Autonomous Chemical Modulation and Unidirectional Coupling in

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Autonomous Chemical Modulation and Unidirectional Coupling in Two Oscillatory Chemical Systems Gábor Holló, and Istvan Lagzi J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b11321 • Publication Date (Web): 04 Feb 2019 Downloaded from http://pubs.acs.org on February 4, 2019

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The Journal of Physical Chemistry

Autonomous Chemical Modulation and Unidirectional Coupling in Two Oscillatory Chemical Systems Gábor Holló1, István Lagzi1,2* 1MTA-BME

Condensed Matter Physics Research Group, H-1111 Budapest, Budafoki út 8, Hungary

2Department

of Physics, Budapest University of Technology and Economics, H-1111 Budapest, Budafoki út 8, Hungary

AUTHOR EMAIL ADDRESS: [email protected]

RECEIVED DATE (to be automatically inserted after your manuscript is accepted if required according to the journal that you are submitting your paper to)

CORRESPONDING AUTHOR FOOTNOTE *Correspondence to: István Lagzi, Department of Physics, Budapest University of Technology and Economics, H-1111 Budapest, Budafoki út 8, Hungary. E-mail: [email protected], Tel.:+361-463-1341, Fax:+ 361-463-4180.

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Abstract Controlling and coupling of out-of-equilibrium reaction networks have great importance in chemistry and biology. We provide an example for the ideal master-slave coupling between two pH oscillators (the sulfite-bromate and the hydrogen peroxide-sulfite pH oscillators operated in continuous flow stirred tank reactors). The coupling between the reactors was realized by transport of the carbon dioxide through a silicon membrane, which is a common chemical species in both systems. We showed that using this strategy, the master system can generate forced pH oscillations in the slave system. We could control the amplitude and frequency of the oscillations in the slave system and reversibly drive the transition in the oscillations between the regular and chaotic regimes. Using this coupling strategy, we could present an example of amplitude modulation in a coupled chemical system.

INTRODUCTION Interaction among autonomous systems operating far from their thermodynamic equilibria is one of the most important features of living (coupling biochemical reaction networks)1-3 and inanimate (coupling reactors in chemical engineering)4 systems. One of the emblematic examples of autonomous systems in chemistry is the chemical oscillators in a continuous-flow stirred tank reactor (CSTR).5 In the past few decades, several phenomena were discovered and investigated in CSTRs such as bistability, perioddoubling bifurcation, and chaotic oscillations.6-9 A higher degree of complexity can be achieved when those chemical oscillators are coupled to each other.10 The coupling can be implemented physically (via transport) and chemically (via common chemical species).10 Such coupling can result in new phenomena such as synchronization (existence of in-phase, anti-phase and phase-death modes) and oscillator death in coupled chaotic chemical oscillators.11-15 Recently, a new design method has been presented and introduced to oscillate “non-oscillatory” chemical species through coupling reaction equilibria (e.g., complexation, precipitation) to pH


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oscillators.16 In this approach, the pH oscillator generates concentration oscillations in a chemical species involved in the equilibrium that would otherwise be non-oscillatory. First examples in the literature showed periodic oscillations in time of non-redox mono-, di- and trivalent cations (fluoride, calcium and aluminum ions).17-19 After the concept was introduced, new oscillatory systems have been engineered driven by pH oscillators: reversible transformation of DNA,20,21 periodic aggregation and disaggregation of nanoparticles,22-24 reversible transformation of vesicle to micelle (and vice versa)25 and periodic contraction of a pH-responsive hydrogel.26 The coupling between two autonomous chemical systems can be manifested in two ways as the master-slave or peer-to-peer coupling. The main characteristic of the master-slave coupling is that the master system drives the phenomenon occurring in the slave system and the slave system does not interfere with the master system in any way. Therefore, the slave system does not affect the main characteristics of the master system. If any feedback exists from the slave system to master system, the coupling can be called as peer-to-peer coupling, in which both systems interact with each other in a mutual way.27 We showed in our recent study that feedback from the slave to the master system exists inevitable in one reactor system.27 In our example, a pH oscillator was coupled to a pH-dependent (carbonate-carbon dioxide) equilibrium. In this coupled system, induced chemical feedback from the slave system to the master system was observed, namely the pH-dependent equilibrium affected the characteristics (time period and amplitude) of the master system. The feedback strength depended on the concentration of the control species (in the study it was hydrogen carbonate). This study showed that a pure master-slave (unidirectional) coupling in one reactor chemical systems through a common (control) chemical species cannot be realized, because this chemical species are involved in both chemical mechanisms of master and slave systems.27


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In this work, we propose and investigate a new experimental setup in which an exceptional masterslave (unidirectional) coupling can be implemented by separating spatially two pH oscillatory systems. The coupling is accomplished with a silicon membrane immersed in the slave CSTR, and the coupling takes place between the effluent solution (waste stream) of the master CSTR and the slave CSTR. This spatial separation ensures no feedback from the slave to the master system. However, due to a transfer of a chemical species (carbon dioxide) through the silicon membrane, the main characteristics of the slave system (amplitude, frequency and the type of the oscillations) can be controlled by the master system. Additionally, we present a concept for autonomous modulation of temporal waves in coupled chemical oscillatory systems.

EXPERIMENTAL We used two chemical oscillators to perform the one-directional coupling. The master system was the bromate-sulfite pH oscillator28-34 and the slave system was the hydrogen peroxide-sulfite pH oscillator (Figure 1).35,36 Both chemical systems have been intensively studied in the literature, and several nonlinear behaviors have been discovered and observed in these systems (e.g., bistability, bifurcation). 37-39

In experiments, we used the following reagent-grade chemicals, NaBrO3 (Sigma-Aldrich),

Na2SO3 (Sigma-Aldrich), H2SO4 (Sigma-Aldrich), NaHCO3 (Sigma-Aldrich) and H2O2 (SigmaAldrich). The aqueous solutions of all chemicals were always prepared freshly before the experiments to avoid any decay of chemicals and the oxidation of sulfite. In experiments, the master and the slave pH oscillations were sustained in CSTRs with a volume of 19.5 mL (at 45.0  0.2 °C) and 10.0 mL (at 20.0  0.2 °C), respectively. The CSTR of the master system was continuously fed by three channels using a peristaltic pump (Ismatec) with constant flows (0.77 mL/min for each channel) of three stock solutions: (i) a solution of acid and bromate ([H2SO4]0 = 8.5 mM and [BrO3−]0 = 0.3 M), (ii) a solution of sulfite ([SO32−]0 = 160 mM) and (iii) a solution of hydrogen carbonate ([HCO3−]0 = ACS Paragon Plus Environment

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6.0 mM). The slave system was fed similarly by two channels (1.83 mL/min for each channel): (i) a solution of hydrogen peroxide ([H2O2]0 = 40 mM) and (ii) a solution of sulfite ([SO32−]0 = 6.0 mM, [HCO3]0 = 0.6 mM, [H2SO4]0 = 0.2 mM). The master and slave CSTRs were fed by three and two channels, respectively. Therefore, the corresponding feed concentrations of chemical species in the CSTRs (𝑐 0𝑖) are three times and twice less than the concentrations of their stock solutions. In both oscillators, the pH was monitored by pH microelectrodes (Mettler Toledo). The coupling between the bromate-sulfite (master) and hydrogen peroxide-sulfite (slave) systems was realized using a peristaltic pump and a plastic tube. The tube - made from polypropylene (Tygon, the tube with a total length of 20.0 cm, an inner diameter of 1.6 mm and a wall thickness of 0.8 mm) - had a middle silicon section (a silicon tube with a length of 94.0 cm, an inner diameter of 0.8 mm and a wall thickness of 0.2 mm). The silicon tube was immersed in the solution of the slave system in the CSTR. The pump continuously transported the effluent solution from the master system with a constant rate through the plastic tube (Figure 1). The silicon membrane is considerably penetrable only for gas phase components such as the carbon dioxide, other chemical species (hydrated ions) cannot cross the membrane. In our approach, the coupling species is the carbon dioxide, which is part of both pH oscillators. We also measured the pH (with another Mettler Toledo pH microelectrode) of the outflow solution from the tube in a small cuvette (the volume of the solution was 1.0 mL). This cuvette contains the effluent solution from the master CSTR went through the silicon part of the tube, where the carbon dioxide exchange took place. Both CSTRs were open to the atmosphere (Figure 1). We carried out several control experiments by flowing sulfite solution at various pH values (pH = 7.0, pH = 6.0 and pH = 5.0) through the silicon tube and observed no change in the characteristics (amplitude and frequency of the oscillations) of the slave system. In order to investigate the effect of carbon dioxide on the hydrogen peroxide-sulfite pH oscillator, we also carried out similar experiments by flowing hydrogen carbonate solution at various pH levels (pH = 7.0, pH ACS Paragon Plus Environment

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= 6.0 and pH = 5.0) through the silicon tube. In this case, we could observe a significant change in the in the characteristics of the slave oscillatory system showing that carbon dioxide can penetrate through the silicon membrane and affect the amplitude and the frequency of the oscillations in the slave system.

Figure 1 The scheme of the master-slave (unidirectional) coupling in the bromate-sulfite pH oscillator and the hydrogen peroxide - carbon dioxide pH oscillator. The master (I) and the slave (II) systems are connected through a silicon membrane, which is fully and selectively penetrable for the produced gas phase carbon dioxide. The pH of the effluent solution from the master system was also measured in a cuvette (III). MM: multimeter; PC: computer.

NUMERICAL MODEL To support our experimental results and observations, we developed a numerical model to investigate the coupling between the master and the slave oscillators. The whole system consists of three subsystems: (i) the master system (bromate-sulfite pH oscillator), (ii) the slave system (hydrogen peroxide-sulfite pH oscillator) and (iii) the plastic tube (tube reactor), which is directly coupled to the slave system via transport of carbon dioxide through the silicon membrane (for simplicity we assume that the tube reactor behaves as a CSTR). In these three systems, the concentration changes of the chemical species in time can be described by the set of ordinary differential equations (ODEs) ACS Paragon Plus Environment

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𝑑𝑐 𝑖 𝑑𝑡

= 𝑅 𝑖 + 𝑖(𝑐 0𝑖 ― 𝑐 𝑖),

(1)

where 𝑐 𝑖, 𝑐 0𝑖, 𝑅 𝑖, and 𝑖 are the concentration of ith chemical species in the CSTRs and tube reactor, the feed concentration of ith chemical species, Ri is the reaction rate, and i is the reciprocal of the residence time, respectively. ODEs were solved numerically by CVODE integration package.40,41 We used the same kinetic mechanism for the bromate-sulfite pH oscillator (Table 1) in the master system and tube reactors. However, in the tube reactor, we neglect the reaction R11 (Table 1), because the system is closed and carbon dioxide cannot be transferred to the gas phase. The kinetic equations for the hydrogen peroxide-sulfite pH oscillator (slave system) can be found in Table 2. The concentration of carbon dioxide in the tube reactor tries to be in equilibrium with the concentration in the slave system. The coupling (carbon dioxide transfer) can be described by the following equation

(

𝑑𝑐 CO2𝑎𝑞 𝑑𝑡

)

𝑠𝑙𝑎𝑣𝑒 = 𝑘 𝑡(𝑐 CO2𝑡𝑢𝑏𝑒 𝑎𝑞 ― 𝑐 CO2𝑎𝑞 ),

(2)

𝑠𝑙𝑎𝑣𝑒

where 𝑘 𝑡 = 10 ―2 1/s, which is the transfer rate constant for carbon dioxide from the tube to the slave system and its value was arbitrarily chosen. A similar equation can be applied to describe the transfer from the slave system to the tube reactor, and its form is the following due to mass conservation

(

𝑑𝑐 CO2𝑎𝑞 𝑑𝑡

)

= ― 𝑡𝑢𝑏𝑒

𝑉 𝑠𝑙𝑎𝑣𝑒 𝑉 𝑡𝑢𝑏𝑒

𝑠𝑙𝑎𝑣𝑒 𝑘 (𝑐 CO2𝑡𝑢𝑏𝑒 𝑎𝑞 ― 𝑐 CO2𝑎𝑞 ),

(3)

𝑡

where 𝑉 𝑠𝑙𝑎𝑣𝑒 and 𝑉 𝑡𝑢𝑏𝑒 are the volumes of the slave CSTR and the tube reactor, respectively. In the simulations, we used the following constant feed concentrations for the master system: 𝑐 0H + = 2.83 × 103 M, 𝑐 0HSO4― = 2.83 × 103 M, 𝑐 0BrO3― = 1 × 101 M, 𝑐 0SO23 ― = 5.33 × 102 M, 𝑐 0HCO3― = 2 × 103 M, 𝑐 0CO2 = 1.32 × 105 M, 𝜅 master = 1.974 × 103 1/s (for all chemical species) and for the slave system: 𝑐 0H + = 4 × 104 M, 𝑐 0SO23 ― = 3 × 103 M, 𝑐 0H2O2 = 2 × 102 M, 𝑐 0HSO4― = 1 × 104 M, 𝑐 0CO23 ― = 3.0 × 104 M, 𝑐 0CO2 = 1.32 × 105 M, 𝜅 slave = 6.1 × 103 1/s (for all chemical species). For the tube reactor, ACS Paragon Plus Environment

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tube we applied a time-dependent feed concentrations (𝑐 tube 0𝑖 ) dictated by the master system (i.e., 𝑐 0𝑖 =

, where 𝑐 master is the time dependent concentration of ith chemical species in the master system) 𝑐 master 𝑖 𝑖 with 𝜅 tube = 3.95 × 102 1/s. The concentration change in the tube reactor can be described similarly to equation (1) as 𝑑𝑐 tube 𝑖 𝑑𝑡

(1)

tube) = 𝑅 tube + tube(𝑐 tube , 𝑖 0𝑖 ― 𝑐 𝑖

tube where 𝑐 tube , 𝑐 tube are the concentration of ith chemical species, the feed concentration of ith 𝑖 0𝑖 and 𝑅 𝑖

chemical species and the reaction rate in the tube reactor, respectively.

no.

Reaction

Rate law

Rate constant

Ref.

R1 SO32 + H+ ⇌ HSO3

k1[SO32 ][H+] k1r[HSO3]

k1=2×1010, k1r=2×103

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R2 HSO3 + H+ ⇌ H2SO3

k2[HSO3 ][H+]  k2r[H2SO3]

k2=1.2×1010, k2r=2×108

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R3 3HSO3 + BrO3 → 3SO42 + Br + 3H+

k3[HSO3][BrO3]

k3= 1.3×101

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R4 3H2SO3 + BrO3 → 3SO42 + Br+ 6H+

k4[H2SO3][BrO3]

k4=3×101

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R5 SO42 + H+ ⇌ HSO4

k5[SO42][H+]  k5r[HSO4]

k5=1011, k5r=1×109

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k6=2

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R6 6H2SO3 + BrO3 → 3S2O62 + Br+3H2O + k6[H2SO3][BrO3] 6H+ R7 H2O ⇌ H+ + OH

k7 k7r[H+][ OH]

k7=1×103, k7r=1×10

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R8 CO32+ H+ ⇌ HCO3

k8[H+][CO32]  k8r[HCO3]

k8=1×1011 k8r=4.8

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R9 HCO3 + H+ ⇌ H2CO3

k9[H+][HCO3]  k9r[H2CO3]

k9=5×1010, k9r=8.6×106

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R10 H2CO3 ⇌ CO2(aq) + H2O

k10[H2CO3]  k10r[CO2(aq)]

k10=1.65×101, k10r=4.3×102

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k11([CO2(aq)]  [CO2(gas)] 

43 k11=5×103 [CO2(gas)] 1.32× 105 M Table 1 Chemical reactions, rate laws and rate constants for the sulfite-bromate pH oscillator (master

R11 CO2(aq) ⇌ CO2(gas)

system (R1-11) and tube reactor (R1-10)) at  = 45 C. ACS Paragon Plus Environment

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no.


Reaction

Rate law

Rate constant

Ref.

R1 SO32 + H+ ⇌ HSO3

k1[SO32][H+]  k1r[HSO3]

k1=5×1010, k1r=3×103

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R2 HSO3 + H+ ⇌ H2SO3

k2[HSO3][H+]  k2r[H2SO3]

k2=6×1010, k2r=109

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R3 H2O2 + HSO3  SO42    

k3[H2O2][HSO3]

k3= 3.5

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R4 H2O2 + HSO3 + H+  SO42     k4[H2O2][HSO3][H+]

k4=5.0×107

adjus ted in this work

R5 H2O2 + SO32  SO42  

k5[H2O2][SO32]

k5=2×101

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R6 SO42   ⇌ HSO4

k6[SO42][H+]

k6=1011 k6r=109

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R7 H2O ⇌ H+ + OH

k7 k7r[H+][OH]

k7=1×103, k7r=1×10

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R8 CO32+ H+ ⇌ HCO3

k8[H+][CO32]  k8r[HCO3]

k8=1×1011 k8r=4.8

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R9 HCO3 + H+ ⇌ H2CO3

k9[H+][HCO3]  k9r[H2CO3]

k9=5×1010, k9r=8.6×106

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R10 H2CO3 ⇌ CO2(aq) + H2O

k10[H2CO3]  k10r[CO2(aq)]

k10=1.65×10, k10r=4.3×102

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k11([CO2(aq)]  [CO2(gas)] 

43 k11=5×103 [CO2(gas)] 1.32× 105 M Table 2 Chemical reactions, rate laws and rate constants for the hydrogen peroxide-sulfite pH

R11 CO2(aq) ⇌ CO2(gas)

oscillator (slave system) at  = 20 C.

RESULTS AND DISCUSSION In our experimental setup, the master system is not directly coupled to the slave system to avoid any feedback from the slave system to the master one. The master system (sulfite-bromate pH oscillator) oscillates independently with an amplitude of ΔpH = 4.0  0.04 and time period of 1613  15 s (Figure 2). Due to the added hydrogen carbonate, there is a production of gas phase carbon dioxide, which ACS Paragon Plus Environment

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concentration is pH-dependent and regulated by the sulfite-bromate pH oscillator. Inevitably, the solution in the tube - which is removed from the master CSTR - contains carbon dioxide as well. In the slave system (hydrogen peroxide - sulfite oscillator) with our experimental conditions, the amplitude and the time period of the oscillations are much less than in the sulfite-bromate pH oscillator. The coupling can be realized by the transfer of carbon dioxide to the slave systems through the silicon tube (membrane). This silicone membrane is part of the plastic tube, which transports the effluent solution from the master CSTR. Therefore, the plastic tube can be considered as another reactor, in which the same reactions take place like in the master system (Table 1). The main difference between them is that the master system is feed by constant fluxes of the reagents. However, in the tube reactor, the feed concentrations of the chemical species always change in time and equal to the actual concentrations of the chemical species dictated by the reaction mechanism of the master system.

Figure 2 Temporal oscillations in the master (sulfite-bromate pH oscillator) and in the coupled slave ACS Paragon Plus Environment

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(hydrogen peroxide - sulfite pH oscillator) systems. The inset shows the first return map in case of an irregular oscillation. The determined maximum Lyapunov exponent for the irregular oscillations ( = 0.24) supports the deterministic chaotic oscillation behavior.

We hypothesized that, when the carbon dioxide penetrates into the slave system, the carbon dioxide affects the main characteristics of the slave oscillatory system due to its pH sensitive nature. Figure 2 shows the detected pH oscillations in the master and slave systems. When the pH is high in the master system the concentration of the carbon dioxide is low, reversible reactions are shifted towards the production of hydrogen carbonate (R8-11 in Table 1), thus less carbon dioxide is transferred to the slave system through the silicon membrane. In this regime, the slave system oscillates regularly with an amplitude of pH = 1.6  0.05 and time period of 40  4 s. However, when the pH decreases in the master system (reversible reactions are shifted towards the production of carbonic acid), more carbon dioxide penetrates into the slave system. This increased concentration of carbon dioxide dramatically changes the nature of the oscillations. Firstly, the amplitude of the oscillations decreased to pH ~ 0.5 and the frequency significantly increased compared to the previously discussed regime; the time period decreased from 40 to 6 s. Most importantly, the behaviour of the oscillations changed from regular to irregular (Figure 2). We can highlight another important feature of this coupled system, namely an experimental realization of chemical modulation. In physics, the modulation is the process of varying properties of a periodic waveform, called the carrier signal, with a modulating signal. There are two types of modulation: frequency modulation and amplitude modulation. In amplitude modulation, the amplitude of the carrier wave is varied in proportion to that of the message signal being transmitted. In our case, the “carrier” signal is a high frequency oscillation in the hydrogen peroxide - sulfite system, and the “modulating” signal is a low frequency oscillation in the sulfite-bromate pH oscillator ACS Paragon Plus Environment

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(Figure 2). There is an also important phenomenon in the coupled system, there is a phase difference between the oscillation of master and the upper envelope of the oscillation in the slave systems (Figures 2 and 3), which is 208  26 s in the experiments. Therefore, as we mentioned in the experimental part, the pH of the effluent from the master system was also measured after the solution went through a silicon tube and interacted with the hydrogen peroxide-sulfite oscillatory system through carbon dioxide membrane transport (Figure 1). It can be seen that there is no phase difference between the oscillation in the effluent solution (tube reactor) and the upper envelope of the oscillation in the slave systems (Figure 2). The phase difference between the master and slave systems (out-ofphase oscillation) exists because these systems are unidirectionally coupled, the slave system cannot influence the master system. However, the tube reactor and the slave system are bidirectionally coupled through carbon dioxide transport. This coupling (between the tube reactor and the slave system) can be considered as a peer-to-peer coupling because the properties of both systems are significantly modified by each other. Therefore, the upper envelope of the oscillation in the slave system and oscillation in the tube reactor are in-phase. Another contributing factor to the phase delay can be the resident/transport time of the solution in the silicon tube. Based on the experimental conditions (inflow rate of the master CSTR) and geometrical constraints (diameter and length of the tube), the resident time of the solution in the tube is 12 s. This is one order of magnitude less than the phase delay observed in experiments (12 s vs 208 s). Therefore, we can conclude that the resident/transport time of the solution in the silicon tube plays a negligible role in the phase delay between the oscillations in master and slave systems.


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Figure 3 Temporal variation of pH in experiments in the sulfite-bromate pH oscillator (master system, I), in the hydrogen peroxide - sulfite pH oscillator (slave system, II) and in the tube reactor (outflow, III), which ensures the coupling between the master and slave systems.

Numerical simulations qualitatively describe the results observed in experiments (Figure 4). Similarly to the experiments, when pH is high in the master system, the slave system exhibits high amplitude and low frequency oscillations. However, when pH is in the acidic range, slave system shows low amplitude and high frequency oscillations. There is a slight difference between the results observed in experiments and obtained in simulations. The kinetic model cannot reproduce the irregular oscillations at low pH regime in the master system. However, it does reproduce the concept of chemical modulation and phase delay in a coupled oscillatory system.


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Figure 4 Temporal variation of pH in numerical simulations in the sulfite-bromate pH oscillator (master system, I), in the hydrogen peroxide-sulfite pH oscillator (slave system, II) and in the tube reactor (outflow, III).

CONCLUSIONS In this study, we presented a new approach to coupling two oscillatory systems in an exclusively masterslave manner. The driving (master) and slave oscillatory systems were coupled by using a tube reactor, which removed the effluent from the master CSTR and had a silicon part (penetrable for gas phase carbon dioxide) submerged in the liquid phase of the slave CSTR. In our setup, the driving system was the sulfitebromate pH oscillator and the slave system was the hydrogen peroxide-sulfite pH oscillator. The coupling was realized by the transfer of carbon dioxide through the silicon membrane, which chemical species was a common species in both, master and slave systems. We showed that using this strategy an ideal masterACS Paragon Plus Environment

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slave coupling can be implemented, in which the master system drives the phenomenon in the slave system, however, the slave system cannot interfere with the master system, thus it cannot affect the main characteristics of the master system. Since the master system remained intact, the properties of the slave system changed significantly: we could control the amplitude and frequency of the oscillations in the slave system and reversibly drive the transition from regular to chaotic regimes. A new frequency arose in the slave system, which was specific to coupling from the master system to the slave system. Using this coupling strategy, we could show an example of the amplitude modulation in a chemical system based on generating forced oscillations via a single chemical species.

ACKNOWLEDGMENT Authors acknowledge the financial support of the Development and Innovation Office of Hungary (NN125752) and the BME-Nanotechnology FIKP grant of EMMI (BME FIKP-NAT). The authors declare no competing financial interest.

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MS title: Autonomous chemical modulation and unidirectional coupling in two oscillatory chemical systems Authors: Gábor Holló, István Lagzi


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TOC Graphic 82x39mm (300 x 300 DPI)


Autonomous Chemical Modulation and Unidirectional Coupling in

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