Presented at the COMSOL Conference 2010 Boston

Simulating the Electrical Double Layer

Guigen Zhang, Ph.D.

Dept. of Bioengineering, Dept. of Electrical & Computer Engineering
Institute for Biological Interfaces of Engineering
Clemson University
 Overview
The drive to make supercharge capacitors
Electrochemical based capacitor
The structure of electrical double layer (EDL)
The effect of EDL structure on electron transfer
The EDL capacitor
Conclusions
 Making Supercharge Capacitors
3.0

Capacitance (F)
2.5

2.0

1.5

Flat
1.0
-600 -400 -200 0 200 400 600 800
Overpotential (mV)

20

Capacitance (F)
16

12

Nano
8
-600 -400 -200 0 200 400 600 800
The Electrochemical Society Interface, Fall 2008 Overpotential (mV)
Electrochemical Based Capacitor
This topic has been a major interest
in electrochemistry for about a D. C. Grahame, 1947.
century

In 1997, the Electrochemical

Society sponsored a symposium on
the double layer to recognize the
50th anniversary of Grahame’s
seminal work

C   0 A / d
 Electrical Double Layer (EDL)
The EDL structure Helmholtz Model Gouy-Chapman Model

x - x
- +
-
- + -
+ -
- -
-
- +
+ - +
-
- -
- -
A -
- C   0 -
- d Diffuse zone

Ψ0 -
OHP

Gouy-Chapman-Stern Model

Diffuse layer
x
-
-
- + +
+ +
- - -
- -
Problems with the classic theories on electrical double layer (EDL): +
+ +
- +
1. No electron transfer across the electrode/solution interface -
- -
2. Boltzmann distributions for ions in the solution - +
3. Electro-neutrality - Stern Plane
Bulk solution
- OHP, PET
- Compact layer Gouy plane
 Modeling the EDL
Using COMSOL
A

Axis of in-plane symmetry

Mass transport by diffusion and 1000 r0

IHP OHP

electromigration r0 v

– Nernst-Planck equation 
r  u 2  v2

ci zF Axis of Electrolyte

 ( Di ci  i Di ci V ) axisymmetry
t RT
Electrostatics ( 0V )   
  l1  l2

– Poisson equation In the compact layer:  0 u

In the solution:    zi ci
i

Reversible/irreversible systems
– Butler-Volmer kinetics
kf

O e 
z
 R z 1

kb
k f  k 0  exp[F ( Et  V  E 0 ' ) / RT ]
kb  k 0  exp[(1   ) F ( Et  V  E 0 ' ) / RT ]
 Modeling Using COMSOL
Dielectric constant inside the compact layer
 1cosh 2 [S1 (r  r0 )], r0  r  r0  l1

   2 cos 2 [S2 (l1  l2  r0  r )], r0  l1  r  r0  l1  l2
 2, r0  l1  l2  r

B
a b c

IHP OHP
2
Dielectric constant (x)

Electrode PET
surface

Electrolyte
1

r0 r0 +Distance
l1 (x)
r0 + l1 + l2
Radial Distance (r)
 The Size Factor of the EDL
-0.25 -0.030 6
A Diffuse Layer
B
-0.025 Diffuse Layer 5
-0.20 1 nm

Concentration (mM)
100 nm Potential
-0.020 4

Potential (V)
Potential (V)

Concentration
-0.15 1 nm
-0.015 100 nm 3

-0.10 -0.010 2

-0.005 1
-0.05

0.000 0
0.00
0.0 0.2 0.4 0.6 0.00 0.05 0.10 0.15 8.00 12.00

r-r0 (nm) (r-r0-)/

1diffuse
nm  1.8 (nm), 100 nm  4.5 (nm)
diffuse

1diffusion
nm  14 (nm), 100 nm  820 (nm)
diffusion

~13% ~0.5%
EDL Effect on Electron Transfer
0.75
0.0 1.10
Insert-1 Insert-2 0.0
0.2 1.05 Diffusion
0.80 0.4 0.2nm
i/idL

0.6 1.00 0.2

Diffusion
0.4nm
0.8 0.95 z=-1 0.7nm
0.85 1.0 z=+1
0.90 0.4
-0.2-0.1 0.0 0.1 0.2 0 100 200
E-E0'(V) Electrode Radius (mm)
0.90

i/idL
-0.12 6
i/idL

1nm 0.6

Concentration (mM)
-0.10 5
10nm

Potential (V)
-0.08 4
0.95 50nm 0.8 -0.06 3
100nm Potential
-0.04 2
200nm Concentration
1.0 -0.02 1
1.00 z = -1 0.00 0
z = +1 0.02
1.05 Diffusion 1.2 0.0 0.5 1.0 100.0
(r-r0-)/

-0.2 -0.1 0.0 0.1 0.2 0.3

-0.25 -0.20 -0.15 -0.10 -0.05 0.00
E-E (V)
0'
E-E (V)
0'

Effect of electrode size Effect of compact layer thickness

Note: “Diffusion” represents the case in which the EDL effect is not considered.
 Effects of EDL
-0.030 5 -0.05 5

-0.025
4 -0.04 4

Concentration (mM)
Concentration (mM)
-0.020

Potential (V)
Potential (V)

3 -0.03 3
-0.015
Potential Potential
Concentration 2 -0.02 Concentration 2
-0.010 0.33nm ES=6
0.44nm ES=12
0.55nm ES=18
-0.005 0.66nm 1 -0.01 ES=24 1

0.000 0 0.00 0
0 1 2 3 4 5 6 0 1 2 3 4 5 6

Distance into Solution from PET (nm) Distance into Solution from PET (nm)

5 -0.04 5
-0.05

4 4
-0.03
-0.04
Concentration (mM)

Concentration (mM)
3
Potential (V)

Potential (V)
3
-0.03
-0.02
Potential 2
Potential
Concentration 2
-0.02 Concentration
0.00 M 0.5 nm
0.05 M 1.0 nm
-0.01 5.0 nm 1
0.50 M
-0.01 5.00 M 1 10.0 nm
Flat

0
0.00 0 0.00

0 2 4 6 8 10 12 0 2 4 6 8 10

Distance into Solution from PET (nm) Distance into Solution from PET (nm)
 EDL Capacitance
Size effect
26

24
 2
Capacitance (F/cm )
2

22 C   0
rE
20

18

16

14

12 y=-4.21+28.66x/(0.42+x)

10
1 10 100 1000
Radius of Electrode (nm)
 EDL Capacitance
Dielectric effect
40

35
Capacitance (F/cm )
2

30

B
25 a b c

IHP OHP
2

Dielectric constant (x)

20 Electrode
surface
PET

Electrolyte

15 1

r0 r0 +Distance
l1 (x)
r0 + l1 + l2
Radial Distance (r)

10
0 6 12 18 24
Dielectric Constant at Electric Satuaration
 EDL Capacitance
Thickness effect
24

22
Capacitance (F/cm )
2

20

18

16

14

12

10

8
0.33 0.44 0.55 0.66
Thickness of Compact Layer (nm)
 EDL Capacitance
Electrolyte effect

17.0

16.5
Capacitance (F/cm )
2

16.0

15.5

15.0

14.5

14.0

13.5

13.0
0 1 2 3 4 5 6
Concentration of Supporting Electrolyte (M)
EDL Capacitance: A Surprise
C   0 A / d

11.385
FEM
Capacitance (F/cm2)

11.380
With ET
without ET
D. C. Grahame, 1947.

11.375

11.370

11.365
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
Overpotential (V)

1.6
MD
Capacitance (C/CPZC)

1.5

1.4

1.3

1.2

1.1

1.0

0.9
-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
Overpotential (V)
 Conclusions
EDL capacitance varies as a function of
– Dielectric constant
– Compact layer thickness
– Electrode size
– Electrolyte concentration
When redox is allowed, the capacitance-potential
curve exhibits a dip feature near the potential of zero
charge
This study shed some new light into enhancing the
supercharge capacitors
 Acknowledgement

National Science Foundation

Bill & Melinda Gates Foundation

Thank You!