Ž .Journal of Electroanalytical Chemistry 443 1998 149–154

On the electrical capacitance of interfaces exhibiting constant phaseelement behaviour

Piotr Zoltowski )

Institute of Physical Chemistry of the Polish Academy of Sciences, ul. Kasprzaka 44r52, 01-224 Warsaw, Poland

Received 20 May 1997; received in revised form 1 August 1997

Abstract

Ž .It is stressed that any attempt to characterise the interfacial double layer exhibiting a constant phase element CPE behaviour byelectrical capacitance is misleading. The only possibility to obtain the value of the interfacial double layer capacitance is to ascribe to thedouble layer a capacitive character. Such a simplifying assumption causes a modelling error and influences all parameters of the system.The problem is analysed for several characteristic examples of an electrode system, where a simple electrode reaction takes place. Thetwo most frequently used definitions of CPE are applied. One of them should be rejected, as intrinsically erroneous. An electrode with adouble layer of CPE character can never be an ‘ideally polarizable electrode’. The conclusions can be qualitatively applied to all systemswith an interface revealing CPE behaviour. q 1998 Elsevier Science S.A.

Keywords: Double layer; Solid electrodes; Electrical capacitance of interface; Constant phase element; Ideally polarizable electrode; Impedancespectroscopy

1. Introduction

At any interface, an electrical double layer exists. It is<present also at the electronic conductor ionic conductor

<interface, i.e., at electrode electrolyte solution interface. Itis characterised traditionally by an electrical capacitancew x1–9 .

Several electrochemical techniques are useful for esti-mation of the double layer capacitance. The most impor-tant are: cyclic voltammetry, ac-voltammetry, bridge meth-

w xods, and impedance spectroscopy 10 . All of them, but thefirst one, are small-signal techniques. In recent years,impedance spectroscopy is the most frequently used and

w xthe most reliable technique 11 .The estimation of the double layer capacitance by

small-signal techniques is relatively simple for systemsinvolving discrete elements, i.e., purely capacitive charac-ter of the double layer. This is the case of the liquid

<mercury solution interface. In order to include the smallesttime constant of the system, high enough frequencies have

<to be used. For the solid electrode solution interface, thesituation is complicated by apparently non-capacitive prop-

) Corresponding author. Fax: q48-22-632-5276, q48-3912-0238; e-mail: pizolt@ichf.edu.pl.

erties of the interface. In modelling this system, the capaci-tance should be replaced by a distributed electrical ele-

Ž .ment. Usually this is a constant phase element CPE . TheCPE is characterised by two parameters, neither of them

w x w xbeing a capacitive one 11–20 21,22 . The origin of CPEbehaviour at the interface has been a subject of many

w xpapers 11–21 . This phenomenon is also called ‘frequencyw xdispersion of capacitance’ 21 . Its physico–chemical

grounds are still not clear, and they are beyond of thescope of this paper. However, in spite of observed CPEbehaviour of an interface, it is frequently characterised by

w xits double-layer capacitance 23,24 . Probably this is be-Ž .cause of an incomplete understanding that i the CPE has

Ž .no capacitive parameter, and ii that a large error resultsfrom this simplification for all parameters in the model.

In this paper, the consequences of the application of theterm ‘double-layer capacitance’ to electrodes revealing aCPE character will be presented and discussed. The resultsof attempts to estimate the value of the capacitance fromthe CPE parameters will be examined, using two defini-

w xtions of the admittance of a CPE 11,22 . The problem will<be analysed for an electrode solution interface in the pres-

ence of a simple electrode reaction, without adsorption atthe electrode surface, and in the absence of faradaic pro-cesses, i.e., for apparently ideally polarizable electrodes.

0022-0728r98r$19.00 q 1998 Elsevier Science S.A. All rights reserved.Ž .PII S0022-0728 97 00490-7

( )P. ZoltowskirJournal of Electroanalytical Chemistry 443 1998 149–154150

However, the conclusions could be applied to other cases,e.g., when complicated processes take place at the elec-trode.

2. Theory

The admittance, Y, of the CPE element can be alterna-tively defined by the phenomenological equationsw x11,19,22,24 :

a fY sQ iv 1aŽ . Ž .CPEŽa. a

a fY s Q iv 1bŽ . Ž .CPEŽb. b

'Ž .where: i is the imaginary unit is y1 and v theŽ .angular frequency vs2p f , f being the frequency . The

Ž a f .coefficient Q or Q Q sQ and fractional exponenta b a b

a are the parameters of the CPE. Generally, y1Fa F1.f fŽ .The definition according to Eq. 1a is recommended by

w x y1IUPAC 19 . The units of Q and Q are respectively Va ba f Ž y1 y2 a f w x. y1ra f a f Žs or V m s 19 and V s orŽ 2 .y1ra f a f .V m s .

A CPE cannot be described by a finite number ofŽ .discrete elements R, C and L with frequency-independent

values. However, for some values of a , it simplifies tofŽdiscrete elements: for a s1, it is a capacitance Q sQf a b

. Ž y1 .sC , for a s0 a reciprocal of resistance Q sR ,f aŽ y1 .and for a sy1 an inductance Q sL . For a s0.5,f a f

the CPE is the Warburg element, used for modelling'Žsemi-infinite linear diffusion Q s 2 =s , where sa W W

. w xis the Warburg parameter 11 .In this paper, we will analyse an electrochemical inter-

Žface, of unit surface area, where the impedance, Z Zsy1 .Y , can be modelled by the electrical equivalent circuit

presented in Fig. 1. The impedance of this system isdescribed by the following equation:

y11Z sR q qY 2Ž .el s CPEŽd .RF

where R represents the ohmic resistance, and R thes F

resistance of reaction taking place at the interface. In thecase of a complicated faradaic process, the latter should besubstituted by a general symbol, Z , representing theF

Ž .impedance of this process. CPE d in Fig. 1 models theelectrical properties of the double layer, and Y in Eq.CPEŽd.Ž . Ž . Ž .2 is its admittance, defined by Eq. 1a or Eq. 1b . If

Fig. 1. Electrical equivalent circuit used in this paper for modelling of the<electrode electrolyte solution interface.

a s1, in both cases Y s ivC , where C is thef CPEŽd. d d

double layer capacitance.<According to the literature, for the solid electrode solu-

w x wtion interface usually 1)a )0.8 12–18,20,23,24 25–fx27 . In this case, the following units can be used as the

most similar to those of capacitance: F my2 syŽ1 ya f . forŽ y2 yŽ1ya f ..1r a fQ and F m s for Q . The often used as-a b

Ž .sumption that C fQ or Q is an error, or at least ad a b

crude simplification.ŽFor infinite value of R no short-circuiting of theF

.double-layer element and a equal to one, the electrode isf

traditionally called ‘ideally polarizable’. There is no dissi-pation of energy. However when a -1, the above namef

can no longer be used, because dissipation of energy is anŽ .intrinsic feature of a CPE see Fig. 2c . Such an electrode

is never ideally polarizable.The impedance spectra, IS, of electrochemical systems

are usually presented and analysed in Bode andror variousw xcomplex plane coordinates 11 . The latter are more appro-

priate for analysis of systems revealing distributed elementŽ Y X.behaviour. The impedance plane Z, i.e., yZ vs. Z is

the most frequently used complex plane plot. TheŽ 2 .impedance units are V or V m . For the model pre-

sented in Fig. 1 and a equal to one, the IS plot in thisf

plane is defined by the equation of the semi-circle. Theresistive elements of the model are the only parameters of

Ž Y XŽ ..this equation yZ sZ R , R . Hence, the plot is resis-s F

tance-sensitive. The values of the resistive elements can beestimated directly from the plot. However, the estimationof C needs calculation from the characteristic frequency,d

v ) , at the top of the semi-circle, i.e., at the maximal valueof yZY :

y1) y1v st s R C 3Ž . Ž .F d

where t is the discrete time constant of the system.There are other immitance functions that are capaci-

Ž .tance sensitive. Among them are elastance vZ and com-Ž y1 . w x 1plex capacitance Yv 12,19,21,28–32 . These func-

tions can be treated as impedance and admittance nor-malised by frequency. The units are Fy1 m2 and F my2 ,respectively. The IS plots in the two planes are more

Ždiversified than in the Z one, hence, more informative for. Ž .elastance plane see Fig. 3 . They consist of i a circular

section starting from the origin, with the circle centre onŽ .the imaginary axis, and ii a straight line section with

Ž . Ž .slope equal to zero. Sections i and ii are low- andŽ .high-frequency ones in the vZ plane see Fig. 3 , and

inversely in the Yvy1 plane. Extrapolation of the high-frequency straight-line section in the vZ plane to vZX s0

1 Two other functions are the modulus function, M, and dielectricpermittivity, ´ . They can be defined as elastance divided by a particularimaginary quantity, and complex capacitance multiplied by this imaginaryquantity, respectively. Both these functions are frequently used for solid

w xelectrolytes 11,22 .

( )P. ZoltowskirJournal of Electroanalytical Chemistry 443 1998 149–154 151

Fig. 2. Complex plane impedance plots of synthetic IS data generated for the equivalent circuit of Fig. 1, for parameter sets given in Table 1 and variousŽ Ž . Ž .. Ž . Ž .a values see Eqs. 1a and 1b . Solid and open symbols are for Y according to Eqs. 1a and 1b , respectively. `, solid lines are for a s1; I,f CPEŽd. f

dotted lines for a s0.98; ^, dot-dashed lines for a s0.95; e, short-dashed lines for a s0.9; \, long-dashed lines for a s0.8. Numbers at the curvesf f f fŽ . Ž X . 2 Ž . Žindicate frequency. a Parameter set A. a Parameter set A recalculated to unit surface area of 1 cm . b Parameter set B solid and open symbols are

. Ž X . 2 Ž .overlapping . b Parameter set B recalculated to unit surface area of 1 cm . c Parameter set C.

gives the reciprocal of C . Hence, this plot can be used ford

direct estimation of C , even when a complicated electroded

process takes place in the system, i.e., when R in theF

model of Fig. 1 is substituted by Z . The vZ plot reducesFŽ . Ž .to i a semi-circle when R equals zero, and ii as

Žstraight-line section when R equals infinity ideally polar-F.izable electrode .

When a -1, the IS data plot in the Z plane is thefŽso-called depressed semi-circle circle section limited to

. w x Ž .the 1st quadrant 11,19 . Its characteristic features are: ithe centre of the circle is depressed into the 4th quadrant,

Ž .and ii the tangent of the section at its high-frequency

limit has a finite slope, deviating from verticality by theŽ . Ž Ž .Ž .. Žconstant phase angle CPA , f fs pr2 1ya seef

.Fig. 2 . Hence, this plot is no longer defined only byŽ .resistive elements, and Eq. 3 holds no longer. t is a

distributed quantity. The information on a can be eventu-fw xally obtained from the highest-frequency data 11,19,22 .

However, no information on the value of Q can be ob-tained from the plot.

The shape of the vZ plot is very sensitive to minordeviations of a from unity, even at medium frequenciesfŽ . Ž .see Fig. 3 . This is observed as i important deformationof the low-frequency arc, especially at higher frequencies,

( )P. ZoltowskirJournal of Electroanalytical Chemistry 443 1998 149–154152

Fig. 3. Complex plane elastance plots of the same IS data as in Fig. 2.Ž . Ž .Symbols and lines as in Fig. 2. a Parameter set A. b Parameter set B.

Ž .c Parameter set C.

Ž .and ii an apparent deviation of the straight-line sectionw xfrom zero slope 28–32 . No information on Q can be

estimated from the plot, similarly as in the case of the dataplot in the Z plane. The information on Q can be obtained

only by fitting computations.Ž .For a -1, Q Q or Q is never equal to C . There isf a b d

no possibility of estimation of C value from Q values.d

The only way is to repeat computations with a fixedf

equal to one. It is equivalent to the assumption that theinterface has capacitive properties. The consequence ofsuch an assumption is presented in Section 3.

3. Numerical examples

IS data were computed for the equivalent circuit of Fig.1, in the frequency range 1–1=105 Hz, with 10 logarith-

Ž .mically distributed frequencies per decade. Eqs. 1a andŽ .1b were applied alternatively in calculations of Y ,CPEŽd.with the following values of a : 1, 0.98, 0.95, 0.9 and 0.8.f

Three sets of R , R and Q values used in datas F

generation are given in Table 1. These sets are for 1 m2

surface area. IS data were also generated for sets A and Bafter their recalculation to 1 cm2 surface area. Theseadditional sets will be named AX and BX, respectively. Set

ŽA represents a moderately fast faradaic reaction R s100F2 .V cm on a perfectly smooth metallic electrode surface

Ž y2 .if a s1, Q sQ sC s20 mF cm in a low resis-f a b dŽ 2 .tance solution R s1 V cm . Set B represents the sames

system after replacement of the smooth electrode with anelectrode of roughness factor equal to five. Set C isequivalent to set A in the absence of faradaic reaction; thevalue of R was increased by six orders of magnitude, inF

order to approximate infinite resistance.The generated data are presented in Fig. 2 in the Z

plane. The increase of the Q or Q parameter of CPEa b

moves the data at a given frequency on the semi-circles toX Žthe left, i.e., to lower Z values compare Fig. 2a and b, or

X X. 2Fig. 2a and b . The recalculation of parameters from 1 mto 1 cm2 surface area has no influence on the data distribu-tion on the depressed semi-circle sections when the YCPEŽd.

Ž . Žwas defined by Eq. 1a compare solid points in Fig. 2aX X.and a , or Fig. 2b and b . This is not the case when YCPEŽd.

Ž . Ž .was defined by Eq. 1b open points . It should beconcluded that the IS data are not directly scalable with

Ž .area when Eq. 1b is used for Y .CPEŽd.When a decreases, the semi-circle is gradually de-f

pressed and the data move at given frequency to the rightX Žhand side, i.e., to higher values of Z solid points, calcu-

Table 1Ž . Ž .Parameter sets of equivalent circuit of Fig. 1, and Eqs. 1a and 1b , used

Žin generating synthetic IS data a values were alternatively: 1, 0.98,f.0.95, 0.9 and 0.82 2 y2 yŽ1ya .fSet R rV m R rV m Q rFm ss F a

y2 yŽ1ya . 1raf fŽ .and Q r F m sb

y4 y2A 1=10 1=10 0.2y4 y2B 1=10 1=10 1y4 4C 1=10 1=10 0.2

( )P. ZoltowskirJournal of Electroanalytical Chemistry 443 1998 149–154 153

Ž . Ž Ž . .Fig. 4. Result of fitting of the equivalent circuit in Fig. 1 with a capacitance in place of CPE d Y according to Eq. 1a , a s1, Q sC to theCPEŽd. f a dŽ . Ž .synthetic IS data see Fig. 2 and Fig. 3 vs. a used in their generation. ` and solid lines for set A of parameters used in data generation Table 1 ; I andf

Ž . Ž . Ž . Ž .dotted lines for set B; ^ and dashed lines for set C. Error bars in b , c and d indicate the standard deviation of the given best-fit parameter, s . apjŽ . Ž . Ž . Ž .Standard deviation of fits, s . b Ratio of the best-fit C C to Q used in data generation. c Ratio of the best-fit R R to R used in datafit d dŽfit. a F FŽfit. F

Ž . Ž .generation. d Ratio of the best-fit R R to R used in data generation.s sŽfit. s

Ž . .lated according to Eq. 1a for Y . The comparison ofCPEŽd.Fig. 2a and c illustrates the effect of the increase of RF

from 1=10y2V m2 to infinity. As a result, the semi-circle

arc evolves into a straight line of the same slope as thehigh-frequency tangent of the original semi-circle arc,being dependent on a .f

IS data generated for sets A, B and C are presented inFig. 3 in the vZ plane. The deviation of the system fromcapacitive behaviour is clearly visible in these plots at

Ž . Ž .frequencies below 1 kHz Fig. 3a , 100 Hz Fig. 3b andŽ .even close to 1 Hz Fig. 3c .

Ž . Ž .Since the CPE d according to Eq. 1b is not linearlyscalable with area, this definition should never be used.

It is interesting to observe the numerical errors createdwhen one tries to estimate interfacial capacitance, and alsoresistive elements of the model consisting only discrete

Ž .elements, despite CPE d behaviour indicated by the ISdata. For this purpose, the model of Fig. 1 was fitted to theabove synthetic IS data, under the assumption of capacitive

Žbehaviour of the double layer a s1, i.e., Q sC ,f a d.Y s ivC , in spite of the value of a -1 used inCPEŽd. d f

data generation. The fittings were performed using the

w xZView software 22 . Proportional weighting of IS dataw xwas applied in fitting 11 . The results are presented in Fig.

4. To distinguish the computed best-fit parameters fromthose used in data generation, the former are named R ,sŽfit.R and C .FŽfit. dŽfit.

In general, a large value of standard deviation of the fitof a model to IS data, s , disqualifies either the data orfit

the applied model. Usually the results of modelling arequestionable when s is higher than ca. 5%. In thefit

present case, s could result only from the modellingfit

error. In Fig. 4a the values of s are plotted against afit f

values used in data generation. In the case of a s1, sf fity4 Ž .was equal to ca. 2=10 % instead of zero . This re-

sulted from the truncation of the synthetic IS data to sixdigits prior to fitting the model. The smaller a became,f

the larger s became. This effect is specially pronouncedfit

in the case of large R .F

The ratios of C , R and R to Q , R and RdŽfit. FŽfit. sŽfit. a F s

are a measure of consequences of simplification of thesystem by substitution of the CPE with a capacitance. Fora s1 all three ratios should be equal to one. They aref

plotted in Fig. 4b, c and d against a used in dataf

( )P. ZoltowskirJournal of Electroanalytical Chemistry 443 1998 149–154154

yŽ1 ya f . Žgeneration. For example, C rQ s0.2 s see Fig.dŽfit. a.4b indicates that the magnitude of the best-fit C is fived

times smaller than the magnitude of Q used in dataa

generation. Error bars indicate the computed standard devi-ation of the given parameter, s .pj

The largest change of parameter value is observed forŽR in the case of an ‘ideally polarizable electrode’ Fig.F

.4c . When a s0.8, R is smaller by almost four ordersf FŽfit.of magnitude than R . R never differs from R moreF sŽfit. s

Ž .than by a factor of about two Fig. 4d . The C rQdŽfit. a

magnitude is reduced to approximately 0.2 when a de-f

creases. No important influence of the Q value on Ca dŽfit.Ž .is noticed Fig. 4b . High values of R decrease theF

influence of a on C .f dŽfit.It should be stressed that the values of s , indicated forpj

some symbols in Fig. 4b, c and d, are much smaller thanthe real deviation resulting from the application of the

Ž Ž ..simplified model i.e., C in place of CPE d . This resultsd

from simplifying assumptions applied generally in algo-rithms for computation of standard deviations of individual

w xmodel parameters 11 . Therefore, the computed parameterstandard deviations should be treated with caution, asunderestimated standard deviations.

Evidently, any attempt at estimation of equivalent cir-cuit parameters directly from the IS data plots gives erro-neous values if a -1. The quantitative relations will bef

dependent on the system, especially on the ratio of R toF

R and the presence of Z in place of R , the frequencys F F

range and number of data per frequency decade, noiselevel of the IS data, and the weighting system used in thefitting computation. However, the picture presented abovewill probably apply qualitatively in all cases.

4. Conclusions

1. If the electrode double layer has a distinctly CPEcharacter, the error in all parameters of a capacitive modelof the system is high. Estimation of capacitance from thevalue of the Q parameter of the CPE is illogical.

2. The interfacial double layer involving CPE is in anycase a dissipative one. Such an electrode is never ‘ideallypolarizable’.

3. The definition of CPE admittance according to Eq.Ž .1b should never be used, since it does not scale linearlywith electrode area.

Acknowledgements

This work has been sponsored by State Committee forScientific Research through grant 2 P303-075-07. Prof. Z.

Borkowska, Dr. M. Opallo and Dr. A. Sadkowski from theInstitute of Physical Chemistry of the Polish Academy ofSciences are gratefully acknowledged for helpful discus-sions.

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