Analysis of Charge Redistribution During Self-discharge of

Analysis of Charge Redistribution During Self-dischargeof Double-Layer Supercapacitors

CHENGLONG HAO,1 XIAOFENG WANG,1,2 YAJIANG YIN,1

and ZHENG YOU1

1.—Department of Precision Instruments, Tsinghua University, No. 1 Tsinghuayuan, HaidianDistrict, Beijing, People’s Republic of China. 2.—e-mail: [email protected]

Self-discharge is an important factor that severely affects the performance ofdouble-layer supercapacitors. This paper studies the self-discharge behavior ofdouble-layer supercapacitors with experimental and modeling methods. Themovement of ions, side-reactions, and instability of the double layer are taken intoconsideration. The influence of various factors, such as the initial voltage, chargeduration, short-term history, and current, on the self-discharge is simulated,showing good agreement with experimental data. The simulation of the ion dis-tribution also gives a detailed explanation of the mechanism of self-discharge andverifies the interpretation of the relaxation process proposed in a recent study. Itfurther clarifies the key role of the charging/discharging current in influencingcharge redistribution during self-discharge, which was neglected in previousstudies. The results show that the relaxation period during which the superca-pacitor loses energy very quickly is due to the unbalanced distribution of ions, andit could be avoided by further charging or by applying a small charging current.

Key words: Double-layer capacitor, modeling, self-discharge, relaxationperiod, charge redistribution

List of symbolsCdl Double-layer capacitance (F/m)cl Electrolyte concentration (mol/L)Dl;eff Effective diffusion coefficient of electrolyteDl Diffusion coefficient of electrolyteF Faraday’s constant, 96,484 C/equivi0 Exchange current density (A/m3)Icell Total current density (A/m2)il Current density in electrolyte phase (A/m2)is Current density in solid phase (A/m2)Lþ Thickness of positive electrode (lm)Le Thickness of separator (lm)L� Thickness of negative electrode (lm)R Universal gas constant, 8.314 J/(mol K)Sf Specific surface area for side-reactions per

unit electrode volume (m�1)Sd Specific surface area for double-layer

capacitance per unit electrode volume (m�1)T Absolute temperature (K)t Time (s)tþ Transport number

U1 Equilibrium potential (V)aa Anodic transfer coefficient of faradaic

reactionac Cathodic transfer coefficient of faradaic

reactionel Volume fraction of electrolyte phasees Volume fraction of solid phaserl;eff Effective conductivity in electrolyte phase (S/

m)rl Conductivity in electrolyte phase (S/m)rs;eff Effective conductivity in solid phase (S/m)rs Conductivity in solid phase (S/m)/l Potential in electrolyte phase (V)/s Potential in solid phase (V)

INTRODUCTION

Electrochemical double-layer capacitors are use-ful storage devices. They have very distinctivecharacteristics, such as high power density andlarge cycle time,1–3 which make them more

(Received October 21, 2015; accepted January 16, 2016;published online February 8, 2016)

Journal of ELECTRONIC MATERIALS, Vol. 45, No. 4, 2016

DOI: 10.1007/s11664-016-4357-0� 2016 The Minerals, Metals & Materials Society

2160

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suitable than batteries in various conditions. How-ever, one of the drawbacks of supercapacitors istheir high self-discharge rate, which is much morepronounced than in batteries.4

To date, only a few studies discussing the mech-anism behind the self-discharge phenomenon havebeen published. Conway et al.5 gave a generaldescription of self-discharge mechanisms and pro-vided three possible routes: electrolyte decomposi-tion, impurities, and internal ohmic leakagepathways. However, Levie et al. believed6 thatConway’s model overlooked charge redistribution,which is a result of an uneven charging processacross the electrode. Ricketts and Ton-That7 dividedthe self-discharge into two stages: the first is adiffusion process during the first 1 h or 2 h whenmore energy is lost, and the second is a slowerleakage process. However, they did not discover thenature of the reactions and guessed that the pres-ence of water in the organic electrolyte might be afactor. Furthermore, Black and Andreas8 demon-strated the effect of charge redistribution in the self-discharge process. More recently, Kowal et al.9

revealed the role of the relaxation period in self-discharge, as inferred from their experiments,which also showed that the self-discharge ratedepends on various parameters such as the initialvoltage, temperature, charge duration, and short-term history. To account for the relaxation effect inself-discharge, Kaus et al.10 proposed an electricalmodel which was able to predict the effects of theseparameters. However, both of them overlooked theimportant role of current in elaborating the mech-anism of self-discharge.

This paper proposes a dynamic mathematicalmodel simulating the self-discharge behavior ofsupercapacitors. It considers the common factorsused by previous researchers.6–8 The novelty of themodel is that it simulates the movement of chargesin the electrolyte as well as in the electronic doublelayer, enabling simulation of the self-dischargemechanism. We repeated some of the experimentsfrom Kowal et al.’s study9 and further investigatedthe effect of current. Through simulation of themovement of ions between the electrode and elec-trolyte as well as across the electrolyte, we finallydemonstrate their conclusion about the effect ofcharge redistribution on self-discharge. Further-more, the simulation and experimental data alsoshow that a critical factor—the charging/discharg-ing current—actually plays a determining role incharge redistribution during self-discharge, whichwas neglected by previous studies.

EXPERIMENTAL PROCEDURES

We repeated some of the experiments from Kowalet al.’s study9 and further investigated the effect ofcurrent. A 360F commercial carbon-based superca-pacitor with organic electrolyte was used for all theexperiments presented in this paper. The activated

carbon (AC) used in the supercapacitor has Bru-nauer–Emmett–Teller (BET) specific surface area of1453 m2/g. The electrolyte was acetonitrile solutionof quaternary ammonium salt with conductivity of0.94 S/m. All experiments were performed using aPARSTAT 2273 (0 A to 2 A) at room temperature.

Effect of Initial Voltage

As shown in Fig. 1, the supercapacitor wascharged to different voltages (1.5 V, 2.0 V, 2.4 V,and 2.7 V) at current of 250 mA and then placed todisplay the self-discharge phenomenon. The resultsshow that the voltage decayed faster for higherinitial voltage. Moreover, the relaxation periodtended to last 10 h or even longer at 2.7 V, whileit lasted for no more than 2 h at 1.5 V.

Effect of Charge Duration

Figure 2 shows the dependence of the self-dis-charge on the charge duration. The supercapacitorwas further charged for 12 h, 2 h, and 0 h, respec-tively, after being charged to 2.7 V at current of250 mA. The results show that such further charg-ing reduced the rate of self-discharge during therelaxation period. However, periods after the relax-ation were almost unaffected: after about 25 h, therate of self-discharge became the same.

Effect of Current

The impact of current on self-discharge has rarelybeen studied. So, in this experiment, the superca-pacitor was charged to 2.7 V at 62.5 mA, 125 mA,250 mA, 500 mA, and 1000 mA. The effect on self-discharge is displayed in Fig. 3. When the currentwas lower than 250 mA, increasing the currentcaused a larger voltage decay during the relaxationperiod. However, when the current exceeded250 mA, the voltage decay became smaller due to

Fig. 1. Voltage decay of supercapacitor in self-discharge aftercharging to 1.5 V, 2.0 V, 2.4 V, and 2.7 V at current of 250 mA(Color figure online).

Analysis of Charge Redistribution During Self-discharge of Double-Layer Supercapacitors 2161

the lower initial voltage. Figure 3 shows that in factthere was a combined effect of current and initialvoltage.

Effect of Short-Term History and Current

Figure 4 shows the influence of the short-termhistory on the self-discharge. Two conditions wereconsidered: firstly, the supercapacitor was chargedto 2.1 V at 250 mA and further charged for 12 h,then charged at 50 mA/2.4 V without further charg-ing; secondly, the supercapacitor was charged to2.7 V at 250 mA and further changed for 12 h, thendischarged at 50 mA to 2.4 V without furthercharging. The results show that the self-dischargewas different under the different conditions. Thesupercapacitor underwent a large voltage decayafter a short charging process, whereas its voltageunderwent a small increase after a short-termdischarging process. Then, the impact of current

on the short-term history effect was investigated bydecreasing the charging/discharging current from50 mA to 5 mA. As shown in Fig. 4, the voltagevariations in the relaxation period became muchsmaller under a small current.

MODEL DESCRIPTION

To investigate the self-discharge behavior of dou-ble-layer supercapacitors, we propose a dynamicmathematical model. In contrast to equivalent circuitmodels,10 this model uses parameters with physicalmeanings and describes the real physical phe-nomenon. Many models have been developed todescribe the behavior of double-layer capacitors.Posey and Morozumi developed a model to simulatethe potentiostatic and galvanostatic charging ofdouble-layer capacitors with porous electrodes.11

Johnson and Newman’s model12 described the ionicabsorption process in the double layers. Pillay andNewman13 went on to study the influence of side-reactions on the capacitor’s performance. More influ-ential variables were studied by Srinivasan andWeidner,14 including ionic and electronic resistance,in the design of supercapacitors. Unfortunately, noneof these models can provide a complete description ofthe self-discharge behavior, although they provide atheoretical basis. Besides, we recently developed amodel of an asymmetric supercapacitor and studiedthe relaxation process during discharge.15

Figure 5 presents a schematic diagram of atypical double-layer supercapacitor, consisting oftwo identical porous electrodes and a separator. Theelectrode is composed of AC and acetylene black(AB). As electrolyte, acetonitrile solution of quater-nary ammonium salt fills the pores in the electrodeand separator. During charge/discharge, the double-layer reaction takes place between the interface ofthe solid phase and the electrolyte phase inside theelectrode.

Fig. 2. Voltage decay of supercapacitor for different charge dura-tions in self-discharge after charging to 2.7 V at current of 250 mA(Color figure online).

Fig. 3. Voltage decay of supercapacitor after charging to 2.7 V at62.5 mA, 125 mA, 250 mA, 500 mA, and 1000 mA (Color figureonline).

Fig. 4. Voltage decay of supercapacitor after short-term historiesperformed with different currents (charging to 2.4 V after charging for12 h to 2.1 V, or discharging to 2.4 V after charging for 12 h to 2.7 V,at 50 mA and 5 mA, respectively) (Color figure online).

Hao, Wang, Yin, and You2162

A one-dimensional mathematical model is devel-oped for the double-layer supercapacitor to describeits self-discharge behavior. The diffusion coefficientin the solid and solution phase and the transportnumber are assumed to be independent of theelectrolyte concentration. All thermal effects areneglected, but the side-reactions concerning impu-rities and electrolyte decomposition are taken intoconsideration.

Considering the porous characteristic of the elec-trode, the conductivity and diffusion coefficientshould be modified in the electrode area16:

Dl;eff ¼ e1:5l Dl;

rl;eff ¼ e1:5l rl;

rs;eff ¼ e1:5s rs;

ð1Þ

where Dl;eff , rl;eff , and rs;eff are the effective diffusioncoefficient of the electrolyte, the effective conduc-tivity in the electrolyte phase, and the effectiveconductivity in the solid phase, respectively. Theiroriginal values are Dl, rl, and rs. el is the volumefraction of the electrolyte phase, and es is thevolume fraction of the solid phase.

At any given time, when the total current densityfor charging is Icell, then for every position in themodel, conservation of charge leads to

Icell ¼ is þ il; 0 ¼ @is@x

þ @il@x

; ð2Þ

where is represents the current density in the solidphase, and il is the current density in the electrolytephase. As the capacitor charges, the current is trans-ferred from the solid phase to the electrolyte phasecontinuously in the positive electrode, with is decreas-ing and il increasing. In the negative electrode, thecurrent is gradually transferred back from the elec-trolyte phase, with is increasing and il decreasing.However, it travels only in the electrolyte phase in thefield of the separator because no electrons are allowedto move in this area. The transfer of the current ispowered mainly by the double-layer charge but isinfluenced by the side-reactions due to impurities andelectrolyte decomposition.

@il@x

¼ SdCdl@ð/s � /lÞ

@tþ Sf jf ; ð3Þ

where Sd is the specific surface area for the double-layer capacitance per unit electrode volume. Cdl

represents the double-layer capacitance. /s and /lare the potential in the solid phase and in theelectrolyte phase, respectively. Sf is the specificsurface area for the side-reactions, and jf is thecorresponding transfer current density.

Fig. 5. Schematic diagram of typical double-layer supercapacitor (Color figure online).


Side-reactions play a very important part inaccounting for the mechanism of self-discharge.They are assumed to occur only in a small area ofthe total electrode area and only when the capacitorvoltage reaches a threshold value, E1; For example,the decomposition voltage of water is 1.3 V. Theinstability of the double layers should also be takeninto account. They are assumed to be more unsta-ble for greater capacitor voltage. In a similar way,only a small part of the double layers tend to bebroken at the same time. The escape of ions tends todecrease the capacitor voltage to another value, E2.Both side-reactions and the instability of the doublelayer are hard to measure. Therefore, we put themtogether and use a single transfer current jf todescribe their combined contribution. This can bedescribed in kinetic form using the Butler–Volmerequation:

jf ¼ i0 exp aaF

RT/s � /l �U1ð Þ

� ��

� exp �acF

RT/s � /l �U1ð Þ

� ��;

ð4Þ

where aa and ac are the anodic and cathodic transfercoefficients of the faradaic reaction, F is Faraday’sconstant, R is the universal gas constant, T is theabsolute temperature, i0 is the exchange currentdensity, and U1 is the equilibrium potential, whichstands for the combined effect of the side-reactionsand the instability of the double layer. U1 is the‘‘driving force,’’ which provides a tendency for thevoltage to drop, referred to in Conway’s study.5 U1 isnot fixed but is expected to vary during the self-discharge process. Because there are not enoughphysical property parameters to determine theequilibrium, it was only evaluated approximatelyaccording to the experimental data.

In the solid phase, /s is correlated to is by Ohm’slaw

is ¼ �rs;eff@/s

@x: ð5Þ

In the electrolyte phase, however, Ohm’s law ismodified to consider the porosity of the electrode17

il ¼ �rl;eff@/l

@x� 2rl;effRT

F1 � tþð Þ @ðln clÞ

@x; ð6Þ

where tþ is the transport number. The concentra-tion of the electrolyte cl is also correlated to thefaradaic redox reaction and the double-layer reac-tion through a material balance on the electrolyteusing concentrated solution theory16:

el@cl

@tþr � �Dl;effrcl þ

iltþF

� �¼ 1

F

@il@x

: ð7Þ

In the separator, porous electrode theory is notapplicable. Equations 6 and 7 can be simplified to

il ¼ �rl;eff@/l

@x; ð8Þ

@cl

@tþr � �Dlrcl þ

iltþF

� �¼ 0: ð9Þ

The boundary conditions of the model include:At x ¼ 0:

@cl

@x¼ 0; il ¼ 0; Icell ¼ is: ð10Þ

At x ¼ Lþ ¼ Lþ þ Ls:

is ¼ 0;@il@x

¼ 0: ð11Þ

At x ¼ Lþ þ Ls þ L�:

@cl

@x ¼ 0; il ¼ 0; Icell ¼ is; /s ¼ 0: ð12Þ

The initial conditions are:At t ¼ 0:

cl ¼ c0; /sjx¼0¼ 0; ð13Þ

where c0 is the initial concentration of theelectrolyte.

The model equations include the double-layerprocess, the side-reactions, the instability of thedouble layer, the porosity characteristic of theelectrodes, and the ion transport inside the porouselectrode. Because the variables are coupled witheach other, they can only be solved simultaneously.The equations of the model were solved usingCOMSOL Multiphysics software using the modelparameters listed in Table I.

RESULTS AND DISCUSSION

Movement of Ions During the Charge/Self-discharge Process

The ion distributions of the positive charges in theelectrolyte were simulated during charging, asshown in Fig. 6. Numerous double layers form atthe interface between the solid phase and theelectrolyte phase of the porous electrode. In theelectrolyte phase of the negative electrode, positivecharges begin to decrease as they are absorbed bythe opposite charges in the solid phase to form thedouble layer. The reaction rate is not the same indifferent areas of the electrode. The transport ofions falls behind and cannot catch up with theabsorption of ions, which leads to an unbalanceddistribution of negative ions along the capacitor, asseen in Fig. 6. There are only a few positive ions leftin the electrolyte phase of the negative electrodebecause most have been absorbed to form the double


layer. The situation in the positive electrode is justthe opposite.

As shown in Fig. 7, the first stage of self-dis-charge is a period of relaxation, powered by the ionconcentration gradients. Positive ions transfer fromthe positive electrode to the negative electrode, andnegative ions transfer in the opposite direction.Some of the unstable double layers also lose theirions in the process. In the positive electrode, forexample, some negative ions leave the double layerto the electrolyte phase, attracted by the excess

positive ions, which is especially pronounced at thehead of the electrode. During the process, thesupercapacitor quickly loses significant capacitydue to the redistribution of ions.

At the end of the first stage, the concentrationgradients of ions along the electrolyte are almosteliminated. For ions in the double layer, theattraction from the opposite ions in the electrolytephase becomes nearly balanced with the bindingforce of the double layer. The self-discharge thenmoves into a second stage where ions escape from

Table I. Parameters used in the simulation

Parameter Physical Meaning Value References

Lþ Thickness of positive electrode 55 lm AssumedLe Thickness of separator 30 lm AssumedL� Thickness of negative electrode 55 lm AssumedCdl Double-layer capacitance 0.08 F/m2 MeasuredSd Specific surface area for double-layer capacitance per unit electrode volume 6 9 108 m�1 MeasuredSf Specific surface area for side-reactions per unit electrode volume 0.1 m�1 AssumedDl Diffusion coefficient of electrolyte 4.1 9 10�10 m2/s 18es Volume fraction of solid phase 0.5 Assumedel Volume fraction of electrolyte phase 0.33 Assumedi0 Exchange current density 10 A/m2 Assumedc0 Initial concentration of electrolyte 2 mol/L MeasuredT Temperature 298.15 K Assumedtþ Transport number 0.363 17aa Anodic transfer coefficient of faradaic reaction 0.5 19ac Cathodic transfer coefficient of faradaic reaction 0.5 19rs Conductivity in solid phase of electrodes 34.2 S/m Measuredrl Conductivity in electrolyte phase 0.94 S/m 20

Fig. 6. Concentration gradients of positive ions across the capacitor during the charging process (Color figure online).


the double layers much more slowly. As shown inFig. 8, there are almost no concentration gradientsacross the electrolyte after 10 h of self-discharge.An increasing number of ions begin to escape fromthe double layer to the electrolyte as time passes,which leads to a slow voltage decay of thesupercapacitor.

Effect of Initial Voltage

The experimental data in Fig. 1 show that thevoltage decays faster for higher initial voltage.Figure 9 shows the simulation results comparedwith the experimental data. According to the sim-ulation and discussion in ‘‘Movement of Ions During

Fig. 7. Concentration gradients of positive ions across the capacitor during the first stage of self-discharge (Color figure online).

Fig. 8. Concentration gradients of positive ions across the capacitor during the second stage of self-discharge (Color figure online).


the Charge/Self-discharge Process’’ section, thereare three possible reasons for this result. Firstly,the ion distribution is more unbalanced when thesupercapacitor is charged to a higher voltage. Thus,more energy is lost in the relaxation period.Secondly, there are more unstable double layers athigher voltage. Thirdly, more side-reactions includ-ing impurities and electrolyte decomposition tend tohappen when the voltage is higher.

Effect of Charge Duration

The experimental data are shown in Fig. 2.Figure 10 shows the simulation results based onthe mathematical model. It shows that the furthercharging reduces the rate of self-discharge, espe-cially at the beginning—the relaxation period. Afterfurther charging for 12 h, the relaxation process isalmost avoided, but the slow voltage decay in thesecond stage of self-discharge is almost unaffected.As predicted in ‘‘Movement of Ions During the

Charge/Self-discharge Process’’ section, furthercharging provides a rest period for ions to moveacross the electrolyte driven by the unbalanceddistribution. After the further charging, the ionswould be more equally distributed than before,which eliminates the driving force of the relaxationprocess. However, it has little effect on the secondstage of self-discharge because it cannot change theinstability characteristic of the double layers.

Effect of Short-Term History and Current

The experimental data in Fig. 4 show that a shortcharging process was followed with voltage decayand a short discharging process was followed withvoltage increase. The simulation results are shownin Fig. 11. The short-term history disrupts thedistribution balance of ions. After further long-termcharging, the ion distribution is balanced. However,short discharging leads to another unbalanceddistribution of ions. Then, just as the short dis-charge ends, the redistribution effect drives someions back to the double layer. At the same time, thevoltage recovers, leading to a small voltage increase.For the same reason, short charging results in asmall voltage drop. Moreover, the experimentaldata and simulation results show the importantrole of current. The charging/discharging currenthas seldom been discussed in related studies, but itplays a decisive role in the relaxation period. Whena small current is applied in the short-term history,the voltage variations in the relaxation period arereduced, as shown in Fig. 11. This proves that alarger current would magnify the effect of short-term charging/discharging, whereas a small currentwould eliminate the effect of the short-term history.A reasonable interpretation is that, when a smallcurrent is applied, the reactions on the electrodebecome much slower. In this case, the movement ofions could almost catch up with the production andconsumption of ions. Therefore, there are not large

Fig. 9. Simulation results of effect of initial voltage compared withexperimental data (Color figure online).

Fig. 10. Simulation results of effect of charge duration comparedwith experimental data (Color figure online).

Fig. 11. Simulation results of effect of short-term history and currentcompared with experimental data (Color figure online).


distribution gradients across the electrolyte in theshort-term charging/discharging process. The relax-ation period would then not take place in the self-discharge.

Mechanism of Self-discharge

This section summarizes the mechanism of self-discharge presented in the former sections. Accord-ing to the simulation results, a normal self-dis-charge has two stages: a fast diffusion stage and aslower voltage decay stage. The redistribution effectdue to concentration gradients of ions is the maincause of the first stage. Side-reactions and theinstability of double layers contribute to the self-discharge in both stages.

Figure 12 displays a schematic of the ion distri-bution during the charging process. Initially, posi-tive and negative ions distribute evenly across theelectrolyte. As soon as a current flows into thesupercapacitor, negative ions begin to be absorbedin the positive electrode, and positive ions begin tobe absorbed in the negative electrode, which formsdouble layers at the interfaces between the solidphase and the electrolyte phase inside the elec-trodes. However, the reaction rate is different acrossthe electrode, resulting in the formation of concen-tration gradients. At the end of charging, the ionsacross the electrolyte and the charges across theelectrode are unevenly distributed. The unevendistribution of ions causes two types of drivingforces: one drives ions to transfer across the elec-trolyte, and the other drives charges to escape fromthe double layer. In the positive electrode, for

instance, negative ions are absorbed to the doublelayers, and there are more positive ions left in theelectrolyte phase. Therefore, under the drivingforces, positive ions transfer from the positiveelectrode to the negative electrode. Negativecharges in the double layer, attracted by the positiveions left in the electrolyte, likely escape back to theelectrolyte phase. However, due to the unevendistribution of double layers across the electrode,areas where the double layer is concentrated aremore likely to lose their charges.

Figure 13 shows a schematic of the self-dischargeprocess. It is divided into two stages. The drivingforces caused by the uneven ion distribution bringabout the first stage of self-discharge—relaxation.The relaxation is a fast diffusion process where ionsmove across the electrolyte and charges escape fromthe double layers, motivated by the driving forces.Therefore, the capacitor suffers a rapid and pro-found loss of energy in relaxation. This processtakes a few hours, and a relative equilibrium iseventually reached. All ions and charges are equallydistributed. In the positive electrode, there arefewer positive ions left in the electrolyte phase.Therefore, the driving force exerted on charges toescape from the double layer is much smaller. Side-reactions also become less active under a lowervoltage. Without the strong driving forces, only aslow voltage decay is observed in the voltage curveof the second stage of self-discharge.

However, if the supercapacitor is further chargedat the end of the charging process, the situa-tion is different, as shown in Fig. 14. When the

Fig. 12. Schematic of ion distribution during the charging process (Color figure online).


supercapacitor is further charged after reaching thetarget voltage, the ions transfer to eliminate theconcentration gradient, just like in the relaxationstage of the self-discharge. However, the differenceis that the lost double layers and reduced capacityare recovered in the further discharge. Finally, thedouble layer across the electrode is in a state ofhomogeneous distribution. Eliminating the unbal-anced distribution during the further dischargeprocess causes the capacitor to skip the relaxationstage of self-discharge, being an important charac-teristic of the further discharge.

With a further charging process, the capacitorreaches a comparative equilibrium in which no fastdiffusion process occurs. However, a short chargingprocess or discharging process breaks the equilib-rium. A short charging process causes unequal iondistributions, as stated above. However, the extentof the imbalance is smaller because the chargingtakes less time. Therefore, the short chargingprocess is followed by a small voltage decay. A short

discharging process, however, acts in the oppositeway. During discharging, charges escape from dou-ble layers and move back to the electrolyte, the rateof which is much faster than self-discharge. There-fore, unlike self-discharge, the transfer of ionscannot catch up with the reaction rate, causing anunbalanced ion distribution. Once the discharging isstopped, the unbalanced ion distribution drives iontransfer across the electrolyte and pushes some ionsback to re-form the double layer. This explains thevoltage recovery phenomenon during the relaxationperiod. After the relaxation period ends, a newequilibrium is reached, followed by slow voltagedecay.

An important factor neglected by former researchis the charging or discharging current, as shown inFigs. 3 and 11. During charging, for example, alarger current increases the reaction rate. Doublelayers form more quickly, and concentration gradi-ents across the capacitor become larger. The largerthe concentration gradient, the more time it takes

Fig. 13. Schematic of ion distribution during the self-discharge process (Color figure online).


for redistribution of ions, and the more energy is lostin the charge redistribution. On the contrary, if thecharging current is small enough, the reaction rateis small and the diffusion rate of ions can catch upwith it. Then, the concentration gradients are smallenough to be ignored. In this case, the chargeredistribution does not happen, even without afurther charging process. In other words, applyinga small charging current plays the same role as afurther charging process, as shown in Fig. 15. Thismakes the effect of the short-term history clearer. Ifthe charging/discharging current is small, then theshort charging/discharging to 2.4 V does not maketoo much difference, as shown in Fig. 11.

CONCLUSIONS

We propose a mathematical model to describe theself-discharge behavior of double-layer supercapac-itors. The movement of ions, side-reactions, and the

instability of the double layer are taken into con-sideration. Many factors, including the initial volt-age, charge duration, short-term history, andcurrent, that influence self-discharge are simulated.The results agree well with experimental data. Thesimulation also gives a detailed explanation of themechanism of self-discharge and verifies the chargeredistribution interpretation proposed in a recentstudy9 based on theoretical calculations. The resultsshow that self-discharge has two distinct stages: afast diffusion stage and a slow voltage decay stage.The fast diffusion stage, during which the superca-pacitor loses energy very quickly, is due to theredistribution effect of the concentration gradients,and it could be avoided by further charging orapplying a small current. In fact, the large currentis responsible for the unbalanced distribution of ionsbecause the transfer of ions across the electrolytecannot catch up with the reaction rate.

Fig. 14. Schematic of ion distribution with a further charging process (Color figure online).


ACKNOWLEDGEMENT

The authors acknowledge financial support fromthe National Natural Science Foundation of China(NNSFC) (Project No. 50905096).

CONFLICT OF INTEREST

The authors declare that they have no conflict ofinterest.

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Fig. 15. Schematic of ion distribution when a low charging current is applied (Color figure online).


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Analysis of Charge Redistribution During Self-discharge of