Theories for anomalous responses in disordered electrodes · DST – Theories of 4G...

Theories for anomalous responses in disordered electrodes

Towards 4G Electrochemistry

RAMA KANT

Complex Systems Group

Department of Chemistry

University of Delhi

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How to predict electrochemical response from surface microscopy?

Electro-

Chemistry:

THEORY Microscopy:

LOCAL & STATISTICAL MORPHOLOGY/WORK FUNCTION

SPACTIALLY RESOLVED & GLOBAL RESPONSE

MICROSCOPY TO ELECTROCHEMISTRY

‘FOURTH GENERATION’ ELECTROCHEMISTRY

Faraday discussions 1973, 56; 1980, 70; 1994, 94; 2002, 121; D. E. Williams

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Geometrical landscape Energetic landscape

Elev

ation

En

erg

etic

diso

rde

r

“…much remains to be done because roughness is absolutely ubiquitous and often is the major obstacle to an understanding and control of reality.” -Mandelbrot

Wolff (1926): Capacitance De Levie (1968) Vetter (1952): Faradic

To understand and develop next generation of devices one needs to develop “Theoretical Electrochemistry in Constraint Environments of Disordered Electrodes``

Self-similar disorder:

Fractal Geometry

Pajkossy(1986)

E

AC

Disordered System

Charge Transfer Adsorption

Diffusion/

Migration

Double Layer

Electrode Stability: Nanoactuation

Bulk

Reaction DtkD /

mm(WE/CE) L

(WE/RE) ~

s

w

R

DR

0

~~

n

E

DTFFM

F

M

f

Hk

D

I ~ ; D Tkr BH

Kant, Kaur & Singh, Nanoelectrochemistry in India, SPR Electrochemistry, 2013, 12, 336–378

ATOMIC nm

nm - m

m

Ohmic

Origin of Complexity & Length Scales

ENERGETICS

ROUGHNESS

STM image: 580Å x 580Å, 22.0V, 0.1 nA

Au/Cu(111) surface

HOW DOES SURFACE MORPHOLOGY INFLUENCES THE WORK FUNCTION OF METAL?

Jia, Inoue, Hasegaw

a, Yang and Sakurai,

Phys. R

ev. B, 58, 1998, 1193.

Spatial variation of the local work function STM Image

steps on metal surface

J. Kaur, R.Kant, J. Phys. Chem. Lett., 2015, 6, 2870-2874

charge transfer, catalytic activity, adsorption, electronic properties, photoelectric threshold

0K)-(H

Energy Willmore

2 S

dSw

4-LOBED WILLMORE TOROUS

Mean: H=(k1+k2)/2

Gaussian: K=k1k2

Durham University, U.K.

local surface shape

Singh & Kant, Proc. Roy. Soc. A 469 (2013) 20130163; Kant & Singh, Phys. Rev. E.88 (2013) 052303 (1-16).

Kant & Rangarajan, J. Electroanal. Chem. 552 (2003) 141,

Arbitrary Geometry: Mean and Gaussian Curvature

Work function on curved surfaces (2015)

Work function on smooth surface (2015)

Work function on a sphere (2015)

Morphology dependent work function

Kaur and Kant, J. Phys. Chem. Lett., 2015, 6, 2870-2874

Thomas – Fermi electronic screening length

“Work function of metal can be tailored through its local shape”

h =0.39 nm

*

E

J. Kaur, R.Kant, J. Phys. Chem. Lett., 2015, 6, 2870-2874

charge transfer, catalytic activity, adsorption, electronic properties, photoelectric threshold

95.0

05.1

3/4 atomic layers thick geometric fluctuation

Gaussian checker surface

Nanoparticles: Surface Plots

50

5.0

Prolate

Oblate

Triaxial

Parameters used

Prolate- x=y=12, z=15, Oblate- x=y=12,

z=9, Triaxial- x=15, y=12, z=10 All lengths are in units of lTF.

Shape induced nonuniform ECD

Scal

ed

EC

D

Height = Radius = 20 lTF ~ 2.5nm

TUNING EXCHANGE CURRENT DENSITY (ECD) THROUGH ELECTRODE SHAPE & ROUGHNESS

enhancement & suppression of ECD

How does roughness influence electrode kinetics?

metal

Relative ECD

)17.0( 1 :electrode Solvated

Kaur, Kant, J. Phys. Chem. Lett. 2015, 6, 2870;

Harinipriya & Sangaranarayanan, J. Phys. Chem. B, 2002, 34, 106

Random roughness characterization

and

AB- INITIO APPROACH for calculating

dynamic response at disordered electrode

Noise & Topographical Power Spectrum Density

Dhillon & Kant; Applied Surface Science, 282(2013)105;

Electroanalysis, 26(2014)2350

3D Reconstruction Denoised 3D Reconstruction

Local 3D Reconstruction Local Gradient

Noise & Topographic PSD

2D SEM Micrograph

3D Reconstruction of SEM Image of Statistical Fractal:

CV and SEM Method

bifractal

fractal flicker

Laplacian

Noise

Filter

(SEM) MICROSCOPY TO ELECTROCHEMISTRY ? Microscopic area from cyclic voltammogram

Electrochemical Response Functions for Disordered Electrode:

I, Q, Y, A, C, E

Diffusion, Debye-Falkenhagen, Gouy-

Chapman, Thomas- Fermi Equations

Electrochemical Constraints :

Boundary Conditions

Green’s Function: Smooth/Curved

Geometries

Expansion of Boundary Condition in Roughness

or Surface Curvature

Integro-Differential

Equation

Expression for Concentration/Potential

Field

Spatially Resolved Local Electrochemical

Response

Global Response Equation for Random &

Finite Fractal Electrode

2nd order

Perturbation in

Height/curvature

Admittance/Impedance

Pulse Voltammetries

Chronocoulometry Chronoamperometry

Chronoabsorptometry

Ergodicity:

Ensemble Average Surface Average

Fourier &

Laplace

Transform

AB- INITIO APPROACH FOR DISORDERED ELECTRODES

Kant, Phys. Rev. Lett. (1993)

Capacitance

Work function/Kinetics

Chronopotentiometry

Structure of Response Equations for Rough Electrodes

Response

of Rough

Electrode I, Q, Y, A

Response

of Smooth

Electrode

Electrode

Roughness

Power

Spectrum

Phenomenological

Length and Time Scale

Dependent Operator

“What attracts me is the unknown. When I am facing a very tangled skein, I cannot help but think that it would be nice to find the main thread.” -P. G. de Gennes

Mechanically & Electrochemically Roughened Au

CV-SEM 3D reconstruction of Au random fractal electrode

Dt

L l

e- lτ

DH

Reversible Charge Transfer Generalized Cottrell and Warburg Problems

ΩL

Kant, Dhillon & Kumar, JPCB, (2015), Parveen & Kant, JPCC, 118 (2014) 26599 ; Srivastav, Kant, JPCC, 117 (2013) 8594; Kant & Islam,

JPCC, 114 (2010) 19357; Kant, JPCC, 114 (2010) 10894; Kant, Kumar & Yadav, J. Phys. Chem. C (Lett.) 112 (2008) 4019; Dhillon &

Kant, Electrochim. Acta,129 (2014) 245; Parveen & Kant, Electrochim. Acta, 111 (2013) 223; Islam & Kant, Electrochim. Acta 56

(2011) 4467; Kant & Rangarajan, J. Electroanal. Chem. 552 (2003) 141; Kant & Rangarajan, J. Electroanal. Chem. 368 (1994) 1; Kumar

& Kant, J. Chem. Sci. 121 (2009) 579;

Phenomenological length scales: Diffusion length ( ); Ohmic length (LΩ)

Fractal length scales: Minute (l), long (L) and Topothesy (lτ); self-similarity index/fractal dimension (DH)

Dt

Chronoamperometry

Impedance Spectroscopy (EIS)

Local EIS

Chronocoulometry

Chronoabsorptometry

• Pulse Voltammetries

SCV, CSCV, DPV, SWV • Cyclic Voltammetry

o Single Potential Step

o Double Potential Step

o Staircase Potential

Models for Electrochemical

Techniques

Huge reaction rate heterogeneity induced by roughness

Diffusion limited charge transfer at a rough Pt electrode

Local admittance density using SEM micrograph– Log scale

Pit on Pt electrode

Kant, Dhillon & Kumar, JPCB, 2015

THEORY FOR AN ANOMALOUS COTTRELL CURRENT OF A ROUGH ELECTRODE

Cottrell Equation (1902)

-O+ne R

Frederick G. Cottrell

(1877-1948) Hermann W Nernst

(1864-1941)

“By viewing the old we learn the new”

0

)()( A

tD

ECDnFtI

O

sOC

Double potential step chronoamperometry , S. Dhillon, R. Kant, Electrochimica Acta 129 (2014) 245–258

4G - 2002

Pt (R1)

R*=1.1

h=60nm

Rough Pt Electrodes: SEM Image

Pt (R2)

R*=5

h=0.6µm

Pt (R3)

R*=13

h=15µm

SEM provides 2D projection

Missing 3rd dimension

Mechanical

Polishing &

Electrochemical

Roughening

Finite fractal nature

Srivastav, Dhillon , Kumar and Kant, J. Phys. Chem C 117(2013)8594

Experimental Validation of Power Spectrum Based Theory in Aqueous Medium Electrolyte

Srivastav, Dhillon, Kumar & Kant, J. Phys. Chem. C 2013, 117, 8594-8603

15 mM Fe(CN)63-/Fe(CN)6

4-

in 3 M NaNO3

Our theory: black lines

De Gennes Scaling: red line

Data : Points

Theory guides. Experiment decides.

Pulse Voltammetric Current for the Rough Electrode(2014)

Parveen & Kant, J. Phys. Chem. C 118(2014)26599, Electrochimica Acta 111 (2013) 223

Current for Arbitrary Potential Sweep and

Generalized Cottrellian Current:

Kant, J. Phys. Chem. C 114(2010)10894

E/V

t/s

))1((...)2()()( 121 NtuEtuEtuEEtE Ni

u(t

-τ)

τ t

Heaviside Unit Step Function

Arbitrary Shape Pulse

Influence of Fractal Characteristics on Voltammograms Fractal Dimension Finest Scale of Roughness Width of Roughness

Peaks become broader and higher with increase in roughness

Cyclic voltammograms of 5mM ferrocene in BmimBF4 ionic liquid performed at rough electrode (Experiment- red

circles, theory-solid black line) at various scan rates ((a)-10 V/s, (b)-9 V/s, (c)-8 V/s), (d)-7 V/s, (e)-6 V/s, (f)-5 V/s, (g)-

4 V/s, (h)-3 V/s,n (i)-2 V/s, (j)-1 V/s, (k)-900 mV/s, (l)-800 mV/s). Fractal characteristics of gold electrode are: Fig.(a)-

DH = 2:45, L = 1.26m, l= 44 nm, l

= 0.39 µ m. Other parameters used in the calculations are as: diffusion coefficient

DO = 2.6 *10-7cm2/s, DR = 3.4 *10-7cm2/s, sampling parameter (a = 0), electrode area (A0 = 0.033 cm2), concentration

(CO = CR = 2.5mM).

Experimental Validation of Theory: Ferrocene in BmimBF4

Mechanically Roughened Gold Electrode

Conclusions

• Work function and surface kinetics of

the nano-particle/structure can be

tailored through its shape and

roughness…!

• Electrode surface microscopy can be

used to predict dynamic response of a

disordered electrode…!!

`CONCLUSIONS

Generalization of Fundamental Equations of Electrode Kinetics

“Nothing is too wonderful to be true if it be consistent with the laws of nature.” -Michael Faraday

Disordered Electrodes

Cottrell

Danckwerts

Anson

Debye- Falkenhagen

Thomas-Fermi Gouy-Chapman

Acknowledgements

Funding:

DST – Theories of 4G Electrochemistry,

DU-DST PURSE and University of Delhi –

Experimental Electrochemistry

Applied Materials India – Modeling of Solid State Batteries

Pierre-Gilles

de Gennes (1932-2007) Won 1991

Nobel Prize for

Physics

Electro-

Chemical

Devices

Graphene Polymer

Supercapacitors

sensors

Electrochemistry: “Taming electricity through chemistry at an electrode”

Current Transient on Finite Fractals: Unequal Diffusivity

Parveen & Kant, J. Phys. Chem. C 118(2014)25699

Kant & Jha, J. Phys. Chem. C 111(2007)14040

“Things should be made as simple as possible, but not any simpler.” -Albert Einstein