Electrochimica Acta 54 (2009) 3568–3574

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Electrochimica Acta

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Effects of charge redistribution on self-discharge of electrochemical capacitors

Jennifer Black, Heather A. Andreas ∗

Department of Chemistry, Dalhousie University, Halifax, NS B3H 4J3, Canada

a r t i c l e i n f o

Article history:Received 10 October 2008Received in revised form 17 December 2008Accepted 3 January 2009Available online 15 January 2009

Keywords:Self-discharge

a b s t r a c t

The effect of charge redistribution on the self-discharge profile of porous carbon (Spectracarb 2225)electrodes is examined. A model pore based on the de Levie transmission line circuit is used to showthat self-discharge due purely to charge redistribution results in the same self-discharge profile as thatexpected for an activation-controlled self-discharge mechanism (the potential falls linearly with log t),thus the linear log time profile is not characteristic of an activation-controlled mechanism. The additionof a hold step reduces the amount of charge redistribution in porous carbon electrodes, although the holdtime required to minimize the charge redistribution is much longer than expected, with electrodes which

Porous electrodesTransmission line modelActivation controlledC

have undergone a 50 h hold time still evidencing charge redistribution effects. The time required for thecharge redistribution through the porous electrode is also much greater than predicted, likely requiringtens of hours. This highlights the importance of the charge redistribution in self-discharge of systems

such

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harge redistribution using porous electrodes,

. Introduction

Electrochemical capacitors (ECs), also called supercapacitors,nd ultracapacitors are charge storage devices. They may storeharge in a Faradaic reaction as pseudocapacitance or may storeharge in the double-layer of an electrode/electrolyte interphase1]. Double-layer ECs typically utilize high surface area electrodesoften, carbon which may have a Brunauer, Emmett and Teller (BET)urface area of up to 2500 m2/g). The high surface area is neces-ary since the charge is stored in the electrode/electrolyte boundaryegion, and a higher surface area results in more charge storage. Byecessity, this requires very small pores in the electrodes, and thesemall, often very tortuous pores, lead to significant diffusion andigration limitations down the pores during charging/discharging

3]. A mathematical evaluation of the potential and current distri-ution in porous electrodes was previously given by de Levie [3],hich showed that the electrode processes do not proceed evenly

hroughout the thickness of a porous electrode, largely due to the IRrop in the solution within the pores, resulting in faster electroderocesses at the mouth of the pore, closer to the counter electrode,ersus those which occur at the base of the pore [2]. In other words,he external surface of the electrode will charge/discharge faster

han the surfaces within the pores. This behavior can be modeledith an RC transmission line [2]. As a result of this pore effect and

ts transmission line characteristics, when a porous electrode isharged there will be a distribution of potentials down the electrode

∗ Corresponding author. Tel.: +1 902 494 4505; fax: +1 902 494 1310.E-mail address: heather.andreas@dal.ca (H.A. Andreas).

013-4686/$ – see front matter © 2009 Elsevier Ltd. All rights reserved.oi:10.1016/j.electacta.2009.01.019

as electrochemical capacitors.© 2009 Elsevier Ltd. All rights reserved.

pores. As the system is switched to open-circuit these potentialswill equilibrate as the charge redistributes evenly over the elec-trode surface [4]. The potential at the tip of the pore, where thepotential is measured, will fall during this charge redistribution,and may appear as a rapid, initial self-discharge.

Self-discharge is the spontaneous decline in voltage with timeof a charged EC left on open-circuit. The rate of self-discharge isdetermined by the mechanisms of the processes by which the self-discharge takes place [5]. When a charged EC is placed on open-circuit there is no external circuit through which electrons maypass and discharge the cell, and therefore self-discharge must takeplace through coupled anodic and cathodic processes. Conway etal. [5] proposed three mechanisms through which self-dischargemay occur, and derived models which describe the predicted self-discharge profile for each mechanism:

(i) An activation-controlled Faradaic process, where the decline ofvoltage (V or Vt) versus log time would give a straight line [5,6]:

V = − RT

˛Fln

˛Fi0RTC

− RT

˛Fln

(t + C�

i0

)(1)

and

Vt = Vi − A log(t + �) (2)

where R is the universal gas constant, T is the absolute temperature,˛ is the charge transfer coefficient, F is the Faraday constant, i0 isthe exchange current density, C is the capacitance, t is the time, � isan integration constant, Vi is the initial charging potential and A isa constant related to the Tafel slope. This model describes the self-

chimica Acta 54 (2009) 3568–3574 3569

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ischarge due to the Faradaic reaction of a species which is eithert high concentration in the cell (e.g. electrolyte decomposition [7])r is attached to the surface (e.g. oxidation/reduction of a carbonurface functionality [8]).

ii) A diffusion controlled Faradaic process where the potentialwould decline with the square root of t [6]:

t = Vi − 2zFAD1/2�1/2c0

Ct1/2 (3)

here D is the diffusion coefficient of the redox species, z is theharge, c0 is the initial concentration and A is the electroactive area.his model is based on the Faradaic reaction of a species which haslow concentration in the EC and relies on transport (diffusion) to

he electrode surface to react (e.g. an Fe shuttle reaction [6]).

iii) An internal ohmic leakage or a ‘short circuit’ due to faultyconstruction. Here a plot of ln Vt versus t would give a linearrelationship [6]:

n Vt = ln Vi − t

RC(4)

here R is the resistance of the ohmic contact.It was previously shown by Niu et al. [6] that the self-discharge

rofile of a highly porous Spectracarb 2225 carbon cloth was lin-ar when voltage was plotted versus log t, suggesting that theelf-discharge proceeded through an activation-controlled Faradaicechanism. It was also shown in Ref. [6] that the slope of the self-

ischarge profile when plotted as a function of log t was dependentpon the initial potential of the electrode, contrary to that predictedy Eq. (2). Niu et al. [6] suggested that the slope dependence on thenitial potential may be due to a charge redistribution effect in theorous electrode, as the amount of charge redistribution experi-nced will be dependent on the polarization potential. In this workhe effect of charge redistribution on self-discharge was elucidated,ncluding an examination of the self-discharge which results purelyrom charge redistribution and a discussion of the time domainsequired for this charge redistribution to complete in highly porouslectrodes.

. Experimental

.1. Electrodes, cell and procedure to reach steady-state

Each working electrode was composed of a ca. 10 mg piecef Spectracarb 2225 carbon cloth (Spectracorp, BET surfacerea = 2500 m2/g) mounted in a Teflon Swagelok system (PFA tubetting) with ionically insulating, electronically conducting currentollectors (supplied by Axion Power International Inc.), and electri-al contact to a Pt wire, sealed in glass tubing. Counter electrodesere fabricated from the same carbon cloth, wrapped with Au wire

or electrical contact. A standard hydrogen electrode (SHE) wassed as a reference electrode, and all potentials in this paper areeferenced to it.

Experiments were performed in a three-compartment, three-lectrode glass cell filled will 1.0 M H2SO4 prepared fromoncentrated H2SO4 (Sigma–Aldrich, 99.999% pure) and 18 M�ater. The reference compartment was separated from the working

lectrode compartment via a luggin capillary. Nitrogen was bubbled

hrough the working and counter electrode compartments prioro and during experiments to remove O2 from the electrolyte. Thexperiments were conducted at 22 ± 3 ◦C. All measurements wereerformed using a Princeton Applied Research VMP3 multipoten-iostat. Data was collected using EC-Lab software.

Fig. 1. Schematic of the hardware transmission line circuit, based on de Levie’s [3]transmission line model of a pore.

Working electrodes were cycled between 0.0 and 1.0 V versusSHE using a sweep rate of 1 mV/s for approximately 1 week (∼300cycles) prior to self-discharge measurements in order to bring theelectrodes to a steady-state whereupon no further changes in theshape or size of the cyclic voltammograms were seen. The 0.0and 1.0 V potential limits were used since at potentials outsideof these limits undesirable, irreversible Faradaic reactions occurwhich destroy the carbon cloth electrode.

2.2. Transmission line circuit

A hardware transmission line circuit was used as a model pore,based on the de Levie [3] transmission line model. A schematic ofthe hardware transmission line circuit used in this work is shownin Fig. 1. It consists of eight parallel sections, with each sectionbeing composed of a resistor and capacitor in series. The resis-tors model the solution resistance down the pore which leadsto the potential drop down the pore during charging/discharging(based on Ohm’s law). The capacitors model the capacitance ofthe double-layer at the electrode/electrolyte interface. The valueof the resistance can be set to a value between 10 and 100 k�in 10 k� increments or to 200 or 300 k�. Each capacitor wasmade up of a parallel combination of ten 10 �F/6.3 V multilayerceramic chip capacitors, for a nominal capacitance of 100 �F.The capacitors in the hardware circuit could be shorted out andwere brought to a zero charge state (0 V) prior to each experi-ment.

A VMP3 multipotentiostat was used to track the potentials ateach terminal (RE 1–8) by connecting the working electrode (WE)leads of the multipotentiostat together with connection to the WEpoint on the hardware circuit (as shown by WE in the schematic

in Fig. 1). Similarly, the counter electrode (CE) lead of terminal 1was connected to the CE point. The reference electrode (RE) leadsof each of eight channels were connected to points RE 1–8 of thehardware circuit.

3570 J. Black, H.A. Andreas / Electrochimic

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to the porosity of the electrode which causes this deviation fromthe model, since the models were developed for planar electrodes,rather than the porous carbon used in this work. They suggested[4] that charge redistribution occurs within the first 100 s of self-

ig. 2. Self-discharge data for a carbon cloth electrode (ca. 10 mg, in 1 M H2SO4)lotted versus log t after charging from 0.0 to 1.0 V at a potential ramp rate of 1 mV/s,eld at 1 V for up to 5 h, and then switched to open-circuit.

.3. Self-discharge measurements

Self-discharge experiments were performed on carbon elec-rodes by first ramping the potential of the working electrode from.5 V to the desired initial potential (typically, 0.0 or 1.0 V), thenolding at this potential for a given amount of time (0 s to 75 h).he initial potential of 0.5 V was chosen as this was close to thepen-circuit potential for the electrodes, although there was somemall variation in open-circuit potentials between electrodes. Theystem was then switched into open-circuit configuration and theotential of the working electrode was monitored over time.

A similar procedure was used for the transmission line circuit,xcept in this setup the master channel (RE 1) was charged from.0 V to a chosen final potential (ranging from 0.1 to 1.0 V) at variousamp rates (between 1 and 50 mV/s) and then held for the desiredime (0 s to 30 min). The circuit was then opened and the voltagesf all eight channels of the transmission line circuit (RE 1–8) wereecorded with time.

.4. Surface area measurements

BET measurements were carried out on a 0.1866 g sample of thepectracarb 2225 carbon cloth using a Micrometrics ASAP 2000,olio Instruments, Inc. (Department of Physics, Dalhousie Univer-ity). The pore size distribution was calculated using slit shapedores and the density functional theory (DFT). Also provided ismore realistic estimation of the pore size distribution curve,herein the sharp drops, which are an artifact of the DFT calcu-

ations, were removed.

. Results and discussion

.1. Effect of initial charging potential on self-discharge profiles ofarbon cloth

When the self-discharge potential of a Spectracarb 2225 carbonloth electrode in H2SO4 was plotted versus log t, a plateau waseen (Fig. 2), such as that predicted by the Conway model [5,6] forn activation-controlled process, where the length of time for thelateau corresponds to the integration constant, �, in the model.fter this plateau, the self-discharge potential fell linearly with log t,gain corresponding to the expected profile based on the activation-

ontrolled Conway model [5,6], suggesting that the self-dischargeroceeds through an activation-controlled Faradaic mechanism.

From Eq. (1) it is predicted that the decline in voltage over logime for an activation-controlled Faradaic mechanism has a slopehich is the negative of the Tafel slope (RT/˛F), and thus, the slope

a Acta 54 (2009) 3568–3574

should be the same for a given Faradaic self-discharge mechanism.For example, it would be expected that an electrode ramped to aninitial potential of 1 V would have a certain rate of self-discharge,and as this electrode reaches 0.9 V, the rate of self-discharge wouldthen match the rate of self-discharge for an electrode was rampedto an initial potential of 0.9 V, based on the assumption that thesame self-discharge mechanism was in effect at each potential,independent of the history of the electrode. However, when theself-discharge profiles from different initial charging potentials areexamined as a function of the log of self-discharge time, Fig. 3 (andthose seen by Conway), it was seen that there is no correlationbetween the self-discharge potential and the slope. This is also con-trary to that seen for redox electrodes, such as NiOx [6] where theslope is dependent on the self-discharge potential and changes withself-discharge mechanism (as the Tafel slope of the mechanismschange).

Rather, it was seen that, when the self-discharge potential wasplotted versus log t for various initial potentials (Fig. 3), the slope ofthe self-discharge curve was dependent upon the initial potential;specifically, the magnitude of the slopes increases with increasingpolarization potential. This is contrary to Eq. (2), where the modelpredicts that the slope of the self-discharge profile, when plottedas a function of log time, should be independent of initial charg-ing potential. Therefore, the slope of the self-discharge profile (asplotted versus log time) cannot simply be the negative of the Tafelslope of a Faradaic self-discharge reaction. Conway and cowork-ers suggested [4] that it may be a charge redistribution effect due

Fig. 3. Self-discharge profile for a carbon cloth electrode (ca. 10 mg, in 1 M H2SO4)plotted as a function of log time for various initial charging potentials, (a) Vi = 0.8,0.9 and 1.0 V, (b) Vi = 0.0, 0.1 and 0.2 V, utilizing no hold time.

J. Black, H.A. Andreas / Electrochimica Acta 54 (2009) 3568–3574 3571

Table 1The hold time required to reach full charge and the time required for charge redis-tribution for transmission line circuits of various resistancesa.

Resistance (k�) Hold time to full charge (s) Charge redistribution time (s)

10 150 3420 315 7030 480 10240 640 13450 810 17060 880 19670 1060 23480 1230 25590 1580 306

100 2030 3302

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Fig. 4. Data taken from the hardware transmission line circuit at various positionsdown the transmission line circuit ( RE 1, RE 3, RE 5, RE 7, RE 8), with50 k� resistances, during (a) charging, (b) self-discharge with no hold, and (c) a hold

00 5600 685

a The time to full charge and to completion of charge redistribution was defineds the time at which the potentials of capacitors 1–8 exhibited a <5 mV difference.

ischarge and that this first initial potential drop due to chargeedistribution is what affects the slope of the activation-controlledrocess which follows. To examine this, the charge redistributionffects in a transmission line circuit (which acts as a model pore)ere examined. Results from the model pore were then comparedith the self-discharge of highly porous carbon electrodes.

.2. Charge redistribution and its effect of self-discharge using aransmission line model pore

The distributed potential within a pore which results fromharging/discharging can be modeled with a transmission line cir-uit (as shown in Fig. 1). The capacitor at the top of the circuit (RE) represents the behavior at the mouth of the pore, while capac-

tors further down the circuit represent the behavior toward thease of the pore. This circuit can be used to show how the chargingesults in a distributed potential down the model pore, and how thentroduction of a hold step after charging can minimize the chargeedistribution effect, as this hold time allows all terminals to reachhe desired potential.

Fig. 4a shows that during charging of the RC-circuit the potentialt RE 1 climbed more quickly than the potential at RE 2, and so on,ue to the added resistance down the length of the circuit. At thend of the charging process the terminals at the “pore mouth” (e.g.E 1 and 2) were at significantly higher potentials than those at thepore base” (e.g. RE 7 and 8). Fig. 4b shows that when the systemas placed in open-circuit configuration the potential of the capac-

tors near the top of the circuit fell, while the potential of capacitorsear the back of the circuit climbed until all potentials have equal-

zed, requiring approximately 150 s. However, Fig. 4c shows that ifhold step of 900 s was added after the charge, the potential of ter-inals RE 2–8 climbed to the desired 1.0 V during this hold step,

nd for long enough hold times (ca. 600 s) all terminals reached.0 V. When this condition has been achieved (all the capacitorst the desired potential), there are no charge redistribution effects.ig. 5 shows that as the length of the hold step increased, the masterhannel (RE 1) underwent less self-discharge due to charge redis-ribution throughout the circuit, resulting in higher equalizationotentials. Thus, the addition of a hold step in the charging profilects as both a diagnostic for the presence of charge redistribu-ion and as a means for minimizing charge redistribution in porouslectrodes.

In a typical porous electrode system, it is difficult, if not impossi-le to determine when charge redistribution (i.e. the time required

o reach the equalization potential) is complete. Data obtained withhe RC circuit suggests that time required for charge redistributionn open-circuit is shorter than the hold time required for the wholeystem to reach the full charging potential (Table 1). In each situa-ion, the potentials were said to be equal when there was less than

step. In each case, the voltage of the first capacitor was ramped from 0.0 V to 1.0 Vat 50 mV/s. (For interpretation of the references to colour in this figure legend, thereader is referred to the web version of the article.)

5 mV difference between the potentials on all capacitors 1–8. Dueto the scale, this small difference in potential is difficult to see inFig. 4b and c. That the time required for charge redistribution isshorter than the hold time required for the whole system to reachfull charge can be justified in terms of the total amount of chargepassed in order to reach completion in each situation. Significantlymore charge is required to fully charge (to a potential of 1.0 V) thecapacitors near the base of the model pore (capacitors 6–8) duringcharging than is passed during the charge redistribution. For exam-ple, with a 50 mV/s ramp and 50 k� resistances used in Fig. 4, ca.

0.61 mC of charge is passed during the hold step to fully charge allof the capacitors, while only ca. 0.16 mC of charge is passed dur-ing charge redistribution. Since more charge is passed during thehold step, a longer time is required to pass this charge. In terms

3572 J. Black, H.A. Andreas / Electrochimic

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of the pore size is shown by the dashed line. Thus, the large cumula-tive solution resistance down these long, narrow pores results in avery slow charge at the base of the pore. In terms of holding times,an extremely long hold time is required for the charge to reach

ig. 5. Charging data from hardware transmission line circuit with various holdime. All resistors were set to a value of 50 k�. The voltage of the first capacitor wasamped from 0.0 to 1.0 V at 50 mV/s.

f a real porous system, the charge built up on the wall of the poreust be countered with an ion in solution. Thus, during the holding

tep during charging, when more charge is passed, more ions mustove through the pores to counter this charge, requiring a longer

ime than during charge redistribution when a smaller charge, andherefore smaller numbers of ions, must pass.

As the resistance in the transmission line is increased the holdingime required to completely remove charge redistribution increasesTable 1), although this is not a directly linear relationship. Con-ersely, there is a directly linear relationship between the timeequired for charge redistribution (Table 1) and the resistance inhe transmission line. The increase in the required time to com-lete these processes is due to the fact that the charge that isassed in each of these situations must pass through the cumula-ive resistance for the transmission line, and therefore, with higheresistances the charge will move slower. Similar to the argumentbove, when the wall is charged in a real pore the ions whichre required to counter this charge move down the pore. A higherolution resistance will result in slower ion transport, and slowerharging.

Previously, Conway predicted that the charge redistribution inhe porous electrodes under study required 100 s to complete [4,5].s will be discussed in Section 3.3, the time constant is likelyuch longer than this and may, in fact, extend well into the region

redicted by Conway to be self-discharge due to an activation-ontrolled mechanism.

Given that the effect of charge redistribution may be affectinghe self-discharge profiles of the carbon cloths at longer times thanxpected, including the linear log t region, the self-discharge pro-le of the hardware transmission line circuit was examined whilendergoing charge redistribution. In this way, the effect of theharge redistribution on the self-discharge profile may be deter-ined. Fig. 6 shows the self-discharge of RE 1 of this circuit during

harge redistribution. It can be seen in this figure that the self-ischarge profile falls linearly with log self-discharge time, afterome plateau. This is the same profile as predicted for activationontrol by Conway. Thus, the linear fall in potential with log time isot characteristic of activation control, and may indeed, arise evenhen only charge redistribution is occurring in the system.

This result throws doubt upon the Conway assignment of theelf-discharge profile of the carbon cloth as being due to anctivation-controlled self-discharge mechanism. It also shows that

he Conway model (at least for activation-controlled mechanisms)annot be utilized directly for highly porous electrodes, as theharge redistribution which may take place with these electrodesives a similar profile as the activation-controlled mechanism.

a Acta 54 (2009) 3568–3574

When different initial charging potentials were used on thetransmission line circuit, it was seen (Fig. 6) that the slope of theself-discharge profile, as a function of log t, varies directly with ini-tial potential, with a larger slope for higher initial potentials. Again,this exactly corresponds to what is seen in the porous electrodes(Fig. 3). This supports that the self-discharge profile exhibited bythe porous carbon cloth is not due to an activation-controlled self-discharge mechanism, but is, instead, due to a very long chargeredistribution step down the pores of this carbon.

3.3. Charge redistribution effects in the self-discharge profile ofhighly porous carbon cloth

In order to detect the presence and degree of charge redistri-bution in the porous carbon the potential of the working electrodewas held at the desired initial potential for some period of timebefore the system was switched to open-circuit. A very long holdstep will allow time for the entire electrode surface to reach thedesired potential, and therefore when the system is switched toopen-circuit there will be no redistribution of the charges necessaryto equalize the potential on the electrode surface.

Fig. 7a shows that for an electrode charged to 1.0 V, as the lengthof the hold step was increased, there was a reduction in the self-discharge due to a decrease in charge redistribution. This showsthat charge redistribution is, indeed, occurring in the porous elec-trodes, this result was expected given the porosity of the carbonelectrode. A major reduction in self-discharge was seen after a holdtime of 1 h, and as the length of the hold time increases, the decreasein self-discharge due to charge redistribution was less noticeable.Nevertheless, even after very long hold times (>50 h) charge redis-tribution was still evident (cf. 75 h hold in Fig. 7a). The hold timesrequired to remove charge redistribution were much longer thanexpected, suggesting that the cumulative solution resistance in thepores is very high. Given that the electrolyte used was 1.0 M H2SO4,meaning there is a high concentration of highly mobile protons inthese pores, it is not ion mobility or diffusion which causes this highresistance. The high resistance is likely therefore due to the largeratio of the length to cross-sectional area of the pores, meaning theyare long and narrow. This correlates nicely with the very narrowpores (<2.5 nm), predicted by BET measurements (Fig. 8), whichgives the Spectracarb 2225 its very high surface area. The sharpdrops in the pore size distribution curve seen in Fig. 8 are artifactsdue to the DFT calculations; therefore, a more realistic estimation

Fig. 6. Self-discharge profile for transmission line circuit, plotted as a function of logself-discharge time, for various initial charging potentials using 50 k� resistors anda 50 mV/s ramp rate to charge.

J. Black, H.A. Andreas / Electrochimic

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ig. 7. Self-discharge profile of Spectracarb 2225 in 1.0 M H2SO4 with various holdimes for an electrode charged to (a) 1.0 V and (b) 0.0 V.

he base of the pore. A material with wider and shallower poresould be expected to have a lower cumulative solution resistance

as resistance is directly related to length and inversely related toross-sectional area), and therefore would be expected to requireess time for the charge to reach the base of the pore, and wouldequire a shorter hold step during charging to completely removeharge redistribution.

The large equivalent resistance has repercussions regarding theength of time expected for charge redistribution, since the timeor charge redistribution is related to the RC time constant (andince our capacitance is constant for this circuit, is related to the

ig. 8. DFT calculation of pore size distribution based on BET measurement of Spec-racarb 2225 carbon cloth.

a Acta 54 (2009) 3568–3574 3573

resistance) for the equivalent circuit [3]. The Conway group sug-gested that charge redistribution would take about 100 s [4]. Thedata from the RC circuit (Fig. 4 and other data) suggest that asolution resistance of 50 k�, which only requires 600 s to removecharge redistribution completely requires approximately 150 s tocomplete charge redistribution. Thus, it is unlikely that the carbonhere, which requires more than 50 h to completely remove chargeredistribution only requires 100 s to allow charge redistribution tofinalize (as predicted by Conway and coworkers [4]). Given the timerequired to achieve full charge for the carbon cloth is a minimum of50 h (180,000 s) of hold time, the charge redistribution time for thecarbon cloth is estimated to be in the tens of hours, which is sig-nificantly longer than the time of 100 s suggested by Conway andcoworkers [4]. This, then, suggests that the charge redistribution infact continues well into the region of self-discharge time the Con-way model predicts is activation controlled, and suggests that theself-discharge profile seen in these porous systems is not purelyactivation controlled, if at all. Again, the extremely long predictedcharge redistribution time is due to the high cumulative solutionresistance in the long, narrow pores. Since even during charge redis-tribution, the charge on the carbon surface must be countered byions in solution, the limitation is still the rate of ion movementdown the pores. The high effective cumulative solution resistancein the long, narrow pores of the Spectracarb 2225 dictates thatcharge redistribution will be much longer than the 100 s predictedby Conway for the same material (where a much smaller solutionresistance was assumed). Materials with wider and shallower poreswould be expected to have smaller charge redistribution times;however, as high surface areas are often a requirement for elec-trodes used in energy storage/conversion applications, the chargeredistribution is likely not negligible. Because the time required forcharge redistribution is so long, the role of charge redistribution inself-discharge profile of carbon cloth electrodes and other highlyporous electrodes should not be ignored.

Fig. 7b shows that the addition of a hold step for an electrodenegatively charged to 0 V has no effect on the self-discharge, indi-cating that charge redistribution was not taking place in this case.There are two possible explanations as to why an electrode chargedto 1 V experiences a charge redistribution effect while an electrodecharged to 0 V does not. The open-circuit potential of the Spec-tracarb 2225 is generally around 0.5 V (this varies depending onthe electrode) so when the potential of the electrode is ramped to0.0 V, the electrode is negatively charged, while an electrode at 1.0 Vwould be positively charged. Since the 0.0 V electrode was nega-tively charged its charge will be balanced by the cations in solution(in this case, H+), and the small size and high mobility of H+ meanthat migration was not a rate limiting step in the system. The 1.0 Velectrode, however, will have its positive charge balanced by thesolution anions (HSO4

−), which are larger and less mobile leadingto migration limitations. It is also possible that 0 V may be close tothe potential of zero charge (PZC) for the carbon cloth. If this werethe case, much less charge would be present on the electrode sur-face at 0 V compared to 1 V, and therefore the charge redistributioneffect would be greatly diminished.

As mentioned previously, when the self-discharge profile for acarbon cloth electrode is plotted versus log t, a linear relationshipis seen after time, � (Fig. 2). Conway’s model [5] predicts that thisis due to an activation-controlled self-discharge mechanism. How-ever, the self-discharge data collected with the RC circuit suggeststhat charge redistribution also gives a linear relationship with log t(Fig. 6), and therefore the linearity seen in the self-discharge profile

versus log t may either be due to an activation-controlled mech-anism, or to a charge redistribution process, or perhaps to somecombination of both.

In the self-discharge profile of the carbon cloth there is a plateaupresent (Fig. 2), in agreement with the Conway model for an

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574 J. Black, H.A. Andreas / Electro

ctivation-controlled process. In the Conway model the length ofhe plateau corresponds to �, an integration constant where [5]:

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here ii is the current at potential, i, and i0 is the exchange currentensity for the Faradaic reaction.

This model then predicts that the integration constant, �, isependent on Vi, the initial charging potential. However, the self-ischarge data obtained with both the carbon cloth electrodesFig. 3), and with the hardware RC circuit (Fig. 6) suggests that thelateau length, and therefore the integration constant, is indepen-ent of the initial charging potential. This data then supports thathe self-discharge mechanism of the carbon cloth is not activationontrolled, and is instead related to a very long charge redistributionithin the pores of the electrode.

Based on the above discussion, it is suggested that the linear log telf-discharge profile for the highly porous carbon cloth material isominated by the charge redistribution in the cloth after charging.lthough this profile shape is characteristic of either an activation-ontrolled self-discharge mechanism or a charge redistributionffect, the long charge redistribution times predicted suggest that aignificant portion of the self-discharge is likely due to charge redis-ribution. Additionally, the effect of the initial charging potential onoth the self-discharge slope (when plotted as a function of log t)nd the length of the plateau, �, both conform to that expected fromcharge redistribution effect, rather than activation control.

. Conclusions

This research has shown that adding a hold step during theharging of the highly porous carbon electrodes used for elec-

[[[[

a Acta 54 (2009) 3568–3574

trochemical capacitors minimizes the charge redistribution effectduring self-discharge of these systems. A hold step of more than50 h is required to minimize this charge redistribution, whichis much longer than had been predicted previously. Addition-ally, the highly porous nature of the carbon means that thecharge redistribution step itself takes much longer than anticipated,likely tens of hours, rather than the 100 s predicted previously byConway and coworkers.

It was also shown that the self-discharge profile of a transmis-sion line circuit, having only charge redistribution and no otherself-discharge mechanism, results in a linear self-discharge profilewhen plotted as a function of log time. This profile is also whatwas predicted by Conway for activation-controlled self-discharge.Thus, the Conway model is not diagnostic for activation con-trol in porous electrodes, and the model cannot be applied toporous electrodes unless charge redistribution has been completelyremoved.

Acknowledgements

The authors would like to acknowledge the support of the Nat-ural Sciences and Engineering Research Council for funding of thisproject, as well as Axion Power International Inc. for supplying thecurrent collector.

References

1] B.E. Conway, Electrochemical Supercapacitors: Scientific Fundamentals andTechnological Applications, Kluwer–Plenum Publ. Co., New York, 1999.

2] M. Yaniv, A. Soffer, J. Electrochem. Soc. 123 (1976) 506.3] R. de Levie, Electrochim. Acta 8 (1963) 751.

5] B.E. Conway, W. Pell, T. Liu, J. Power Sources 65 (1997) 53.6] J. Niu, B.E. Conway, W. Pell, J. Power Sources 135 (2004) 332.7] B. Pillay, J. Newman, J. Electrochem. Soc. 143 (1996) 1806.8] K. Kierzek, E. Frackowiak, G. Lota, G. Gryglewicz, J. Machnikowska, Electrochim.

Acta 49 (2004) 515.