Pattern Formation in Electrochemical Systems

Nonlinear dynamics in electrochemical systems Corrosion

Electropolishing

Yuzhakov et al. PRB 56 (1997) 12608

O. Steinbock et al., Univ. of Florida, USA

Fuel Cells

Electroplating Krastev and Koper, Physica A 213, 199 (1995)

Zhang and Datta, J. Electrochem Soc. 149,A1423 (2002)

Pattern Formation in Electrochemical Systems: The Role of Nonlocal and Global Coupling Katharina Krischer Non-equilibrium Chemical Physics Physik-Department Technische Universität München

Hydrogen oxidation in the presence of poisons Potential probe

I Pt-Ring electrode

x x x I x

x

Controlling pattern formation by tuning the spatial coupling

Basic electrochemical set-up U

I e

Ireac = f(DL) e

e

M M+

Cations Anions

(x,y,z)

X X

e

Local charge balance  DL C t

WE double layer

iF

  z

 DL : potential drop

across the interface

z :

electrolyte   0

electrostatic potential in the electrolyte

CE basic equation for the potential evolution:

 DL  C   iF    f  DL , c   t z z WE

        z w  z  WE migration coupling

Modeling Electrochemical Pattern Formation Mass balance for reacting species:  Charge balance:

c  g DL , c   D c t

 DL  C   iF    f  DL , c   t z z WE

        z w  z  WE migration coupling

Migration coupling: • nonlocal • coupling range depends on the cell geometry and can be tuned experimentally

Integral Formulation of the Nonlocal Coupling            H   x  x '  DL  x ', t   DL  x, t   dx '  z w  z  WE el

J. Christoph, Berlin

= w/L : aspect ratio of the cell

L

double layer

electrolyte

H(x-x’) H(x-x’)

WE

decreasing 

w 0

CE

-L/2

x-x’

-L/2

Experiments: Different coupling ranges Case 1:nonlocal

Case 2:local

WE U

CE

2:local

1:nonlocal Varela, Beta, Bonnefont, KK, PRL 94, 174104 (2005)

Experiments: Defect turbulence Evolution of the oscillation phases (a)and amplitudes (b)

local

nonlocal Space-time defect

The larger the range of the coupling, the lower the density of defects Varela, Beta, Bonnefont, KK, PRL 94, 174104 (2005)

Defect turbulence in oscillatory reaction-diffusion systems Complex Ginzburg-Landau Equation (CGLE):

 t A  A  (1  ic1 ) A  (1  ic3 ) A A 2

2 x

BF

Shraiman et al., Physica D 57, 241 (1992)

Phase Chaos

3

Defect Chaos c1 2

u( x , t )  A( X , T )Ue 

A( X , T) 

 

Aqe

i ( qX  q T )

e

BC

1

i0t qT

0

1

c3

2

3

Nonlocal CGLE for Electrochemical Systems Center manifold reduction of the electrochemical model  Nonlocal CGLE

 t A  A  (1  ic2 ) A A  (1  ic1 )  H   x  x '  2

 A( x ', t )  A( x, t ) dx '

WE

 three parameters: c1, c2,     0 (local or diffusive coupling limit): Nonlocal CGLE  CGLE  Instabilty of uniform oscillation: c1c

increasing  in BF unstable region: turbulence  coherent structures

V. Garcia-Morales and KK, PRL 100, 054101 (2008)

Defect turbulence for different coupling ranges  = 0.2

 = 1.0 The larger the range of the coupling, the lower the density of defects


Stability of the uniform oscillation


Spatially Coherent Structures for Large Coupling Range c1 = 3; c2 = 1

V. Garcia-Morales, R. Hölzel and KK PRE 78, 02615 (2008)

time

time

Symmetrie of the coherent structures

x

V. Garcia-Morales,R. Hölzel and KK, PRE 78, 02615 (2008)

x

time

Stable heteroclinic orbits

Phasespace projection on Fourier coefficients a1, a1 and b2. V. Garcia-Morales,R. Hölzel and KK, PRE 78, 02615 (2008)

Effect of the coupling range  in the NCGLE: • stability range of uniform oscillation unchanged compared to the CGLE:   1  c c  0 1 2

• increasing the coupling range in the BF- unstable regime:



larger structures with a smaller density of space time defects

coherent structures

Global coupling in electrochemical systems through an external control

global coupling for: • galvanostatic control or external ohmic resistance: > 0 • close reference electrode: < 0

     DL      DL   DL  C  f  DL , c         z w  z  WE t migration

global coupling

Experiments with global coupling: Turbulence  2-phase clusters

U

Intermittency-type transition from a modulated limit cycle to an oscillating cluster state H. Varela, C. Beta, A. Bonnefont and KK, PCCP 7, 2429 (2005)

The extended Nonlocal CGLE center manifold reduction

global coupling is weak!

 t A  A  (1  ic2 ) A A  (1  ic1 )  H  ,  x  x '  2

 A( x ', t )  A( x, t ) dx '

WE

V. Garcia-Morales and KK, Phys. Rev. E 78, 057201 2008

Standing waves for small negative global coupling


Two phase clusters for negative global coupling and large 




Heteroclinic networks through NLC and their modification through GC   0.005

 0.003

0

0.12

0.14 V. Garcia-Morales and KK, Phys. Rev. E 78, 057201 2008

Phase clusters vs. subharmonic clusters H. Varela, C. Beta, A. Bonnefont, KK, PCCP 7, 2429 – 2439 (2005)

Analytical Signal 

Im()

Re()

Phase clusters vs. subharmonic clusters H. Varela, C. Beta, A. Bonnefont, KK, PCCP 7, 2429 – 2439 (2005)

Im()

Re()

Silicon electrooxidation Electrochemical treatment of Si surfaces: • allows for  surface preparation (smoothing/polishing)  tailoring optical and physical surface properties Porous silicon

Ph.D. Thesis, Stefan Frey Group H. Föll, Univ. Kiel Germany

Si electrodissolution Oxidation

h

n-type Etching

3-phase domain type clusters

Snapshots successive maxima of ξ

Power spectra of the local time series 0

Phase, 

Amplitude, |a|

0

0/2 /3 /3 2 0

2

2 2

I. Miethe, V. García-Morales, KK, PRL 102, 194101 (2009)

Nonlinear Global Coupling Averaging ….

t W 

Nonlinear global coupling introduces conservation constraint

 t W  i W W  W0   e

i t

Motivation: Globally conserved phase ordering dynamics  Real Ginzburg-Landau equation with nonlinear global constraint M. Conti et al., PRE 65, 046117 (2002)

2-phase subharmonic cluster pattern

  0.66 c1  0.2; c2  0.58

 1

V. Garcia-Morales, A. Orlov and KK, PRE 82, 065202 (R), 2010

Further experimental examples CO-Oxidation on Pt(110)

M. Bertram, C. Beta, M. Pollmann, A.S. Mikhailov, H.H. Rotermund, and G. Ertl Phys.Rev. E 67, 036208, 2003

H2 Electrooxidation

H. Varela, C. Beta, A. Bonnefont, KK, PCCP 7 (2005)

BZ reaction

V. K. Vanag, A. M. Zhabotinsky and I. R. Epstein J. Phys. Chem. A, 2000, 104

Labyrinthine patterns on n-type silicon at the maxima of ξ

Labyrinthine structures I. Miethe, V. García-Morales, KK, PRL 102, 194101 (2009)

Local time series (x1,y1)

(x2,y2)

(x3,y3)

Global signal:



x, y

t 

Power spectra of the local time series Phase, 

Amplitude, |a| 0

0

0/2 0/2

 0

2

Irregular clusters with additional 1:1 resonant forcing CGLE

1:1 Resonance

Nonlinear global coupling MODIFICATION

I. Miethe, V. Garcia-Morales, and KK, Phys. Rev. Lett. 102, 194101 (2009)

Simulation results Local oscillators with complicated modulations  spatial irregularity Local extrema locked to average signal Two main peaks in power spectrum 1:1 homogeneous distribution of amplitude and phase 2:1 labyrinthine pattern with two antiphase domains Ising walls and Phase balance

I. Miethe, V. Garcia-Morales, and KK, Phys. Rev. Lett. 102, 194101 (2009)

Summary • Nonlocal CGLE – Larger characteristic length, coherent structures

x

• Nonlocal CGLE + global coupling

x

– 2-phase clusters t

• CGLE + Nonlinear global coupling

x

– Subharmonic 2-phase clusters

t

• CGLE + Nonlinear global coupling + 1:1 forcing – Irregular subharmonic cluster patterns

x

t

Thanks to Theory

Funding:

Vladimir Garcia Morales Robert Hölzel Alexander Orlov EU (DYNAMO project)

Experiments Hamilton Varela (H2 oxidation) Antoine Bonnefont (H2 oxidation) Carsten Beta (data analysis) Iljana Miethe (Si oxidation) Andreas Heinrich (Si oxidation)