T plane so that the theory can be quantitatively tested and compared with experimental measurements. This is the primary objective of our present work. The kinetic scheme previously proposed is slightly modified.The glow and explosion limits are calculated with composition of the mixture and wall efficiencies of the vessel varied. The boundaries of the oscillatory regions in the P-T plane are delineated. Computed limits with water content pressure and temperature varied in wide ranges ([H,O] -0.001 -10% P -1-400 Torr and T -400-1 1OOOC) are compared with the existing data. The well-known observation that explosion limits are sensitively affected by traces of water 194-6 is theoretically verified. Wide agreement between the data 7-9 and the theoretical predictions argues convincingly for the validity of the currently proposed kinetic mechanism for the low temperature oxidation of carbon monoxide. 114 C. H. YANG 115 KINETIC MECHANISM AND MATHEMATICAL MODEL The kinetic scheme contains all reactions proposed previously with the addition of only one new reaction (reaction (18)) kl c02+ 02 + c02+ 0 (1) k2 CO+O+M 3CO,"(CO,)+M (2) k3 co:+o 3CO+O (3) k4 CO:+M -+ C02+M (4) ks CO+OH 4CO,+H (5) k7 O+H20 -+20H (7) k8 H -+ wall destruction k9 0 + wall destruction kio HO + wall destruction ki1 H+02+M -+ HO2+M kl2 H02+ HO2 + H202 + 02 k13 H2023-M + 20H+M k14 H02 -+ destruction on wall k15 H2023destruction on wall kl6 H+HzOz 4 H2O+OH k17 O+OH -+ 02+H k18 H+HO + 20H 116 OSCILLATORY AND EXPLOSlVE OXIDATION OF CARBON MONOXIDE y9 = Yg”+!f(Yl-Y8 -Y; f y7-YZ -Y4) -Y5 -Y6* (27) [CO] represents the sum of the concentrations’ of both ground state CO and excited CO; molecules produced in reaction (2) and yo2 represents the initial concentration of the species yi.All chain termination reactions are assumed in the kinetic region. The third body efficiencies for all species are assumed to be unity with only one exception; a value of 2.3 is selected for C02 molecules. Other details of the calculation method are reported in ref. (3). Selected values of the rate constants for all gas phase reactions are listed in table 1. No systematic attempts had been made to iterate the numerical values of rate constants to achieve the optimum fitting of the data. TABLE 1 reaction activation energy/kcal mol-1 Arrhenius factor */cm3 mol-1 s-1 remarks (1) co+o2 (2) CO+O+M (3) co;+o (4) CO;+M’ (5) CO+OH (6) H+02 (7) 0+H20(11) H+02+M (12) HO2+HO2 (13) H202+M (16) H202+H (18) H+HOz (17) OH40 -60 0 0 0 1.08 16.8 19.5 13 0 45 0 9 0 2.25 x loll LOX 1014 5.0~1O1O 1.25 x 105 4.2~10l1 2.24~1014 4.2~1014 5.o~1015 3.18~1014 1.3 x 1013 3.o~1013 1.8 x 10l2 1.17~1O1O ref.(3) ref. (10) ref. (3) ref. (3) ref. (10) ref. (10) ref. (10) ref. (10) ref. (10) ref. (10) ref. (10) ref. (10) (4 * Arrhenius factor for terniolecular reactions has the dimensions of cm‘ mol-2 S-’. (a)A value smaller by a factor of two than the one suggested in ref. (11) is selected. COMPUTATION RESULTS The definitions of the various phenomena such as glow oscillation etc. are generally vague. In the laboratory they are used to differentiate the different characteristics displayed by the bluish light emission from the reacting gases.In general if the pulse of emission is visible for an extended period of time a few seconds or more it is taken to be a glow. The temperature and the pressure of the mixture is then called the “glow limit”. When the emission is intense and brief lasting no longer C. H. YANG 117 than a split second it is identified as an explosion and the corresponding temperature and pressure is denoted as the “explosion limit”. Oscillations are usually referred to a series of glows that appear periodically. The limits of oscillation however had not been systematically measured in previous work. Before the presentation of our results it is desirable to define these terms a little more precisely.Glow-a single stroke of visible emission that lasts a few seconds. After the passage of a glow only a small portion of the reactants in the mixture is consumed. Invisible reactivities are supported for a long time after the glow. In the P-T plane the point where the glow occurs is marked with the sign “2’as the glow limit in our Presentation of data that follows. Sustained glow-an extended stroke of emission which lasts from a second to several minutes. All reactants are consumed after the passage of a sustained glow. The sustained glow limit is marked with “x”. Oscillation-periodic glows with virtually undamped amplitudes. The pressure and temperature at which this occurs is marked with the sign “0”in the P-T plane. After glow-a sustained glow which is preceded by a glow or several cycles of oscillation.It will be shown that there is no real difference between the solutions of sustained glow and explosion. The criterion to differentiate them by the length of their derivation is arbitrarily chosen. It will also be demonstrated that glows and oscilla- tions are all represented by periodic solutions. For a glow the oscillatory solution is sharply damped with only the first one of its emission peaks visible. In all cases calculated trajectories for all reactants and all intermediate species are obtained. As expected the concentration of reactants decays with time. The trajectory of either O2 or CO can be conveniently used to indicate the degree of completion of the reaction at any point in time.0 atom concentration is always the highest among all radicals. The implication is that 0 atoms are involved in the slowest reaction path in the oxidation process. Water concentration is always 180” out of phase with 0 concentration. Hydrogen containing radicals or intermediates are assumed to form water molecules in the gas phase as soon as they are terminated on the walls. According to a bimolecular association theory (CO + 0 + CO +hv) proposed by Broida and Gaydon l4 the emission intensity from the reacting gas is proportional to the concentrations of CO and 0. A threshold intensity must exist below which the emissions become undetectable. An 0 atom concentration of 1O-I’ mol ~m-~ selected previously 2*3 to be the threshold value and it has been was retained in the present work.Since the visible emission always coincides with the 0 atom concentration peaks the presentation of the 0 atom concentration trajectory usually is sufficient to convey the complete kinetic behaviour of the system. The wall efficierxies are assumed to have the values listed in table 2. The first set of limits is calculated with CO +202 mixtures ; results are plotted in fig. 1. Curves A B C D and E correspond to mixtures with water contents of 10 I 0.1 0.01 and 0.001 % by volume respectively. The computing procedure follows the same sequence of steps used in the heating method for determining limits in the laboratory. Trajectories of eqn (19)-(27) are integrated for mixtures with fixed water content composition and pressure.Bath temperatures are raised with increments of 1 or 2 K each time until the desired limits are reached. In fig. 1 solid curves join the points of the lowest temperature at which either a sustained glow or an explosion is calculated. On many occasions these calculated limits are preceded by glows oscillations or both at slightly lower bath temperatures. The boundary of the oscillation region in the P-T plane is drawn with a closed curve of broken lines. 118 OSCILLATORY AND EXPLOSIVE OXIDATION OF CARBON MONOXIDE No oscillation has been obtained for mixtures containing 10 % water or above the pressure of 120Torr. It is interesting to note that the highest pressure at which Dove l2 observed oscillation is 110 Torr. TABLE 2.-wALL EFFICIENCIES reaction wall efficiency (8) H 10-4 (9)0 10-4 (10) OH 10-4 (14) HOz 10-3 (15) Hz0* 10-3 Fig.1. shows that the largest pressure range (40-1 10 Torr) within which oscilla- tions are calculated is for mixtures with 1 % water content. This does not mean however that oscillations are more likely to be found in wetter mixtures experimen- tally. It may be that the reverse is true. Thus for a wet mixture with the pressure fixed the bath temperature range within which oscillation can be calculated is extremely narrow often no more than 1 or 2 K. Unless the heating rate is very slow the heated mixture passes this range and reaches sustained glow or explosion T/"C FIG. 1.-Explosion limits and oscillation region of CO + 202 mixtures.Calculated curves A Water content = 10 % B Water content = 1 % C Water content = 0.1 % D Water content = 0.01 % E Water content = 0.001 %. Experimental results :G Hadman Thompson and Hinshel- woody7Q Gordon and Knipey6 L Lewis von Elbe and Roth,8 NDickens. temperatures before oscillations are fully developed. In contrast with the drier mixtures that contain water from 0.1 to 0.01 % the oscillatory bath temperature range extends up to 20 K in span. To observe oscillations experimentally in these mixtures will probably not be difficult even with an average heating rate. Dove's l2 discovery of oscillations in the CO system is thus probably linked with his use of very dry mixtures at low pressures. Measured axplosion limits by several authors 6-9 are C.H. YANG 119 replotted in dotted lines though the exact water content for the experimental data is unknown. Dickens apparently used the driest mixtures. His data fall in between the calculated limits with water contents of 0.01 and 0.1 %. This seems to be high for his supposedly well dried mixtures. Gordon and Knipe’s data match the limits calculated for mixtures containing 0.1 % water which is one or two orders of magnitude higher than their estimate; one is dubious about the meaning of these discrepancies when the water content in the mixture has not been determined accurately. It should be noted that the experimental limits are merely straight lines drawn through sets of scattered data points. There is no reason to expect the actual limits being straight lilies in the P-T plane.The slopes of the limits are generally steeper for the wetter mixtures. Figures around the points where oscillations are calculated indicate the period of oscillation in seconds. Dove l2 observed no clear dependence of period of oscillation on composition pressure and temperature in his experiments. Our results in fig. 1 show the same random character. EXPLOSION AND SUSTAINED GLOW LIMITS In experimental studies explosion and glow limits are obtained by heating the mixture to a critical temperature. Below this temperature the reactivity of the system is usually unmeasurable. We found this to be the case in our calculations with very wet mixtures or at very high pressures. Two trajectories are plotted in fig.2. They 6 7Q-4-:-j-7794~ r Curve A 1 timeis FIG.2.-Trajectories of explosion and sustained glow. Water content = 10 % composition = COf 20,. Calculated trajectories :A temperature = 441°Cand pressure = 10 Torr B temperature = 400°C and pressure = 3 Torr. are calculated with C0+20 mixtures containing 10 % water. The pressure for trajectory A is 10Torr and the temperature is 441°C. The concentration of 0 atoms rises to exceed the threshold after an induction time of 73 s. Within a period of 0.3 s CO concentration is reduced to 10 % of its initial value. The trajectory describes an explosion according to our definition. If the calculation is repeated with the bath temperature one degree lower than the preceding case the calculated 0 atom con- centration remains indefinitely many orders of magnitude below the threshold.120 OSCILLATORY AND EXPLOSIVE OXIDATION OF CARBON MONOXIDE No reactivity will be detectable in correspondence with this trajectory. The explosion and non-explosion regions in the P-T plane are thus sharply divided by the limit. Trajectory B is calculated at a pressure of 3 Torr and a temperature of 400°C. The 0 atom concentration exceeds the threshold after an induction period of 516 s. It is reduced to 5 % of its initial value in 10 s. The visible emission will probably last 5 to 6 s. It is a sustained glow limit. Again results show the reactivity of the system to be unmeasurable at a temperature one degree lower than the limit. GLOW AND SUSTAINED GLOW TRANSITION The glow and sustained glow phenomena cannot be differentiated in experiments if the reactant consumption is not measured.Hoare and Walsh reported that if stationary conditions of pressure and temperature were used a glow would appear and then slowly fade away in time ; but on reducing the pressure slightly the glow would brighten again. Obviously this could not have happened at a sustained glow limit where the reactants would be exhausted in one stroke. In calculated trajectories glow and sustained glow are clearly different. The trajectory for a glow is an oscillatory solution and the trajectory for a sustained glow is not. A series of trajectories that maps the transition from glow to sustained glow is presented in fig. 3. Mixtures of time/s FIG.3.-Glow and sustained glow transition.Composition = CO+2O2 water content = 0.1 % pressure = 20 Torr. Calculated trajectories A temperature = 575"C B temperature = 577"C C temperature = 580"C D temperature = 581°C. CO +20 with 0.1 % of water are used and the pressure is fixed at 20 Torr. Trajec-tories A B C and D correspond to bath temperatures of 575 577 580 and 581"C respectively. The 0atom concentration represented by trajectory A never exceeded the threshold. It is therefore undetectable. The first peak of trajectory B and the first and second peaks of trajectory C are above the threshold level. Visible glows C. H. YANG will accompany these peaks. The results also show that after the passage of the first peak of trajectory B only 0.4 % of CO concentration is consumed and after the passage of the first and second peak of trajectory C the CO consumption is increased to 3 %.When the bath temperature is raised only one degree to 58 1"C,a sustained glow trajectory D is attained. CO concentration is consumed to 8 % of its initial value in 20 s. The visible glow will last 15 to 20 s and then fade away as reactants are completely exhausted. GLOW OSCILLATION SUSTAINED GLOW AND EXPLOSION TRANSITION Calculations show that as the mixtures become drier the temperature interval in which the transitions take place (glow 3 oscillation -+sustained glow -+ explosion) becomes wider. A series of trajectories is calculated and presented to show such transitions in fig. 4. CO+2O2 mixtures containing 0.1 % of water are used and the tinie/s FIG.4.-Glow oscillation sustained glow and explosion transitions.Composition = CO + 202 water content = 0.1 % pressure = 60 Torr. Calculated trajectories A temperature = 646°C B temperature = 648°C. C temperature = 657°C D temperature = 662°C E temperature = 682°C. pressure is fixed at 60Torr. Trajectories A B C D and E correspond to bath temperatures of 646 648 657 662 and 682"C respectively. Trajectory A is presumably undetectable and a glow must accompany the first peak of trajectory €3. Within the temperature range from 649 to 658°C all solutions are oscillatory and trajectory C (T = 675°C) represents a typical example. The period of oscillation decreases from 18 to 10 s as the bath temperature is raised from one end to the other.Oscillations of this type may continue for many cycles without appreciable change in their period and amplitude. In ref. (3) one hundred cycIes were calculated for one 122 OSCILLATORY AND EXPLOSIVE OXIDATION OF CARBON MONOXIDE case and McCaffrey and Berlad observed over two hundred cycles of oscillation in their experiments with the CO system. In a binary system this type of behaviour is generally associated with oscillations about a limit cycle. Trajectory C represents a sustained glow the visible emission in this case will last about 20 s. The surging second peak of 0 atom concentration is reminiscent of the " pic d'arret " l5 phenomenon in hydrocarbon oxidation. The kinetic mechanism for " pic d'arret "is still unclear but the surge of 0 atom concentration in the final phase of CO oxidation such as shown here may be attributed to the depletion of CO concentration in the mixture.Reactions (2) and (3) are the basic inhibitive steps in the kinetic scheme. The depletion of CO weakens both reactions and chain carriers are no longer pre- vented from a temporary divergence which is finally checked by complete fuel exhaustion. The duration of the sustained glow becomes shorter as the temperature is raised higher above 662°C. At T = 682°C the sustained glow is compressed into a flash or an explosion. AFTER GLOW Linnett et all6 reported the observation of after glows which followed explosions and lasted as long as 20 s. Trajectory A in fig. 5 is computed with a C0+20 mixture containing 1 % water.The pressure is 80Torr and the temperature is 605°C. The flash of emission that coincides with the first peak of 0 concentration is a glow by definition. Seven seconds later the 0 concentration rises again and exceeds the threshold for 10 s. An event like this is naturally identified as an after 20 time/s Fro. 5.-Trajectories of after glow A water content = 1 % pressure = 80Torr temperature = 605"C composition = C0+2OZ. B water content = 1 % pressure = 30 Torr temperature = 525"C composition = CO+502. C. H. YANG 123 glow in the laboratory. The first peak will probably be identified as an explosion due to its brief duration (4s). After glows may also follow many cycles of oscillation. A typical example is calculated with a CO+ 502 mixture containing 1 % water and plotted in fig.5 (trajectory B). The bath temperature and pressure are 525°C and 30 Torr respectively. The glow will last over 20 s. Like the second peak of trajectory C in fig. 4 after glows are caused by the depletion of CO concentration in the final phase of oxidation. EFFECTS OF PRESSURE Egerton and Warren l7 reported that using a withdrawal method a blue glow appeared as the explosion limit of their moist CO and O2mixtures was approached? that on crossing the explosion limit at a fast rate the glow suddenly increased to a brilliant flash ;but that with sufficiently slow evacuation no such transition from glow to flash occurred? the glow rising to a constant intensity throughout the explosion region.Their experiment evidently dealt with the transition between a sustained glow and an explosion. Three trajectories are calculated to show the transitions timeis FIG.6.-Transition among oscillation sustained glow and explosion with pressure varied. Compos-tion = CO+2O2 water content = 0.1 % temperature = 652°C. Calculated trajectories A(A’) pressure = 45 Torr B(B’) pressure = 50 Torr C(C‘ and D) pressure = 45 Torr. among oscillation sustained glow and explosion by varying the pressure. The composition of the mixture is CO +20 and its water content is 0.1 %. The tempera- ture is fixed at 652°C. In fig. 6 trajectory A B and C are calculated at pressures 45 50 and 55 Torr respectively. The fuel consumption rate in terms of CO concentration over its initial value [CO]/[CO],,is also presented for all three cases (curves A B’ and C’).Under 45 Torr pressure trajectory A represents a marginal explosion with a 124 OSCILLATORY AND EXPLOSIVE OXIDATION OF CARBON MONOXIDE duration of about one second. The corresponding fuel consumption curve A’ shows that the fuel concentration falls rapidly one second after the reaction started. Trajectory B (P= 50Torr) represents a typical sustained glow which probably would be visible for about 8 or 9 s. Under the pressure of 55 Torr trajectory C is an oscillatory solution. The corresponding fuel concentration curve C’ shows the steps of falling off that coincide with the peaks of 0 atom trajectory. Water concen- tration over its initial value for the case P = 55 Torr is represented by curve D in fig.6. It shows that 8-12 % of the total water molecules are dissociated at the peaks of 0 atom concentration. Linnett et a1.16 defined a quasi limit according to the time required for the visible emission to build up to the peak intensity. If the required time is equal to or less than 0.15 s then the system is said to be on or to have surpassed the quasi explosion limit. A series of trajectories is calculated with pressures across the entire explosion peninsula B in fig. 1. The composition for the mixture is CO +20 and the water content is 1 %. The bath temperature is fixed at 624°C which was also the value used in the experiments of Linnett ef al. TrajectoriesA B C D E F and G corresponding to pressures of 10 15,20,40 80,90 and 110 Torr respectively are presented in fig.7. Trajectory A (P= 10 Torr) appears to satisfy the criterion for the “ quasi limit ” but the measured limit for the same mixture by Linnett et al. was 20Torr. The dis-crepancy may be attributed to many factors the uncertainties involved with surface efficiencies the numerical values adopted for the rate constants etc. The sustained glow limit for this mixture under 10Torr pressure is 480°C (curve B in fig. 1). Between the temperatures of 480 and 624°C (with pressure fixed at 10Torr) the timels FIG.7.-Trajectories across explosion peninsula. Composition = CO+ 202 water content = 1 % temperature = 624°C. Calculated trajectories A pressure = 10 Torr B pressure = 15 Torr C pressure = 20Torr D pressure = MTorr E pressure = 80Torr F pressure = 90 Torr G pressure = 110 Torr.calculated 0 atom trajectory starts as a sustained glow and accelerates gradually to become a “ quasi explosion ” at 624°C. The transition is always smooth without sudden disruption at any point when the temperature is raised. This is consistent with their conclusion that a lower explosion limit does not exist inside the glow limit. C. H. YANG 125 The mixture is most reactive at the pressure of 40Torr (trajectory D). The ffash of emission in this case lasts no longer than 0.04 s. Curve B in fig. 1 shows that the upper Iimit for the mixture at the temperature of 624°C is 120 Torr. The rate of reaction is gradually decreasing as the pressures are raised to approach the upper limit as shown in fig.7 (trajectories E F and G). EFFECTS OF COMPOSITION Experimental results l2 showed that the reaction rates of lean CO mixtures are usually greater than that of the stoichiometric mixture. Limits for stoichiometric mixtures are calculated and presented in fig. 8. No oscillations can be obtained for T]"C FIG. S.-Explosion limits and oscillation region of stoichiometric mixtures. Calculated curves A water content = 10 % B water content = 1 % C water content = 0.1 % D water content = 0.01 % E water content = 0.001 %. Experimental results from Hoare and Walsh :A' water content = 5 % B' dry. such mixtures with water content in excess of 0.5 %. Hoare and Walsh's data for mixtures of the same composition are replotted as dotted lines (curves A' and B').Their curve A' (5 % water content) appears to fall on the calculated curve A with a water content of 10 %. Results for their dry mixtures (curve B') fit the calculated curve B (1 % water) quite well. It is noteworthy that the peculiar curvature of the measured upper limit at low pressures is closely matched by the calculated one. The shape of the peninsula tip calculated with different water contents does not change appreciably. Hoare and Walsh4 noted that upper limits for the dry and wet mixtures merge together and become one at very low pressures. Our calculated results show that the effect of water on the temperature part of the limit is still distinct only the pressure scale between the different limit curves is compressed at low pressures.It must be pointed out that the glow limits reported by Hoare and Walsh are probably a combina-tion of gtow and sustained glow limits as the two are indistinguishable as long as fuel consumptions are not measured. Limits for very lean and rich mixtures are plotted in fig. 9. Curves A B and C 126 OSCILLATORY AND EXPLOSIVE OXIDATION OF CARBON MONOXIDE are calculated with mixtures containing 1 % water while curves D E and F are for drier mixtures with only 0.01 % of water. The experimental results of Dickens et a2.l are reproduced in dotted lilies A’ B’ and C’. Theoretical results are at least consistent with the experimental measurements in predicting higher reaction rate for the leaner mixtures. The glow and oscillation limits for curves B and E are not shown in fig.9 to avoid crowding. It can be concluded from the results that oscillations are less likely for rich mixtures and the period of oscillation is shorter for leaner mixtures. T/”C FIG.9.-The effects of composition on explosion limits. Calculated curves A 3 C water content = 1 % D E F water content = 0.01 % A D composition = CO+5O2 B E composition = CO+2O2 C F composition = 4CO+O2. Experimental results from Dickens et a/.’ A’composi-tion = C0+902 B’ composition = CO+2O2,C composition = 2CO+O2. EFFECTS OF WALL EFFICIENCIES It is assumed that the kinetic role of the walls is limited to the destruction of chain carriers and intermediates in this work. Wall efficiencies (listed in table 2) which have been used for all calculations so far are intended for mildly reflective vessel surfaces.Limit curves B and D of fig. 1 are replotted in fig. 10 with the labelling retained for comparison with limits curves A and C. Curves A and C are calculated exactly as for curves B and D respectively except all the wall efficiencies are one order of magnitude greater. These higher efficiencies are selected to simulate efkctive vessel surfaces. Curves A and C show the hits shifted to higher temperatures at low pressures where heterogeneous termination is most effective. One surprise result is that the difference between the limits for reflective and effective surfaces are greater at higher pressures than at intermediate pressures. Curves A‘ and C’ represent data from Dickens et aZ.l for CO+202 mixtures in two different vessels (vessel No.1 and 3). Qualitative features of the experimental ~neasurements im clearly followed by the calculated ones. One intriguing question remains namely why the C. H. YANG 127 wall efficiencies for the two vessels which were made of same materials (quartz) and had been treated the same way before tested are so different. The calculated period of oscillation is generally shorter for effective surfaces. The pumping mechanisms due to wall site saturation proposed previously 2* are not invoked in all the calculations presented here. Our other calculations show that the oscillatory region in the P-T plane is enlarged towards the higher and lower pressures if the wall site saturation mechanism is imposed on the scheme.At the present time this region in the P-T plane has not been mapped out by measurements. It is probably premature to either accept or reject this pumping mechanism. T/"C FIG. 10.-The effects of surface on explosion limits. Calculated curves B D reflective surface (wall efficiencies listed in table 2) A C effective surface (lox wall eficiencies listed in table 2). Experimental results from Dickens :A' vessel No. 1 C' vessel No,3. DISCUSSION It was suggested ' that the explosive reaction of CO oxidation is retarded by some product formed in the reaction. Earlier Jon0 found that the reaction rate of a CO and O2mixture fell off sharply after a fast initial start. This was confirmed later by Knipe and Gordon.20 Minkoff and Tipper proposed that in the early phase of oxidation carbon suboxides such as C20and C302may have been produced which inhibit the reaction by removing the chain carrier 0 atoms from the system.Using C20 as an intermediate Gray 21 constructed a theory quite similar in mathematical structure to that of ref. (2) to explain the oscillations and glow limits in CO oxidation. The calculated trajectories for the cases of explosion and sustained glow in fig. 2 3 4 and 7 clearly confirm this observation. The reaction rate which is proportional to 0 atom concentration decreases by as much as several orders soon after the reaction starts. Excited CO; molecules are clearly the inhibitors. Reaction (3) the rate retardation step becomes effective once sukient CO; concentration is accumul- ated shortly after commencement of the reaction.The oscillation phenomenon is of course another product of this autocatalytic and inhibitive reaction tncchanism. 128 OSCILLATORY AND EXPLOSIVE OXIbATION OF CARBON MONOXIDE Dove l2 raised the question :is the supposed effect of water merely to depress an explosion which already exists at high temperature or is the limit entirely dependent on the presence of water? Our answer is affirmative to the second part. It was shown in ref. (2) that water plays a key role in the autocatalytic chain. In the absence of the autocatalytic reaction both the oscillation and explosion phenomena disappear. Our calculations for a mixture with 0.0003 % water have yielded only glow with long lasting slow reaction or extended sustained glow at very high temperature.Dove l2 made similar observations in his experiments with extremely dry mixtures. Dove l2 also reported that intensive use of one vessel caused the limits to become very indistinct or to be replaced by a relatively slow reaction accompanied sometimes by a glow occurring at temperatures above the limit. A possible explanation for his observation may be proposed as follows if we assume that water molecules can cling on the vessel walls so tightly as to defy the drying process they may become active and play a significant role when explosion limits for extremely dry mixtures are measured. After each experiment the wall may readsorb the water from the product gases and the dry mixtures are continually made to appear slightly wetter than they really are.Only after intensive use of the vessel do water molecules manage to escape and when they do the kinetic behaviour of the extremely dry mixture becomes evident as described by Dove. The calculated explosion limits appear to be consistently higher than the experi- 41 mental measurements.” 6* l6 This may mean that either the rate constants selected for the autocatalytic chain reactions are too low or the termination reaction rate constants are too high. One can probably achieve a better fit between the calculated results and data by using a smaller activation energy for reaction (2) or a smaller numerical value for the rate constant k2. Actually a more comprehensive calculation program is needed to optimize the numerical values of all key rate con- stants (such as k3,kq,etc.) in the CO system.P. G. Dickens J. E. Dove J. E. Harold and J. W. Linnett Trans. Faraduy Soc. 1964,60 539. C. H. Yang Comb. Flame 1974 23 97. C. H. Yang and A. L. Berlad J.C.S. Faraday 1 1974 70 1661. D. E. Hoare and A. D. Walsh Trans. Faraday SOC.,1954,50 37. A. S. Gordon J. Chem. Phys. 1952,20 340. A. S. Gordon and R. H. Knipe J. Phys. Chem. 1955,59 1160. ’G. Hadman H. W. Thompson and C. N. Hinshelwood Proc. Roy. SOC.A 1932 137 87; 1932 138,297. B. Lewis G. von Elbe and W. Roth 5th Zrrt. Symp. Combustion (Thc Combustion Institute Pittsburgh Pennsylvania 1955) p. 610. P. G. Dickens Dissertation (Oxford 1956). ’*D. L. Baulch D. D. Drysdale D. G. Hoare and A. C. Lloyd High Temperature Reaction Rute Data (Leeds University 1968) No.1 and 2. ’I R. R. Baldwin L. Mayer and P. Dorao Trans. Faraduy Soc. 1967 63 1665. ’’ J. E. Dove Dissertation (Oxford 1956). B. McCaffrey personal communication. l4H. P. Broida and A. G. Gaydon Trans. Fnruday Soc. 1953 49 1120. Is M. Lucquin J. Chim. Phys. 1968 55 827. l6 J. W. Linnett B. G. Reuben and T. F. Wheatley Comb. Flame 1968 12 325. A. C. Egerton and D. R. Warren Nature 1952 170,420. I8 G. J. Minkoff and C F. H. Tippcr Chemistry of Contbustioii Reactions (Butterworth London 1962). W. Jono Rev. Phys. Chem. Japan 1941 15 17. 2o R. H. Knipe and A. S. Gordon J. Chern. Phys. 1957 27 1418. 21 B. F. Gray Trans. Faraday Soc. 1970 63 1 118.

ISSN:0301-5696    

DOI:10.1039/FS9740900114

出版商:RSC    

年代:1974

数据来源: RSC