Recent advances in the temporal and spatiotemporal dynamics induced by bromate–sulfite-based pH-oscillators

The bromate–sulfite reaction-based pH-oscillators represent one of the most useful subgroup among the chemical oscillators. They provide strong H+-pulses which can generate temporal oscillations in other systems coupled to them and they show wide variety of spatiotemporal dynamics when they are carried out in different gel reactors. Some examples are discussed. When pH-dependent chemical and physical processes are linked to a bromate–sulfite-based oscillator, rhythmic changes can appear in the concentration of some cations and anions, in the distribution of the species in a pH-sensitive stepwise complex formation, in the oxidation number of the central cation in a chelate complex, in the volume or the desorption-adsorption ability of a piece of gel. These reactions are quite suitable for generating spatiotemporal patterns in open reactors. Many reaction–diffusion phenomena, moving and stationary patterns, have been recently observed experimentally using different reactor configurations, which allow exploring the effect of different initial and boundary conditions. Here, we summarize the most relevant aspects of these experimental and numerical studies on bromate–sulfite reaction-based reaction–diffusion systems.


Introduction
The BrO 3 − and SO 3 2− are the most frequently used initial constituents in oscillatory chemical reactions. The BrO 3 − is a strong oxidant, it is capable of oxidizing many organic and inorganic compounds in oscillatory manner. When the BrO 3 − reacts with the strong reductant SO 3 2− in a CSTR, we can observe two types of oscillations 1 3 depending on the environment in which the reaction takes place. In a strong acid medium ([H 2 SO 4 ] ~ 1-2 M), the instability arises from the chemistry of BrO 3 − . In the oscillatory cycle the positive feedback is the autocatalytic formation of intermediate HBrO 2 , the negative feedback species is Br − , which removes HBrO 2 in a fast reaction. In some systems, a catalyst is needed for the oscillations to occur. The oscillators, composed of BrO 3 − , reductant (e.g., SO 3 2− [1]), and H 2 SO 4 (and catalyst) are referred to as Belousov-Zhabotinsky (BZ)-type systems.
When the reaction between excess BrO 3 − and SO 3 2− takes place in an unbuffered solution, we can encounter H + -autocatalysis in batch reactor, bistability between high and low pH-states or large amplitude pH-oscillations (∆pH ~ 3-4) in a CSTR [2]. The overall stoichiometry in the oxidation of SO 3 2− to SO 4 2− by BrO 3 − is identical to that it is in the highly acidic solution, but the source of nonlinearity is rather different, it originates from the sulfur chemistry. In neutral or slightly basic mixture of BrO 3 − and SO 3 2− the reaction starts very slowly but becomes selfaccelerating when the more reactive HSO 3 − and H 2 SO 3 intermediates are formed and oxidized to H 2 SO 4 with increasing rate. To observe oscillations, the original pH should be restored. This can be achieved by adding a second substrate to the system that reacts with BrO 3 − in a H + -consuming process and it is silent at high pHs but dominant in the acidic state. The two-substrate pH-oscillators in which the oxidant is BrO [4]. Under special condition, the negative feedback can develop in the BrO 3 − -SO 3 2− system without the presence of a third component. In this situation, a new route-accompanied with H + -consumption-opens in the BrO 3 − oxidation of SO 3 2− . It is worth to mention that the highly alkaline Na 2 SO 3 input solution contributes significantly to the overall negative feedback in each oscillators. The BrO 3 − -SO 3 2− pH-oscillator (abbr. BS) can be viewed as minimal version in the subgroup of SO 3 2− -based pH-oscillatory reactions. Its extended versions show pH-oscillations in a wider range of the experimental parameters, and they exhibit richer dynamics than the simplest one. The composite reactions (Eqs. 1-4), the brutto process in the H + -autocatalytic BrO 3 − oxidation of SO 3 2− (Eq. 5) and the negative feedback reactions in the BS (Eq. 8), BSF (Eq. 6) and the BSM (Eqs. 7a and 7b) systems are listed in Table 1. Table 1 The composite and overall chemical reactions that play essential role in the pH-oscillations observed in the BS, BSF and BSM systems In this report, we demonstrate that some pH-oscillators, especially the BS, BSF, and BSM systems can provide strong enough H + -pulses to stimulate repetitive responses in such combined systems where pH-sensitive chemical or physical equilibria are linked to the pH-oscillator. Induced oscillations are generated in the coupled subsystem if the primary oscillator is able to alter the equilibrium between both fore and back directions periodically. In this case the concentration of all components of the equilibrium are forced to oscillate with the frequency of the core oscillator. Bearing this approach in mind, the number and variety of the species that can be incorporated in oscillatory process can significantly be extended.

Induced oscillations in the BS, BSF and BSM systems
The BS pH-oscillator and its variants are perfectly suitable to play the role of the primary oscillator in the combined systems. They exhibit the largest pH-changes, their operation is practically resistant to the presence of the components of the coupled subsystem, the time the oscillator spends in the high and low pH-states can be tuned to the desired range by varying the experimental parameters. Fig. 1 shows the pH-oscillations in time recorded in the BSM CSTR system.
The simplest manifestation of the induced oscillations are the periodic color changes of an acid-base indicator in a pH-oscillatory mixture. To see more complex induced oscillations, the indicator should be replaced with an appropriate pH-dependent complexation or precipitation reaction [5]. Here we list interesting examples. It was possible to bring about rhythmic changes in the concentration of some non-redox ions. Periodic Ca 2+ pulses were generated in the BSF-CaEDTA flow system [6]. The BSF oscillator does not react with CaEDTA, however this complex undergoes dissociation at low pHs but stable in high pHs, which results in two orders of magnitude changes in the concentration of the free Ca 2+ during an oscillatory cycle. Oscillations in the free [Al 3+ ] were observed in the BSF-Al(NO 3 ) 3 flow system due to the formation of highly insoluble Al(OH) 3 above pH 4 and its complete dissolution at pH < 4. Periodic pulses in the [F − ] were measured in the BSM-Al(NO 3 ) 3 -NaF combined system where two equilibria, a precipitation and a complex formation, were coupled to each other and to the BSF oscillator [7]. In this case, at low pHs, the [F − ] is low as it is bound in the complex (Al 3+ + 4F − → AlF 4 − ), at high pHs, the free [F − ] is high due to the formation of highly insoluble Al(OH) 3  . Beside visual observation of the periodic appearance and disappearance of the free M 2+ , if M 2+ = Ca 2+ or Cd 2+ their concentration can be followed by ion selective electrode as well [8]. In this example, the oxidant BrO 3 − and the reductant SO 3 2− participate directly in bringing about induced oscillations. Even more complicated forced periodical phenomenon appeared when the BS oscillator was coupled to the highly pH-dependent stepwise complex formation reaction between Ni 2+ and histidine (abbr. His) [9]. In this case, periodically changing species distribution in the aqueous [Ni 2+ ], the [NiHis + ] and the [Ni(His) 2 ] were observed and recorded by simultaneous measurement of the pH and the light absorbances at the characteristic wavelengths of these species.
Periodic changes in the oxidation number of some redox ions may be generated if the ion in question is involved in the oscillatory mechanism, as Ce(IV) and Mn(II) do in the BZ-reaction. Similar dynamical phenomenon was found to occur in the BS system coupled to complex formation reaction between Co 2+ and histidine [10]. During the pH-oscillatory cycle, no reactions take place at pH < 4 but above this value the Co(II) undergoes SO 3 2− -induced air oxidation to Co(III), which is stabilized in the highly inert Co(III)-histidine chelate. The original composition is restored at the acidic pHs when the Co(III) complex is washed out from the reactor and the reagents are replenished by the continuous flow.
There are several examples of using BS systems to induce oscillations in physical properties of coupled subsystem. When a piece of poly(acrylic acid) cryogel was suspended in the BSF pH-oscillatory mixture, threefold changes in the volume of the gel were observed. The transitions occurred fast, the repetitive shrinking and swelling did not destroy the gel structure. These features make this system suitable to be used as a dynamical unit in molecular motors [11]. Reversible adsorption-desorption of Ag nanoparticles (obtained by reduction of AgNO 3 ) was detected on the surface of a honeycomb-patterned pH-sensitive hydrogel when a film of this gel was fixed in the BSF batch-like pH-oscillator. The phenomenon was supposed to arise from the formation of different morphology on the surface of the gel at low and high pH [12].

Reaction-diffusion patterns in one-side-fed gel reactors
The spatiotemporal dynamics of the activatory-inhibitory reactions based on the IO 3 − -SO 3 2− and the H 2 O 2 -SO 3 2− reactions have been thoroughly studied after clarifying the origin of the patterns in the IO 3 [13,14]. The Swinney group has observed a large variety of FIS patterns in the early'90 s [15,16]. However, the explanation of the origin of the differential diffusion that leads to the formation of the stationary patterns was missing. It turned out that the presence of low mobility weak acid functions plays a crucial role as they can induce an effective slowing down of the H + ions by immobilizing these species through fast reversible protonation of these functions [13,14]. Based on this finding, the pattern forming capacity of several similar systems has been successfully explored. Turing and other stationary and dynamic patterns have been found and studied experimentally in the IO 3 [19,20]. These experiments were made in open one-side-fed gel reactors (OSFR), where a piece of hydrogel, typically agarose, is in contact through a single surface with the content of a CSTR. The sketch of an OSFR is presented in Fig. 2a. Different reactors have been designed to follow the 3D patterns along the direction of the diffusive feed (Fig. 2b) and perpendicular to that (Fig. 2c). The pH-change is monitored by bromocresol green indicator (pK a = 4.8). In the pictures (panels b and c) at the dark regions, the pH is above 6, while at the light regions, it is below 4. For details see Reference [23] Study of the spatiotemporal dynamics of the core BS reaction-diffusion (RD) system in an OSFR resulted in observation of spatial bistability between two different spatial states and spatiotemporal oscillations [21]. When the gel is fed from the CSTR with a mixture of the reactants, at high pH and low extent of reaction the gel content can be in a quasi-homogeneous (high pH) F (flow) state or in a mixed M state characterized by a color switch of the pH-indicator that settles parallel to the CSTR/gel interface. These stationary spatial sates can also be observed in the extended versions of BS system, as it is illustrated in Fig. 2b and c. The stability domain of these spatial states may overlap, which defines spatial bistability. The appearance of spatiotemporal oscillations in the core BS system is originated from the kinetics of the reaction. It is known that the partial oxidation of SO 3 2− to dithionate ions (Eq. 8) creates a H + -consuming pathway, which provides an effective negative feedback. Numerical simulations have revealed this explanation [21].
Recently, we have studied how the introduction of another H + -consuming reaction, which is independent of the oxidation of SO 3 2− ions, enriches the BS reaction dynamics [22,23]. We have explored experimentally and numerically the behavior of the BSF reaction and its coupling to the equilibrium reaction between Ca 2+ and EDTA [24].
The primary negative feedback process in the BSF system is the reaction between BrO 3 − and [Fe(CN) 6 ] 4− . The BSF RD system shows spatial bistability between the F and M states (Fig. 2b and c) at low input feed concentration of [Fe(CN) 6 ] 4− . An appropriate perturbation of the system in the domain of spatial bistability induces a moving front. However, above a critical input feed concentration of [Fe(CN) 6 ] 4− , spatiotemporal oscillations, pH waves can develop [22]. During this phenomenon, first an acidic M state-like region appears, which expands. Due to the negative feedback process, the middle part of this M state-like region recovers to the original high pH-state after some delay. At this point the cycle can restart. Snapshots of the Fig. 3 Snapshots of sustained spatiotemporal pH-waves in the BSF system (a-e) and Ca 2+ waves in the coupled BSF-CaEDTA system (f-j). The period time of the spatiotemporal oscillations is about 20 min. The pH and Ca 2+ concentration changes are monitored by bromocresol green and arsenazo III indicator, respectively. In snapshots a-e in the dark regions, the pH is above 6, while in the light regions, it is below 4. In snapshots f-j at the dark regions, the Ca 2+ concentration is high. For details see References [22] and [24] corresponding experiments are shown in Fig. 3a-e. Numerical simulations successfully reproduced the experimental observations [23].
The robust formation of pH-patterns with a relatively large pH step makes the BSF reaction a perfect candidate to create chemically coupled RD systems. In this case, the spatiotemporal pH-variation is used to drive a pH-sensitive equilibrium reaction, the complexation of Ca 2+ ions by EDTA [24]. The calcium patterns are visualized by using arsenazo III indicator (Fig. 3f-j). The dynamics of this coupled system seem to be very sensitive to small changes in the thickness of the gel, which results in the localization of the waves. Besides the wave phenomena, stabilization of spots has also been observed [24]. This behavior indicates that the feedback of the driven CaEDTA equilibrium on the driving BSF RD system cannot be neglected. Differential diffusion may result in slowing down the effective diffusivity of H + ions by the interaction of the protonated CaEDTA with the agarose matrix.
In RD systems, stationary patterns can form in the presence of long-range inhibition, e.g., slow diffusion of the activator compared to that of the inhibitor. Sodium polyacrylate (NaPAA), a large molecular weight molecule, has been used to ensure long-range inhibition in the BSF reaction. The different observed pattern formation scenarios are shown in Fig. 4.
A labyrinthine pattern may form when the M state becomes unstable. The repulsive interaction between the remaining high pH-islands avoids their fusion, and finally a labyrinthine pattern settles (Fig. 4a and e). Similar patterns also form by the instability of the F state ( Fig. 4b and f), when at the beginning M state islands and filaments appear. At higher input feed concentration of [Fe(CN) 6 ] 4− ions, spatiotemporal oscillations develop from the F state. The waves create stationary filaments or spots when they approach another existing stationary stripe. The oscillatory region shrinks and vanishes at the expense of the stationary pattern ( Fig. 4c and g). At last, we have observed standard Turing instability: the For details see Reference [23] 1 3 formation of the low pH-spot pattern follows a hexagonal symmetry (Fig. 4d and  h). The typical wavelength of these patterns is λ = 2.7 mm.

Reaction-diffusion patterns in gel reactors with separated sources of reactants
In an OSFR, the reactions take place both in the CSTR and in the gel. The natural operation method (i.e., the use of an appropriately large flow rate) keeps the CSTR on an "unreacted" steady state, limiting the available parameter space. A simple way to fully avoid the progress of reactions in the feeding reservoirs is the separation of the reactants and the use of different sources for the primary reactants. Different reactors have been designed for this purpose. The two-side-fed gel reactors (TSFR) consist of two open tanks, which are separated by a gel, and the gel is fed from its two opposite sides (Fig. 5a). This configuration creates strong cross-gradients for the reactants. Another way is to soak the two gels with reactants and push them together (Fig. 5b). As the reactants diffuse toward each other, they can react, and transient dynamical phenomena may occur. Recently, a new design has been proposed, where channels are made in a hydrogel (Fig. 5c). The solutions of the reactants flow through the channels and diffuse into the gel. A significant difference between this construction and the TSFR configuration is that here the diffusion field is inhomogeneous. Before our investigations, none of these reactors was used to create and study pattern formation in H + -autocatalytic reactions, like the BS reaction-based ones.
During the last three years, we have started to explore systematically the RD dynamics of H + -autocatalytic reactions in reactors with separated sources of reactants [25][26][27]. In our experiments, the BS-based reactions-above all the BSF system-were preferred to use. We separated the reactants of the autocatalytic process (i.e., acid and oxidant) on different sides, while the species responsible for the negative feedback (i.e., SO 3 2− and [Fe(CN) 6 ] 4− in particular cases) were placed symmetrically. The TSFR and the multi-channel reactors are of open configurations, where stationary spatial states can be sustained. These states are indicated by F and M (like in an OSFR). In the F state, the whole RD domain is in a low-extent-of-reaction state  [25][26][27] ( Fig. 6a and c). The M state, in which the gel has a localized high-extent-of-reaction zone, can be reached at higher values of the boundary acid concentration (Fig. 6b  and d). Here, contrary to an OSFR (Fig. 2b), the high-extent-of-reaction zone is located between two low-extent-of-reaction zones (Fig. 6b). In the two-channel configuration, the sources of the reactants are framed by the gel, so the RD field is not uniform in direction z, and the high-extent-of-reaction zone of the M state widens on the top and bottom (Fig. 6d). The boundary concentrations of the separated species and the gel thickness (or channel distance) in direction x are good control parameters to tune the width and position of the middle zone of the M state of the gel. In the autocatalytic subsystem and at low concentrations of [Fe(CN) 6 ] 4− , spatial bistability was found between the F and M states in both open geometries. The twolayer gel reactor is a closed configuration; thus, we cannot sustain time-independent stationary states. In the autocatalytic subsystem, a fast autocatalytic front propagates through the gel, starting from the contact zone of the two halves (Fig. 6e). In the presence of [Fe(CN) 6 ] 4− (negative feedback), localized high-extent-of-reaction zones develop in the contact zone and in a separated region (Fig. 6f) (separation of the two reaction zones begins after 4 h). The middle one is quasi-stabilized, but the upper one moves towards the upper edge of the reactor.
After finding spatial bistability, the negative feedback was increased in order to look for more complex dynamic phenomena. In the TSFR and the two-channel gel reactor, we created sustained, localized pH waves above a critical concentration of  (Fig. 7a). The waves propagate in the middle of the two-side-fed gel domain with a period of about 15 min. The flow-through gel reactor can be extended with further channels, thus we can create neighbouring RD zones that may diffusively interact. The time-evolution of two oscillatory zones, being fed through one common channel (B) and two distinct outer channels (A1, A2), can be seen in Fig. 7b. In a TSFR, Turing instability was induced by adding NaPAA. Due to the oscillatory destabilization of the M state, a localized row of stationary, low-pH spots emerged ( Fig. 7c and d). Similar Turing pattern developed starting from the F state, by the consecutive appearance of the spots. In a two-layer gel reactor, the existence of a complex spatiotemporal BSF pattern is highly nontrivial, because of the lack of the continuous supply of the reactants. However, simulations made by the Rábai model of pH-oscillators suggested transient spatiotemporal oscillations in this geometry (Fig. 7e). The reason behind this phenomenon is the fact that the relatively large gel halves can act as transitory reagent sources of the initial reactants, providing Simulated space-time plot of the spatiotemporal oscillations in the two-layer gel reactor by using the model of two-substrate pH oscillators (e). A and B indicate the separation of the acid and the BrO 3 − reactants, respectively. For details see references [25][26][27] fuel for several oscillatory cycles. Being very sensitive to the kinetic parameters, this behavior was not produced experimentally neither with the BSF, nor with other pH-oscillators. But it should be noted that the two-layer gel reactor was successful in creating patterns with a ClO 2 -I 2 -malonic acid oscillator [28].
The price of the flexibility gained by using separated sources of the reactants is the presence of sharp cross-gradients. When TSFR was used to study the spatiotemporal dynamics of the ClO 2 − -I − -malonic acid (CIMA) reaction in a TSFR, it was recognized that in this feeding mode, different instabilities might develop at different positions in the gel. This property made difficult the interpretation of the observed patterns [29]. It is essential to recognize that the CIMA reaction has much more complex kinetics than the BS-based reactions. The capability of the CIMA reaction to show temporal oscillations in a batch reactor demonstrates it. The relative simplicity of the BS-based reactions, especially the fact that one of the reactants is nearly totally consumed in the reaction zone, avoids this problem. Patterns develop only in a well-defined region of the gel as gradients naturally localize them. The interaction of the gradients and the nonlinear kinetics is a paradigm of developmental biology since the seminal work of Wolpert [30]. This perspective also makes it challenging to explore the dynamics of H + -self-activated reactions in these reactors.

Conclusions
Information was collected and experiments were carried out to establish the usability of the bromate-sulfite-based pH-oscillators for producing novel temporal and spatiotemporal dynamical phenomena. From these knowledges and results it was concluded that both BS system and its variants are good-may be the best-candidates among all pH-oscillators for generating rhythmic changes in the concentration of the species or in the states of such a pH-dependent equilibrium reaction that is coupled to the pH-oscillator through H + . Many examples are presented where a BS based primary oscillator forces the linked equilibrium to respond periodically with the frequency of the core system. This kind of induced oscillations may represent a way as oscillations occur in living organisms. The BS based pH-oscillators were successfully used to produce spatial dynamical phenomena as well. So far spatial bistability and Turing structure were observed in a TSFR. The use of other kind of gel reactors, like two-layer closed gel reactor or the recently designed device which is fed from two or three flow-through channels provides best chances to bring about more sophisticated spatial structures.

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