# Belousov-Zhabotinsky reaction

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The Belousov-Zhabotinsky (BZ) reaction is a family of oscillating chemical reactions. During these reactions, transition-metal ions catalyze oxidation of various, usually organic, reductants by bromic acid in acidic water solution. Most BZ reactions are homogeneous. The BZ reaction makes it possible to observe development of complex patterns in time and space by naked eye on a very convenient human time scale of dozens of seconds and space scale of several millimeters. The BZ reaction can generate up to several thousand oscillatory cycles in a closed system, which permits studying chemical waves and patterns without constant replenishment of reactants (Field and Burger, 1985; Epstein and Showalter, 1996; Epstein and Pojman, 1998; Taylor, 2002).

## Basics

Oscillations can arise in a macroscopic medium if the system is sufficiently far from the state of thermodynamic equilibrium (Nicolis and Prigogine, 1977). Oscillating chemical reactions have been known for about three hundred years, but these were mainly heterogeneous reactions. By the beginning of the 20th century, two excellent examples of heterogeneous oscillating reactions had been discovered: the so-called "iron nerve"- the periodic dissolution of an iron wire in nitric acid, and the "mercury heart" - the oscillatory decomposition of hydrogen peroxide on the surface of metallic mercury (Zhabotinsky, 1991). On the other hand, the principle of detailed balance forbids oscillations in the vicinity of thermodynamic equilibrium (Nicolis and Prigogine, 1977). The latter result made a very strong impression on the majority of chemists, who interpreted it as forbidding oscillations in all homogeneous, closed chemical systems. Therefore, the unfortunate custom prevailed until the mid-1960s to ascribe all concentration oscillations observed in chemical and biochemical systems to some undetermined but important heterogeneous processes or simply to technical errors (Zhabotinsky, 1991). This situation has changed after the discovery and study of the BZ reaction.

Belousov has discovered the first reaction of this class with the Ce3+/Ce4+ couple as catalyst and citric acid as reductant. He observed that the color of the reaction solution oscillated between colorless and yellow and found that the frequency of oscillations increased with rise of temperature (Belousov, 1959; 1985). Zhabotinsky replaced citric acid with malonic acid (MA) to create the most widely used version of the BZ reaction (Zhabotinsky, 1964a). He has shown that the oscillations in the solution color were due to oscillations in concentration of Ce4+ (Fig. 1). He further found that oxidation of Ce3+ by HBrO3 was an autocatalytic reaction and self-sustained oscillations of Ce4+ concentration arose after accumulation of bromomalonic acid (BMA). He demonstrated that Br- ion was an inhibitor of the autocatalytic oxidation of Ce3+. He suggested that the BZ reaction consisted of two main parts: the autocatalytic oxidation of Ce3+ by HBrO3 and the reduction of Ce4+ by MA and its bromoderivatives, which were produced during the overall reaction. In his scheme, the Ce4+ reduction is accompanied by the production of Br- from the bromoderivatives of MA. Br- is a strong inhibitor of the autocatalytic oxidation of Ce3+ because of its rapid reaction with the autocatalyst, which is presumably HBrO2 (Zhabotinsky, 1964a,b).

Figure 1: Oscillations of concentration of Ce4+ and phase-shifting caused by pulsed injections of Br- (a), Ag+ (b), and Ce4+ (c).

An oscillatory cycle can be qualitatively described in the following way. Suppose that a sufficiently high Ce4+ concentration is present in the system. Then, Br- will be produced rapidly, and its concentration will also be high. As a result, autocatalytic oxidation of Ce3+ is completely inhibited, and the [Ce4+] decreases due to its reduction by MA and BMA. The Br- decreases along with that of [Ce4+]. When [Ce4+] reaches its lower threshold, the bromide ion concentration drops abruptly. The rapid autocatalytic oxidation starts and raises [Ce4+]. When [Ce4+] reaches its higher threshold [Br-] increases sharply and inhibits the autocatalytic oxidation of Ce3+. The cycle then repeats. The reader can check this description by tracing a limit cycle generated by the Oregonator model and shown in Figure 3a. Phase resetting experiments ( Figure 1) validate this scheme (Vavilin et al., 1973). One can see that pulse injections of Br- or Ce4+ during the rising of [Ce4+] produces an immediate switch to the phase of the [Ce4+] decrease ( Figure 1a, c). An injection of Ag+ , which removes Br- by forming AgBr, switches the system from a declining to an increasing [Ce4+] phase ( Figure 1b).

Vavilin and Zhabotinsky (1969) showed that HOBr was the final product of the oxidation of Ce3+ to Ce4+ by HBrO3. Vavilin put forward the simplest mechanism of the autocatalytic oxidation of Ce3+ or ferroin by bromate and its inhibition by bromide ion (Vavilin and Zaikin, 1971):

Figure 2: A core scheme of the BZ reaction with Ce and BMA.

$\mbox{HBrO}_3 + \mbox{HBrO}_2 \rightarrow \mbox{2BrO}_2^{\bullet} + \mbox{H}_2\mbox{O}$ $\mbox{H}^+ + \mbox{BrO}_2^{\bullet} + \mbox{Fe(phen)}^{2+}_3 \rightarrow \mbox{Fe(phen)}^{3+}_3 + \mbox{HBrO}_2$ $\mbox{HBrO}_2 + \mbox{H}^+ + \mbox{Br}^- \rightarrow \mbox{2HOBr}$

Addition of production of Br- during oxidation of BMA by Ce4+ to this mechanism results in the core scheme of the BZ reaction with cerium ions as catalyst and BMA as reductant ( Figure 2).

Figure 3: Phase portraits of the two-variable Oregonator models, which show nullclines and trajectories of relaxation limit cycles. H is the higher [Ce4+] switching threshold, L is the lower one.

Field, Koros and Noyes (1972) performed systematic and detailed thermodynamic and kinetic analysis of the basic quasi-elementary reactions involved in the BZ reaction and suggested a detailed mechanism of the reaction responsible for oscillations. This meticulous paper stimulated numerous chemists to study the BZ reaction. On the basis of the FKN mechanism, Field and Noyes (1974) have developed a mathematical model of the BZ reaction named Oregonator. The variables of Oregonator are concentrations of HBrO2, Br-, and Ce4+. The relevant scheme is an extension of the scheme in Figure 2. It includes additionally reaction of HBrO3 with Br- that produces HBrO2, and disproportionation of HBrO2. It replaces complexity of production of Br- via the oxidation of MA and its bromoderivatives by Ce4+ with a quasi-stoichoimetric factor, which is the ratio of the rate of Br- production to that of Ce4+ consumption. Oregonator properly models oscillations and excitability in the BZ reaction. Tyson (1977, 1979) reduced Oregonator to two two-variable versions with the fast variable being [Br-] or [HBrO2], and the slow variable [Ce4+]. These versions are variants of the generalized Rayleigh-Van-der-Pol equation. They give excellent presentation of thresholds and switches in the phase plane. Figure 3 shows qualitative drawings of nullclines and relaxation limit cycles for both models.

However, the original Oregonator is not a quantitative model of the BZ reaction. The total concentration of metal-ion catalyst is not incorporated into its parameters, it poorly reproduces the shape of the oscillations, and it does not reproduce the observed oscillatory domains in its parameter space. These deficiencies can be corrected by taking into account reversibility of reactions of the catalyst with the reductant or oxidant (Rovinsky and Zhabotinsky, 1984; Nagy-Ungvarai et al., 1989a,b; Zhabotinsky et al., 1993; Vanag et al., 2000). In particular, the correct shape of the [Ce4+] oscillations shown in Fig. 1 has been obtained with the improved version of Oregonator developed by Nagy-Ungvarai et al. (1989b).

## The BZ reaction variants

The only irreplaceable initial reagent is the oxidant bromate. Ce and Mn ions can be used as the catalysts as well as complex ions of Fe, Ru, Co, Cu, Cr, Ag, Ni, and Os; each usually with two or more different ligands. A plethora of various reductants give rise to oscillations (Zhabotinsky, 1964b; Field and Burger, 1985).

## The BZ dynamics in well-stirred systems

### Closed systems

Oscillations have been found in large ellipsoidal domains in the space of initial reactant concentrations: cerium, bromate, and MA or BMA. The long axes of the domain sections at constant concentrations of total Ce are directed approximately along the diagonal of the bromate-MA(BMA) concentration plane, where the values of projections of the end points differ about three orders of magnitude. The difference in [Ce] is almost four orders of magnitude in the MA system and about three orders in the BMA system. The period-1 oscillations were mostly observed in the BZ reaction, while period-2 oscillations can arise during evolution of oscillations (Zhabotinsky, 1964b; Vavilin et al., 1967a, 1967b). Excitability (Ruoff, 1982) and bistability (Ruoff and Noyes, 1985) arise outside the oscillatory domain. Bistability requires continuous Br- production from a source other than reactions of Ce4+.

### Open systems

Stationary oscillations can continue indefinitely in a continuous-flow, stirred tank reactor (CSTR). More complex regimes such as bursting (Vavilin et al., 1968) and chaotic (Schmitz et al., 1977) oscillations have been found when the BZ reaction was run in CSTR. Later various modes of complex periodic and chaotic BZ oscillations have been studied. The CSTR allows fine tuning of parameters of a chemical oscillator that makes it possible to trace bifurcation sequences, which connect such regimes (Epstein and Pojman, 1998).

## Chemical waves and patterns

Concentration waves can propagate in reaction-diffusion systems with oscillatory or excitatory local chemical kinetics (Field and Burger, 1985; Epstein and Pojman, 1998; Taylor, 2002). Using a thin layer of unstirred solution with the ferroin-catalyzed BZ reaction, Zaikin and Zhabotinsky (1970) observed periodic propagation of concentric chemical waves generated by point pacemakers, which formed target patterns ( Figure 4). These trigger waves are composed of pulses of excitation followed by refractory zones (Field and Noyes, 1974). Collision of such waves leads to mutual annihilation due to the presence of non-excitable refractory zones ( Figure 4c, d) and ( Figure 4f, g).

Figure 4: Generation of concentric waves by pacemakers and entrainment by the fastest pacemaker resulting in a single target pattern in the BZ reaction-diffusion system.

If such waves are broken the excitation fronts curl around their refractory tails forming spiral waves ( Figure 5) (Zhabotinsky and Zaikin, 1971, 1973; Winfree, 1972).

Figure 5: Development of spiral waves after hydrodynamic breaking of a concentric wave (Zhabotinsky and Zaikin, 1971).

Two-dimensional wave fronts can propagate in thicker layers of solution. Breaking such fronts results in formation of three-dimensional scroll waves (Winfree, 1973). Concentric, spiral and scroll waves attracted much attention (Field and Burger, 1985; Epstein and Pojman, 1998; Taylor, 2002; Mikhailov and Showalter, 2006). Many other spatio-temporal patterns have been discovered in various BZ reaction-diffusion systems. Use of 1,4-cyclohexanedione as reductant leads to emergence of negative dispersion of the wave speed. As a result, multiple chemical waves group into packets (Manz et al., 2000; Hamik et al., 2001). The BZ reaction run in a microemulsion leads to emergence of a plethora of new patterns. It generates Turing and short-wave instabilities of the spatially uniform steady state, and produces Turing structures, standing waves, localized structures, waves propagating towards their sources, and segmented waves (Vanag and Epstein, 2001a, b; Mikhailov and Showalter, 2006)

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