1

Negative Differential Resistanceof Extended Viologen.

Oscillations with Odd HarmonicFrequencies.

Stochastic Resonance.

J. Heyrovský Institute of Physical Chemistry, AS CR, Prague

Institute of Organic Chemistry & Biochemistry, AS CR, Prague

Lubomír Pospíšil, Magdaléna Hromadová

2

Redox reactions of organic radicalsExample of negative impedance causing oscillations

-1.5 -2.0 -2.5 -3.00.0

0.2

0.4

FA-i

/

A

E / V vs Fc

-10k -5k 0 5k

0

5k

10k

D

-6k -4k -2k 0 2k

0

2k

4k

6k

8k

F

-3k -2k -1k 0

-1k

0

1k E

0 2k 4k 6k

0

2k

4k B

0 2k 4k 6k

0

2k

4k A

-2k 0 2k

0

2k

C

-Z

''/o

hm

.cm

2

Z' / ohm.cm2

aromatic nitro-compounds

This presentation: extended viologenNegative differential resistance near E0

3

4

Why to deal with electrochemicaloscillators?

Good models for system dynamics

Many controlling parameters

Electrical current or potential directly monitorsthe rate of kinetics

5

Why to model oscillations andchaos?

Climate & geophysics(7 years polar motion, El Nino, sea temperature)

Economic & Social(business cycle, generation gaps, news cycle)

Astronomy(helioseismology, cosmological cyclic model, neutron stars)

6

Common chemical reactions:reactants → products

The first electrochemical oscillator:1828 Fechner

Periodic release of gas bubblesfrom iron in HNO3

G. Th. Fechner: Schweiger’s J. Chem. Phys. 53 (1828) 129

7

Important oscillating processes:

Circadian clock, or circadian oscillatorbiochemical mechanismoscillates with a period of 24 hours

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Important oscillating processes:

Alpha waves: neural oscillations in brainfrequency 8 – 12 Hz1931 Berger inventor of electro-encephaloraphy

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Belousov–Zhabotinsky reaction

1950 Boris Belousov1961 Anatol Zhabotinsky1968 congress in Prague and dissemination to the West

Classical example of non-linear thermodynamicsmix of potassium bromate, Ce(IV) sulfate, propanedionic acid and citric acidin dilute H2SO4

concentration of the cerium(IV) and cerium(III) ions oscillatescolor oscillates: yellow ↔ colorless

reduction Ce(IV) → Ce(III) by propanedionic acid oxidation back to Ce(IV) → Ce(III) ions by bromate(V) ions.

10

Two driving forces of opposite sense

Mass on a string & gravity

Charging a capacitor in parallel with inductance

Energy dissipation as heat, friction,… damping

External energy source: driven oscillator

Damped & driven oscillators

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Oscillating reactions

Corrosion of metals in acids

Oxidation of organic compounds on Cuelectrodes

Electrocatalysis by adsorbed ions

Oxidation of gaseous CO on Pt

Bio-processes

12

Outline of our recent results

Molecular conductors –

viologen oligomers

Driven oscillations yield only odd harmonics

Stochastic resonance of a driven system

13

Molecular conductorsocto- and deca-cations of “extended viologens”

N NAcHN NHAcN N N N N N N N

Two electrons for each viologen unit

n = 2, 3, 4, 5, 6

14

DC polarograms (low current damping)Reversible 3-electron reductionCurrent instability near potential E0

0.7 mM 0.07 mMConcentration

of extended viologen

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Mechanism & models

Positive feed-back loop:

disproportionation re-generates reducible parent form

1)-(n1)-(n2)-(nn

2)-(n1)-(nn

WWWW

...WWW ee

Negative feed-back loop:change of concentration gradient & mass transport

16

e-

e-

e-

18+18+

17+17+

16+16+

15+15+

+e-

-e--

-e--

+e-

electrode solution of 18+

masstransport

heterogeneouselectron transfer

homogeneousbimolecular

electron transfer

Coupling of heterogeneous and homogeneouselectron transfers & mass transport

Oscillations caused by periodic changesof rates of heterogeneous electron transfer,

homogeneous electron transferand mass transport rate

Extended viologen:Negative differential resistance near E0

17

18

Extended viologen:AC polarograms with admittance < 0

Low frequency AC polarogram

0.32 Hz

Im

Re

19

Extended viologen:Currenttime series & FFT

Fourier transform

Odd harmonics: 3rd, 5th, 7th, 9th, 11th…

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Extended viologen:Odd harmonics due to stochastic resonance

Stochastic resonance (SR)cooperative action of fluctuation andperiodic driving in bi- & multi-stable systems

SR can trigger the transfer of powerto the periodic signal and enhance it.

Driving by pickup of ~1 mV & 50 Hz

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• 1980’s: Bartussek and co-workers: theory of higherharmonics for non-linear systems

• cases with potential-symmetric, potential-asymmetric,and additive or multiplicative fluctuations

• potential-symmetric systems suppress 2nd harmonicsand retain the odd harmonics

• Fast and reversible viologens are potentialsymmetric systems.

Theory of stochastic resonancein non-linear systems

Examples of stochastic resonancefrom other fields

GeoscienceLaser opticsAcousticsEnzymatic reactions

M. Hromadová, M. Valášek, N. Fanelli, H. N. Randriamahazaka, L. PospíšilStochastic Resonance in Electron Transfer Oscillations of Extended Viologen.J. Phys. Chem. C, 2014, dx.doi.org/10.1021/jp501608b

23

Co-authors:

Magdaléna HROMADOVÁMichal VALÁŠEKNicolangelo FANELLIHyacinthe N. Randriamahazaka

Acknowledgement:

Grant Agency Czech Rep.,

Joint project C.N.R.-AS CR

Joint project C.N.R.S.-AS CR

24Thank you for your attention

25

Chaos is lifePeriodicity is boring

Stability is death

Thank you for your attention

26

Can we predict a change:

stability – periodicity – deterministic chaos

???

27

Deterministic chaos random fluctuations

Observation at time tn depends on values at tn-1, tn-2, …

Functional dependence exists - may not be known

Random events do not correlate with previous eventsNo functional law

The proof of a hidden functionaldependence:

Do the observation in timeform an attractor ?

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Attractor

An attractor can be a point, a finite set ofpoints, a curve, a surface,

or even a complicated set with a fractalstructure known as a strange attractor.

Observed time series: i(t1), i(t2),…i(tn-1), i(tn)

Plotting i(t) vs i(t - 2) vs i(t - 4 )

30

Extended viologen:Time delay analysis of experimental series

31

Deterministic chaos random fluctuations

Diagnostic criteria

5 10

-0.5

0.0

0.5

i/A

t / s

Current-time series → Time-delayed plot →→ Phase space diagram → Attractor orbit →→ Exponential evolution → Ljapunov exponent λ

0 10 20 30 40 50

-8

-6

-4

-2

S(

n)

n

exponential divergence of nearbytrajectories for three embedded

space dimensions → λ=0.41 > 0 !!!Reconstructed attractor

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Ljapunov exponent :quantifies the stregth of chaos

Adjacent trajectories diverge in time

Chaos: exponentially fast rate of separation

δo distance between two points

δΔt = δo e λ Δt

Ljapunov exponent : averaged exponent λ

Type of motion: λ<0 … fixed point

λ=0 … stable orbit periodicityλ>0 … deterministic chaosλ= … noise

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Universal theoryof deterministic chaos

Mitchell Feigenbaum (*1944)

Properties of iterating equations

Feigenbaum universality constant

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Stability depends on parameter a

Quadratic iterator: xn+1 = a xn(1-xn)(Peitgen, Juergens, Saupe: Chaos and Fractals)

a

lim xn

n

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Feigenbaum constant

lim = k-1 / kk

k-1

k

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Feigenbaum constant stability – oscillation – deterministic chaos

lim = k-1 / kk

Experiment: = 4.68Theory: = 4.6692…

Universal constant : similar to universality of π

Experimental verification of only in few casesFeigenbaum constant for prediction of chaos

37

Other examples ofelectrochemical oscillators:

Cationic electrocatalysisAnionic electrocatalysis

38

Reduction of aromatic radical anionRepulsion is size-dependent

-1.5 -2.0 -2.5

0.0

0.5

1.0

1.5

2e-

1e-

3e-

HexBuPrEt

Me

-i/

A

E / V (Fc/Fc+)

Different tetra-alkyl ammonium salts as electrolytesDC polarograms:

Ar-NO2−●

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Acceleration effect :Cationic catalysis

Ar-NO2−● + K+ ↔ [ Ar-NO2

−●, K+ ]

Ion- pair formation eliminates electrostatic repulsion

[ Ar-NO2−●, K+ ] + 3e- → Ar-NHOH + K+ ≈ -1.6 to -1.9 V vs Fc

" cationic catalysis "

Ion pairs are reduced at less negative potentials by 800 mV:

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Termination of cationic catalysis

Acceleration effect is terminated at potentials of K+ reduction

K+ + e- ↔ K(Hg) ≈ -2.3 V vs Fc

-1.5 -2.0 -2.5

0.0

0.1

0.2

0.3

10 M KPF6

20 M KPF6

2

-i

/

A

E / V vs Fc

DCBA

50 M nitrobenzene

0.5 mM THexA PF6

-1.5 -2.0 -2.5

0.0

0.1

0.2

20 M KPF6

Ar-NO2 /Ar-NO2−●

K + / K

41

Negative impedance & Impedance discontinuity

-1.5 -2.0 -2.5 -3.00.0

0.2

0.4

FA-i

/

A

E / V vs Fc

-10k -5k 0 5k

0

5k

10k

D

-6k -4k -2k 0 2k

0

2k

4k

6k

8k

F

-3k -2k -1k 0

-1k

0

1k E

0 2k 4k 6k

0

2k

4k B

0 2k 4k 6k

0

2k

4k A

-2k 0 2k

0

2k

C

-Z

''/o

hm

.cm

2

Z' / ohm.cm2

42

Bursting oscillations at Hg dropping electrode

5 10 150

10

20

30

-i/

A

time/ s10 15 20

0

10

20

30

-i/A

time / s

10 15 200

10

20

30

-i/A

time / s

43

Chrono-amperommetry(reduction current – time, E=const.)

octo-cation

44

Small changes of potential:large effect on current oscillations

0 50-10

0

10

i

-0.560 V

0 50-10

0

10

-0.565 V

0 50-10

0

10

-0.570 V

0 50-10

0

10

i

-0.575 V

0 50-10

0

10

i

-0.590 V

0 50-10

0

10

i

-0.580 V

0 50-10

0

10

i

-0.595 V

0 50-10

0

10

i

-0.585 V

0 50-10

0

10

i

-0.610 V

curr

ent

time (0 – 100 sec)

deca-cation