Charging and Discharging Electrochemical Supercapacitors in the Presence of Both Parallel Leakage Process and Electrochemical De

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Electrochimica Acta 90 (2013) 542–549

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Electrochimica Acta

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Charging and discharging electrochemical supercapacitors in the presence of

both parallel leakage process and electrochemical decomposition of solvent

a a,∗ a a a b

Shuai Ban , Jiujun Zhang , Lei Zhang , Ken Tsay , Datong Song , Xinfu Zou

National Research Council of Canada, Vancouver, BC, Canada V6T 1W5

Department of Applied Mathematics, University of Western Ontario, London, ON, Canada N6A 5B7

a r t i c l e i n f o a b s t r a c t

Article history: Simple models for electrochemical supercapacitors are developed to describe the charge–discharge

Received 5 November 2012

behaviors in the presence of both voltage-independent parallel leakage process and electrochemical

Received in revised form

decomposition of solvent. The models are validated by experimental data collected using a symmetric

14 December 2012

two-electrode test cell with carbon powder as the electrode layer material and stainless steel as the

Accepted 15 December 2012

Keywords:

Electrochemical supercapacitor

Charge and discharge

Parallel leakage Self-discharging

Solvent decomposition

1. Introduction and ionic liquid solutions which have wider electrode potential

windows [4,7–9]. From the energy density equation mentioned

Electrochemical supercapacitors (ESs) are considered important above, it is obvious that increasing cell voltage to enhance energy

energy efﬁciency devices for rapid energy storage and deliv- density is more effective than increasing the speciﬁc capacitance

ery. Among the advantages of ESs are high power density, long because the voltage is squared in the formula. However, if the cell

lifecycle, high efﬁciency, wide range of operating temperatures, voltage is too high, the electrochemical decomposition of solvent

environmental friendliness, and safety. ESs also serves as a bridg- may occur due to the limited electrode material stability and/or

ing function for the power/energy gap of traditional dielectric solvent’s thermodynamic decomposition windows. This solvent

capacitors. These characteristics have made ESs very competitive electrochemical decomposition could produce gaseous products,

for applications in electric hybrid vehicles, digital communication leading to the pressure build-up inside the cell, causing safety

devices such as mobile phones, digital cameras, electrical tools, concerns, and self-discharge [10–17]. Another challenge of super-

pulse laser technique, uninterruptible power supplies, and storage capacitors is the parallel leakages, causing a fast self-discharge,

of the energy generated by solar cells [1–3]. Unfortunately, there reducing the shelf-life. The main contributors to the parallel leak-

are still some challenges, such as low energy density, high cost, and ages of the supercapacitors can be summarized into four processes

high self-discharge rate, which have limited wider applications of [7]: (1) Faradaic reaction of electrolyte impurities; (2) parasitic

ESs [3,4]. redox reactions involving impurities (oxygen groups and metals);

In order to increase energy density, according to the energy den- (3) non-uniformity of charge acceptance along the surface of elec-

sity expression E = 1/2CV , where E is the ES’s energy density, C is trode material pores; and (4) possible short-circuit of the anode and

the speciﬁc capacitance, and V is the cell voltage, respectively, two cathode from improperly sealed bipolar electrodes. Due to these

major approaches have been taken. One is to increase the speciﬁc parallel leakages, the shelf-life could be signiﬁcantly reduced when

capacitance (C) of the electrode material [3–6], and the other is to compared to those of electrochemical batteries [7].

increase the cell voltage (V) using solvents such as non-aqueous To better understand the effects of parallel leakage process

and/or electrochemical decomposition of solvent on supercapa-

citors’ performance, some experimental and theoretical studies

∗ have been carried out to investigate the charge and discharge

Corresponding author at: 4250 Wesbrook Mall, Vancouver, B.C. V6T 1W5,

behaviors [18–23]. Since the parallel leakage process and the sol-

Canada. Tel.: +1 604 221 3087; fax: +1 604 221 3001.

E-mail address: [email protected] (J. Zhang). vent decomposition are mainly related to the electrochemical

S. Ban et al. / Electrochimica Acta 90 (2013) 542–549 543

Nomenclature o

Vn standard electrode potentials for negative electrode

Symbol, Meaning reaction (V)

electron transfer coefﬁcient in the rate determining

n Vp positive electrode potential of the solvent electro-

step of negative electrode reaction

chemical decomposition (V)

˛ electron transfer coefﬁcient in the rate determining

p Vp standard electrode potentials for positive electrode

step of positive electrode reaction

reaction (V)

b Tafel slope of the negative electrode reaction

n Vsc supercapacitor voltage (V)

b Tafel slope of the positive electrode reaction max

p Vsc designed charging voltage of the supercapacitor (V)

−2 o

C capacitance (F cm ) V

sc maximum supercapacitor voltage (V)

−2

Cdl double-layer capacitance (F cm )

−1

Cm speciﬁc capacitance (F g )

−3

Cs mole concentration of the solvent (mol cm )

processes occurring inside the supercapacitor cells, describing the

o o o o o

V = Vp − Vn − bp ln ip − bn ln (in)

charge and discharge behaviors by electrochemical parameters,

e− electron

−2 such as equivalent series resistance, capacitance, and several elec-

E energy density (Wh cm )

trode kinetic constants, seems more appropriate, in particular for

(Em)max maximum energy density of the supercapacitor

−1 electrochemical decomposition of the solvent.

(Wh kg )

−1 In this paper, for fundamental understanding of supercapacitor

F Faraday’s constant (96,487 C mol )

charging and discharging behaviors, through experiment validation

iF current density induced by the solvent electrochem-

−2 we present some simple mathematical models incorporating the

ical decomposition (A cm )

voltage-independent parallel leakage process and electrochemical

in current density of the negative electrode reaction

−2 decomposition to describe the supercapacitor charge and dis-

(A cm )

charge behaviors. Using these models, the experiment data could

ip current density of the positive electrode reaction

−2 be simulated to obtain the desired values of parameters such as

(A cm )

speciﬁc capacitance and equivalent series resistance from which

isc charging or discharging current density of the

−2 both the energy and power density can be calculated. The sim-

supercapacitor (A cm )

ulated parameter values related to both parallel leakages and

Icell constant current density for charging or discharging

−2 solvent decomposition could be used to predict supercapacitor’s

the supercapacitor (A cm )

self-discharge and shelf-life, performance/efﬁciency losses, as well

kn reaction constant of the negative electrode reaction

−1 as the limiting workable cell voltage of the supercapacitors. There-

(cm s )

fore, the proposed models are useful in evaluating and diagnosing

kp reaction constant of the positive electrode reaction

−1 supercapacitor cells. We hope these simple models could be used as

(cm s )

a tool for understanding the charging–discharging behavior of the

m mass of the electrode material (g)

supercapacitors. Furthermore, we believe that these models could

n˛n electron transfer number in the rate determining

also be useful in obtaining the essential parameters of the supercap-

step of negative electrode reaction

acitors such as equivalent series resistance and capacitance based

n˛p electron transfer number in the rate determining

on the recorded curves of charging–discharging.

step of positive electrode reaction

nn overall electron transfer number in the negative

2. Experimental

electrode reaction

np overall electron transfer number in the positive

2.1. Electrode layer preparation

electrode reaction

pn product of the negative electrode reaction

For electrode layer materials, Carbon black BP2000 (Cabot Cor-

pp product of the positive electrode reaction

poration), conducting carbon Super C45 (from TIMCAL), and 60 wt%

(Pm)max maximum power density of the supercapacitor

−1 Polytetraﬂuoroethylene (PTFE)-water dispersion (Aldrich) were

(W kg )

−1 −1 used as received without further modiﬁcation.

R universal gas constant (8.314 J K mol )

For electrode layer preparation, the procedure has been

Resr equivalent series resistance of the supercapacitor

2 described in our previous publication [3], and is repeated as below.

( cm )

2 Carbon and conducting carbon powders were ﬁrst mixed with a

Rlk equivalent leakage resistance ( cm )

Vortex Mixer (Thermo Scientiﬁc) for 30 min to form a uniformly-

S reactant of the solvent

mixed powder, which was then transferred into a beaker containing

t time (s)

both PTFE binder and ethanol solution under constant stirring to

tfd time needed for a supercapacitor being fully dis-

form a suspension. After the solvent was evaporated at high tem-

charged (s)

perature, the paste formed was collected on a glass slide. This paste

T temperature (K)

was manipulated by repeatedly folding and pressing using a spatula

Ts time needed when the supercapacitor voltage goes

until a sufﬁcient mechanical strength was achieved. Then this paste

down to 95% of its maximum voltage (s)

was rolled to a thin electrode sheet with the required thickness

V voltage (V)

using a MTI rolling press. Finally, this electrode sheet was placed

Vcell voltage of the supercapacitor cell (V) ◦

V max into a vacuum oven at 90 C under active vacuum for at least 12 h.

maximum voltage of the supercapacitor cell (V)

cell ×

The formed dry electrode sheet was cut into two 2 2 square cen-

VF voltage induced by the solvent electrochemical 2

timeters (cm ) for electrode layers, which were then sandwiched

decomposition (V)

␮

with a 30 m thick porous PTFE separator (W. L. Gore & Associates,

Vn negative electrode potential of the solvent electro-

Inc.) in the middle, forming an electrode separator assembly (ESA).

chemical decomposition (V)

The assembly was then impregnated with an aqueous electrolyte

◦

containing 0.5 M Na2SO4 inside a vacuum oven at 60 C for at least

544 S. Ban et al. / Electrochimica Acta 90 (2013) 542–549

one hour. This electrolyte-impregnated ESA was then assembled

into the two-electrode test cell.

2.2. Two-electrode test cell measurements

In the two-electrode test cell [24], the active electrode sur-

face for both positive and negative electrodes was 4.0 cm . The

electrolyte was 0.5 M Na2SO4 aqueous solution, and the cur-

rent collectors were made of stainless steel. The additional

effect of electrolyte concentration on supercapacitor’s perfor-

mance was carefully studied in pour previous publication [24]. For

electrochemical measurements, both cyclic voltammograms and

galvanic charge–discharge curves were recorded using a Solartron

1287 potentiostat. For cell internal resistance measurements, AC

impedance spectra at a frequency range of 0.1 Hz–10 kHz were

collected using a Solartron FRA analyzer. For a more detailed

description about the measurement and calculation procedures,

please see our previous publication [24].

For recording charging/discharging curves of supercapacitors,

there are mainly two options, one is to control a constant

charging–discharging current, then record the change of cell volt-

age with time, and the other is to control a constant cell voltage,

then record the change of cell current with time. The most popular

option is the former. In this paper, we used the former option to

obtain the charging–discharging curves.

All measurements were made at room temperature and ambient

pressure.

3. Results and discussion

In general, for charging and discharging a supercapacitor, there

are two major options. One is the charging or discharging at a con-

Fig. 1. Proposed equivalent circuits of a double-layer supercapacitor at a constant

stant cell voltage to record the cell current change with time, and current charging (a) and discharging (b) in the absence of both parallel leakage

the other is charging or discharging at a constant current to record process and electrochemical decomposition of the solvent. K: electric switcher; Resr:

equivalent series resistance; Icell: constant current for charging or discharging the

that cell voltage change with time. In this paper, we will only focus

supercapacitor; Vcell: supercapacitor cell voltage; Cdl: double-layer capacitance; Vsc:

on the charging and discharging at a constant current.

voltage across the Cdl; and isc: current used to charge or discharge Cdl; respectively.

3.1. Charging at a constant cell current in the absence of parallel

Eq. (3) suggests that the cell voltage is linearly proportional to the

leakage process and electrochemical decomposition of solvent

charge time. If the supercapacitor is charged to a designed cell volt-

max

age of , a discharge process at a constant current (Icell) can be

Fig. 1(a) depicts a proposed equivalent circuit for a double- cell

started immediately, as shown in Fig. 1(b). Note that at this desired

layer supercapacitor with a constant current in the absence of both

V max

cell voltage (or maximum cell voltage) of , the voltage across

cell

voltage-independent parallel leakage process and electrochemical o

the double layer capacitance (Cdl), sc, should be given by:

decomposition of the solvent, where K is the electric switcher, Resr

is the equivalent series resistance, Icell is the constant current for o max

V = V − I R

sc cell cell esr (4)

charging or discharging the supercapacitor; Vcell is the superca-

pacitor cell voltage; Cdl is the double-layer capacitance; Vsc is the in the case of discharging, this sc will be the load to discharge

voltage across the Cdl; and isc is the current used to charge or dis- the supercapacitor. The cell voltage (Vcell) can be expressed by the

charge, respectively. Note that in the case shown in Fig. 1(a), the following equation

current used to charge or discharge the double layer capacitance

o Icell

V = V − I R − t

(isc) is equal to the charging current (Icell), which has a constant cell sc cell esr (discharge process) (5) Cdl

value:

As an example, Fig. 2 shows the charge–discharge curves using a

Icell = isc (1)

two-electrode test cell in which two electrodes are identical (sym-

assuming that, before the charging starts, the supercapacitor is at metric cell). Based on both Eqs. (3) and (5), several parameters

zero-charge state, that is, the voltage across the supercapacitor (Vsc) such as the double-layer capacitance (Cdl), the voltage across the

is equal to zero. When the switch is turned on at t = 0 (t is the charg- double layer capacitance at the end of charging ( sc), as well as

ing time), the charge process is then started and the voltage across equivalent series resistance (Resr) can be simulated, respectively,

the supercapacitor can be expressed as: using both the charge and discharge curves. In the case shown in

t=t t=t Fig. 2, parameters obtained using the simulation are: capacitance

1 1 Icell −2

V C = 0.127 F cm or 0.509 F for the entire cell with electrode area

sc = isc dt = Icell dt = t (2) dl

Cdl Cdl Cdl 2 −1

t=0 t=0 of 4 cm , or the speciﬁc capacitance (Cm) of 34 F g for the carbon

loading (m) of 0.015 g for each electrode; the voltage across the dou-

therefore, the supercapacitor cell voltage can be expressed as: o

ble layer capacitance Vsc = 0.991 V; as well as the equivalent series

Icell 2

resistance Resr = 2.36 cm or 0.589 for the entire cell, respec-

Vcell = IcellResr + Vsc = IcellResr + t (charge process) (3)

Cdl

tively. Note that the speciﬁc capacitance obtained here was also

S. Ban et al. / Electrochimica Acta 90 (2013) 542–549 545

Fig. 2. Charge–discharge curves recorded using a symmetric supercapacitor cell

with a geometric area of 4.0 cm for each electrode, in 0.5 M Na2SO4 aqueous solu-

tion at ambient conditions. Stainless steel sheets as both the positive and negative

−2

current collectors. Charge and discharge current density: 0.025 A cm , and total BP

−2

2000 carbon loading of electrode layers: 3.8 mg cm . Note that the Resr showing

in the ﬁgure should be the sum equivalent series resistances of both positive and

negative electrodes [24]. Note that the curve corresponds to the simulation and the

points to experimental data.

conﬁrmed by using cyclic voltammetry measurements. Using these

values, the maximum energy and power densities ((Em)max and

(Pm)max, respectively) can be calculated according to the equations

o 2 Fig. 3. Proposed equivalent circuits of a double-layer supercapacitor by constant

presented in our previous publication [24]: (Em) = 1/2Cm(Vsc) ,

max current charging (a) and discharging (b) in the presence of parallel leakage process.

o 2

P = / m V /R and ( m)max 1 4 (( sc) esr ). The obtained (Em)max and (Pm)max K: electric switch; Rlk: equivalent parallel leakage resistance; Resr: equivalent series

− −

1 4 1 resistance; Icell: constant current for charging or discharging the supercapacitor;

are 4.64 Wh kg , and 2.78 × 10 Wh kg , respectively.

Vcell: supercapacitor cell voltage; Cdl: double-layer capacitance; Vsc: voltage across

the Cdl; and isc: current used to charge or discharge Cdl; respectively.

3.2. Charging at a constant cell current in the presence of a

voltage-independent parallel leakage process V o

as sc which is equal to IcellRlk as t tends to inﬁnity. If there is no

→ ∞

leakage current, Rlk , for small t, Eq. (8) can be approximated by

Fig. 3 shows the scheme of a supercapacitor charging at a con-

stant current density. Compared to Fig. 1, there is an additional V

= I Resr + I (charge process) (9)

cell cell cell C

equivalent leakage resistance (Rlk) which represents a voltage- dl

independent parallel leakage process. Note that for simplicity, we

∞

note that if charging an ideal supercapacitor (Resr = 0, and Rlk = )

treat all voltage-insensitive leakage processes as an equivalent par-

using a constant current, the cell voltage will become:

allel leakage (R ), suggesting that the magnitude of this R would

lk lk t

V = I not signiﬁcantly change with the charge or discharge process. How- cell cell (10)

Cdl

ever, we do need to keep it in mind that this Rlk may change with

cell voltage in practice. In a later section, we will give a detailed discussion about the

Again, we assume that before the charge process starts, the effect of solvent electrochemical decomposition on the superca-

supercapacitor is at zero-charge state, that is, the voltage across pacitor charge and discharge behaviors.

the supercapacitor is equal to zero. When the switch shown in After the supercapacitor cell is fully charged with a maximum

≥

Fig. 3(a) is closed at t = 0, the charge process is started. When t 0, voltage of ( sc), a discharge process using a constant current (Icell)

the supercapacitor charging voltage can be expressed by will be started for which the equivalent circuit can be illustrated in

Fig. 3(b). The integration equation used to describe this situation is:

dVsc Vsc

C + − I = 0 (6) dl dt R cell t=t

lk 1 o

i dt + i R − V + I R =

C sc sc lk ( sc cell lk) 0 (11)

solving Eq. (6) with the initial condition t = 0, Vsc = 0, the following t=0

solution can be obtained:

solving this equation, we can get: t

V = I R − − sc cell lk 1 exp (charge process) (7) (Vsc + IcellRlk) t RlkCdl isc = exp − (12)

Rlk RlkCdl

accordingly the cell voltage can be expressed as:

and the voltage across the double-layer capacitance can be t

V = I expressed as:

R + I R − − cell cell esr cell lk 1 exp R C (charge process) lk dl t=t (8) V o 1 sc = Vsc − iscdt Cdl

t=0

∞ Eq. (8) shows that at t = 0, Vcell = IcellResr, and when t = ,

o o t

Vcell = IcellResr+IcellRlk. In this case, the voltage across the superca-

= V − (V + I R ) 1 − exp − (13)

sc sc cell lk R C

pacitor (Vsc) increasingly approaches its maximum value, deﬁned lk dl

546 S. Ban et al. / Electrochimica Acta 90 (2013) 542–549

Fig. 4. Theoretically calculated charge–discharge curves, Rlk is changed from 100 to

2 2 T 2 2

= .

10,000 cm with ﬁxed Resr = 0.5 .cm , dl 0 4 F/cm , and Icell = 0.05 A/cm .

the cell voltage (Vcell) can be expressed by:

o o

Vcell = −IcellResr + Vsc − (Vsc + IcellRlk)

× 1 − exp − (discharge process) (14)

RlkCdl

→ ∞

if there is no parallel leakage process (Rlk ), for limited t, Eq.

(14) can be approximated by

o t Fig. 5. Proposed equivalent circuits for a double-layer supercapacitor at a constant

Vcell = −IcellResr + V − Icell (15)

sc current charging (a) and discharging (b) in the presence of electrochemical decom-

Cdl

position of the solvent. K: electric switcher; Resr: equivalent series resistance; Icell:

constant current for charging or discharging; Vcell: supercapacitor cell voltage; iF:

the supercapacitor full discharge time, denoted by tfd, can be

current of solvent electrochemical decomposition; Cdl: double-layer capacitance;

derived by setting V = 0 and solving this equation for t, giving

sc isc: current used to charge or discharge the double-layer capacitance; VF: electrode

potential of the solvent electrochemical decomposition; and Vsc(=VF): voltage across

I R

cell lk the double-layer capacitance, respectively. tfd = −RlkCdl ln o (16)

Vsc + IcellRlk

Assuming that the solvent (S) decomposition reactions at the

As an example, Fig. 4 shows the theoretically calculated charge

positive and negative electrodes are expressed by the following

and discharge curves according to Eqs. (8) and (14). It can be seen

Reactions (I) and (II), respectively:

that when the parallel leakage resistance becomes smaller (larger

2 −

leakage current) from 10,000 to 100 cm , both the cell voltage → npe + Pp (I)

and the maximum supercapacitor voltage become smaller, suggest-

S −

+ nne → Pn (II)

ing that the performance of the supercapacitor cell is reduced.

Furthermore, due to the parallel leakage process, self-discharge

here np is the overall electron number of S oxidation to produce

will become signiﬁcant. For example, if a fully charged supercapac-

P , and n is the overall electron number of S reduction to produce

V o p n

itor with a voltage of sc is put on the shelf (Icell = 0 in this case),

Pn, respectively. Assuming that both reactions are irreversible, the

according to Eq. (13), the voltage will be gradually reduced due to

current density for Reaction (I) (ip) and that for Reaction (II) (in) can

the self-discharge through Rlk. When the Vsc goes down to 95% of

be expressed by Eqs. (18) and (19), respectively:

V max

sc , the time (Ts) for Vsc to reach this level is given by:

o o

˛pn˛pF(Vp − V ) Vp − V

i p o p

p = npFkpCS exp = ip exp (18)

T = . R C s 2 996 lk dl (17) RT bp

2 −2 o o

˛ n V − V in the situation of even R = 10,000 cm , and C = 0.4 F cm , n ˛nF V − Vn n

lk lk i = n ( n ) o n

n nFknCS exp = in exp (19)

Ts = 3.33 h. Therefore, self-discharge is one of the major challenges RT bn

for supercapacitors. o o

i = n

Here p pFkpCS and in = nnFknCS can be deﬁned as the

standard current densities, p = RT/˛pn˛pF and bn = Rt/˛nn˛nF can

3.3. Charging at a constant cell current in the presence of be deﬁned as the Tafel slopes for Reactions (I) and (II); kp and

˛ ˛

electrochemical decomposition of solvent kn are the reaction constants, p and n are the charge transfer

coefﬁcients, n˛p and n˛n are the electron transfer numbers in the

V o o

Fig. 5(a) shows the proposed equivalent circuit for a double- reaction determining steps, p and n are the standard electrode

layer supercapacitor in the case of constant current charging in the potentials for Reactions (I) and (II); Vp and Vn are the electrode

presence of electrochemical decomposition of the solvent. Com- potential of the positive and negative electrodes, respectively. CS is

pared to Fig. 1, there is an additional parallel electrochemical the mole concentration; F is Faraday’s constant; R is the universal

process included with a potential dependent current (iF) and volt- gas constant; and T is the temperature; respectively.

age (VF) induced by the solvent electrochemical decomposition. Note that because both Reactions (I) and (II) represent the

From the equivalent circuits in Fig. 5, it can be seen that Vsc = VF. solvent decomposition, the solvent concentration should not

S. Ban et al. / Electrochimica Acta 90 (2013) 542–549 547

be changed with its consumption by electrochemical reactions. Using Eq. (28), the maximum voltage (Vsc) across the double-

Therefore, there is no mass transfer limitation involved in the elec- layer capacitance can also be derived by setting t → ∞ (the moment

trochemical reactions. when the supercapacitor is fully charged):

From Eqs. (18) and (19), their electrode potentials can be derived o

o V

as Eqs. (20) and (21), respectively: Vsc = (bp + bn) ln Icell exp (30)

bp + bn

o o

Vp = Vp − bp ln(ip) + bp ln(ip) (20)

max

V The maximum supercapacitor cell charging voltage ( ) can

o o cell

= V + b i

V − b i

n p n ln( n) n ln( n) (21) be expressed as the sum of the voltage drop across the equivalent

series resistance (Resr) and the maximum voltage across the double-

from Fig. 5, it can be seen that the voltage drop across the double- V o

layer capacitance ( sc):

layer capacitance (Vsc) can be expressed as:

max o

V = I R + V

o o o o cell cell esr sc

Vsc = Vp − Vn = Vp − Vn − bp ln(ip) − bn ln(in) + bp ln(ip) + bn ln(in) (22)

o V o

= V + bp ln(ip) + bn ln(in)

= IcellResr + (bp + bn) ln Icell exp (31)

bp + bn where

o o o o o V o

= .

V = Vp − Vn − bp ln(ip) − bn ln(in) (23) for example, if the solvent is water, one has p 1 23V

and Vn = 0.00 V vs. NHE. Assuming that the kinetic param-

− − −

for a decomposition reaction, i = i , they should also be equal to i , 7 2 3

p n F eters bp = bn = 0.0514 V/dec., ip = 1.0x10 A cm , in = 1.0x10

↑ +

as is shown in Fig. 5, that is, iF = ip = in. Substituting this into Eq. (22), −2 1 −

→ + +

A.cm- for water decomposition (H2O 2 O2 2H 2e ), and

the following equation for iF can be obtained: −2 o

Cdl = 0.125 F cm , sc can then be calculated to be 2.034 V at a

−2

o charging current density of 0.025 A cm .

Vsc − V

i =

F exp (24) b For the discharge case shown in Fig. 5(b), the integration equa-

p + bn

tion can be written as:

t=t

In the case of constant current charging as shown in Fig. 5, we o o

1 Vsc − V

iscdt + isc − I − =

ln( ) 0 (32)

assume that before the charging starts, the voltage across the super- b + b cell

( p n)Cdl bp + bn

t=0

capacitor (Vsc) is equal to zero. Then the charging current (isc) used

to charging double-layer capacitance (Cdl) can be expressed as: Solving this equation, we can get isc from which the voltage

dV across the double layer capacitance (Vsc) can be derived respec- i = C sc sc (25) dl dt tively:

o o

Icell(exp(Vsc − V /bp + bn) + Icell)

in Fig. 5(a), the constant charging current can be expressed as: i =

sc o o

Icell + exp(Vsc − V /bp + bn)(1 − exp(−(Icell/(bp + bn)Cdl))t)

Icell = isc + iF (26) (33)

o o o o

V − V /b + b + I I / b + b C t − V − V /b + b

o exp( sc p n) cell exp( cell ( p n) dl) ) exp( sc p n)

Vsc = Vsc − (bp + bn) ln (34) Icell

Eq. (34) indicates that when t = 0, Vsc = Vsc. The supercapacitor

cell charging voltage (Vcell) can be expressed as: V = −I R + V cell cell esr sc

o o

o o (discharge process) (35)

V − V /b + b + I I / b + b C t − V − V /b + b

o (exp( sc p n) cell) exp(( cell ( p n) dl) ) exp( sc p n)

= −IcellResr + Vsc − (bp + bn) ln

Icell

As an illustration, Fig. 6 shows the charge and discharge curves

calculated according to Eqs. (29) and (35). It can be seen that the sol-

Combining Eqs. (24), (25) and (26), the following differential o

vent decomposition plateau voltages ( V Eq. (23)) are all higher

equation can be obtained:

than the thermodynamic voltages because of the limited kinetic

dVsc −V Vsc processes. However, for a safe operation of supercapacitor, it is

Cdl + exp exp − Icell = 0 (27)

better not to charge the cell to a thermodynamic voltage of sol-

dt bp + bn bp + bn

vent decomposition. For example, if using water as the solvent, the

maximum charging voltage should not be higher than 1.23 V.

With the initial condition Vsc = 0 at t = 0, Eq. (27) can be solved

to obtain:

V cell

= b + b

sc ( p n) ln o o (charge process) (28)

Icell − exp(−V /bp + bn)) exp(−Icell/(bp + bn)Cdlt) + exp(−V /bp + bn)

The supercapacitor cell charging voltage (Vcell) should be the

It is worthwhile to point out that the model shown in Fig. 5(a)

sum of the voltage drop across the equivalent series resistance (Resr)

and its associated equations discussed above do not include the

and the voltage across the double-layer capacitance (Vsc):

reactant mass transfer limitation since there may not be a limited

cell = IcellResr + Vsc

Icell (charge process) (29)

= I R + b + b

cell esr ( p n) o o

Icell − exp(−V /bp + bn)) exp(−Icell/(bp + bn)Cdlt) + exp(−V /bp + bn)

548 S. Ban et al. / Electrochimica Acta 90 (2013) 542–549

Fig. 6. Calculated charge–discharge curves according to Eqs. (29) and (35) using

−7 −2

× the following assumed parameters: bp = bn = 0.0514 V/dec., ip = 1.0 10 A cm ,

−3 2 −2 −2

in = 1.0 × 10 A cm , Cdl = 0.125 F cm , and Icell = 0.025 A cm at three different

thermodynamic cell voltages (Vp − Vn) (1.23, 2.20, and 3.20 V, respectively).

supply of reactants. However, please also note that the build of

product gasses in a closed cell, such as O2 and H2 if using an aqueous

electrolyte, may lead to mass transport limitations and also pres-

sure building-up, which will have some impact on the accuracy of

Fig. 7. Proposed equivalent circuits for a double-layer supercapacitor at a constant

the model.

current charging (a) and discharging (b) in the presence of electrochemical decom-

position of the solvent. K: electric switcher; Resr: equivalent series resistance; Rlk:

3.4. Charging at a constant cell current in the presence of both parallel leakage resistance; ilk: parallel leakage current; Icell: constant current for

voltage-independent parallel leakage process and electrochemical charging or discharging; Vcell: supercapacitor cell voltage; iF: current of solvent

electrochemical decomposition; Cdl: double-layer capacitance; isc: current used to

decomposition of solvent

charge or discharge the double-layer capacitance; VF: electrode potential of the sol-

vent electrochemical decomposition; and Vsc(=VF): voltage across the double-layer

Fig. 7(a) shows the proposed equivalent circuit of a double-

capacitance, respectively.

layer supercapacitor in the case of constant current charging in

the presence of both parallel leakage process and electrochem-

For the discharge process, the cell voltage in this ﬁgure is

ical decomposition of the solvent. The integration equation at a

expressed as:

constant charging current (Icell) can be written as: t=t −V o t=t 1 o 1

i + i dt

sc exp exp sc V = −I R + V = −I R + V − i dt

b b cell cell esr sc cell esr sc sc (39)

p + bn ( p + bn)Cdl Cdl t=0 t=0 t=t

+ iscdt − Icell = 0 (36) RlkCdl

t=0

and for the discharge process at a constant current (I ): cell

o o t=t V − V i sc 1

sc − exp exp − isc dt

bp + bn (bp + bn)Cdl t=0 t=t

− isc dt − Icell = 0 (37) RlkCdl

t=0

Unfortunately, both of these two equations cannot be solved

analytically, and numeric solutions will be sought using the soft-

ware Matlab.

Fig. 8 shows the numeric solutions for both charge and discharge

curves with several sets of assumed parameters in the presence of

Fig. 8. Charge–discharge curves plotted by numeric solutions of Eqs. (36)–(39)

both voltage-independent parallel leakage resistance and solvent

using the following assumed parameter sets: set #1 bp = bn = 0.0514 V/dec.,

decomposition. The cell voltage (Vcell) for the charge process in this o 2 −2 −2

V = 2.413 V, Rlk = 10,000 cm , Cdl = 0.125 F cm , and Icell = 0.025 A cm ,

max

2 o

ﬁgure is expressed as: Vsc = 2.034 V, Resr = 2 cm ; set #2 bp = bn = 0.0514 V/dec., V = 2.413 V,

2 −2 −2 max

V = . t=t Rlk = 100 cm , Cdl = 0.125 F cm , and Icell = 0.025 A cm , sc 2 034 V,

2 o 2

Resr = 2 cm ; set #3 bp = bn = 0.0514 V/dec., V = 2.413 V, Rlk = 100 cm ,

1 −2 −2 max 2

V = .

Vcell = IcellResr + Vsc = IcellResr + isc dt (38) Cdl = 0.125 F cm , and Icell = 0.1 A cm , sc 2 034 V, Resr = 2 cm ; set #4

Cdl o 2 −2

bp = bn = 0.0514 V/dec., V = 2.413 V, Rlk = 100 cm , Cdl = 0.125 F cm , and

t= −2 max 2

0 Icell = 0.01 A cm , Vsc = 2.034 V, Resr = 2 cm , respectively.

S. Ban et al. / Electrochimica Acta 90 (2013) 542–549 549

Acknowledgements

The authors would like to thank the support from National

Research Council of Canada (NRC) and Transport Canada. Tech-

nical support from and discussion with Dr. Lucie Robitalille, Dr.

Alexis Laforgue, Dr. Dongfang Yang, Dr. Yves Grincourt, Mr. Francois

Girard, and Mr. Ryan Baker are highly appreciated.

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