Electric Double-Layer

Prévia do material em texto

Journal of The Electrochemical Society, 151 ~4! A571-A577 ~2004!
0013-4651/2004/151~4!/A571/7/$7.00 © The Electrochemical Society, Inc.
A571
Complex Capacitance Analysis of Porous Carbon Electrodes
for Electric Double-Layer Capacitors
Jong H. Jang and Seung M. Oh* ,z
School of Chemical Engineering and Research Center for Energy Conversion and Storage, Seoul National
University, Seoul 151-744, Korea
A new analytical methodology, complex capacitance analysis, is developed for porous carbons that are employed as electric
double-layer capacitor~EDLC! electrodes. Based on the transmission line model, the imaginary capacitance profiles (Cim vs.log f)
are theoretically derived for a cylindrical pore and further extended to multiple pore systems. The theoretical derivation illustrates
that two important electrochemical parameters in EDLCs can be estimated from the peak-shaped imaginary capacitance plots: total
capacitance from the peak area and rate capability from the peak position. The usefulness of this analysis in estimating EDLC
parameters is demonstrated by applying to two sets of practical porous carbon electrodes. In addition, the penetrability distribution
curves that are derived from the experimental imaginary capacitance data using the log-normal assumption and discrete Fourier
transform allow us to estimate the pore structure of carbon electrodes.
© 2004 The Electrochemical Society.@DOI: 10.1149/1.1647572# All rights reserved.
Manuscript submitted July 14, 2003; revised manuscript received October 28, 2003. Available electronically February 20, 2004.
e
ctric
cou
orab
ce is
high-
des.
n for
elec-
com
nder
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etrat
not b
DLC
nce
por
e
ture
ly-
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nden
not
eems
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with
aper
sys-
n of
de-
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des
odel
solid
re-
pore
an
red to
DLC
c con-
,
e
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deally
ap-
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i-
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a
-
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ies,
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ated
The electric double-layer capacitor~EDLC! utilizes the doubl
layer formed at the electrode/electrolyte interface where ele
charges are accumulated on the electrode surface and ions of
tercharge are arranged in the electrolyte side. The charge st
capacity of EDLCs that is normally expressed by capacitan
roughly proportional to electrode surface area, such that
surface-area carbons are commonly utilized for EDLC electro
The rate capability of EDLCs is another important consideratio
their practical application, because a complete utilization of
trode surface for charge storage is not always guaranteed. In
plete utilization arises particularly at high-current conditions, u
which ionic motions in the electrolyte are slower than the cur
transient. This undesirable situation is more pronounced when
are narrow, long, and irregularly connected, as ions cannot pen
into the deeper region, thereby the surface exposed there may
utilized for charge storage. In short, the pore structure of E
electrodes is an important consideration because the capacita
determined by surface area and the rate capability affected by
size, length, and connectivity.1-6
Electrochemical impedance spectroscopy~EIS! has proven to b
a useful tool for analysis of EDLC performance and pore struc
of porous carbon electrodes.7-10 In previous studies, however, ana
ses were made mostly employing Nyquist plots (2Zim vs. Zre) that
are too featureless to obtain detailed information on pore stru
Sophisticated nonlinear fitting is also required to obtain EDLC d
Recently, there were several efforts to characterize the EDLC
rameters using complex capacitance plots or frequency-depe
capacitance, but the detailed theoretical derivation was
given.6,11,12
In this paper we present a new analytical method that s
useful to estimate the EDLC performance of porous electrodes
as total capacitance and rate capability that are deeply related
the pore structure of porous electrodes. In the first half of this p
the complex capacitance for a cylindrical pore and multipore
tems is theoretically derived. Also, two methods for estimatio
penetrability distribution patterns from experimental data are
scribed. In the second half of this paper, application of the com
capacitance analysis made on two sets of porous carbon electro
provided.
Theory
Complex capacitance analysis for a cylindrical pore.—Porous
electrodes have been analyzed with the transmission line m
which assumes the porous electrode surface as a distributed
* Electrochemical Society Active Member.
z E-mail: seungoh@plaza.snu.ac.kr
n-
le
-
s
e
e
is
e
.
t
,
is
,
/
liquid interface impedance with a variable ionic and electronic
sistance according to the position of the interface from the
opening.7-10,13-24The overall impedance is derived as Eq. 1 with
assumption that the electrode resistance is negligible as compa
the ionic resistance in the electrolyte13
Z 5 ~Rion,LZi,L!
1/2 coth@~Rion,L /Zi,L!
1/2L# @1#
whereZi,L is the interfacial impedance per unit pore length,Rion,L is
the electrolyte resistance per unit pore length, andL is the pore
length. The elimination of electrode resistance is valid in an E
system, where common carbon materials have a higher electri
ductivity ( . 102 S cm21! than electrolyte solutions~for example
2.0 M sulfuric acid:,1 S cm21!.25,26 The interfacial impedanc
(Zi,L) can further be simplified as a purely capacitive one afte
noring the faradaic charge-transfer process that is pertinent to i
polarized electrodes, for example, carbon electrodes for EDLC
plication. TheZi,L andRion,L for a cylindrical pore are then given
Zi,L~ f ! 5
L
j 2p f C0
5
1
2pr
1
j 2p f Cd
, Rion,L 5
1
kpr 2
@2#
The impedance of porous electrodes is then expressed as
Z~a! 5 R0 3 aA2/j coth@~aA2/j !21# @3#
a 5 a0v
20.5 5
a0
A2p f
, where a0 5
1
2
A k
Cd
Ar
L
5
1
A2R0C0
@4#
Here,C0( 5 Cd 3 2prL ) and R0( 5 Rion,L 3 L) are the capac
tance and electrolyte resistance for the entire pore, respectivel
dimensionless penetration depth~a! is defined in this work as
function of frequency~f!, and the penetrability (a0) that is in turn
defined by the pore structure~r: pore radius,L: pore length! of
electrodes and electrochemical parameters~k: electrolyte conductiv
ity, Cd : double-layer capacitance per unit area! in pores. The dimen
sionless penetration depth reflects the degree of ion penetratio
pores according to the current transient; thereby it is an impo
parameter to characterize the rate capability of EDLC electr
When the current transient is rapidly varied at high ac frequenca
becomes smaller because ions cannot penetrate deep inside
due to the limited ionic motions. A smallera is also expected
either the pore radius is smaller or pore length is longer becau
degree of ion penetration becomes lower.
The complex capacitance for a cylindrical pore is calcul
from Eq. 3 as
ce is
n a
e is
e,
fea-
s dif-
5 is
spec
s
-
is
en-
hen
n be
e re-
s-
ted
eak
Eq. 8.
s a
x
n s-
Journal of The Electrochemical Society, 151 ~4! A571-A577 ~2004!A572
C~a! 5
1
j vZ~a!
5 C0 3 aA2/j tanh@~aA2/j !21# @5#
Then, the real and imaginary part of the complex capacitan
derived as follows
Cre~a! 5 C0 3 S a sinha21 1 sina21
cosha21 1 cosa21
D @6#
Cim~a! 5 C0 3 S 2a sinha21 2 sina21
cosha21 1 cosa21
D 5 C0Cim0 ~a!
@7#
In Fig. 1a, the Nyquist impedance plot that is simulated o
cylindrical pore using Eq. 3 is presented. The curve shap
changed with an increase ina from a straight line with 45° slop
intermediate region, and vertical line. As seen, the profile is so
tureless that a visual estimation of the EDLC properties seem
ficult. The complex capacitance plot that is derived using Eq.
displayed in Fig. 1b, and the real (Cre) and imaginary part (Cim) of
complex capacitance that are calculated using Eq. 6 and 7, re
tively, are plotted as a function ofa in Fig. 1c and d.C increase
Figure 1. Simulated Nyquist plots for a cylindrical pore:~a! impedance,~b!
complex capacitance,~c! the real part, and~d! the imaginary part of comple
capacitance as a function of dimensionless penetrability~a!. The numbers i
the figures area values.
re
-
monotonicallyfrom 0 toC0 with a ~Fig. 1c!, whereasCim shows a
peak-shaped curve with a maximum atap 5 0.4436~Fig. 1d!. The
total capacitance can be estimated from theCre value at a suffi
ciently high a ~low-frequency extreme!. In practice, however, it
difficult to obtain data at such a low frequency due to the instrum
tal limitation and extremely long measuring time. In contrast, w
the imaginary capacitance plot is utilized, total capacitance ca
estimated by simply integrating the peak area. The quantitativ
lationship between the peak area~A! and total capacitance (C0) is
obtained as
A 5 2 E
2`
`
Cim~a!d log f 5 0.682C0 @8#
The factor, 0.682, obtained by a numerical integration ofCim(a) in
Eq. 7 is identical top/~2 ln 10! that is predicted from Kramer
Kronig transforms.27,28 The imaginary capacitance plots calcula
with a variation inC0 are presented in Fig. 2a. As seen, the p
area is proportional to the total capacitance as expected from
The peak frequency (f p) where the imaginary capacitance ha
maximum value can be obtained by substitutinga with ap in Eq. 4
as
Figure 2. ~a! Relationship between the total capacitance (C0) of a cylindri-
cal pore and peak area in the imaginary capacitance plot~penetrability is
assumed the same at 1 s20.5!. ~b! Relationship between the penetrability (a0)
and peak position in an imaginary capacitance plot~total capacitance is a
sumed the same at 1 F!.
re
er a
e
s
nuni-
ility
. In
le-
of th
ry ca
e
ance
-
ility
on o
that
he
etra-
r,
vant
in
aram
The
s
g Eq.
aci-
work.
tri-
gree
sump-
l dis-
e
poly-
this
s a
three
ent
he ex-
ion
Journal of The Electrochemical Society, 151 ~4! A571-A577 ~2004! A573
f p 5
1
2pap
2
a0
2 5 0.809a0
2 @9#
The change off p according toa0 is demonstrated in Fig. 2b, whe
it is seen that the two parameters are correlated to each oth
described in Eq. 9. The peak of the same area, asC0 is assumed th
same, moves to the high-frequency direction whena0 become
higher.
Complex capacitance analysis for multiple pore systems.—As
the porous carbons used in practical EDLCs contain many no
form pores that differ in both radius and length, the penetrab
should have a certain distribution instead of a single value
multiple-pore systems, the capacitance (C0,i) and penetrability (a0,i)
of the ith pore are
C0,i 5 Cd,i~2pr iL i!, a0,i 5
1
2
A k i
Cd,i
Ar i
L i
@10#
wherek i , Cd,i , r i , andL i are the electrolyte conductivity, doub
layer capacitance per unit area, pore radius, and pore length
ith pore, respectively. Then the frequency-dependent imagina
pacitance is calculated as
Cim~ f ! 5 (
i
Cim
0 ~a0,i , f !C0,i
5 E
0
`
Cim
0 S a0A2p f D Ctot@p~a0! da0#
5 E
2`
`
Cim
0 S a0A2p f D Ctot@p8~a0! d log a0# @11#
p~a0! 5
d@C~a0!/Ctot#
da0
, p8~a0! 5
d@C~a0!/Ctot#
d log a0
@12#
whereC(a0) represents the sum ofC0,i for pores having the sam
penetrability ofa0 , whereasCtot is the sum ofC(a0) for entire
pores, andp(a0) and p8(a0) are thea0 distribution function in
linear and logarithmic scale, respectively. The total capacit
(Ctot) is related to the peak area by integrating Eq. 11 with logf as
A 5 2 E
2`
`
Cim~ f !d log f
5 2 S E
2`
`
Cim
0 ~a!d log f D S E
0
`
Ctot p~a0!da0D
5 0.682Ctot @13#
In this work, the characteristic penetrability (a0* ) has been de
fined as thea0 value having a maximum population in penetrab
distribution curves. This parameter is expressed as a functi
peak frequency and an empirical parameterb as
a0* 5 A2papb f p0.5 5 1.11b f p0.5 @14#
The b is inserted because the peak frequency is different from
calculated by Eq. 9 witha0* due to the unsymmetric nature of t
Cim
0 (a) function.b varies according to the broadness of the pen
bility distribution curves. As the distribution becomes narroweb
approaches unity and Eq. 14 is simplified to Eq. 9, which is rele
to a single-pore system.
Total capacitance (Ctot) and penetrability (α0) distribution in
multiple-pore systems.—As described in Eq. 4, the penetrability
multipore systems has a certain distribution because the four p
s
e
-
f
-
eters~r, L, k, andCd) are variable according to pore structure.
total capacitance anda0 distribution pattern in multipore system
can be obtained from experimental capacitance data by solvin
11, whereCim( f ) is the experimentally observed imaginary cap
tance data. To solve Eq. 11, two approaches are made in this
In the first approach,a0 is assumed to have a log-normal dis
bution; thereby the ultimate goal here is the extraction of the de
of dispersion as the distribution pattern is preassumed. This as
tion is made as porous materials frequently show a log-norma
tribution in pore size.8,9 The log-normala0 distribution function is
expressed by two parameters: standard deviation~s! that reflects th
degree of dispersion and characteristic penetrability (a0* ) as
p8~a0! 5
d@C~a0!/Ctot#
d log a0
5
1
A2ps
expF2 1
2s2
~ log a0 2 log a0* !
2G @15#
To obtain the standard deviation from experimental data, the
nomial interpolation method was utilized. As the first step in
method, log-normala0 distributions are derived using Eq. 15 a
function of s with a0* ~1 s
20.5! being fixed ~Fig. 3a!. Then the
imaginary capacitance is calculated using Eq. 11. In Fig. 3b,
imaginary capacitance plots that are obtained with three differs
are represented. In the profiles, the peak width increases at t
Figure 3. ~a! Log-normala0 distribution as a function of standard deviat
~s! and ~b! the resultant imaginary capacitance plots.
rea
ak
plots
ed
oly-
peri-
m
y
g
ak
cy
q.
based
i-
men-
ained
e
m
s
to any
ll as
simu-
sing
,
ction,
ans-
a-
-
aci-
en as
e pro-
e
xam-
ieving
ca-
lyno-
Journal of The Electrochemical Society, 151 ~4! A571-A577 ~2004!A574
pense of peak height whens increases. Also note that the peak a
remains unchanged asCtot is fixed at a constant value. The pe
frequency slightly moves to a higher value due to a decrease inb as
s increases. The next step is the location of the peak width~the full
width at half maximum, fwhm,PW), peak height (PH0), and b
from the calculated imaginary capacitance plots. ThePW, PH0,
and b that are collected from sixteen imaginary capacitance
that are made with sixteen differents are indicated as the clos
circles in Fig. 4. The data points are best-fitted with 5th-order p
nomial functions
PW~s! 5 1.24532 2.213 1023 s 1 22.0s2 2 58.5s3
1 72.0s4 2 29.2 s5 @16#
PH0~s! 5 0.41732 0.012s 2 5.11s2 1 19.5s3 2 31.8s4
1 19.7s5
Figure 4. ~a! The peak width (PW), ~b! peak height (PH0), and ~c! b in
imaginary capacitance plots as a function of standard deviation~s! in log-
normal a0 distribution. The data points are derived from 16 imaginary
pacitance plots and the lines indicate the best fitting with 5th-order po
mial functions.
b~s! 5 1.001 3.483 10-3 s 2 0.936s2 2 0.889s3
1 6.54s4 2 6.10s5
Now the standard deviation can be obtained from given ex
mental impedance data. To this end, the peak frequency (f p), peak
height (PH), and peak width (PW) should roughly be decided fro
the experimental imaginary capacitance plot. Thens is obtained b
interpolating thePW value in Fig. 4a.Ctot is calculated by dividin
the peak height (PH) by PH0(s), which corresponds to the pe
height whenCtot 5 1 F. a0* is calculated from the peak frequen
andb~s! using Eq. 14. Thea0 distribution is then derived using E
15 with as-obtaineds anda0* values. TheCtot anda0 distribution
that are obtained so far still need refinement because they are
on the roughly estimatedf p , PH, andPW values from the exper
mental data points. The refinement is made by fitting the experi
tal data with the simulated imaginary capacitance profiles obt
from the as-obtainedCtot anda0 distribution with Eq. 11. The thre
parameters (Ctot , s, a0* ) are extracted from the best fitting, fro
which the finalCtotp8(a0) curves are derived using Eq. 15.
The second approach to obtainCtot and thea0 distribution is the
use of the discrete Fourier transform~DFT!. This method seem
more versatile than the previous one because it can be applied
type of a0 distribution,but a noise reduction procedure as we
complicated transformations is required to obtain a reasonable
lation. Equation 11 can be converted to a convolution form u
two defined functions:x 5 ln(2pf) andy 5 ln a0
2
g~x! 5 E
2`
`
h~x 2 y! f ~y!dy 5 ~h* f !~x! @17#
with
f ~y! 5 Ctot
d@C~a0!/Ctot#
d ln a0
2
, g~x! 5 Cim~x!,
h~x 2 y! 5 Cim
0 ~a 5 e2~x2y!/2! @18#
where f (x), g(x), andh(x) represent thea0 distribution function
experimental imaginary capacitance, and known transfer fun
respectively. The DFT and inverse transform are defined as29,30
Fk 5
1
N (2N/211
N/2
f ne
2i2pnk/N, f n 5 (
2N/211
N/2
Fke
i2pnk/N @19#
where f n andFk are the original data set and discrete Fourier tr
formed one, respectively, andN is the number of data points. Equ
tion 17 is discrete Fourier transformed to
Gk~v ! 5 NDxHk~v !Fk~v ! @20#
whereDx is the data interval andv is the reciprocalx. Then, thea0
distribution is calculated according to the following order:~i! dis-
crete Fourier transforming experimentally determinedg(x) and
known h(x), (i i ) dividing Gk(v) by Hk(v) to yield Fk(v), and
( i i i ) inverse transforming to determinef (x). The procedure to ob
tain thea0 distribution from a given experimental imaginary cap
tance is as follows. The three profiles shown in Fig. 3b are tak
the experimental data. The experimental imaginary capacitanc
files are discrete Fourier transformed toGk(v) ~Fig. 5a!, which are
divided by NDxHk(v) to yield Fk(v) as shown in Fig. 5c. Th
inverse transformation ofFk(v) producesa0 distributions, which
are identical to the initial log-normal ones~Fig. 3a!.
Experimental
The EDLC performance of porous carbon electrodes was e
ined using the complex capacitance analysis. The molecular-s
Co.
ile
o-
e
s, bu
f
tribu
g as
es, a
2
ount
ronic
furic
ge of
.
2000
erted
and
tively.
e the
er the
tive
that
C25
made
o
l im-
imagi-
points.
Journal of The Electrochemical Society, 151 ~4! A571-A577 ~2004! A575
carbon~MSC25! was provided by Kansai Coke and Chemicals
and new mesoporous carbon~NMC! was prepared using the Mob
Composite Material 48 as a template.4,31 The other series of mes
porous carbons were prepared by using a silica-sol template.28 The
average pore diameter of these carbons is similar~ca.9 nm! becaus
identical silica sol particles were used to generate mesopore
the pore volume differs as the silica-sol content~the mole ratio o
silica to carbon precursor: S15 0.77; S25 1.16; S35 1.54) was
adjusted to different values. The surface area and pore size dis
tion were calculated from the Nadsorption isotherms~Micromet-
Figure 5. DFT profiles for ~a! experimental imaginary capacitance,~b!
transfer function, and~c! a0 distribution calculated from~a! and ~b!. The
inverse transformation of~c! generates ana0 distribution that is identical t
Fig. 3a, thus omitted.
2
t
-
rics ASAP 2010! using the Brunauer, Emmett, and Teller~BET!, and
Barrett, Joyner, and Halenda~BJH! method, respectively.
A three-electrode configuration was used with a platinum fla
the counter electrode and saturated calomel electrode~SCE! as the
reference electrode. For the preparation of working electrod
mixture of carbon power, polytetrafluoroethylene~PTFE! binder,
and carbon additive for conductivity enhancement~Ketjenblack
ECP-600JD! was coated on a stainless steel mesh~apparent area
cm2!. The weight of carbon powder was 10 mg. The loading am
of carbon additive was controlled to give a comparable elect
conductivity for each electrode. The electrolyte was 2.0 M sul
acid. EIS measurements were made over the frequency ran
106 2 5 3 1023 Hz ~Zahner, im6e! with ac amplitude of 10 mV
Experimental data were processed and analyzed with Mathcad
Professional~MathSoft, Inc.! software.
Results and Discussion
In Fig. 6a, the imaginary capacitance data that are conv
from the experimentally observed impedance data for MSC25
NMC are indicated as the closed and open circles, respec
Even with the raw data points, one can compare at first glanc
EDLC performance of the two carbon electrodes. The area und
peak-shaped profile of MSC25 is larger than that of NMC, indica
of a larger total capacitance in the former. It is also predictable
the NMC electrode has a better rate capability than that of MS
as the peak frequency is higher. A more quantitative analysis is
Figure 6. ~a! Imaginary capacitance plots converted from experimenta
pedance data for MSC25 and NMC electrodes. The lines indicate the
nary capacitance curves that are best-fitted to the experimental data
~b! Ctotp8(a0) curves for MSC25 and NMC by log-normal assumption~solid
line! and DFT~dashed line!.
es
eth-
C25
pears
n
S25
C,
ngth
ithe
ore
4, su
elec
e,
e pe
face
say
in th
and
of the
e ex-
rived
gely
oints
sing
r can
e
LC
nab
. The
rface
i-
hat
ies o
te. In
car
iately
arge
s pre
magi
l dat
s
he
ente
il-
e II.
whic
dsor
d
ee
s
bility
may
uctiv-
(
ize is
er
ental
dicate
al data
aci-
Journal of The Electrochemical Society, 151 ~4! A571-A577 ~2004!A576
after a derivation ofa0 distribution curves for the two electrod
~Fig. 6b!. Two features are apparent in Fig. 6b. First, the two m
ods give largely the same penetrability distribution for the MS
electrode and also for the NMC electrode, even if there ap
some mismatch. This implies that the log-normal assumption ia0
distribution is valid for the two porous carbons. Secondly, MC
gives rise to smallera0 values as compared to those for NM
illustrating that the pores of either smaller radius or longer le
are more populated in MSC25. The narrowera0 distribution in
NMC suggests that the pores in this carbon are more uniform e
in radius or length. The penetrability is a function of both p
structure and electrochemical parameters as expressed in Eq.
that the prediction of pore structure froma0 distribution is only
possible when the electrochemical parameters are known. The
trolyte conductivity~k! in pores differs to that of bulk; furthermor
it varies according to the pore size. The double-layer capacitanc
unit area (Cd) is also varied according to the pore size and sur
properties of pores. Even with this uncertainty, it may be safe to
that two electrochemical parameters~k and Cd) are similar in the
two electrodes as their pore size is comparable to each other
range of 1-2 nm. It should further be noted that the pore radius
length cannot be predicted separately but as a combination
two.
The imaginary capacitance profiles that are best-fitted to th
perimental data are indicated as lines in Fig. 6a. The profiles de
from the log-normal assumption and Fourier transform look lar
the same and are reasonably fitted to the experimental data p
The total capacitance (Ctot) and characteristic penetrability (a0* ) are
obtained from the best-fitted imaginary capacitance profiles u
Eq. 11 and 15, and the results are listed in Table I. The forme
also be calculated from the peak area~A! and the latter from th
peak frequency (f p) using Eq. 13 and 14, respectively. The ED
data derived from the complex capacitance analysis are reaso
matched to those predictable from gas adsorption analysis
higher total capacitance in MSC25 is ascribed to its larger su
area. The smallera0 in MSC25 is due to the dominance of m
cropores~,2.0 nm!, which is compared to the NMC electrode t
contains mesopores~average diameter 2.3 nm!.
The complex capacitance analysis is made for another ser
carbon electrodes that were prepared using silica-sol templa
Fig. 7a, the experimental imaginary capacitance data for three
bon electrodes are indicated as circles or triangles. It is immed
apparent even before the fitting that the S3 electrode has the l
total capacitance and best rate capability among the three a
dicted from its peak area and peak frequency. The simulated i
nary capacitance profiles that are best-fitted to the experimenta
with an assumption of log-normala0 distribution are indicated a
lines in Fig. 7a, and theCtot p8(a0) curves that are derived from t
best-fitted imaginary capacitance profiles using Eq. 11 are pres
in Fig. 7b. The totalcapacitance (Ctot) and characteristic penetrab
ity (a0* ) that are obtained with the best-fitting are listed in Tabl
S3 shows the largest total capacitance among the three, from
it is predicted that S3 has the largest surface area. The gas a
Table I. The best-fitted EDLC parameters obtained by complex
capacitance analysis and nitrogen gas adsorption data for MSC25
and NMC carbon.
MSC25 NMC
Ctot ~F g
21! 188.96 2.2 92.86 0.6
s 0.4406 0.011 0.2186 0.005
a0* ~s
20.5! 0.2886 0.008 0.8316 0.003
ABET ~m
2 g21!a 1970 1257
Dpore,aver.~nm!
b ,2.0 ca. 2.3
a Surface area calculated by BET method.
b Average pore diameter calculated by BJH method.
r
ch
-
r
e
.
ly
f
-
st
-
-
a
d
h
p-
tion data confirm this prediction. Also, theCtot values are in goo
accordance with those obtained from the cyclic voltammetry~S1, 59
F g21; S2, 70 F g21; S3, 96 F g21!. The pore structure of the thr
carbons can also be compared using thea0 distribution curve
shown in Fig. 7b, where it is seen that the characteristic penetra
becomes higher with an increase in pore volume. This feature
be explained by the assumption that both the electrolyte cond
ity ~k! in the pores and double-layer capacitance per unit areaCd)
are largely the same for the three carbons, as their pore s
comparable to each other~ca.9 nm!. The higher population of larg
Figure 7. ~a! Imaginary capacitance plots converted from the experim
impedance data for silica-sol-templated carbon electrodes. The lines in
the imaginary capacitance curves that are best-fitted to the experiment
points. ~b! Ctotp8(a0) curves derived from the best-fitted imaginary cap
tance profiles.
Table II. The best-fitted EDLC parameters obtained by complex
capacitance analysis and nitrogen gas adsorption data for silica-
sol-templated mesoporous carbons.
S1 S2 S3
Ctot ~F g
21! 60.46 0.9 74.66 3.1 98.06 2.3
s 0.1416 0.014 0.1496 0.037 0.1766 0.019
a0* ~s
20.5! 0.2816 0.007 0.2976 0.019 0.4346 0.011
ABET ~m
2 g21! 457 626 746
Dpore,aver.~nm! ca. 9 ca. 9 ca. 9
Vpore ~cm
3 g21!a 0.68 1.08 1.50
a Pore volume calculated by BJH method.
pore
s not
orter
tiple
plex
elec
rang
pore
apa-
d by
paci
over
quis
an
hen
rodes
re rea
data
Cen-
ssel-
d H.
ring
.,
.,
te
del-
,
-
Journal of The Electrochemical Society, 151 ~4! A571-A577 ~2004! A577
a0 but with a comparable pore size manifests itself in that the
length becomes shorter with an increase in pore volume. It i
difficult to explain this in that the distance between pores is sh
as the pore volume increases.
Conclusion
The complex capacitance for a cylindrical pore and mul
pores is theoretically derived and the penetrability (a0) distribution
obtained either by log-normal assumption or DFT. The com
capacitance analysis is made on two sets of porous carbon
trodes: one series of carbons differs in the pore size in 1–2 nm
and the other differs in the pore volume but with a comparable
size ~ca. 9 nm!. It is found that the total capacitance and rate c
bility of the porous carbon electrodes can be easily estimate
displaying the experimental impedance data in the imaginary ca
tance plots. This is the strong point of this new methodology
conventional ones where analyses were made employing Ny
impedance plots (2Zim vs. Zre). The pore structure of carbons c
also be predicted from the penetrability distribution curves. W
the EDLC performance of the two series of porous carbon elect
is examined using complex capacitance analysis, the results a
sonably matched with those predicted from the gas adsorption
Acknowledgment
This work was supported by KOSEF through the Research
ter for Energy Conversion and Storage.
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