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