Chemical Resonance, Beats, and Frequency Locking in Forced Chemical Oscillatory Systems

Resonance, beats, and synchronization are general and fundamental phenomena in physics. Their existence and their in-depth understanding in physical systems have led to several applications and technological developments shaping our world today. Here we show the existence of chemical resonance, chemical beats, and frequency locking phenomena in periodically forced pH oscillatory systems (sulfite–hydrogen peroxide and sulfite–formaldehyde–gluconolactone pH oscillatory systems). Periodic forcing was realized by a superimposed sinusoidal modulation on the inflow rates of the reagents in the continuous-flow stirred tank reactor. The dependence of the time period of beats follows the relation known from classical physics for forced physical oscillators. Our developed numerical model describes qualitatively the resonance and beat phenomena experimentally revealed. Application of periodic forcing in autonomously oscillating systems can provide new types of oscillators with a controllable frequency and new insight into controlling irregular chemical oscillation regimes.

Step R6 (Table S1) is negative feedback, which consumes the produced H + ions.

Brusselator model
We used the Brusselator model to demonstrate the effect of the periodic inflow in the simulations 3 : B + X  Y + D where A and B are the reagents in the model. Reaction rate constants in reactions (1)  and [Y]0 = 0. The system can be described with ordinary differential equations (ODEs): changed periodically in the simulations as f = 0 (1 + num sin( f )), where 0 was set to 0.03, and f were varied in the simulations. The time period of the system without perturbation ( num = 0) was 21.3. The ODEs were solved in MATLAB with the sundials solver (relative tolerance = 10 10 , absolute tolerance = 10 14 ).

Kinetic model of the sulfite-hydrogen peroxide pH oscillatory system
To support our experimental and numerical results, we used a detailed kinetic model of the sulfitehydrogen peroxide pH oscillatory system to investigate the existence of the resonance and beats. 2 The concentration changes of the chemical species in time can be described by the set o f ODEs where , 0 , and  are the concentration of i th chemical species in the CSTRs, the concentration of i th chemical species in the input feed, Ri is the reaction rate and i is the reciprocal of the residence time, respectively. The kinetic equations with the appropriate reaction rate constants for the sulfite-hydrogen peroxide pH oscillator can be found in Table S1. The reciprocal residence time for all chemical species was changed periodically in the simulations as i = 0 (1 + num sin( f )), where 0 and num were set to 2×10 −3 s −1 and 10 −3 , respectively, f were varied in the simulations. The time period of the system without perturbation ( num = 0) was 80 s. The ODEs were solved in MATLAB with the CVODE solver (relative tolerance = 10 10 , absolute tolerance = 10 14 ). We used the sulfite-formaldehyde-gluconolactone pH oscillator in a continuous stirred-tank reactor (CSTR) with a volume of 4.5 mL and a stirring rate of 800 rpm at 25.0  0.5°C. 4,5 In experiments, we used the following reagent-grade chemicals, Na2SO3 (Sigma-Aldrich), NaHSO3 (Sigma-Aldrich), formaldehyde (Sigma-Aldrich), and D-(+)-Gluconic acid δ-lactone (Sigma-Aldrich). Two solutions were allowed to flow simultaneously with the same flow rate into the reactor using two programmable syringe pumps. The solutions contained (i) sodium sulfite/sodium bisulfite buffer ([SO3 2 ]0 = 4.6 × 10 3 M, [HSO3  ]0 = 4.6 × 10 2 M) and (ii) 1.28 × 10 1 M formaldehyde and 9.4 × 10 3 M gluconolactone, respectively. The aqueous solutions of all chemicals were prepared freshly prior the experiments. The chemical resonance and beats were investigated by forcing the chemical oscillatory system, we superimposed a sinusoidal modulation on the inflow similarly used in the sulfite-hydrogen peroxide system: f = 0 (1 + sin( f )), where 0 , , and f are the constant inflow rate, the relative amplitude of the inflow rate, and the angular frequency of the forcing, respectively. The pH in CSTR was monitored by a pH microelectrode (Mettler Toledo). In this chemical system, the pH changes slowly due to the buffer capacity of the sulfite/bisulfite. When the buffer is consumed, the pH sharply increases up to pH ~ 9 due to the formaldehyde-sulfite reaction. Lactone hydrolysis is pH dependent and its reaction rate is low in acidic, neutral, and slightly alkaline environments. When the pH reaches alkaline values the rate of hydrolysis becomes more pronounced (negative feedback), and the product of the hydrolysis (H + ) slowly decreases the pH back to acidic.

Figure S1
The sketch of the experimental setup. Two solutions were allowed to flow simultaneously with the same flow rate into the reactor using two programmable syringe pumps. In case of periodic forcing, we superimposed a sinusoidal modulation on the inflow rate. The pH was monitored in the CSTR by a pH microelectrode (Mettler Toledo). The peristaltic pump was used at constant but high flow rate (the outflow rate exceeded the sum of the highest inflow rates of syringe pumps) and the position of the outlet was fixed keeping the volume constant in the CSTR.

Figure S2
Dependence of the peak-to-peak amplitude of the resonant oscillations (the forcing frequency is equal to the natural frequency of the oscillatory chemical system) on the relative amplitude of the forcing ( ) in the sulfite-hydrogen peroxide pH oscillatory system ( 0 = 15.5 µL/s) and in numerical model simulations (inset) using sinusoidal forcing function for the inflow of the reagents ( 0 = 0.03).

Figure S13
Application of the resonance and chemical beat phenomenon to determine the unknown natural time period of the aperiodic/irregular oscillation ( 0 = 13.3 µL/s) in the sulfite-hydrogen peroxide pH oscillatory system. The Fourier power spectrum of the aperiodic oscillation has three main components at = 28, 48, and 70 s. When the periodic forcing was applied at = 28 and 48 s, the beat phenomenon was clearly seen. However, when the periodic forcing was applied at the highest time period component from the Fourier spectrum ( = 70 s), the resonant oscillation (close to the resonance) became evident indicating 70 s is the natural time period of the irregular oscillation.

Figure S14
Existence of a) the chemical resonance and b) chemical beats in the sulfite-formaldehydegluconolactone pH oscillatory system using sinusoidal periodic forcing of the inflow of the reagents with two driving frequencies ( 0 = 11.0 µL/s, = 0.909); a) f = 85 s (resonance) and b) f = 70 s (beats). The natural time period of the system is 85 s. S20 Figure S15 Application of a sinusoidal periodic forcing of the inflow of the reagents to obtain pH oscillations with alterable time periods ( 0 = 11.0 µL/s, = 0.909) in the sulfite-formaldehydegluconolactone pH oscillatory system with various driving frequencies (the time period of the pH oscillations are equal to the forcing time periods); a) f = 200 s, b) f = 300 s, and c) f = 400 s. The natural time period of the system is 85 s.