중앙대학교융합공학부나노소재전공

윤성훈

Theoretical consideration of electrochemical

impedance spectroscopy based on 2-D

transmission line model in the porous

electrodes and its application into various

mesoporous carbon materials

I. 수퍼캐패시터 특성 분석을 위한 transient 방법 EDLC analysis Pseudocapacitor analysis

Capacitor

2 2

sp A(F/g) (μF/cm ) (m /g) C C SS

Cd

Schematic Presentation of Composite Electrode

• Inter-particle macropore

– Ionic movement similar to that of bulk phase(similar resistivity)

– Small contribution to total ESR

• Intra-particle meso and micropore

– Ionic movement different from that of bulk phase(high resistivity ~ 108Wcm)

– Major contribution to ESR

macropore

meso and micropore

Rbulk

Reference Electrode

Working Electrode

Porous Carbon Particle

Current Collector

Description of Electrochemical Capacitor Electrodes

Flat and smooth

electrodes

1-D transmission

line model (TLM)

2-D transmission

line model

R1-C1 series R1-C1 trans. R1-C1 trans.

/R2-C2 trans.

Oversimplification

bulk behavior of E-Cap.

Resistance

- R1 : ESR (Rb + Re+ Rct+ Rp)

- Rp : total pore resistance

C1: total capacitance

Easy to measure

: simple equations

Representative pores

: Intra-particle pores

Total capacitance

Complicate equations

: analytical solution exists

Representative pores

: Intra-particle pores +

Inter-particle pores

Capacitance at inner pores

: disregard cap. at outer pores

Very complicate equations

: very hard to acquire analytical

solution

Time/s

0 5 10 15

Curr

ent d

ensity/m

Acm

-2

0

1

2

3

4

5

6

7

8

MSC25-7%KBNMC

ESR =

Relec trode

Rbulk

Rpore

ESR C

I(t) = (E/R) e -t/

r

c

r

c

r

c

Ri

Ri

Ri

electron

ion

Carbon

Electrodes

carbon

current collector

Flat electrode

: simple RC circuit

Thin electrode

Pores: parallel connection

1-D TLM

Common electrode

Pores : not parallel connection

2-D TLM

thin composite

electrode thicker electrode

1-D TLM

Impedance of electrode

carbon

current collector

Thin electrodePores: parallel connection

1-D TLM

Z

Z

Z

Z

p

p

p

( )( )

1coth 2 j

2

p

p

Z fZ f

N

Rf

m n fj

pore

r

c

r

c

r

c

(a) (b)

(c)

Rs

Zp : impedance of one pore

Np: total pore number

m : particle number

n : pore number of one particle

• Initial condition

– V(y,0) = 0 : zero initial over-potential

• Boundary condition

– dV(y,t)/dyy=1 = 0 : no potential gradient at y = 1 (end of pore)

– Different resulting equations according to individual BCs (potential/current step)

• In Laplace domain

r

c

r

c

r

c

t

tyV

y

tyV

),(),(2

2

y = l/lp, = RpCp

Rp = r x lp, Cp = c x lp

]coth[)(

)()( s

s

R

sI

sVsZ

p

p

Differential equation for pore impedance

lp : length of pore

: time constant

Rp : pore resistance

Cp: pore capacitance

Ref) R. de Levie, in Advances in Electrochemistry and Electrochemical Engineering, Vol. VI, P. Delahay, Editor, p. 329, John Wiley & Sons, New York, 1967.

Nyquist plot of ac-impedance experiment

]coth[)(

jj

RZ

2cos2cosh

2sin2sinh

2

RZreal

2cos2cosh

2sin2sinh

2

RZimag

f 2 RC

Zreal

*

0.0 0.2 0.4

-Zim

ag

*

0.0

0.2

0.4

0.6

0.8

1.0

Nyquist Plot of Simple TLM

Z’ = Rp/345o slope: Warburg like behavior

Z’

jj

RjZ coth)(

Imaginary Capacitance Analysis

2cos2cosh

2sin2sinh

2)(

CCim

)(

1)(

ZjC

2cos2cosh

2sin2sinh

2)(

CCreal

Imaginary capacitance plot (Cim): s-TLM

• fpeak

– Related with time constant

• Peak area (A)

– Proportional to capacitance

(Ctot)f/Hz

0.001 0.01 0.1 1 10 100 1000 10000

Cim

(f)

0.0

0.1

0.2

0.3

0.4

0.5

peak frequency (fpeak)

p p

peak

0.4p C R

f

im tot( ) log 0.682A C f d f C

2cos2cosh

2sin2sinh

2)(

CCim

Ref) Jong H. Jang, Songhun Yoon, Bok H. Ka, and Seung M. Oh*, Journal of the Electrochemical Society 152 A1418 (2005).

II. 2-D TLM- Theoretical consideration

of Nyquist plot

Double TLM for Composite Electrode

Zp

Zp

Zp

Zp

Re

Re

Re

Re

Parallel connection : Ztot = (Zp + Re)/m

Complicate connection: Ztot = f(Zp,Re,t)/m

thickness increase

Zp Zp

Re Re

t=2

Zp Zp

Re Re

Zp

Re

t=3t=1

m=4

Theoretical Development for Double TLM

here, A = (k2+4k+2-(k+2)(k2+4k)2)/2

p = Zp(k +(k2+4k)1/2)/2, q = Zp(k-(k2+4k)1/2) /2,

k = Re/Zp

t : number of particles (thickness)

t

t

ptot

A

qAptZfZ

1),R,( e

k : governing factor to determine Ztot k(Re/Rp)

Governing Equation

Double TLM Simple TLM

Breakage of parallel connection with thickness increase

– Critical thickness (tc) ?

Zp Zp

Re Re

Zp

Re

Zp

Re

Zp Zp ZpZp

When Re << Rp

Ztot= f(Zp, Re, t) complicate connection

: too complicate for fitting

Ztot = Zp/t parallel connection

: general consideration

Simulation of Effects by Electrode Thickness

Effects of electrode thickness on the ionic resistancer = 1, R = 0.01 ~ 100

number of paticles

0 2 4 6 8 10 12

Resis

tance

0.0

0.5

1.0

1.5

2.0

2.5

Ze =0.01

0.1

1

Ze ~ Zpt : similar pore resistance to bulk

less decrease of resistance

Zpt = 1

2D Graph 4

Zreal

0 100 200 300 400

-Zim

ag

0

100

200

300

400

500

t = 1

t = 2

t = 3

Ze << Zpt : larger pore resistance than bulk

distinct decrease of resistance No change with thicknessZe : independent of thickness

Zreal

*

0.0 0.2 0.4

-Zim

ag

*

0.0

0.2

0.4

0.6

0.8

1.0

Nyquist Plot of Simple TLM

Z’ = Rp/345o slope: Warburg like behavior

Z’

jwjw

RjwZ

p

coth)(

Thickness Effects : Nyquist Plots (Re/Rp =10-3)

Zreal

*

0.0 0.1 0.2 0.3 0.4

-Zim

ag

*

0.0

0.1

0.2

0.3

0.4

t=1

t=5

t=10

Zreal

*

0.00 0.02 0.04 0.06 0.08 0.10

-Zim

ag

*

0.00

0.02

0.04

0.06

0.08

0.10

t=50

t=100

t=200

• ESR

– Decrease and increase

– More distributed resistance with thickness

• Intercept of Zreal

– constant

Critical thickness (tc)

• t < tc

– Parallel connection : Ztot = Zp /t

• t > tc

– Non-parallel connection : Ztot = f(Zp, Re,t)

Thickness(t)

20 40 60 80 100 120 140

Zre

al(f=

10

-3)

0.00

0.02

0.04

0.06

0.08

0.10

Thickness(t)

20 40 60 80 100 120 140

t xZ

real(f=

10

-3)

0

2

4

6

8

10

12

14

16

tc

II. 2-D TLM- Imaginary capacitance

analysis- CMK-3 carbon analysis

• fpeak

– Related with r parameter

• Peak area (A)

– Proportional to capacitance

(Ctot)

Imaginary capacitance plot (Cim): 2D-TLM

Log ( f /Hz )

0.001 0.01 0.1 1 10 100 1000 10000

Cim

( f

)/C

tot

0.0

0.1

0.2

0.3

0.4

0.5

r =10-3

10-110

0

101

102

p = 1 sec

p p

p p p

2 21 1 ( )( ) tanh

2 coth 2 coth 2

r r

n

tot

f j f j C fC f

f j Cf j f j

2 i

p

Rm

Rr tot p C m n C

im tot( ) log 0.682A C f d f C

Ref) Jong H. Jang, Songhun Yoon, Bok H. Ka, and Seung M. Oh*, Journal of the Electrochemical Society 152 A1418 (2005).

Peak frequency (fp) and peak area (A)

r

0 10 20 30 40 50 60

1/(

1 f

p)

0

50

100

150

1 p

0.41

f r

im 0( ) log 0.682A C f d f C

Empirical relation

between fp and r

Relation between A and C0

by numerical integration

Resulting equations using m parameter

• Calculation of m and n

– Possible for carbons having well-defined pore structure

• Separation between R0 and Ri

– R0 : decrease, Ri : increase according to increase of electrode thickness (m)

• Separation between 1 and 2

– Find dominant factor affecting tot by experiments !

2

pore pore inter pore

p

tot 0 1 2 tot

0.4

R C R C mf

R C

pore intertot 0 i

R mRR R R

m n n

0 pore C m n C 1 0 0 R C 2 i 0 R C

Description of electrode

a0

Current

collector HMC particle

R

CA

Rinter

Z0=

Intra-particle

pore

Inter-particle

pore

interinter 0

0

pore

0 1

1

1( ) ( ) coth

( )

( ) coth 22

RZ f R Z f m

n Z f

RZ f j f

j f

Describing equation

m = 10, n = 4 case

Ref) Songhun Yoon, Jong H. Jang, Bok H. Ka, and Seung M. Oh, Electrochimica Acta 50 2255-2262 (2005).

CMK-3 pore structure

100 nm

100 nm

(b)

(a)

ao

b /2

HMC particle

lpore

Carbon rodIntra-particle pore

Carbon wire

ao (nm) 10.7

ABET ( m2g-1 ) 943

b (nm) 6.7

lpore (mm) 2.5

(b)

5 mm

(a)

10 mm

2 /degree

1 2 3 4 5

Inte

nsity/

A.U

.

Pore diameter/nm

5 10 15 20

dV

dD

-1 /cm

3g

-1

0.0

0.5

1.0

1.5

S. Yoon, J. H. Jang, B. H. Ka, S. M. Oh*, Electrochim. Acta 50 2255-2262 (2005). (29 times cited)

lpore

a0

t (t

hic

kness)

Aelectrode

Current collector 0

(electrode thickness)m

a

electrode

0 pore

(1 )A

na l

m ( number of intra-particle pores in electrode layer)

n ( number of independent electric paths )

27

• SBA-15 templated OMC

– Model electrode material for EDLC

Analysis of CMK-3 electrodes

• According to increase of

thickness

– Increase of A

– Decrease of fp

– Coincide to theoretical

prediction

log ( f /Hz)

0.01 0.1 1 10 100

Cim

( f

)/F

-0.3

-0.2

-0.1

0.0

44 mm

71 mm90 mm

148 mm

m2/10

8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

1/

f p

0.0

0.5

1.0

1.5

2.0

2.5

3.0

2

pore pore inter pore

p

0.4R C R C m

f

14

pore

12

pore

5

inter

(2.0 0.1) 10 F

(3.6 0.9) 10

(2.9 0.1) 10

C

R

R

W

W

• From least square fitting

– Cpore : (2.0±0.1) x 10-14 F

– Rpore : (3.6±0.9) x 1012 W

– Rinter : (2.9±0.1) x 105 W

From

2

pore pore inter pore

p

0.4 R C R C m

f

0pore

CC

m n

28S. Yoon*, C. W. Lee and S. M. Oh, J. Power Sources. 195 4391-4399 (2010). (6 times cited)

Analysis of Rtot with thickness

Thickness/mm

20 40 60 80 100 120 140 160 180

Re

sis

tan

ce

/W

0.0

0.2

0.4

0.6

0.8

1.0

R0

Ri

• R0

– Decrease with thickness

– Increase of pore number

• Ri

– Increase with thickness

– Extension of inter-particle

pore length

– Dominant factor in Rtot for

thick electrode

Analysis of tot with thickness

• 1

– Constant

– Reflect properties of

intra-particle pores

• 2

– Increase with thickness

– Dominant factor in tot for

thick electrode

Thickness/mm

20 40 60 80 100 120 140 160 180

tim

e c

on

sta

nt/

s

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1

2

II. Analysis of pore length effect

MCM-41 synthetic mechanism

CTAB in solution rod-like SSM formation

agglomeration MCM41 formation

~ a few hours Above several hours

CTAB

TEOS

pH >14

lengthening

~ a few min

SSM in solution : template (porogen)

ref) J. Zhang, Z. Luz, H. Zimmermann and D. Goldfarb, J. Phys. Chem. B, 104, 279 (2000).

Control of pore length

Concentration of CTAB : 1, 2, 5 and 10 wt%

at 40 oC, 5 hr reaction time

Low concentration

1) RF adding 2) carbonization3) silica etchig

CTAB in solution

CTAB

High concentration

TEOS

pH >14

5 hr

5 hr

SSM

S. Yoon*, S. M. Oh, C. W. Lee, J.-W. Lee, Journal of the Electrochemical Society, 157, A1229-A1235 (2010)

SEM of carbons

1 wt%

10 wt%

2 wt%

5 wt%

TEM of carbons

• Highly mesoporous and wormhole-like pore morphology

– Similar pore morphology irrespective of CTAB concentration

50 nm

10 wt%1 %1 wt%

100 nm

PSD of carbons

• Similar pore size : ~ 3 nm

– Pore size : depend on reaction

time of SSM

– SSM reaction time : fixed as 5 hr

– Dominant pores

0.05

D/nm

0 2 4 6 8 10 12 14

dV

/dD

10 %

5 %

2 %

1 %

CTAB concentration / wt%

0 2 4 6 8 10

BE

T s

rufa

ce a

rea/m

2g

-1

500

550

600

650

Imaginary capacitance analysis

f/Hz

0.01 0.1 1 10 100

Cim

(f)/

F

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

46 mm

170 mm

125 mm

83 mm

f/Hz

0.01 0.1 1 10 100

Cim

(f)/

F

0.0

0.1

0.2

0.3

32 mm

135 mm

119 mm

57 mm

f/Hz

0.01 0.1 1 10 100

Cim

(f)/

F

0.0

0.1

0.2

0.3

35 mm

148 mm

113 mm

63 mm

f/Hz

0.01 0.1 1 10 100

Cim

(f)/

F

0.0

0.1

0.2

0.3

37 mm

143 mm

94 mm44 mm

1 wt%

10 wt%5 wt%

2 wt%

Estimation of 1

t2/(10mm)

2

0 5 10 15 20 25 30 35

tot/s

0.0

0.1

0.2

0.3

0.4

0.5

0.6

t2/(10mm)

2

0 2 4 6 8 10 12 14 16 18 20

tot/s

0.1

0.2

0.3

0.4

0.5

t2/(10mm)

2

0 5 10 15 20 25

tot/s

0.1

0.2

0.3

0.4

0.5

0.6

t2/(10mm)

2

0 5 10 15 20 25

tot/s

0.0

0.5

1.0

1.5

1 = 0.028 s 1 = 0.15 s

1 = 0.30 s1 = 0.18 s

1 wt%

10 wt%5 wt%

2 wt%

2

1 tot

p

0.4 kt

f

1 vs. CTAB concentration

• Increase of CTAB

concentration

– Pore length increase

– 1 increase

CTAB concentration/wt%

0 2 4 6 8 10 12

tim

e c

onsta

nt o

f in

tra p

art

icle

pore

s (

1)/

s

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

1 1>

thickness/mm

20 40 60 80 100 120 140 160 180

Rto

t/W

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

thickness/mm

20 40 60 80 100 120 140 160 180

Rto

t/W

0

1

2

3

4

5

6thickness/mm

20 40 60 80 100 120 140 160 180

Rto

t/W

0.0

0.2

0.4

0.6

0.8

1.0

R0

Ri

thickness/mm

20 40 60 80 100 120 140 160 180

Rto

t/W

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Rtot vs. thickness plots

1 wt%

10 wt%5 wt%

2 wt%

III. Analysis of pore size effect

Control of pore size

Reaction time of silicate : 2, 5, 7 and 16 hr

at 40 oC, 10 wt% CTAB

CTAB in solutionrod-like SSM formation

and lengtheningSSM condensation MCM41 formation

~ a few hours Above several hours

CTAB

1) RF adding 2) carbonization3) silica etchig

TEOS

pH >14

Songhun Yoon*, Seung M. Oh and Chulwee Lee, Material Research Bulletin, 44, 1663-1669 (2009).

SEM images of carbons

• Multi-faceted tubule

morphology

• Inhomogeneous

morphology after 7 hr

5 hr2 hr

16 hr7 hr

Scale bar: 5 mm

7 hr

TEM images of carbons

• Wormhole-like pores

• Increase of pore size

5 hr

16 hr

2 hr

7 hr

Scale bar: 50 nm

f/Hz

0.01 0.1 1 10 100

Cim

(f)/

F

0.0

0.1

0.2

0.3

37 mm

94 mm

143 mm

44 mm

f/Hz

0.01 0.1 1 10 100

Cim

(f)/

F

0.0

0.1

0.2

0.3

61 mm

112 mm

120 mm 86 mm

f/Hz

0.01 0.1 1 10 100

Cim

(f)/

F

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

47 mm

81 mm118 mm

66 mm

f/Hz

0.01 0.1 1 10 100

Cim

(f)/

F

0.0

0.1

0.2

0.3

50 mm

99 mm110 mm

70 mm

Imaginary capacitance analysis

2 hr

16 hr7 hr

5 hr

t2/(10mm)

2

0 2 4 6 8 10 12 14 16

tot/s

0.0

0.2

0.4

0.6

t2/(10mm)

2

0 5 10 15 20 25

tot/s

0.0

0.5

1.0

1.5

t2/(10mm)

2

0 2 4 6 8 10 12 14

tot/s

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t2/(10mm)

2

2 4 6 8 10 12 14 16

tot/s

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1 = 0.23 s 1 = 0.30 s

1 = 0.11 s1 = 0.14 s

Estimation of 1

2 hr

16 hr7 hr

5 hr

2

1

p

0.4 kt

f

1 vs. reaction time

• 1 increase until 5 hr

– Pore length increase

– SSM lengthening

• 1 decrease after 5 hr

– Pore size increase

– SSM agglomeration

Reaction time/hr

0 2 4 6 8 10 12 14 16 18

tim

e c

onsta

nt

of in

tra

pa

rtic

le p

ore

s (

1)/

s

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

1 1<

thickness/mm

0 20 40 60 80 100 120 140 160

Rto

t/W

0

1

2

3

4

5

6

thickness/mm

20 40 60 80 100 120 140 160

Rto

t/W

0

1

2

3

4

5thickness/mm

20 40 60 80 100 120 140 160

Rto

t/W

0.0

0.2

0.4

0.6

0.8

1.0

1.2

R0

Ri

thickness/mm

20 40 60 80 100 120 140 160

Rto

t/W

0.0

0.5

1.0

1.5

2.0

2.5

Rtot vs. thickness plots

2 hr

16 hr7 hr

5 hr

TLM in LIB by Toyota

18650 cell thickness effect

Symmetric cell thickness effect

Conclusions

I. 1-D TLM

I. Simple and easy to use

II. Application into very thin electrode case

II. 2-D TLM

I. Consideration of inter-particle electrolyte

resistance with thickness increase

II. Higher contribution in total ESR for thick electrode

III. More versatile equation in ICA than Nyquist form

III. Application into LIB electrode simulation

I. Advanced TLM model !!