Synchronization of large number of nonidentical electrochemical oscillators of S-NDR type Adrian Birzu University of A. I. Cuza, Iaşi, Romania Vilmos Gáspár University of Debrecen, Debrecen, Hungary 1st Workshop, Haslev, Denmark, May 2-5, 2007 Hungarian Research Found 60417, Romanian-Hungarian S&T Programme

Motivation During the last few decades, the motivation for studying nonlinear chemical dynamics has originated – partially – from our hope to model similar behavior of living systems (rhythm of heart, neural activity in the brain, etc.). However, it is characteristics of biological tissues that they are built of large number of cells global and/or local coupling of the units must play an essential role in generating the collective dynamics With the present project we plan to investigate nonlinear dynamics of coupled chemical systems, learn about the general laws governing the emergence of coherent dynamics develop algorithms for achieving synchronized (controlled) behavior. To reach these goals coupled electrochemical systems are studied experimentally and numerically.

Previous results with an HN-NDR type electrochemical oscillator Potenciostat Rcoll Rext 1 2 3 4 5 6 7 8 Rind C R Pt electrode Counter electrode Ni wires Working electrodes Hg/Hg2SO4 Reference electrode Synchronization and Control of Chaos on Coupled Electrochemical Oscillators I. Z. Kiss, V. Gáspár, J. L. Hudson: J. Phys. Chem. B, 2000, 104, 7554.

Polarization curve of one Ni electrode in H2SO4 electrolyte (284 K, Rcoll = 0 W ) HN-NDR: HN-type of Negative Differential Resistance N I (mA) V (V)

Polarization curve of one Ni electrode in H2SO4 electrolyte (284 K, Rcoll = 200 W ) SL H - Hopf C – Chaos SL – Saddle-Loop I (mA) V (V)

Chaotic current oscillations of 8 Ni electrodes (weak global coupling) t /s i (mA)

Chaotic current oscillations of 8 Ni electrodes (weak global coupling + local feedback) i (mA) electrode t /s Individual resistors are varied as: Synchronized chaos

An S-NDR type electrochemical system Anodic deposition of Zn from ZnCl2 solution S M. G. Lee, J. Jorné: J. Electrochem. Soc., 1992, 139, 2843.

An S-NDR type electrochemical oscillator S-NDR type systems may oscillate only at large Cd’ values Recursive derivative control: I. Z. Kiss, Z. Kazsu, V. Gáspár: J. Phys. Chem. A, 2005, 109, 9521.

Synchronization First observed and described by Christiaan Huygens in1665 “I finally found that this happened due to a sort of sympathy” 1629-1695

Coupled pendulum clocks

Simple modes of synchronization in-phase anti-phase

Synchronization cronoz - chronos (time)  - syn (same, common) “synchronous” - „sharing the common time”, „occurring in the same time” Universal behavior occurring in physical, chemical, biological, economical etc. systems. SYNC: adjustment of rhythms of oscillating objects due to their weak interactions.

Anodic deposition of Zn Zn2+ (aq) + 2e- ⇌ Zn (s) Mechanism: Zn2+ + e- ⇌ Zn+ad (1) Zn+ad + e- ⇌ Zn (2) Kinetic study proved that the first step is autocatalytic Zn2+ + Zn+ads + e- ⇌ 2Zn+ads M. G. Lee, J. Jorné: J. Electrochem. Soc., 1992, 139, 2843.

Detailed mechanism of Zn electrodeposition M. G. Lee, J. Jorné: J. Electrochem. Soc., 1992, 139, 2843. K1 H+ + e- Had K2 H+ + Had + e- H2 K3 Zn2+ + Zn+ad + e- 2Zn+ad K’3 K4 Zn+ad + Had H+ + Zn K5 Zn+ad + e- Zn K6 Zn2+ + Had Zn+ad + H+

M. G. Lee, J. Jorné: J. Electrochem. Soc., 1992, 139, 2843. Had Zn+ad 1 and 2: fractional surface coverage 1 and 2: surface capacities (mol cm-2) A1 … A6: complex functions of the potential through K1 … K6

Circuit of an array of Zn electrodes (n) The electrolyte (through which the global coupling occurs) is not shown

Model equations: n electrodes + global coupling local charge balance Had Zn+ad Faradaic current density current of the i-th electrode

Strength of global coupling (κ) The strength of global coupling κ is varied by changing the individual (Rind) and/or collective resistances (Rcoll) For simplicity, we consider unit surface area (A = 1,0 cm2) for each electrode.

Characterizing synchronization phase diagram (Hilbert transform) sH(t) Pk(t) s(t) order parameter: r(t) = |Z(t)|

Order parameter From the book “SYNC” with the permission of the author S. Strogatz

Order parameter vs. coupling strength From the book “SYNC” with the permission of the author S. Strogatz

Two nonidentical Zn electrodes κ = 0 Ω-1 cm-2 Independent oscillations (V = -0.0642 V and Cd = 10 F cm-2, A  B)

Two nonidentical Zn electrodes κ = 0.8 Ω-1 cm-2 In-phase oscillations (V = -0.0642 V and Cd = 10 F cm-2, A  B)

Two nonidentical Zn electrodes κ = 1.1 Ω-1 cm-2 Anti-phase oscillations (V = -0.0642 V and Cd = 10 F cm-2, A  B)

Order parameter vs. coupling strength

Two nonidentical Zn electrodes κ = 0.2 Ω-1 cm-2 Partial synchronization (V = -0.055 V and Cd = 15 F cm-2, A  B)

Two nonidentical Zn electrodes κ = 0.2 Ω-1 cm-2 Partial synchronization (V = -0.055 V and Cd = 15 F cm-2, A  B)

Two nonidentical Zn electrodes κ = 0.5 Ω-1 cm-2 Period-2 synchronization (V = -0.0522 V and Cd = 10 F cm-2, A  B)

Two nonidentical Zn electrodes κ = 0.5 Ω-1 cm-2 Period-2 synchronization (V = -0.0522 V and Cd = 10 F cm-2, A  B)

128 nonidentical oscillators time evolution vs. κ (Ω-1 cm-2) V = - 0.06 V Cd = 10 F cm-2

128 nonidentical oscillators < r > vs. κ (Ω-1 cm-2) V = - 0.06 V Cd = 10 F cm-2

128 nonidentical oscillators κ = 0 Ω-1 cm-2 V = - 0.06 V Cd = 10 F cm-2 Amplitude Phase

128 nonidentical oscillators κ = 1.0 Ω-1 cm-2 V = - 0.06 V Cd = 10 F cm-2 Amplitude Phase Partial synchronization

128 nonidentical oscillators time evolution vs. κ (Ω-1 cm-2) V = - 0.0642 V Cd = 10 F cm-2

128 nonidentical oscillators < r > vs. κ (Ω-1 cm-2) V = - 0.0642 V Cd = 10 F cm-2

128 nonidentical oscillators κ = 0.3 Ω-1 cm-2 V = - 0.0642 V Cd = 10 F cm-2 Amplitude Phase clusters

128 nonidentical oscillators κ = 4.2 Ω-1 cm-2 V = - 0.0642 V Cd = 10 F cm-2 Amplitude Phase clusters

128 nonidentical oscillators time evolution vs. κ (Ω-1 cm-2) V = - 0.0521 V Cd = 15 F cm-2

128 nonidentical oscillators order parameter vs. time V = - 0.0521 V Cd = 15 F cm-2

128 nonidentical oscillators < r > vs. κ (Ω-1 cm-2) V = - 0.0521 V Cd = 15 F cm-2

128 nonidentical oscillators κ = 0.15 Ω-1 cm-2 V = - 0.0521 V Cd = 15 F cm-2 Amplitude Phase “travelling waves” “1D spiral”

128 nonidentical oscillators κ = 0.5 Ω-1 cm-2 V = - 0.0521 V Cd = 15 F cm-2 Amplitude Phase “travelling waves” partial SYNC

128 nonidentical oscillators κ = 3.0 Ω-1 cm-2 V = - 0.0521 V Cd = 15 F cm-2 Amplitude Phase SYNC + swinging

No summary but … SYNC + SWINGING = SYMPATHY

Extras X

To be continued ...

A letter of Huygens to his father

Exploring the phase space Two parameter bifurcation diagram of the Lee-Jorné model for Zn electrodeposition showing the locus of Hopf-bifurcation.

The electrodes were made nonidentical by decreasing and increasing the individual surface capacity (mol/cm2) by 25 % as follows:

Two nonidentical Zn electrodes κ = 0 Ω-1 cm-2 Independent oscillations (V= -0.0642 V and Cd = 10 F cm-2, A  B)

Two nonidentical Zn electrodes κ = 0.8 Ω-1 cm-2 In-phase oscillations (V= -0.0642 V and Cd = 10 F cm-2, A  B)

Two nonidentical Zn electrodes κ = 1.1 Ω-1 cm-2 Anti-phase oscillations (V= -0.0642 V and Cd = 10 F cm-2, A  B)

128 nonidentical oscillators κ = 0 Ω-1 cm-2 V = - 0.0521 V Cd = 15 F cm-2 Amplitude Phase