Synchronization of large number of nonidentical electrochemical oscillators of SNDR type

1
Synchronization of large number of nonidentical
electrochemical oscillators of S-NDR type

• Adrian Birzu
• University of A. I. Cuza, Iasi, Romania
• Vilmos Gáspár
• University of Debrecen, Debrecen, Hungary

1st Workshop, Haslev, Denmark, May 2-5, 2007
Hungarian Research Found 60417,
Romanian-Hungarian ST Programme
2
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.

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Previous results with an HN-NDR type
electrochemical oscillator
Potenciostat
Rext
Rcoll
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.

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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)
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Polarization curve of one Ni electrode in H2SO4
electrolyte (284 K, Rcoll 200 W )
H - Hopf C Chaos SL Saddle-Loop
I (mA)
V (V)
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Chaotic current oscillations of 8 Ni electrodes
(weak global coupling)
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Chaotic current oscillations of 8 Ni electrodes
(weak global coupling local feedback)
Individual resistors are varied as
Synchronized chaos
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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.
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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.
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Synchronization

• First observed and described by Christiaan
  Huygens in1665

I finally found that this happened due to a sort
of sympathy
1629-1695
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Coupled pendulum clocks
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Simple modes of synchronization
in-phase anti-phase
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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.
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Anodic deposition of Zn
Zn2 (aq) 2e- ? Zn (s)
(1)
(2)
M. G. Lee, J. Jorné J. Electrochem. Soc., 1992,
139, 2843.
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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 Znad e-
2Znad
K3
K4
Znad Had
H Zn
K5
Znad e-
Zn
K6
Zn2 Had
Znad H
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M. G. Lee, J. Jorné J. Electrochem. Soc., 1992,
139, 2843.
Had
Znad
?1 and ?2 fractional surface coverage ?1 and
?2 surface capacities (mol cm-2) A1 A6
complex functions of the potential through K1 K6
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Circuit of an array of Zn electrodes (n)
The electrolyte (through which the global
coupling occurs) is not shown
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Model equationsn electrodes global coupling
local charge balance
Had
Znad
Faradaic current density
current of the i-th electrode
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Strength of global coupling (?)
The strength of global coupling ? 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.
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Characterizing synchronization

• phase diagram
• (Hilbert transform)

sH(t)
Pk(t)
s(t)

• order parameter r(t) Z(t)

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Order parameter
From the book SYNC with the permission of the
author S. Strogatz
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Order parameter vs. coupling strength
From the book SYNC with the permission of the
author S. Strogatz
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Two nonidentical Zn electrodes ? 0 O-1 cm-2
Independent oscillations (V -0.0642 V and Cd
10 F cm-2, ?A ? ?B)
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Two nonidentical Zn electrodes ? 0.8 O-1 cm-2
In-phase oscillations (V -0.0642 V and Cd 10
F cm-2, ?A ? ?B)
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Two nonidentical Zn electrodes ? 1.1 O-1 cm-2
Anti-phase oscillations (V -0.0642 V and Cd
10 F cm-2, ?A ? ?B)
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Order parameter vs. coupling strength
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Two nonidentical Zn electrodes ? 0.2 O-1 cm-2
Partial synchronization (V -0.055 V and Cd 15
F cm-2, ?A ? ?B)
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Two nonidentical Zn electrodes ? 0.2 O-1 cm-2
Partial synchronization (V -0.055 V and Cd 15
F cm-2, ?A ? ?B)
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Two nonidentical Zn electrodes ? 0.5 O-1 cm-2
Period-2 synchronization (V -0.0522 V and Cd
10 F cm-2, ?A ? ?B)
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Two nonidentical Zn electrodes ? 0.5 O-1 cm-2
Period-2 synchronization (V -0.0522 V and Cd
10 F cm-2, ?A ? ?B)
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128 nonidentical oscillators time evolution vs.
? (O-1 cm-2)
V - 0.06 V Cd 10 F cm-2
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128 nonidentical oscillatorslt r gt vs. ? (O-1
cm-2)
V - 0.06 V Cd 10 F cm-2
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128 nonidentical oscillators ? 0 O-1 cm-2
V - 0.06 V Cd 10 F cm-2
Amplitude
Phase
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128 nonidentical oscillators ? 1.0 O-1 cm-2
V - 0.06 V Cd 10 F cm-2
Amplitude
Phase
Partial synchronization
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128 nonidentical oscillators time evolution vs.
? (O-1 cm-2)
V - 0.0642 V Cd 10 F cm-2
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128 nonidentical oscillatorslt r gt vs. ? (O-1
cm-2)
V - 0.0642 V Cd 10 F cm-2
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128 nonidentical oscillators ? 0.3 O-1 cm-2
V - 0.0642 V Cd 10 F cm-2
Amplitude
Phase
clusters
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128 nonidentical oscillators ? 4.2 O-1 cm-2
V - 0.0642 V Cd 10 F cm-2
Amplitude
Phase
clusters
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128 nonidentical oscillators time evolution vs.
? (O-1 cm-2)
V - 0.0521 V Cd 15 F cm-2
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128 nonidentical oscillatorsorder parameter vs.
time
V - 0.0521 V Cd 15 F cm-2
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128 nonidentical oscillatorslt r gt vs. ? (O-1
cm-2)
V - 0.0521 V Cd 15 F cm-2
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128 nonidentical oscillators ? 0.15 O-1 cm-2
V - 0.0521 V Cd 15 F cm-2
Amplitude
Phase
travelling waves 1D spiral
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128 nonidentical oscillators ? 0.5 O-1 cm-2
V - 0.0521 V Cd 15 F cm-2
Amplitude
Phase
travelling waves partial SYNC
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128 nonidentical oscillators ? 3.0 O-1 cm-2
V - 0.0521 V Cd 15 F cm-2
Amplitude
Phase
SYNC swinging
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No summary but
SYNC SWINGING SYMPATHY
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Extras
X
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To be continued ...
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A letter of Huygens to his father
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Exploring the phase space
Two parameter bifurcation diagram of the
Lee-Jorné model for Zn electrodeposition showing
the locus of Hopf-bifurcation.
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The electrodes were made nonidentical by
decreasing and increasing the individual surface
capacity (mol/cm2) by 25 as follows
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Two nonidentical Zn electrodes ? 0 O-1 cm-2
Independent oscillations (V -0.0642 V and Cd
10 F cm-2, ?A ? ?B)
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Two nonidentical Zn electrodes? 0.8 O-1 cm-2
In-phase oscillations (V -0.0642 V and Cd 10 F
cm-2, ?A ? ?B)
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Two nonidentical Zn electrodes ? 1.1 O-1 cm-2
Anti-phase oscillations (V -0.0642 V and Cd 10
F cm-2, ?A ? ?B)
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128 nonidentical oscillators ? 0 O-1 cm-2
V - 0.0521 V Cd 15 F cm-2
Amplitude
Phase