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Electrochimica Acta 53 (2007) 575–583

Analytical solution for the voltage distribution in one-dimensionalporous electrode subjected to cyclic voltammetric (CV) conditions

Ravinder Reddy 1, Ramana G. Reddy ∗,2

Department of Metallurgical and Materials Engineering, The University of Alabama, Tuscaloosa, AL 35487-0202, USA

Received 15 March 2007; received in revised form 10 July 2007; accepted 11 July 2007Available online 19 July 2007

bstract

An analytical solution for the voltage distribution in one-dimensional porous electrode subjected to cyclic voltammetric (CV) conditions waseveloped by considering the resistivity of both electrode and electrolyte. Voltage distribution inside the electrode as a function of dimensionlessistance was generated at different electrode phase resistances. The analytical solution of voltage distribution for cyclic voltammetric conditions wasmployed in deriving the charging currents. Capacitance studies were conducted on carbons derived from catechol-formaldehyde gels prepared atH conditions of 3, 6, and 7.5 in 30 wt.% H2SO4 electrolyte using CV and constant current charge/discharge studies. Charging currents derived from

oltage distribution were fitted with experimental anodic part of CV curves and equivalent series resistance (ESR) was generated. Thus, generatedSR values were compared with that obtained using constant current charge–discharge curves. The model fitted resistance values matched closelyith resistance values obtained from galvanostatic constant current charge–discharge method.2007 Elsevier Ltd. All rights reserved.

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eywords: Porous electrodes; Voltage distribution; Cyclic voltametry; Catecho

. Introduction

Electrochemical capacitors are charge storage devices andhey are classified as electrochemical double layer capacitorsnd pseudocapacitors depending on charge storage mechanism.he origin of capacitance in the EDLC is charge separation at

he electrode–electrolyte interface, whereas pseudocapacitancerises from fast, reversible faradiac redox reactions taking placen or near the surface of the electrode. Most of the commercialpplications are being developed for EDLCs as back up andulse power sources for many electronic devices. Recently, theyave become of interest in hybrid electric vehicles as an auxiliaryower source in combination with a fuel cell or battery. The rolef the EDLCs in hybrid electric vehicles is load leveling, start-up

nd acceleration [1–6].

EDLC consists of two porous carbon electrodes sandwichedith a separator consisting of ionically conducting electrolyte.

∗ Corresponding author. Tel.: +1 205 348 4246, fax: +1 205 348 2164.E-mail address: rreddy@eng.ua.edu (R.G. Reddy).

1 Research Post Doctoral Fellow.2 ACIPCO Professor.

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013-4686/$ – see front matter © 2007 Elsevier Ltd. All rights reserved.oi:10.1016/j.electacta.2007.07.027

alde gels; Porous carbon spheres

he performance of an EDLC depends on many parameters suchs the material properties, cell design and operating conditions.o fabricate EDLCs for a particular application, it is necessary

o analyze the performance and optimize the system parame-ers. Mathematical modeling has been proved to be useful forhis purpose. One of the quick ways of screening an electrode

aterial for EDLC is employing cyclic voltammetric techniquesing three-electrode method. Very few studies have been donen simulating CV conditions to characterize energy storage sys-ems. This paper reviews the models of voltage distribution inne-dimensional porous electrodes subjected to constant currentnd linear increase in voltage perturbations. The purpose of thisork is to derive voltage distribution in one-dimensional porous

lectrode by considering the effect of conductivity of electrodend electrolyte.

Resorcinol-formaldehyde (RF) gels prepared by sol–gelethod have attracted lot of attention as a precursor for car-

on materials [7–9]. The resulting carbon structure is sensitiveo various syntheses and processing conditions of RF gels and

his leads to a remarkable potential for designing and tailor-ng the resulting carbon to fit to one’s specific need [8]. The

ajor disadvantage of RF polymers is high cost of resorcinol.ence there is a need to find alternative to resorcinol and still

5 ochim

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76 R. Reddy, R.G. Reddy / Electr

e able to prepare a polymer by sol–gel method. A low costatechol, 1,3-dihydroxy benzene can replace resorcinol for eco-omic production of carbon. We prepared highly porous carbonaterials from catechol-formaldehyde gels and were tested for

heir energy storage characteristics for EDLC [10].

. Models describing charging of double layer

Comprehensive review of theory of potentiostatic and gal-anostatic charging of double layer in porous electrodes is giveny Posey and Morozumi [11]. These authors developed potentio-tatic and galvanostatic transients that occur during the chargingf the double layer in porous electrodes. They presented solu-ions for transients due to charging processes in the interior ofdealized one-dimensional porous electrodes of finite length sub-ected to single pulse methods, constant voltage, square waves ofurrent or voltage, and for triangular current or voltage sweeps.n addition, they also considered the effect of voltage drop whichccurs in the solution between the tip of the reference elec-rode capillary and front surface of the porous electrode. Theartial differential equation (PDE) that governs the potential dis-ribution in idealized one-dimensional porous electrode of finiteength is given by [11]:

∂2η(χ, t)

∂χ2 = τ1∂η(χ, t)

∂t(2.1)

here

(χ, t) = �φ(χ, t) = φM − φS(χ, t) (2.2)

1 = RSC (2.3)

= x

l(2.4)

(χ,t) is the potential difference between the matrix phase andlectrolytic phase and τ1 is time constant of double layer charg-ng, a product of electrolyte resistance (RS) and capacitance (C).

is the dimensionless distance. In deriving Eq. (2.1) it wasssumed that the conductivity of matrix phase is higher thanhat of solution phase. If this condition is relaxed and resistancef matrix phase (RM) is also considered Eq. (2.1) becomes [11]:

∂2η(χ, t)

∂χ2 = τ∂η(χ, t)

∂t(2.5)

here

(χ, t) = �φ(χ, t) = φM(χ, t) − φS(χ, t) (2.6)

= (RS + RM)C (2.7)

here φM(χ,t) is the potential of electrode, φS(χ,t) is the poten-ial of electrolyte. τ is time constant of double layer charging,product of (RM + RS) and capacitance (C). The governing Eq.

2.5) can be employed for obtaining potentiostatic transients

hat occur during charging of double layer for the conditionsf constant external voltage, linear increase in voltage (cyclicoltammetry) and sinusoidal voltage change (impedance spec-roscopy).

η

T

ica Acta 53 (2007) 575–583

.1. Constant voltage condition

Posey and Morozumi presented voltage and current distri-ution for the case of constant external voltage for differentases [11]. These different cases are categorized into no lossodel, external loss model and internal loss model followed

rom reference [12].

.1.1. No losses modelThis model considers conductivity of matrix is much higher

han that of electrolyte and employs governing Eq. (2.1).

.1.2. External losses modelThis model considers conductivity of matrix is much higher

han that of electrolyte and additionally considers voltage dropn the electrolyte (IR losses). This model utilizes governing Eq.2.1).

.1.3. Internal losses modelThis model considers comparable conductivity of electrode

nd electrolyte. This model uses governing Eq. (2.5).In order to validate models, it is convenient to calculate and

ompare the resulting charging currents arising during polariza-ion of electrode. The charging current is related to the changen the potential at the front side of the electrode (say χ = 0).ccording to Ohm’s law charging current i(t) is defined as

(t) = − 1

RS

(∂η(χ, t)

∂χ

)χ=0

(2.8)

here RS is the solution resistance. Table 1 presents theoltage distribution and charging currents resulting from one-imensional porous electrode subjected to constant voltageerived by Posey and Morozumi [11] for three different models.

.2. Cyclic voltammetric conditions

Austin and Gagnan developed charging transients for one-imensional porous electrode subjected to cyclic voltammetriconditions (linear potential sweep) for no loss and external lossodel conditions [13]. Table 2 provides the summary of voltage

nd current distribution for these models. The details of theirodels are given below.

.2.1. No losses modelThe potential distribution (Eq. (2.1)) was solved using an

nitial condition and set of boundary conditions. The potentialn the front side of the electrode was set equal to the externallypplied linear increase in voltage and for the backside of thelectrode a Neumann (no flux) condition was applied [13].

(1, t) = kt;

[∂η(χ, t)

∂χ

]χ=0

= 0 (2.9)

he initial condition was taken as zero.

R. Reddy, R.G. Reddy / Electrochimica Acta 53 (2007) 575–583 577

Table 1Voltage distributions and charging currents resulting from different models under constant voltage condition [11]

Model: No losses model, governing Eq.: (2.1)Boundary conditions η(0, t) = ηext

[∂η(χ,t)

∂χ

]χ=1

= 0

Potential distribution η(χ,t)ηext

=

1 − 4π

∞∑p=0

(−1)p

(2p+1) cos((

p + 12

)π(1 − χ)

)exp

{−(p + 1

2

)2π2 t

τ1

}

Transient current i(t) = 2ηext

RS

∞∑p=0

exp

{−(

p + 1

2

)2

π2 t

τ1

}Model: External losses model, governing Eq.: (2.1)

Boundary conditions η(0, t) = ηext − i(t)Re

[∂η(χ,t)

∂χ

]χ=1

= 0

Potential distributionη(χ, t)

ηext= 1 − 2

∑m

cos[(m(1 − χ)] exp{−m2(t/τ1)}m[(1 + λ) sin(m) + λm cos(m)]

, where values of

m are the positive roots of m tan(m) = 1

λ

Transient current i(t) = 2ηext

Re

∑m

sin(m) exp{−m2(t/τ1)}(1 + λ) sin(m) + λm cos(m)

Model: Internal losses model, governing Eq.: (2.5)Boundary conditions

[∂η(χ,t)

∂χ

]χ=0

= −i(t)RS

[∂η(χ,t)

∂χ

]χ=1

= i(t)RM

Potential distributionη(χ, t)

ηext= 1 − 2

∑m

γ cos(mχ) + cos[(m(1 − χ)]

m[sin(m)]exp

{−m2 t

τ

},

where values of m are the positive roots of cos(m) = −γ

s

η

Ti

i

2

eiptc

η

T

η

wwt

χ

i

2

mcTip

Transient current

The analytical solution for voltage distribution for the aboveet of conditions given by [13]:

(χ, t) = kt + kτ1

2(χ2 − 1)

+16kτ1

π3

∞∑n=0

(−1)n

(2n + 1)3 cos

((n + 1

2

)πχ

)

× exp

{−

(n + 1

2

)2

π2 t

τ1

}(2.10)

he charging current is calculated using Eq. (2.8) with χ = 1 ands given by:

(t) = kC − 8kC

π2

∞∑n=0

1

(2n + 1)2 exp

{−

(n + 1

2

)2

π2 t

τ1

}.

(2.11)

.2.2. Liquid ohmic resistance: external lossesIf ohmic resistance (Re) exists between reference and working

lectrode, part of the potential will be lost. The actual driv-ng force for charging the electrode is lower than the appliedotential difference, which in turn effect the potential distribu-ion. This problem was solved using the following boundary

onditions [13]:

(1, t) = kt − i(t)Re;

[∂η(χ, t)

∂χ

]χ=0

= 0 (2.12)

edde

i(t) = 2ηextRS

∑m

exp{

−m2 tτ

}

he resulting voltage distribution is given by:

(χ, t) = kt + kτ1

2h(hχ2 − h − 2)

+2hkτ1

∑m

cos(mχ)

m2[(h2 + m2) + h] cos(m)

× exp

{−m2 t

τ1

}(2.13)

here the values of m are the positive roots of m tan(m) = h,here h is the ratio of electrolyte resistance (RS) to ohmic resis-

ance between reference and working electrode (Re).The charging current is calculated using Eq. (2.8) and keeping

= 1 and is given by:

(t)=kC

[1 −

∑m

2h2

m2[h2 + m2 + h]exp

(−m2 t

τ1

)](2.14)

.2.3. Internal losses modelAustin and Gagnon modeled CV conditions for: (i) no loss

odel conditions and (ii) external loss model conditions forharacterizing double layer capacitance of porous electrodes.hese researchers did not consider the effect of conductiv-

ty of electrode on voltage distribution in one-dimensionalorous electrode. In EDLCs resistivity or conductivity of

lectrode is comparable to that of electrolyte. Hence, it isesirable to incorporate resistivity in simulating CV con-itions for studying EDLC electrodes. In this paper anxpression for voltage distribution under internal loss model

578 R. Reddy, R.G. Reddy / Electrochimica Acta 53 (2007) 575–583

Table 2Voltage distributions and charging currents resulting from different models under cyclic voltammetric condition [13]

Model: No losses model, governing Eq.: (2.1)

Boundary conditions η(1, t) = kt[

∂η(χ,t)∂χ

]χ=0

= 0

Potential distribution η(χ, t) = kt + kτ12 (χ2 − 1) +

16kτ1

π3

∞∑n=0

(−1)n

(2n + 1)3cos

((n + 1

2

)πχ

)exp

{−(

n + 1

2

)2

π2 t

τ1

}

Transient current i(t) = kC − 8kC

π2

∞∑n=0

1

(2n + 1)2exp

{−(

n + 1

2

)2

π2 t

τ1

}Model: External losses model, governing Eq.: (2.1)

Boundary conditions η(1, t) = kt − i(t)Re

[∂η(χ,t)

∂χ

]χ=0

= 0

Potential distribution η(χ, t) = kt + kτ12h

(hχ2 − h − 2) +2hkτ1

∑m

cos(mχ)

m2[(h2 + m2) + h] cos(m)exp

{−m2 t

τ1

}, where values

of m are the positive roots of m tan(m) = h[ ( )]

cc

m

(

((

3i

3

edt

m

((

((

(

(

(

3

pmdgiven shown in Eq. (2.5) and is given by:

∂2η(χ, t)

∂χ2 = τ∂η(χ, t)

∂t(3.1)

Transient current

onditions is developed to simulate cyclic voltammetriconditions.

The objectives of the modeling of simulating cyclic voltam-etry are

i) Development of analytical solution for voltage distributionin the electrode.

i) Derivation of charging current from the voltage distribution.i) Comparison of model derived resistance (ESR) to

that obtained from experiments (constant currentcharge/discharge).

. Voltage distribution for one-dimensional electrode fornternal losses condition

.1. Assumptions

Three-electrode cell consists of working electrode, referencelectrode and counter electrode. Fig. 1 shows the modelingomain consisting of only working electrode and reference elec-rode.

The following assumptions were made while developing theodel.

1) Electrode consists of simplified one-dimensional pores.2) Concentration of species dissolved in the electrolyte flood-

ing the pores of the electrode was constant over the plane xin the electrode.

3) Potential gradients were one dimensional (along x).4) The electrolyte was assumed to be concentrated, so that the

diffusive double layer effects can be neglected.

5) The material properties (σ, κ) were assumed to be constant

through out the porous electrode.6) Capacitance of the carbon material was arising from only

one fourth of the active material. This is a reasonableFr

i(t) = kC 1 −∑

m

2h2

m2[h2+m2+h]exp −m2 t

τ1

assumption as carbon material was mixed with acetyleneblack conductive filler and PTFE binder. Most of the car-bon material surface area is arising from micropores andtheir pore size is being less than 2 nm and this will causeadditional carbon being inaccessible to electrolyte.

7) The open circuit potential was set equal to zero.

.2. Governing equation

Porous electrode theory is used to formulate the governingotential distribution differential equation for the mathematicalodel [14–21]. This is the fundamental relation governing the

ouble layer formation in one-dimensional porous electrode as

ig. 1. Problem domain of three-electrode cell involving working electrode andeference electrode.

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c

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β

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Analytical solution in Eq. (3.13) can be converted to dimen-sionless voltage as given by:

η(χ, t)

kt= 1 − 1

(t/τ)

⎡⎣ ∞∑

p=1

2sin αpχ

α3p

R. Reddy, R.G. Reddy / Electr

here χ is dimensionless distance defined in Eq. (2.4) and τ (s)s time constant in terms of solution phase conductivity (κ) and

atrix phase conductivity (σ).

= acl2(

1

σ+ 1

κ

)(3.2)

here a interfacial surface area per unit volume (cm2/cm3), timeonstant (RC) τ, can be written as C (RM + RS) as given inq. (2.7), where R is equivalent series resistance comprisingf resistance of electrode and electrolyte (RM + RS). Resistancef electrode (RM, ) is given by

M = l

σA(3.3)

here A is cross-sectional area of electrode (cm2). Resistancef electrolyte (RS, ) is given by

S = l

κA(3.4)

is capacitance of electrode and is given by

= scl (3.5)

here s is interior surface area of electrode per unit lengthcm2/cm), c is capacitance per interfacial area (F/cm2) and ls thickness of electrode (cm).

.3. Initial and boundary conditions

.3.1. Initial conditionAccording to assumption 7

(χ, 0) = 0. (3.6)

.3.2. Boundary conditionsThe governing Eq. (3.1) is subjected to a constraint of lin-

arly increasing voltage η(0,t) = kt and the following boundaryonditions (Eqs. (3.7) and (3.8)). These boundary condition aredopted from Posey and Morozumi [11] and applied for linearlyncreasing voltage condition

At χ = 0(∂η(χ, t)

∂χ

)χ=0

= −I(t)RS (3.7)

At χ = 1(∂η(χ, t)

∂χ

)χ=1

= I(t)RM (3.8)

here I(t) is charging current (A).

.4. Analytical solution for potential distribution

Analytical solution for potential distribution η(χ,t) as a func-ion of dimensionless less distance (χ) and time (t) is derivedor governing Eq. (3.1) employing initial condition (Eq. (3.6))nd boundary conditions (Eqs. (3.7) and (3.8)). Since voltage

ica Acta 53 (2007) 575–583 579

hange with time (cyclic voltammetric condition), the analyticalolution is obtained using Duhamel‘s theorem utilizing the pro-edure given by Carslaw and Jaeger [22]. The analytical solutions given by:

(χ, t) = kt − kτ

⎡⎣ ∞∑

p=1

2sin αpχ

α3p

−∞∑

p=1

2sin αpχ

α3p

exp(−α2

p

t

τ

)⎤⎦ (3.9)

here αp are the positive roots of

os(αpχ) = −β (3.10)

here β is ratio of resistance of matrix phase to solution phaser conductivity ratio of solution to matrix phase.

= RM

RS= κ

σ(3.11)

he validity of solution (Eq. (3.9)) can be checked by substitut-ng initial and boundary conditions. Another way of checkinghe validity of Eq. (3.9) is by keeping β as zero then it becomes aase where conductivity matrix is higher than that of electrolytecase discussed in Section 2.2.1 (No losses model). By keeping= zero, τ becomes τ1, and Eq. (3.10) becomes:

p =(

n + 1

2

)π (3.12)

ombining Eqs. (3.9) and (3.12) leads to

(χ, t) = kt − kτ1

(χ2

2− χ

)+ kτ1

∞∑n=0

2 sin(n + (1/2))πχ

((n + (1/2))πχ)3

× exp

{−

(n + 1

2

)2

π2 t

τ1

}(3.13)

ustin and Gagnon [13] derived the analytical solution presentedn 3.13 for identical conditions (no loss model, Eq. (2.10)). How-ver, they derived the voltage distribution for boundary conditioniven in Eq. (2.9) which are opposite boundary conditions inerms of starting and ending of electrode to that derived in Eq.3.9).

−∞∑

p=1

2sin αpχ

α3p

exp(−α2

p

t

τ

)⎤⎦ . (3.14)

580 R. Reddy, R.G. Reddy / Electrochim

Fpc

4

4

s2vobitl

4

cecc(

g

I

T

[

Fv

I

H(c(Cc

I

wmvupc

4e

4c

tdasoarkatbp

(

ig. 2. Influence of the electrode phase resistance at β (a) 0.99 and (b) 0.1 on theotential distribution as a function of dimensionless distance at different timeonstants inside the bed.

. Result and discussions

.1. Effect of electrode phase resistance

Using Eq. (3.14), dimensionless voltage versus the dimen-ionless distance at different times (t = xτ, x = 0.005, 0.05, 0.25,) is plotted in Fig. 2 for two different values of β. At higheralues of β, it can be seen from the upper graph that the backsidef the bed is charged faster than the interior of the bed. This isecause two resistances prevent the transport of charge to thenterior of the pores. It can also be seen from figure that if t ∼ 2τ

he potential inside the bed has become uniform through out theength of the electrode.

.2. Validating model: charging current

In order to validate the model, it is most convenient to cal-ulate the resulting charging currents during polarization of the

lectrode. The charging current I(t) (amp, A) is related to thehange in potential at the front of the electrode. The chargingurrent is obtained by differentiating the voltage distributionEq. (3.14)) according to Eq. (2.7) and keeping χ as 0 and is

tccf

ica Acta 53 (2007) 575–583

iven by:

(t) = kC[1 + β]∞∑

p=1

2

α2p

− kC[1 + β]∞∑

p=1

2

α2p

exp(−α2

p

t

τ

)(3.15)

he first term in Eq. (3.15) is kC given by:

1 + β]∞∑

p=1

2

α2p

= 1 (3.16)

rom Eqs. (3.15) and (3.16), the charging current can be con-erted to

(t) = kC − kC[1 + β]∞∑

p=1

2

α2p

exp(−α2

p

t

τ

)(3.17)

ere αp are the positive roots of Eq. (3.10). When β is 0, Eq.3.17) becomes Eq. (2.11) derived by Austin et al. [13] for no lossondition. R is the total resistance, equivalent series resistanceohm) and C is the capacitance (farads). The product of R and

is time constant τ (s). Since product of kC is equivalent tourrent, Eq. (3.17) can be recast to

(t) = Ic

⎡⎣1 − [1 + β]

∞∑p=1

2

α2p

exp(−α2

p

t

RC

)⎤⎦ (3.18)

here Ic is constant double layer current resulting cyclic voltam-etry (amp, A). Eq. (3.18) can be used to simulate cyclic

oltammetric (CV) curves by fitting I(t) versus t of Eq. (3.18) bysing best fit of R with experimental CV curves. CV curves arelotted as current as a function of voltage, however the voltagean be converted to increment of time (V = kt).

.3. Comparison of model charging currents withxperimental values

.3.1. Electrochemical characterization ofatechol-formaldehyde gel derived carbons

Catechol-formaldehyde (CF) aqueous organic gels were syn-hesized by polycondensation of catechol with formaldehyde atifferent pH conditions. After gelation, the samples were driednd pyrolyzed in order to obtain carbon samples. The synthe-is variables are the catechol/catalyst molar ratio (C/C), the pHf the precursor aqueous solution. Sodium carbonate was useds the catalyst, and the C/C (catechol/sodium carbonate) moleatio was fixed at 50:1. pH of the precursor aqueous solution wasept 3.0, 6.0 and 7.5. These gels were subsequently heat treatedt 750 ◦C to prepare carbon materials. The striking feature ofhese carbon materials is their highly porous nature as depictedy SEM micrograph of carbon material derived from CF gelrepared at pH 6 (Fig. 3).

Electrochemical experiments including cyclic voltammetrythree-electrode method) and galvanostatic charge–discharge

echnique were carried out on carbon materials processed fromatechol-formaldehyde gels that were prepared at different pHonditions of 3, 6 and 7.5. The details of preparation of catechol-ormaldehyde gels and their processing to prepare carbon

R. Reddy, R.G. Reddy / Electrochimica Acta 53 (2007) 575–583 581

sed a

mpcc7tr

Fprc

03d

Fig. 3. (a and b) Carbon materials thermally proces

aterials and the electrochemical characterization (CV) wereresented in our earlier publication [10]. Figs. 4(a), 5(a) and 6(a)ompare the anodic portion of CV curve of carbon obtained

atechol-formaldehyde prepared at pH conditions of 3, 6 and.5, respectively, in 30 wt.% H2SO4 at 10 mV/s scan rate withhe charging currents obtained using Eq. (3.18) for the best fitesistance (R) values. Conductivity (κ) of 30 wt.% H2SO4 is

ig. 4. (a) Cyclic voltammetric curve of carbon prepared from CF gel pre-ared at pH 3 in 30 wt.% of H2SO4 at 10 mV/s scan rate (—), charging currentesulting according to Eq. (3.18) for R value of 4.5 (- - -). (b) Galvanostaticharge–discharge curve recorded at 10 mA current.

cecc

Fprc

t 750 ◦C from catechol-formaldehyde gels at pH 6.

.81 S cm−1 and conductivity of carbon electrode (σ) is taken

.3 S cm−1 (based on conductivity of acetylene black). β con-uctivity ratio of electrolyte to electrode is 0.25.

Table 3 presents the parameters used in the calculatingharging currents arising from carbon material subjected lin-

ar increase in the voltage condition. Constant double layerurrent (Ic) was taken at voltage of 0.1 V of experimental CVurves. Capacitance (C) value was taken from experiments and

ig. 5. (a) Cyclic voltammetric curve of carbon prepared from CF gel pre-ared at pH 6 in 30 wt.% of H2SO4 at 10 mV/s scan rate (—), charging currentesulting according to Eq. (3.18) for R value of 3.2 (- - -). (b) Galvanostaticharge–discharge curve recorded at 10 mA current.

582 R. Reddy, R.G. Reddy / Electrochimica Acta 53 (2007) 575–583

Table 3Parameters used in calculating the charging currents for carbon derived from CF gels prepared at pH conditions of 3, 6 and 7.5 in 30 wt.% H2SO4 at 10 mV/s scanrate and charging currents are obtained using Eq. (3.18) for the best fit resistance (R) values

pH Ic (mA) Capacitance (F) Corrected capacitance (F) Resistance from model ( ) ESR obtained from galvanostaticcharge–discharge curves ( )

3 5.76 0.91 3.64 4.5 467

E at 10 m

ttcfvwvedtog

Farc

co3sati

6.86 1.09 4.36.5 6.65 1.01 4.04

SR values are obtained from galvanostatic charge–discharge curves recorded

his capacitance is corrected according to assumption 6. For fit-ing of charging currents using Eq. (3.18) to the experimental CVurves, corrected capacitance, constant current values are takenrom the experiments and Eq. (3.18) is solved for different Ralues to obtain best fit. These best fit ESR values are comparedith that obtained from galvanic charge–discharge curves. Gal-anic charge–discharge tests were conducted in 30 wt.% H2SO4lectrolyte at 10 mA current and ESR is obtained from voltage

rop at the beginning of the discharge curve. It can be seen fromhe table that resistance values derived by fitting the anodic partf CV curve by Eq. (3.18) for carbon materials derived from CFels in 30 wt.% H2SO4 electrolyte are in between 3 and 5 .

ig. 6. (a) Cyclic voltammetric curve of carbon prepared from CF gel preparedt pH 7.5 in 30 wt.% of H2SO4 at 10 mV/s scan rate (—), charging currentesulting according to Eq. (3.18) for R value of 3.8 (- - -). (b) Galvanostaticharge–discharge curve recorded at 10 mA current.

btpYfff(dcim(p

5

dtctt(epibtt

lfrcalcH

3.3 3.13.8 3.2

A current (β = 0.25). Capacitance was corrected according to assumption 6.

Figs. 4(b), 5(b) and 6(b) display galvanostaticharge–discharge curves recorded at 10 mA of carbonbtained catechol-formaldehyde prepared at pH conditions of, 6 and 7.5, respectively, and the corresponding equivalenteries resistance values of these carbons are presented in Table 3nd they are in the range of 3–4 . A sudden potential drop athe very beginning of the constant current discharge observedn figures is usually observed for carbons, and this drop haseen designated as the IR drop. This potential drop is dueo the resistance of electrolyte due to ions migration in theores of carbon and also results from resistance of electrode.oon et al. observed equivalent series resistance of 4.9

or carbon material using galvanostatic discharge techniqueor material in 2 M H2SO4 electrolyte [23]. As can be seenrom the table resistance values obtained from by fitting Eq.3.18) and the resistance values obtained from constant currentischarge curves matching very accurately. The model derivedharging current can accurately predict the ESR values presentn any electrode and electrolyte system. The advantage of this

odel is its prediction of ESR values for varying range of β,conductivity ratio of electrolyte to electrode) values whichrevious models lacked.

. Conclusions

This work deals with double layer charging of one-imensional porous electrode and electrolyte system subjectedo linear increase in voltage perturbation, by considering theonductivity of electrode and electrolyte. An analytical solu-ion for voltage distribution through out the electrode at variousimes equivalent to fractions of time constants is derived in Eq.3.14). In short periods of time, the resistances of electrode andlectrolyte prevent the transport of charge to the interior of theores, hence building of uniform external set voltage. When times equivalent to two times of time constant, externally set voltageecomes uniform through out the electrode. This indicates thathe increment in the scan rate should be at least equal to twoimes of time constant.

Charging currents arising due to the formation of doubleayer at the interface of electrode and electrolyte is derivedrom voltage distribution as shown in Eq. (3.18). Carbon mate-ials were prepared by high temperature decomposition ofatechol-formaldehyde gels prepared at pH conditions of 3, 6

nd 7.5 and these materials showed ideal electrochemical doubleayer capacitor characteristics as displayed in cyclic voltametricurves and constant current charge/discharge curves in 30 wt.%2SO4 electrolyte. The charging currents derived from volt-

ochim

acCdcvc

efisd

A

b

A

aAcCh

IkLRRRs

tVx

Gβ

χ

φ

φ

�

κ

σ

η

τ

τ

R

[[[[[[[

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[[[[21] V.R. Subramanian, S. Devan, R.E. White, J. Power Sources 135 (2004)

361.

R. Reddy, R.G. Reddy / Electr

ge distribution Eq. (3.18) were fitted with experimental CVurves and equivalent series resistances of carbons derived forF gels prepared at pH conditions of 3, 6 and 7.5. The modelerived resistance values were compared with that obtained fromonstant current charge/discharge curves. Model derived ESRalues matched closely with that obtained from galvanostaticonstant current charge–discharge method.

Cyclic voltammetry is widely used in characterizinglectrode–electrolyte systems. The present model can be usedor prediction of ESR values for varying range of β, (conductiv-ty ratio of electrolyte to electrode) of an electrode–electrolyteystem involved in double layer charging by fitting the modelerived currents with experimental CV curves.

cknowledgement

The authors are thankful for the financial support providedy National Science Foundation grant no. ECS-0099853.

ppendix A. Nomenclature

interfacial surface area per unit volume (cm2/cm3)cross-sectional area (cm2)capacitance per unit area (F/cm2)capacitance (F)ratio of electrolyte resistance (RS) to ohmic resistancebetween reference and working electrode (Re)

(t) charging current (A)scan rate (mV/s)thickness (cm)

e Ohmic resistance ( )M matrix phase resistance ( )S electrolytic phase resistance ( )

interior surface area of electrode per unit length(cm2/cm)time (s)voltage (V)position coordinate

reek lettersconductivity ratio of electrolytic phase to matrix phaseconductivity

[

[

ica Acta 53 (2007) 575–583 583

dimensionless distanceM potential of matrix phase (V)S potential of electrolytic phase (V)φ potential difference between the matrix phase and elec-

trolytic phase (V)electrolytic phase conductivity (S cm−1)matrix phase conductivity (S cm−1)potential difference between the matrix phase and elec-trolytic phase (V)time constant, a product of total resistance, RM + RSand capacitance, C (s)

1 time constant, a product of electrolyte resistance, RSand capacitance, C (s)

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