Production in Oscillating Chemical Systems Bengt Å. G. Månsson Physical Resource Theory, Chalmers University of Technology, S-41296 Göteborg, Sweden

Z. Naturforsch. 40 a, 877-884 (1985); received April 29, 1985 The entropy production in oscillating homogenous chemical systems is investigated by analyz- ing the difference between the average entropy production rate in a stable periodic oscillatory mode and in the corresponding unstable stationary state. A general analytical expression for this difference in the neighborhood of a Hopf bifurcation is derived. The entropy production in two typical models of chemical systems with unstable stationary states and stable periodic is investigated, using fixed as control parameters. The models exemplify both posi- tive and negative entropy production rate differences. One of the investigated models has four free concentrations, the other three. The rate expressions are given by second order mass action kinetics with reverse reactions taken into account. The flows of reactants and products are con- trolled so that only the free concentrations vary, and the entropy of mixing associated with these flows is discussed.

I. Introduction older fields of thermostatics and linear thermo- dynamics, but some general statements valid for Systems that exhibit structure in space and/or large classes of systems have been formulated [1-3]. time (e.g. stationary geometrical patterns, travelling The entropy production in a class of homogenous waves, or periodical oscillations) if they are forced and isothermal chemical systems is investigated away from the thermostatic equilibrium state [1] using mathematical models representative of the have begun to yield some of their hitherto unknown class. The models are constructed as compromises properties in recent years. Systems with such struc- between closeness to physical realism and computa- tures produce entropy, they are dissipative. This can tional tractability. A basic assumption is that the be viewed as a cost for the environment, and thus local equilibrium approximation is valid. Further- raises questions as to why such structures are so more all concentrations are kept homogenous by common in nature and what factors contribute to assuming that the diffusion processes or the effects their high survival ability. In other words, which of the stirring device are very fast compared to evolutionary principles govern the behavior of open effects of the chemical reactions. The reaction rates systems? If the entropy production is a cost, may not are assumed to follow the simplest form of the law minimization of it be favorable for the system? The of mass action in mixtures (or dilute solu- present paper is a contribution to the study of ques- tions). and since true three-body collision-reactions tions such as these for dynamical structures, i.e. are extremely improbable only second order reac- structures in time. tions are used. All reactions are reversible. (The Dissipative structures typically involve parts of one-way reactions of e.g. the "Brusselator" [1] are the state space so far from the thermostatic equilib- thus excluded, as well as the third order reaction.) rium that the macroscopic dynamical equations The basic reason for this approach is that calcula- cannot be linearized in terms of quantities which tion of the entropy production using the thermo- describe deviations from this equilibrium. If the static expression for the becomes dynamics nevertheless can be satisfactorily (in some possible. All fugacity (activity) coefficients are set sense) described by thermal variables we speak of equal to one. "nonlinear ". Nonlinear thermo- The entropy production associated with the intro- dynamics is considerably less developed than the duction and extraction of substances is also anal- yzed. Such flows are intrinsic features of the model since the concentrations of some of the substances are assumed to be kept constant. Variations in the Reprint requests to Bengt Mänsson, Physical Resource Theory Group, Chalmers University of Technology, reaction rates show up as variations in these flows S-41296 Göteborg. Sweden. and the associated production of entropy of mixing.

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Dieses Werk wurde im Jahr 2013 vom Verlag Zeitschrift für Naturforschung This has been digitalized and published in 2013 by Verlag Zeitschrift in Zusammenarbeit mit der Max-Planck-Gesellschaft zur Förderung der für Naturforschung in cooperation with the Max Planck Society for the Wissenschaften e.V. digitalisiert und unter folgender Lizenz veröffentlicht: Advancement of Science under a Creative Commons Attribution Creative Commons Namensnennung 4.0 Lizenz. 4.0 International License. 878 B. Mänsson • Entropy Production in Oscillating Chemical Systems

The entropy production in the stirring and tem- perature control devices is not considered here, nor = J J {(*(/)(6) the entropy production in the devices controlling the flows of reactants and products. + y (x(t)-(x*)) M(*(/)-(**)) + ...! dt,

where the matrix M is the Hessian matrix evaluated II. Entropy Production Rate Difference at x*.

Let the state of a general system be given by a M=\\o(x) x=x<. (7) vector .v (/). and let the equations of motion be The integral over the first term in (6) vanishes when x=f(q-x), (1) the expression (5) is inserted. To second order

2 where q is a scalar control parameter [4] characteriz- A = (e]Me]+e2Me2)r /4 . (8) ing f. Denote the stationary state x* and the entropy production rate a(x(t)). For values of q where Thus the sign of A in the immediate neighborhood closed trajectories solve (1) define of q* is determined by this sum of quadratic forms. If M is positive or negative definite for q close to q* A (q) = ä(q) — erstat (q) , (2) then this obviously determines the sign of A in this neighborhood. There is no guarantee that it will be where astat is the entropy production rate in chemi- always positive or always negative. The two models cal reactions in the stationary state, o (x*), and presented below exemplify both positive and nega- tive values. The derivation above is only valid in an ä = -^Jadr, (3) 1 c immediate neighborhood of q*, for parameter values further away A (q) may for instance change i.e. the average entropy production rate on a closed sign several times. trajectory C with period T. In the neighborhood of a Hopf bifurcation [5] there are two complex conjugate eigenvalues to the jacobian matrix (evaluated at **), which can be written III. The Models

/-i = e (q) + ico (q) and /.2 = e (q) — i co (q) , (4) The models are simple variations of one of a class where e(q*) = 0, q* being the Hopf bifurcation of two-dimensional models with exactly soluble parameter value. (The stationary state is unstable elliptical limit cycles, thoroughly analyzed by and thus oscillations possible if e(q) > 0.) Except in Escher [6-11]. A common feature of these two- pathological cases there exist two corresponding dimensional models is the autocatalytic reaction mutually orthogonal eigenvectors which are also A + 2X-+3X. (9) orthogonal to all other eigenvectors. These two vectors define a plane in the state space, and using This reaction is replaced in each model here by a them two orthogonal unit vectors, say e, and e2, set of mono- and bimolecular, reversible reactions. lying in this plane, can be defined so that for suffi- One set contains two new freely varying substances, ciently small positive e(q) (i.e. q close to q*) the Z and W, which makes the model four-dimensional. closed trajectory is given by the circle The reason for the increased dimensionality is that Hopf bifurcations cannot occur in two-dimensional .V (t) = .v* + r (q) (e} cos co (q) t + e2 sin co (q) t), (5) systems with second order rate expressions [12]. The where r {q*) = 0. other set contains one extra substance and therefore gives a three-dimensional model. Although oscilla- Expansion of the entropy production rate differ- tions can in principle occur in second order two- ence around the stationary state yields dimensional systems where the stationary state is a 1 J saddle point, it is much simpler to create oscillating A = — ] !<7(JC(/))-<7(JC*): d/ 1 0 second order systems in three or more dimensions. B. Mänsson • Entropy Production in Oscillating Chemical Systems 879

The letters A, B, C, and D denote the substances The rate expressions are whose concentrations are kept constant (reactants/ X = J7 + 2JS-J4-2J5 (13a) products) and X, K, Z, and W the substances whose concentrations are allowed to vary freely. The corre- V — — J4 + J 5 — J (, , (13b) = = - JR , sponding small letters are used to denote concentra- (13c) tions. The flows of A, B, C, and D pass through perfect semi-permeable membranes or other devices where the indices 7 and 8 corresponds to reaction through which X, Y, Z, and PT cannot pass. (12a) and (12 b). respectively. The four-dimensional model is given by the reaction scheme IV. Entropy Production in the Models A+X ^ Z, (10a) Z + X ^ W+ X, (10b) There are two kinds of contributions to the entropy production in the described models, one W ^ 2X, (10c) due to mixing and one due to chemical reactions. X + Y ^ B , (lOd) The rate of entropy production due to the reactions 2X ^ Y-hC, (10e) is given by

Y ^ D . (10f) (14) i The reactions (lOa-c) add up to reaction (9). To get elliptical limit cycles in the A'-F-plane one must where T is the temperature and A, is the affinity of exclude the reverse reaction in (9) and (lOe) and reaction /', defined by choose ^-5-C-D-concentrations and rate constants Ai = - Z l'uMj, (15) appropriately. The individual reaction rates are given in where z is the stoichiometric coefficient of j Table 1, with forward and reverse reaction rate (/ ( j = x, y, z, vv) in reaction /, and fj.j is the chemical constants denoted by kt and &_,(/ = 1,..., 6), respec- potential of j, here assumed to obey the ideal gas/ tively. The rate expressions are solution relation

= — J\ + 2J3 - J4 2 Js, (Ha) = RTlnxj, (16) = — J4 + J5 — J6 , (lib) where /.if is the standard state chemical potential at Z = J \ J 2 i (11c) the actual temperature and , and Xj denotes VV = J2~J3. (Hd) the molar fraction. Since the affinities and the reac- With X, Y, and Z as the free substances the three- tion rates are zero in the thermostatic equilibrium dimensional model is given by replacing the reac- state, the affinities are as given in Table 1 (setting tions (lOa-c) by here and in the following the gas constant R = 1 for simplicity). Obviously all terms in the sum in (14) A + Z X + z, (12a) are non-negative, and they are zero only if the reac- Z 2X. (12b) tion rates are zero. The entropy of mixing when one mole of j is Table 1. Reaction rates 7, and affinities A,. introduced into a system is generally

J, exp (Aj/RT) Reaction •^mix — in Xj . (17)

1 k i ax -k-xz k] ax/(k_\z) (10a) Since it is assumed that a, b, c, and d are kept con- 2 k2ZX — k-2 W'.Y k2z/(k.2w) (10b) 2 stant there are necessarily rates of entropy produc- 3 A'3 IV --k. 3-V2 k w/(k_ x ) (10c) 3 3 tion due to entropy of mixing in the in- and out- 4 k4 x v -k^b k4xy/(k_4b) (10d) 5 k5 A-2-~ A"_5 C V' ksxVik.scy) (lOe) flows of A, B, C, and D. These rates are 6 k6y- k-6d ' k6y/(k.6d) (lOf) 7 k-jaz -k-nxz k7a/(k.7x (12a) + J, ln (a/N) - J4 ln (b/N) 2 2 8 k%z —k- 8-V kiz/(k. 8x ) (12b) -J5 ln (C/N)-J6 ln (d/N) (18) 880 B. Mänsson • Entropy Production in Oscillating Chemical Systems in the four-dimensional model and where x is given by the roots of the fourth order polynomial <7mjx = + y7 In (a/N) - J4 ln (b/N) N />, + N PI = 0 (24) -J5\n(c/N)-J6\n(d/N) (19) ] 2 in the three-dimensional one. N is the total molar with . x = A2/A_], (25) It is simple to change the reaction schemes so that ß=k- /ki, an integral over the in- and outflows on any entire 2 (26) closed trajectory is zero. Since the fixed concentra- TV, = 1 +(y. + ß)x, (27) tions can be set equal without changing the dynami- N2 = A6 + A_5c + A4.V , (28) cal behaviour if the rate constants are changed appropriately (keeping the products constant), it is />, = A_4 b(k6 + 3 A_5 C) + A_6 d (2 A_5 c -k4x) 2 always possible to make the integral of (jmix over an -k5x (3k,x + 2k6), (29) entire period vanishing for any closed trajectory. As 2 P2 = x (a A', a- M-3-Y) • (30) a special case it is then also zero in the stationary state. In fact the sign and magnitude of ermix is Generally. (24) can yield more than one stationary arbitrary for entire classes of models with identical state, but in the part of parameter space of current dynamical behavior. Thus it is pointless to make a interest there is only one non-negative real root of thorough investigation of the entropy of mixing in (24). i.e. the stationary state is unique. models of the kind studied here. In real chemical The rate expression of reaction (9) can be re- systems, however, where reactions and rate con- trieved by assuming that r = w % 0, so that (11a) stants are given by nature, this is not so. can be written The sum of the two contributions to the entropy production is x = P2/N\ . (31)

tfsum = a + rrmix. (20) When (x + ß) x 1. expansion of the right-hand side of (31) in powers of x yields The total molar concentration N can be much X greater than the sum of the concentrations of the x = xA,ax2 + X (-AT (32) reacting molecules, indicating the presence of large n = 3 amounts of a nonreacting solvent. By an appropriate •[ßk- (y + ß)n~3 + yk a(y +ß)n~2} , choice of parameters it is possible to make the sum 3 ] either greater or smaller than zero on an entire limit and the rate expression of reaction (9) (with reverse cycle. For the four-dimensional model with the reaction) is found by discarding all terms of order reference parameter values given below this can x4 and higher. Escherts system is in fact reached 40 only happen for N larger than approximately lO , a exactly in the limit number that would make the solution very dilute indeed! For reasonable values of N, say 104— 105, x 0, ß-+0, A_5 = 0. (33a) the sum is positive in this model. xki a constant (i.e. A] a -*• oo), (33 b)

where the entire sum in (32) disappears. If A_5, x V. Stability Analysis and ß are chosen small but non-zero the limit cycles are still roughly planar and elliptic, but they do not To find the interesting regions of the parameter lie in a plane parallel to the .x-y-plane. as can be space where a system oscillates the stability of the seen in the projections into x-v-r- and x-v-w-space. stationary state(s) is investigated. The stationary The time constants for movement perpendicular to states of the dynamical system (11) are given by this plane are roughly five orders of magnitude smaller than those associated with movement paral- y = (A'_4b + k x2 + A_ d)/N , (21) 5 6 2 lel to it. 2 f = {k\ a(\+ ßx) + ßÄ.--3X ) x/{k-\ TV]). (22) The stability of the stationary state is determined 2 ü' = (A_3 + A', a x + i k-2x) x /(k3N\), (23) by the signs of the real parts of the eigenvalues of 881 B. Mänsson • Entropy Production in Oscillating Chemical Systems

the jacobian matrix

4 k-3 x - k4y - 4 ks x - k4x + 2 k-5 c

- k4y + 2 k5.\ - k4x - k-5c - k6 (34) k\a- k2z + L2h' 0

,k2z - k-2 w + 2A'_3x 0

evaluated at the stationary state. The eigenvalues /.,• (/ = 1-4) are the roots of the fourth-order poly- 0.1, b = c = d= 10. The reference value of a is 100 nomial for the four-dimensional model and 490 for the three-dimensional. From projections of the trajec- Det (./-//) = 0. (35) tories onto appropriate planes in A-v-r-vv-space it is Similarly, for the three-dimensional model the sta- clear that the limit cycle is almost planar in both tionary states are given by models and that it has an elliptic character. This is not surprising since the reference case values are y = 9 (v). (36) deliberately chosen to be very close to the values used by Escher. The properties of the models are f = h (x), (37) here found by letting the concentrations o, b, c, and d vary from zero to 104, one at a time, keeping all where other parameters at the reference values. (The refer- 2 g (x) = {k-4b + k.6d+ k5x )/(k4x + k-5c + k6), ence case was also chosen so that the stationary state /7(A) = k-gx2-/kg, (39) would be unique during these variations.) The range of parameter values where qualitative changes and where x is a root of the fourth-order poly- in the dynamical behaviour occur are in all cases nomial close to the reference case.

(kjci - k-7A") h (x) - k4xg (A) -I- k-4b In the four-dimensional model two of the roots of 6 2 (35) have a negative real part of the order of 10 ~ 2 (KS x - A'_5 C g (A)) . (40) throughout, indicating a very fast approach to the One of the roots is real and negative for all values of surface in which the limit cycle/stationary state a, b, c, and d. The jacobian matrix is

/- k-7 z — 4 k-% x - k4y - 4 k5 x - k4x + 2 k-5 c kjü k.7x + 2kt (41) J = I — k4y + 2 k5 A — k4x - k-5 c - kf, 0

\2A-_8a 0 -k*

The eigenvalues are here the roots of the third order polynomial corresponding to (35). As pointed out above, a positive real part of any resides. The locus of the other two roots as a func- eigenvalue to the jacobian matrix evaluated in the tion of the concentration b is shown in Figure 1. stationary state indicates instability of the stationary Since the locus is symmetric with respect to the real state. axis only one root is shown, Im (z (b)) > 0. The locus shows two Hopf bifurcations, one at Im (z) = 0.8007 for b = 9.7887 and one at Im (/) = 3.134 for VI. Simulation Results b = 63.4. The corresponding periods of the oscilla- tions are 7.85 and 2.00 (time units). (The period of The numerical investigations [13] are based upon the is 7.66 for the reference case.) This a reference case with the following parameter val- range is simple to treat numerically. Since B also 6 ues: kx = 100, k.6 = 10 , k2 = 700, k-2 = k4 = k6 = 1.0, participates in a reaction which contains both the 6 4 k3 = 10 , £-3 = 0.01, k-4 = 0.15. ks = 2.5, k.s= 10" , main variables X and F, the properties of the 3 3 k-6 = 0.275. A-7=10 , AT_7 = 10" , A:8 = 7000, A-_8 = system for different b values are investigated in Reft) I f 1 1 1 1 1 1 1 1 • i < • i * ' ' i ' ' ' i • • • ' • i ' i • i • i > 0 8.00 16.0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

Fig'. F Root-locuses /.(b), showing only the branch with Fig. 3. Entropy production rate in the chemical reactions non-negative imaginary parts of the close-to-the-origin as a function of time on the four-dimensional reference eigenvalues of the stationary state jacobian matrix. The case closed trajectory. The solid line indicates the time- three and four-dimensional models are indicated by "3D" average on the trajectory, the dashed line the production and "4D", respectively. The circles mark the reference rate in the unstable stationary state. cases. The locuses return to the real axis for b= 1521 (4D) and b= Mil (3D) and stay on the negative real axis there- after.

more detail than for variations of a, c, and d. The reference case is close to one of the Hopf bifurca- tions (where the complex conjugate roots cross the imaginary axis) where the limit cycle oscillation comes into existence. The locus /. (a) also has two Hopf bifurcations, one at Im (/.) = 0.834 for a = 99.96 and one at Im(/.) = 419 for «=133.1. The corre- sponding periods of the oscillations are 7.53 and 0.015. Due to difficulties in handling the smaller period time numerically this range has not been investigated in detail. The locus /. (c) has one Hopf bifurcation, at Im (/.) = 0.823 for c = 2.5. The corre- sponding period time of the oscillation is 7.63. The locus /. (d) also has one Hopf bifurcation, also at Im (/.) = 0.823 for d= 10.04. In this case the stability increases with increasing d and the oscillations exist also when d= 0. i.e. when the back reaction in (lOf) is excluded. From the time-variation of the flow rates of A. B. Fig. 2. Flows of A. B. C. and D as functions of time on the C. and D on the reference case trajectory (see Fig. 2) four-dimensional reference case closed trajectory. Positive values indicate positive outflow, i.e. production inside the it is clear that A is invariably consumed (reactant) system. while B. C. and D are produced (products). 883 B. Mänsson • Entropy Production in Oscillating Chemical Systems

decreases the chemical entropy production rate by leaving the unstable steady state and going into the oscillating mode for all values of b. The three-dimensional model is dynamically similar, with one eigenvalue of the jacobian matrix being real, negative and of the order of 105. The locus /. (b) (see Fig. 1) also shows two Hopf bifurca- tions, one at Im (/.) = 0.59 for b = 8.96 and one at Im(/.) = 3.517 for 6=88.8, with corresponding pe- riod times of 10.65 and 1.79, respectively. Note- worthy is that the locus /. (a) has only one Hopf bifurcation in this model, and (c) has none (Re (/. (c)) > 0 for all values of c). The entropy production rate difference, also shown in Fig. 4, is different from the one for the four-dimensional system by becoming zero and then positive as b is increased toward the second Hopf bifurcation.

VII. Discussion

It is certainly worthwhile to find out whether there are simple evolution principles governing the behavior of open chemical systems, and especially principles with a clear connection to thermodynam- ics and . It might then become possible to describe evolution in thermodynamical terms, maybe using such concepts as "structure" and "order" which could be quantified with the Fig. 4. The differences A (b) between the chemical entropy help of information theory. It is also possible to production rate in the stationary state and the time average analyze the evolutionary properties of systems using on the limit cycle as a functions of the concentration parameter b. All other parameters are reference case information-theoretical methods [14]. values. The circles indicate the reference cases (b =10). Previously some thermodynamic evolution cri- The difference is negative everywhere for the four-dimen- teria have been found to apply to certain classes sional model (curve labelled "4D") but changes sign for the three-dimensional (3D) one. of systems, e.g. the "minimum entropy production theorem" in linear thermodynamics [1]. The pre- vious studies have mostly concerned themselves The result of the reference case simulation for the with the thermodynamic properties in the neighbor- chemical contribution to the entropy production hood of non-equilibrium stationary states. The pres- rate as a function of time is shown in Fig. 3, with ent investigation differs by studying open, homo- the stationary state value rrstat and the time average genous, non-linear and oscillating chemical systems. <7 also indicated. It is obvious that the entropy One of the simplest rules possible for such systems production rate difference A is quite small com- is that a chemical oscillator has a lower average pared to the variations during a cycle. (Another entropy production rate in the oscillating state than feature, not shown in the figure, is that the entropy in the unstable stationary state. production rate can be given as a simple function of It has been proved here that this rule is not valid. .v with good accuracy.) The derivation in Sect. II of the properties in the Figure 4 shows A as a function of b. It is consis- neighborhood of a Hopf bifurcation indicates that tently negative, i.e. the four-dimensional system both signs of the entropy production rate difference 884 B. Mänsson • Entropy Production in Oscillating Chemical Systems are possible, and the three-dimensional model that the algorithm is local both in state space and proves that there are actually systems where the sign parameter space.) of the difference depends on a parameter. Thus the In conclusion we have seen that an evolution prin- rule is disproved by counterexample. (The same ciple which favors low entropy production rate obviously applies to the version with "higher" would not necessarily favor an oscillating mode instead of "lower".) (although it might do so, as in the four-dimensional The derivation shows that a simple algorithm can model). It remains to be seen whether there are any be used to find out which way the entropy produc- rules that can identify certain classes of chemical tion rate difference turns as the oscillating mode is oscillators as negentropy-savers. This is a stimulat- created in a Hopf bifurcation. The algorithm in- ing topic for further investigation. volves the jacobian matrix J and the hessian matrix M at the stationary state, and it is therefore not Acknowledgement necessary to integrate the equations of motion to My sincere thanks go to Thor A. Bak and Preben find the sign of A in the neighborhood of the bifur- G. Sorensen at the H. C. Orsted Institute, University cation. Nothing is said about the behavior away of Copenhagen, for both initiating this work and from the bifurcation, and the three-dimensional encouragement in many fruitful discussions during model shows that A can change sign. (One can say my visit in the summer of 1984.

[1] G. Nicolis and I. Prigogine, Self-organization in Non- [9] C. Escher, Z. Phys. B 40,137 (1980). equilibrium Systems, John Wiley, New York 1977. [ 10] C. Escher, Chem. Phys. 63, 337 (1981). [2] A. R. Peacocke, The Physical Chemistry of Biological [11] C. Escher, Chem. Phys. 67,239 (1982). Organization, Oxford University Press. Oxford 1983. [12] J. J. Tyson and J. C. Light, J. Chem. Phys. 59, 4164 [3] S. R. de Groot and P. Mazur, Non-equilibrium Ther- (1973). modynamics, North-Holland, Amsterdam 1962. [13] The equations of motion were solved using the [4] H. Haken. Advanced Synergetics, Springer-Verlag, Merson variable-steplength integration method and Berlin 1984. also the procedure described in L. Lapidus, J. H. Sein- [5] J. E. Marsden and M. McCracken, The Hopf Bifurca- feld, Numerical Solution of Ordinary Differential tion and its Applications, Springer-Verlag, Berlin Equations, Academic Press, New York 1971, pp. 1976. 64-65. [6] C. Escher, J. Chem. Phys. 70,4435 (1979). [14] M. Eigen and P. Schuster, The Hypercycle, Springer- [7] C. Escher, Z. Phys. B 35, 351 (1979). Verlag. Berlin 1979. [8] C. Escher, Ber. Bunsenges. Phys. Chem. 84,387 (1980).