Annals of BiomedicalEngineering,Vol. 20, pp. 269-288, 1992 Printed in the USA. All rights reserved.

0090-6964/92 $5.00 + .00 Copyright 9 1992Pergamon Press Ltd.

Linear and Nonlinear Electrode Polarization and Biological Materials H.P. Schwan Department of Bioengineering University of Pennsylvania Philadelphia, PA and Biomedical Engineering and Science Institute Drexel University Philadelphia, PA

(Received 4/16/91; Revised 10/9/91) Electrode polarization is a major nuisance while determining dielectric properties o f cell and particle suspensions and tissues, particularly at low frequencies. Understanding of these interfacial phenomena and appropriate modelling are essential in order to correct for its distortion o f the dielectric properties of the sample o f interest. I survey the following topics, concentrating on contributions from our laboratory: Linear properties of electrode polarization and relevant models Effects of electrode polarization on sample impedance Effects of sample on polarization impedance Techniques o f correction Extension of linear to nonlinear models Harmonics generated in the nonlinear range. Keywords--Bioelectrodes, Electrodes, Linear, Nonlinear, Impedance, Polarization.

INTRODUCTION This b r i e f review is c o m p o s e d o f t h r e e p a r t s : P a r t 1 deals with electrode p o l a r i z a t i o n p r o p e r t i e s a n d states o u r , a n d s o m e o t h e r previous, attempts to model the p o l a r i z a t i o n interface. T h e m o d e l considerations were guided b y m e a s u r e m e n t s over a b r o a d f r e q u e n c y range extending f r o m a few m H z to m o r e t h a n 100 k H z . M o d e l c o n s i d e r a t i o n s m a y be either m a t h e m a t i c a l f o r m u l a t i o n s o f e x p e r i m e n t a l o b s e r v a t i o n s , o r t h e y m a y be b a s e d o n p h y s i c a l insight. B o t h h a v e b e e n used in t h e past. P a r t 2 discusses the effects o f electrode p o l a r i z a t i o n o n biological i m p e d a n c e s a n d the effects o f b i o l o g i c a l samples on the i n t e r f a c e i m p e d a n c e . M e t h o d s to c o r r e c t f o r e l e c t r o d e effects o n s a m p l e d a t a are p r e s e n t e d . Address correspondence to H.P. Schwan, Department of Bioengineering, University of Pennsylvania, Philadelphia, PA 19104.

269

270

H.P. Schwan

Part 3 extends results from the linear to the nonlinear range with voltages up to approximately 1 volt applied to the interface. Our nonlinear interests were motivated by the fact that electrodes can be easily driven into this range, particularly in many biological and clinical applications. POLARIZATION PROPERTIES Kohlrausch (12) was probably the first to recognize the effects of electrode polarization on electrolyte conductivities. In order to reduce these effects he found the recipe for covering platinum with a layer of Pt-black, thereby reducing the electrode surface impedance by orders of magnitude. It was also recognized that the electrode surface impedance has a capacitive imaginary component. Following tradition, like Warburg, we represent the electrode surface impedance (Zp) as a series combination of a resistance (Rp) and capacitance (Cp) (1)

Zp = Rp - j/o~Cp

with co = 2 ~rf angular frequency. At about the same time Warburg (31,32) developed the first theory for the electrode polarization phenomenon, based on diffusion considerations. He stated that the electrode surface polarization capacitance and resistance are both proportional to f-0.5 and that the impedance has a frequency independent phase angle of 45 ~. At the time, it was not possible to check this law over an extended frequency range. Later, as data extending over a few frequency decades became available, Fricke (5) formulated his extension of the Warburg law

C p - f-m;

Rp - f -(l-m)

and

6=mTr/2

(2)

where 6 is the phase angle defined by tan 6 Rpo~Cp. Fricke's law was not based on physical insight, but a description of then available data and the realization that a power function of either component must yield the phase angle rule: The relationship between Rp and Cp expressed in Eqs. 1 and 2 can be shown to be demanded by the Kramers-Kronig relations (13) if a power law frequency dependence of either quantity is assumed. During the fifties, we checked the Fricke relationships over a broader frequency range (21). Our measurements extended from 10 Hz to 200 kHz. They revealed that m is not constant, but changes from a value near 0.3 to 0.5 as the frequency increases from below 1 to more than 10 kHz. Nevertheless, the Fricke prediction for the phase angle turned out to be quite good. Figure 1 shows a typical example o f many measurements. These results enabled us to improve our technique to correct for electrode artifacts in our biological impedance work as discussed below. Jaron et al. (1 I) modeled this behavior as a Cole-Cole function (2) =

c* = c' -jc"

Co

= 1AoZp = 1 + (jwT) ~

(3)

A similar expression quoted by Buck (1) is

zp0 Zp-

1 + (floTo) ~

(4)

Linear and Nonlinear Electrode Polarization

271

IOOC

--

L

Rp (OHM)

I00

Cp(~F) TAN S

IOC,-

~

~

'

~

~

~

--

I0

--

I

Rp

i IO

0.01

_

-

0.1

I

I0

I00

I000

FIGURE 1. Frequency dependence of Cp,apand tan ~ from 20 Hz to 2 0 0 kHz. Pt-electrode area 1.4 mm 2. Medium coverage of Pt-black, physiological saline solution. Dashed lines indicate values calculated from Fricke's law at low and high frequencies. Fricke's law is valid even though tan 6 changes. The slight change in the slope of the Rp-curve at high f is not necessarily significant as the measurement is sensitive to minor changes in the electrolyte series resistance. (Reprinted by permission of Academic Press [21].)

Equation 3 can be represented by the series combination of a frequency independent blocking capacitor with a constant phase element (Warburg element). This model is similar to the well known combination of a Helmholtz capacitor in series with a G o u y - C h a p m a n diffuse layer. Equation 4 in turn puts a frequency independent resistor in parallel with a constant phase angle element. Equation 3 reduces for DC to a capacitor, Eq. 4 to a resistor. The Jaron et al. equation, while approximating data above 10 Hz, is inadequate at lower frequencies because it demands a frequency independent Cp. Above 1 Hz the models given by Eqs. 2, 3, 4 appear to be identical since they reduce at higher frequencies to a constant phase angle behavior. Only the model given by Eq. 4 provides for the finite DC impedance, which is experimentally observed. 1 Measurements at frequencies extending into the m H z range became possible with the introduction of some impedance analyzers. Figures 2 and 3 show results obtained with Pt-electrodes. The polarization capacitance increases strongly as the frequency decreases below 1 Hz. This is necessary if the polarization impedance is to remain finite at DC. The data also illustrate that the locus in the impedance plane is anything

1Measurement of the DC impedance requires great care. Since the linear electrode impedance changes down to mHz frequencies, correspondingtime constants range up to an hour. Thus application of a change in DC potential yields stable current only after considerable time. The determination of the linear DC impedance furthermore requires recognition that the onset of nonlinearity occurs at very small current densities in the #A/cm 2 range (See Fig. 7 of [17]). This has been indeed our experience as we studied the electrode interface in the time domain (18).

272

H.P. Schwan i

Tll 1

1

l ul I

-

105

u

niT I

u

inn

I

Ulrl--rIu1104

r

Zp

104

1o

Rp ,Zp

~

Rp IoJffF)

(Y/') ~o3

I02

10

0,0011

I nll

I I III~

0.01

n lll

I Lall

0.1 I 10 FREQUENCY (Hz)

t I lll

100

I I

1000

FIGURE 2. Frequency dependence of Cp,Rpand Zp from 0.001 to 1000 Hz. R-electrode. Electrode area 0 . 0 8 5 5 c m 2. 0 . 9 % NaCI. pH 6.5, 25~ (Reprinted by permission of Medical & Biological Engineering & Computing News [17].)

but a straight line through the origin, representing constant phase angle behavior. The locus at low frequencies instead may be approximated by a circle as predicted from Eq. 4. Asymmetrical deviations from circular behavior can be better described by a Davidson-Cole like function (3), reflecting the existence of a logarithmic asymmetric T-distribution.

Xp

l ..--"-~.~F7o'~ 63-. 4o F .J .oo4~. j_ ,-'[" .04 .002~.001 201- J " low frequencies I" I-."001Hz o0, r ,20, , ,40, , , 60 , , , ,80, , ,100 140 , , , Rp (k&"~,)

8~Xp (kL'~,)

Xp(k..Q,) 0.8

/" /', 6

6 F /'2~ ~ "4 4 ~ .~,,~6/'5medium frequencies 2 ~-/'/ " 40--1Hz 40 0

~-

0

I

2

I

I

I

I

4 6 Rp (k..Q,)

I

I

8

I

06

, .. 2~

/

/

/ //*'40

/_ ;63

/ ../loo

0.4 60 hrleghuencies 0.2 /'400 999-*25 Hz 0

919'~ _ I

0

I

I

I

1

0.2 0.4 0.6 Rp(k,Q,)

FIGURE 3, Impedance locus for the frequency range from 1 mHz to 1 kHz. Electrode as in Fig. 2. (Reprinted by permission of Medical & Biological Engineering & Computing News [17].)

Linear and Nonlinear Electrode Polarization

273

Zpo Zp-

(1 + j o l T ) ~

(5)

A Davidson-Cole distribution might sometimes be more suitable to characterize the processes involved in generating this behavior than the assumption of a logarithmic symmetrical distribution function yielding Eq. 4. 2 Both Sun and Onaral (30) and Sun et al. (29) discussed models which include further generalizations of the Cole-Cole and Davidson-Cole approaches to better describe experimental data (29). Fractal and electrochemical mechanisms account predominantly for observed behavior at high and low frequencies as discussed in this issue by Buck (1) and de Levie (4). POLARIZATION EFFECTS AND BIOLOGICAL SAMPLES Influence o f Electrode on Sample

Biological samples such as cell suspensions and tissues are highly conductive. Therefore their tan &values increase as the frequency decreases and the resistive component of their impedance dominates. Under such circumstances the effects o f the electrode polarization can be approximated by the following equations (21)3: R = Rs + Rp + R s ( R w C ) 2

(6)

C = Us + 1/o~2R2Cp

(7)

Here, Cs and Rs are sample capacitance and resistance as indicated in Fig. 4, C and R the measured values, 00 the angular frequency. The last two terms of Eq. 6 are usually small compared to the first. However, the second term of Eq. 7 can be large. Thus, the effect of the electrode is usually more pronounced on the capacitance than on the resistance. Figure 5 illustrates this behavior. Figure 6 demonstrates the very small effect on the same sample's resistance and the high resolution often necessary to demonstrate the electrode effect on the resistance. Influence o f Sample on Electrodes

Electrode impedance values are influenced by the concentrations o f cells or particles in suspension. This is indicated in Figs. 7 and 8. Here, Cp and Gp are plotted against the volume fraction of erythrocytes, with the 100~ level representing the most concentrated sample. The curves appear to merge on the axis at a level which we believe to represent the true 100070 packing case. The effect is caused by the erythrocytes 2I have demonstrated elsewherethat any logarithmic symmetricdistribution of time constants will generate in excellent approximation a circular arc with depressed center in the complex impedance or dielectric plane (20; see pp. 154-157). 3Assumptions necessaryto yield the equations are: RtoC< 1 and Rp < R. This is usually the case if the polarization impedance values are smaller than those of the sample (21; see pp. 344-345).

274

H.P. Schwan

@

@

9 • Rp

ELECTRODES~/

I

//3

ram--

i I

I

L__

IR

I

.,i

I

~

7

I I

Ic,T SAMPLE

_L

/

/

I

J

FIGURE 4. a) Sample in contact with electrodes, b) Equivalent circuit. The polarization impedance of the electrodes Zp = Rp + 1/j~Cp is in series with the sample admittance Ys = 1/Rs +JcoCs. c) Circuit of observed total admittance Y = 1/R + j u C . (Reprinted by permission of Academic Press [21].)

I!,,

I0'

'~ pF 8

i

SC(

4

'~ i I0 0.01

2

R2Cp

4

" FREQUENCY (KHz) 6 BO.I 2 4 S S I

Z

I' 4

I 6 e I0

FIGURE 5. Effect of electrode polarization on the apparent capacitance C of a concentrated suspension of blood cells. Pt-electrode with heavy black cover. Area 5 cm 2, distance 8 cm. The capacitance C is the sum of t w o terms, one the sample capacitance Cs and a term 1/~2R2Cp caused by the electrode-sample interface (see Eq. 7). The large electrode area reduces stray field errors and permits measurement at a large electrode distance, thereby minimizing the electrode effects. (Reprinted by permission of Academic Press [21].)

275

Linear and Nonlinear Electrode Polarization 791.6

1.4

1.2

1.0

t90.8

i

0,6

0.~

0.01

--[- J

J

I

,

2

5

0,1

2

FRErQU ENIC Y (KHz) 5

I

Z

,

I

,

5

I0

Z

~

IO

FIGURE 6. Effect of electrode polarization on the apparent resistance R of a concentrated suspension of blood cells9 Electrode parameters and sample as in Fig. 5. The corrected sample resistance Rs is frequency independent below 1 kHz to 1 part in 10 s. This would not have been recognized without correction for electrode polarization, even though the change in apparent R due to polarization is only 0 . 0 4 % at 20 Hz. A comparison of Figs. 5 and 6 demonstrates that the effect of electrode polarization is orders of magnitude larger on the sample capacitance than on its resistance. Theory predicts that the ratio of the two relative errors is given by tan 6 = Rso~Cs if the sample is highly conductive so that its phase angle is much lower than that of the electrode9 (Reprinted by permission of Academic Press [2113

which are poor conductors at low frequencies: The current bypasses the erythrocytes. Thus, the current passing through the sample reaches only part of the electrode surface because the cells shield part of it. This shielding effect is the more pronounced the higher the particle concentration, causing the decrease of the polarization parameters with increasing volume fraction. Deviations of the curves from linearity are shown in Fig. 9 and appear to depend strongly on the degree of preparation o f the electrode surface (21,22). This complex behavior has remained a mystery. In the light of modern knowledge a possible suggestion about fractal systems comes to mind: A suspension of poorly conducting biological cells or particles will conduct current primarily t h r o u g h its interstitial spaces between the cells. The entry of current into such a suspension is followed by subsequent divisions similar to current entering a fractal system. Thus the interaction of a suspension with a rough or fractal electrode has much in c o m m o n with two different fractal systems in series. However in the case of suspensions, recombinations of currents occur after each layer of cells, compensating for current splitting.

276

H.P. Schwan 1:5

I

G(p~176

\\ \ P ( R ELATIVE) %

I 0

20

40

60

80

I00

120

140

FIGURE 7. Polarization conductance Gp = llRp as a function of erythrocyte concentration p. A value of 100% was assigned to the highest concentration achieved. The extrapolations (dashed lines) suggest that 125% may correspond to the true 100% packing case. These curves and those in Fig. 8 suggest that studies of polarization impedances as function of concentration may be useful to establish true packing factors (a problem which often causes concern in work with biological suspensions).

Methods to Correct f or Electrode Effects Figure 5 demonstrates how one can correct for the electrode effect in order to obtain true sample capacitance values. The technique assumes validity of Fricke's power function and extrapolates the low-frequency behavior where polarization dominates. Subtraction of the extrapolated values from the measured data yields true sample values as indicated by the circles. As the frequency decreases comparable numbers are subtracted from each other, limiting accuracy. The technique extends the frequency

Linear and Nonlinear Electrode Polarization

277

S.=O3 -

Cp(pf) 7

6

5

4

\

"N, \

0

I

I

I

I

0.2

0.4

0.6

0.8

-

p

FIGURE 8, Polarization capacitance Cp as a function of erythrocyte concentration p. Same sample and electrode arrangement as in Fig. 7, Again, all curves appear to intercept at the same point (now chosen as p = 1) with the abcissa. (Reprinted by permission of Academic Press [21].)

range of correct sample values by about a decade, provided that the electrode polarization impedance does not change during the frequency run. Figure 10 demonstrates a sophistication of the technique. Measurements at a small and a large electrode distance are taken. The small distance data are largely influenced by electrode polarization and thereby demonstrate the last term in Eq. 6. The curve is then shifted downwards by the squared ratio of the two resistances before being subtracted from the curve for large distance. The technique no longer assumes validity of Fricke's law. However it demands that sample distance variation does not change the electrode surface properties. Both techniques to correct for electrode effects can be performed graphically or numerically. A related distance variation technique plots the measured impedance against distance. The intercept with the ordinate gives the polarization impedance and the slope of the impedance per unit sample length. Distance variation is one of the best techniques provided that constancy of the electrode impedance is checked. In dielectric spectroscopy, 0.1 to 1 volt is typically ap-

278

H.P. Schwan

~F

(a)

I0" Cp

0.05-I/Rp M H0

\ \ P

I 0.8

P

1_ 0.8

(b)

so~

(b)

i,UF

I/Rp

Cp

\ \

P

I 0.8

_ _

_

P

_

08

)JF I000

(c)

Cp

I/R p MHo

P 0

I 0.8

P 0

I o J3

FIGURE 9. Polarization capacitance and conductance as a function of erythrocyte concentration. The effect of electrode preparation is demonstrated. The strongly curved plots were obtained with Pt free of any platinization layer, the straight plots with heavily Pt-black coated electrodes and the intermediate plots with a medium Pt-black coverage. Frequency near 1 kHz.

plied to the sample. Sample impedances are of the order of some hundred ohms with biological cells suspended in a physiological medium. Thus current densities are of the order of m A / c m 2, corresponding to the onset of nonlinear electrode behavior discussed below. Hence, electrode impedances may be different for different distances unless care is taken to check either for constant current or use of currents sufficiently low to assure linear electrode behavior.

Linear and Nonlinear Electrode Polarization

279

J

iO:' 6 4

2

104 8 6

T

I0 8 6 4

O~ 0

o

0

t2~

8

',

+

I

6 4

2

Io

o.ol

0.1

F r e q u e n c y (KHz) I I

I0

FIGURE 10. Correction technique based on electrode distance variation. Measurements were conducted at electrode distances of 15 and 0 . 8 cm. The resistance and capacitance ratios for the t w o distances was 19 at high frequencies, The line approached by the upper curve is shifted downwards by a factor of 192, as demanded by Eq. 7. It is then subtracted from the lower curve in order to obtain the correct sample capacitance indicated by open circles, (Reprinted by permission of the Annals of the New York Academy of Sciences [23],)

One frequently used technique to correct for electrode polarization substitutes the suspension by the medium surrounding the cells or particles in suspension. The electrode impedance obtained with the medium is assumed to be identical to that of the suspension since, so it is argued, in both cases it is the same fluid compartment in contact with the electrodes. The data presented in Figs. 7 and 8 demonstrate that this argument is false and, therefore, the technique is not correct unless the concentration of particles is low. Next, we discuss the importance of electrode preparation. Clearly, from Eqs. 6 and 7, electrode effects on C and also R decrease as Cp increases. Hence large values of Cp are desired. For a long time, Kohlrausch's recipe for coating Pt-electrodes by electrolysis was considered the rule to follow. We undertook a systematic study to find the best combination of current density and time during electrolysis to achieve the re-

280

H.P. Schwan Sondblosled

Cp ( j u f / c m 2) 50,O00

I0 m A / c m 2

40,00(] --

~

I0 m A / c m 2 (Emery Clolh)

30,000 - -

20 m A / c m 2 20,000 --

50 m A / c m 2

5 mA/cm 2

..=_._

I 0,000

o ~--------~ O

IO

I

20

l

30 D u r o t i o n of

I

40 Plotinizin

I

50 9 (minutes)

I

60

I

70

FIGURE 11. Polarization capacitance per cm z of t w o identical electrodes separated by physiological saline solution at 20 Hz as function of platinization. Parameter: current density.

suits summarized in Fig. 11 (21,22). The optimal values of about 50,000 # F / c m 2 at 20 H z probably belong to the highest demonstrated so far. They are significantly higher than those obtainable using the Kohlrausch recipe at the same frequency. However, it has been our experience that Pt-electrodes with such high polarization admittances decline during use more rapidly than electrodes covered with a more moderate thickness of Pt-black. Another technique to reduce electrode artifacts is to reduce the conductivity of the medium surrounding the particles or cells of interest. This would correspondingly increase R in Eq. 7 and decrease its last term. Unfortunately changes in the medium's fluid can affect the suspended phase properties. Furthermore, we and others observed the electrolyte's ion concentration to influence polarization parameters. 4 A theoretically anticipated square root dependence on the electrolyte concentration c, is observed to be only approximately true.5 At low concentrations, the polarization capacitance Cp appears to decrease more rapidly with decreasing concentration than predicted

4Relevant data are not reviewed since they do not agree about the extent of this deviation from theory. Variations between data may be caused by nonlinearity effects. 5Diffusion controlled processes depend on the square root of concentration. If, however, the electrode properties are caused by the fractal nature of the surface, an exact or approximate square root dependence on the inverse conductivityis suggested, with the conductivityof the electrolyte being about proportional to its concentrations.

Linear and Nonlinear Electrode Polarization

281

from theory. Therefore, suspending particles or biological cells in a weaker electrolyte does not reduce the electrode effect on the sample capacitance significantly, because the gain from increasing the R-value in Eq. 6 is offset by the decrease of the polarization capacitance Cp. The most elegant technique to eliminate electrode artifacts uses different current and potential electrodes (21,22,24). A typical arrangement is indicated in Fig. 12. If the potential sensing electrodes are connected to a very high impedance amplifier, no current passes through them, hence, no AC electrode polarization potential can develop, since the polarization potential is given by the product of electrode impedance and current through the electrode-electrolyte interface. Thus the potential across the sample confined by the equipotential surfaces connecting with the sensing electrodes is correctly recorded. The current electrode impedances do not disturb the sample impedance since they are part of the output impedance of the current generator. Care must be taken that no current enters or leaves the sensing electrodes in order to exclude the possibility that imbalance between the potentials generated will lift the electrodes to a wrong potential. This can be achieved by various means, including removing the sensing electrode surface from the sample by use of a salt bridge (21,22). Difficulties arise when stray field components are not well under control. This can lead to apparent relaxation effects which are not caused by the sample. Some commercial equipments appear to suffer from this problem even when used in the two-

CURRENT SOURCE

~

~HIG

H

V

#

,

Ve

Z

|

@

9

FIGURE 12. Four electrode technique. Figure (a) indicates the principle: No potentials are induced at the potential electrode surfaces if no current is drawn by the amplifier. Hence the potential across the sample defined between the equipotential surfaces touching the electrodes is correctly measured. (b) and (c) show how, nevertheless, currents through the potential electrodes may cause erroneous results caused by electrode polarization. This is the case when polarization impedances, at points where currents enter and leave the sampling electrodes, are not equal. (Reprinted by permission of the Annals of the New York Academy of Sciences [23].)

282

H.P. Schwan

terminal mode, with "high" and "low" voltage and current terminals connected. We experienced such effects repeatedly and suspect that they are caused by capacitive components across the series resistor which is used in modern digital equipment to measure the sample current. So far, commercial impedance analyzers are almost useless as 4-terminal devices to eliminate electrode polarization. Finally, we list the use of strong currents as a means of decreasing electrode effects. The electrode impedance decreases with increasing current density as discussed below. Therefore, high current densities eliminate most, if not all, electrode problems. This technique is frequently used in clinical practice: Many cardiac pacemakers use strong signals in order to avoid having most of the energy supplied for pacing being consumed by polarization processes at the electrode. However, operation at high current densities enhances corrosion processes and can decrease electrode life expectancy. Table 1 summarizes the techniques discussed above to eliminate electrode artifacts. NONLINEAR PROPERTIES Nonfinearity

Curiously, most electrode impedance studies have been concerned with linear aspects as detected by currents sufficiently low to validate Ohm's law. This is surprising since many applications require that electrodes be used at higher current densities. Furthermore, as we shall demonstrate, the "limit current density of linearity" decreases with frequency and reaches values well below 0.1 m A / c m 2 below 1 Hz. This indicates that some published data may have been obtained in the nonlinear range while thought to be typical of linear results. This is more likely the lower the frequency. Unfortunately the relevant literature only infrequently specifies applied current densities. Schwan et al. (22,23,25) and Geddes et al. (6,7,8) appear to be the first to study nonlinear effects. Figure 13 is a plot of the polarization capacitance as function of

TABLE 1. Techniques to correct for electrode effects on sample properties.

Techniques 1. Extrapolation technique 2. Distance variation technique 3. Substitution technique 4. Reduction of Zp 5. Reduction of medium conductivity 6.4-electrode technology 7. High current density

Disadvantages (assumes constant m) (allows for variable m, requires constancy of Zp) (exchange of suspension by medium, limited to small concentration) (heavy Pt-black cover less stable) (may affect suspended particle, e.g. cytoplasma, surface properties) (some technical problems, no good commercial equipment yet) (electrode corrosion, hysteresis changes electrode during use)

The table summarizes techniques useful for the elimination or reduction of electrode effects on sample properties. Each technique has its own limitation or disadvantage as briefly indicated in parentheses. For details, see text.

Linear and Nonlinear Electrode Polarization

283

1:30

3 0 Hz

120

/

~

I10 I00 90 80 70' 6(

i(mA) I

I

I

I

I

I

I

I

I

I

2

4

6

8

I0

12

14

16

18

20

FIGURE 13. Polarization capacitance Cp as a function of current density. Uncoated Pt-tipelectrode against large reference electrode. 0.1 m KCI. Area - 1 cm 2. (Reprinted by permission of the Annals of the N e w York A c a d e m y of Sciences [23].)

current density (23). It demonstrates linear dependence up to current density values where Cp is about twice as large as its limit for small current density. This linear dependence indicates that the AC current imposed polarization potential can not exceed a limit value V= as the current density increases. Observations indicate that this limit potential seems to be frequency independent and that it is identical for Cp and gp. It is of the order of 1 to 1.5 volt (10,17,18). From this, the following limit law of linearity in frequency and time domain was derived (22,23,25). it _f(1-m);

it ~ t -(l-m)

(8)

with rn the Fricke power factor in Cpo f-m. Here/~ is the "limit current density of linearity" where Cp and Rp deviate by a small percentage, say 10~ from their small current density limit values Cp0 and Rp0 for i = 0. The derivation does not assume frequency independence of the Fricke power factor m. It is based on two assumptions, one very obvious and the other typical for limited categories of systems: -

The onset of nonlinearity can be characterized by a simple linear relationship such as stated below in Eq. 9. 2. The system under consideration has a current voltage relationship which for increasing current appears to approach a saturation level well above the beginning of nonlinearity. For electrodes it may simply mean that the Butler-Volmer relationship can be used for AC as well as DC. ~

284

H.P. Schwan

Thus, the limit of linearity rule applies to any system which satisfies these two conditions irrespective of whatever mechanisms are involved. McAdams and Jossinet (14) were first to propose a physical model which justifies the nonlinearity rule for electrode-electrolyte interfaces. Schwan and Maczuk (25) first reported experimental confirmation of the nonlinearity relations Eq. 8 for the frequency-domain at frequencies extending from 10 Hz to 100 kHz, Onaral and Schwan (17,18) for both frequency- and time-domain in the range from 1 mHz to 1 kHz and corresponding times. Figures 14 and 15 present examples for frequency and time dependence of i~, demonstrating the validity of Eq. 8. Limit currents of linearity correspond to AC induced electrode potentials near 0.1 volt. Such potentials are frequently reached during measurements.

I00 8

6 4

2

I0 8 6

~

I 0.01

I 2

I-m=05.5 I I 4

1 6

I 80.

f (KHz) I

I 2

I 4

I 6

I 8

FIGURE 14. Limit current of linearity i/(open circles) and low current density polarization capacitance Cp(closed circles] as a function of frequency. Same electrode as in Fig. 14. The slopes of the t w o lines add to unity as demanded by Eq, 8. (Reprinted by permission of the Annals of the New York A c a d e m y of Sciences [23].)

Linear and Nonlinear Electrode Polarization

285

-

d : 0.14 m A / c m 2

-

~

--.

................

_

I--I

o~,-~ "~"

I

I

I

I

0.I

I

I0

I00

I

103

I

104

t (rnsec) FIGURE 15. Polarization impedance (cathodic overpotential normalized with respect to current density) as a function of time. Parameter current density. Electrode area 0.0855 cm 2. 0.9% NaCI. pH 6.5. (Reprinted by permission of Medical & Biological Engineering & Computing News [18].)

Harmonics

Nonlinear electrode impedances imply rectification and generation of harmonics. Simpson et aL (28), Geddes et al. (9) and Ragheb and Geddes (19) investigated rectification and Moussavi et al. (16) harmonics generated by a voltage clamp across the interface in the nonlinear region. Nonlinearity raises questions about the definition of impedances under conditions where Ohm's linear law no longer prevails. It is necessary to specify if the applied voltage or current is "clamped" on as a sine function. For a voltage clamp the current will be distorted, generating a DC component and harmonics. For a current clamp the voltage across the interface is distorted in the nonlinear range. In both cases, it is the ratio of the clamped sin-signal and the measured fundamental component of the response which define operationally the nonlinear impedance. We consider this situation for distortions where higher terms of a series development rp = 1 / Z p = i/v

= ro + ci + . . .

(9)

can be neglected as indicated. Here Yp is the polarization admittance magnitude, 6 Vo the alternating electrode potential imposed on its DC bias, i the alternating current, Y0 the electrode admittance magnitude for small currents and c = 1/Vo~ a constant. Its inverse 1/c is equal to the potential Vo, approached as i increases, provided there are no higher order terms in Eq. 9. Sin our earlier work we reported that onset of nonlinearity occurs at the same current densities for polarization capacitance and resistance and that RcoCp= tan a remains fairly constant, well into the nonlinear range. Thus the c-coefficients for magnitude, real and imaginary component of Y are the same.

286

H.P. Sch wan

Equation 9 may be rewritten for a sinusoidal voltage clamp V sin wt i Yo

V sin wt 1 - cVsin wt

(10)

i sin o:t 1 + ci sin wt/Yo

(11)

or a sine current clamp i sin wt VYo =

Thus the harmonics for i and Vas presented in Eqs. 10 and 11 are the same if the denominator coefficients V and i/Yo are identical. However, V = i / Y . This suggests that the choice between voltage or current sine clamp has no significant influence on the harmonic content of the output in the weakly nonlinear range where Y and Y0 do not differ significantly. During earlier work, it was not always clear to what extent voltage and current were distorted in the nonlinear range. Since this work was done at frequencies above 10 Hz, polarization impedances were usually smaller than those of the series electrolyres, suggesting current clamp conditions. Consistency between the earlier and later work may well be due to the insensitivity of the harmonic distribution to a change from a voltage to a current clamp in the weakly nonlinear range. The harmonic distribution of Eqs. 10 and 11 will be briefly given in the appendix. Deviations from this simple model can be expected at current densities above the limit current of linearity. We also stress that the simple model is based on the assumption that all c-coefficients for magnitude, real and imaginary part of Y, coincide as pointed out in footnote 6. The study of harmonics should aid in refining models for the nonlinear range. This is particularly true at current densities well above the limit current of linearity where deviations from the simple behavior in the weakly nonlinear range can be expected. Porous and Fractal Surfaces and Nonfinearity

Investigations of the effects of porous or fractal surfaces on electrode polarization appear to assume implicitly that Ohm's law is valid, i.e., that the electrode behavior is studied in the linear range. However limits of linear behavior have been reported to be as low as a few/~A/cm z in the m H z frequency range (17). Therefore, parts of the electrode may operate in the nonlinear region even though average current densities appear to be "linear". This is especially indicated for electrodes with irregular surfaces and sharp edges where the current density is well above the average. Under such conditions parts of the electrode surface experiences nonlinear behavior at average current densities much below the range quoted above. These "nonlinear" regions will grow as the average current density is increased until the total area is operating under nonlinear conditions. Thus the onset of nonlinearity varies spatially, with some parts of the surface still operating in the linear range and others to varying degree in the nonlinear part. As the total current is increased, a greater part of the surface will shift from linear to nonlinear behavior. Elements of the fractal or porous structure near the electrode surface will be first subjected to higher current density and their admittance decreases accordingly. They

Linear and Nonlinear Electrode Polarization

287

will t h e r e f o r e p r e f e r e n t i a l l y r e c e i v e c u r r e n t . A s t h e t o t a l c u r r e n t i n c r e a s e s this p r e f e r e n t i a l state s p r e a d s d e e p e r . T h u s it a p p e a r s , t h a t t h e h i e r a r c h y o f a f r a c t a l s y s t e m can change with current. The implications of this spreading behavior deserve consideration. T h e c o n c e p t o f s p r e a d i n g n o n l i n e a r i t y in f r a c t a l s y s t e m s loses i m p o r t a n c e as t h e f r e q u e n c y i n c r e a s e s b e c a u s e t h e limit o f n o n l i n e a r i t y i n c r e a s e s w i t h f r e q u e n c y . T h u s c o n c e r n a b o u t its p o s s i b l e e f f e c t s o n p r o p e r t i e s o b t a i n e d b y i m p e d a n c e s p e c t r o s c o p y a c c o r d i n g l y d i m i n i s h e s . H o w e v e r , this is n o t t r u e in c l i n i c a l a p p l i c a t i o n s w h e r e injected current densities are frequently high.

REFERENCES 1. Buck, R.P. Impedances of thin and layered systems: Cells with even or odd numbers of interfaces. Ann. Biomed. Eng. 20:363-383; 1992. 2. Cole, K.S.; Cole, R.H. Dispersion and absorption in dielectrics. I. Alternating current characteristics. J. Chem. Phys. 9:341-351; 1941. 3. Davidson, D.W.; Cole, R.H. Dielectric relaxation in glycerine. J. Chem. Phys. 18:1417; 1950. 4. de Levie, R. The admittance of the interface between a metal electrode and an aqueous electrolyte solution: Some problems and pitfalls. Ann. Biomed. Eng. 20:337-347; 1992. 5. Fricke, H. The theory of electrolytic polarization. Phil. Mag. 14:310-318; 1932. 6. Geddes, L.A. Electrodes and the measurement of bioelectric events. New York: Wiley Interscience; 1972. 7. Geddes, L.A.; Baker, L.E. Principles of applied biomedical instrumentation. (2nd ed.) New York: Wiley Interscience; 1975. 8. Geddes, L.A.; DaCosta, C.P.; Wise, G. The impedance of stainless steel electrodes. Med. Biol. Eng. Comput. 9:511-521; 1971. 9. Geddes, L.A.; Foster, K.S.; Reilly, J.; Voorhees, W.D.; Bourland, J.D.; Ragheb, T.; Fearnot, N.E. The rectification properties of an electrode-electrolyte interface operated at high current density. IEEE Trans. Biomed. Eng. 34:669-672; 1987. 10. Jaron, D.; BriUer, S.A.; Schwan, H.P.; Geselowitz, D. Nonlinearity of pacemaker electrodes. IEEE Trans. Biomed. Eng. 16:132-138; 1969. 11. Jaron, D.; Schwan, H.P.; Geselowitz, D.B. A mathematical model for the polarization impedance of cardiac pacemaker electrodes. Med. Biol. Eng. 6:579-594; 1968. 12. Kohlrausch, F.; Holborn, L. "Das Leitvermoegen der Elektrolyte." Leipzig: Teubner; 1898. 13. Kronig, R. On the theory of dispersion of x-rays. J. Opt. Soc. Am. 12:547; 1926. (See also Kramers, H.A. Atti Congr. dei Fisici. Como. 1927: p. 545.) 14. McAdams, E.T.; Jossinet, J. DC nonlinearity of the solid electrode-electrolyte interface-impedance. Innovation et Technologie en Biol. et Med. 12:329-343; 1991. 15. McAdams, E.T.; Jossinet, J. A physical interpretation of Schwan's limit current of linearity. Ann. Biomed. Eng. 20:307-319; 1992. 16. Moussavi, M.; Sun, H.H.; Schwan, H.P.; Richter, A. Nonlinear phenomenon of interfacial polarization immittance of a Pt electrode. Ann. Biomed. Eng. 18:505-518; 1990. 17. Onaral, B.; Schwan, H.P. Linear and nonlinear properties of platinum electrode polarization. Part I: Frequency dependence at very low frequencies. Med. Biol. Eng. Comp. 20:299-306; 1982a. 18. Onaral, B.; Schwan, H.P. Linear and nonlinear properties of platinum electrode polarization. Part II: Time domain analysis. Med. Biol. Eng. Comp. 21:210-216; 1982b. 19. Ragheb, T.; Geddes, L.A. Electrical properties of metallic electrodes. Meal. Biol. Eng. Comp. 28:182186; 1990. 20. Schwan, H.P. Electrical properties of tissues and cells. In: Lawrence, J.H.; Tobias, C.A., eds. Advances biological medical physics, Vol. 5. New York: Academic Press; 1957: pp. 147-209. 21. Schwan, H.P. Determination of biological impedances. In: Nastuk, W.L., ed. Physical techniques in biological research, Vol. 6. New York: Academic Press; 1963: pp. 323-406. 22. Schwan, H.P. Alternating current electrode polarization. Biophysik 3:181-201; 1966. 23. Schwan, H.P. Electrode polarization impedance and measurements in biological materials. Ann. New York Acad. Sci. 148:191-209; 1968.

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24. Schwan, H.P.; Ferris, C.D. Four electrode null techniques for impedance measurement with high resolution. Rev. Sci. Inst. 39:481-485; 1968. 25. Schwan, H.P; Maczuk, J.G. Electrode polarization impedance: Limits of linearity. Proc. 18th Ann. Conf. Med. Biol. IEEE-ISA. Philadelphia, PA: 1965. 26. Schwan, H.P.; Onaral, B. Linear and nonlinear properties of platinum electrode polarization. Part III: Equivalence of frequency-and time domain behavior. Med. Biol. Eng. Comp. 23:28-32; 1985. 27. Schwan, H.P.; Sittel, K. Wheatstone bridge for admittance determinations of highly conducting materials at low frequencies. Trans. AIEE. (Comm. & Elec.) May: 114; 1953. 28. Simpson, R.W.; Berberian, J.G.; Schwan, H.P. Nonlinear AC and DC polarization of platinum electrodes. IEEE Trans. Biomed. Eng. 27:166-171; 1980. 29. Sun, H.H.; Charef, A.; Tsao, Y.; Onaral, B. Analysis of polarization dynamics by singularity decomposition method. Ann. Biol. Eng. 20:321-335; 1992. 30. Sun, H.H.; Onaral, B. A unified approach to represent metal electrode polarization. IEEE Trans. Biomed. Eng. 30:399-406; 1983. 31. Warburg, E. Ueber das Verhalten sogenannter unpolarisierbarer Elektroden gegen Wechselstrom. Ann. d. Physik 67:493-499; 1899. 32. Warburg, E. Ueber die Polarisationskapazitaet des Platins. Ann. d. Physik. 6:125-135; 1901.

APPENDIX We d e v e l o p Eqs. 10 a n d 11

i/Yo = Vsin[1 + r s i n + r 2 sin 2 + r 3 sin 3 + r 4 sin 4 -I- . . . ]

(A1)

VYo = i s i n [ 1 - r s i n + r 2 sin 2 - r 3 sin 3 + r 4 sin 4 + . . . ]

(A2)

with r = V/V= o r r = i/V=Yo respectively a n d all sin a r g u m e n t s ~ot. Next we d e c o m p o s e the higher o r d e r sin-terms to linear t e r m s o f h a r m o n i c s . This yields the results given in the Table A 1 . These results s i m p l i f y in t h e w e a k l y n o n l i n e a r r a n g e where r is a b o u t 0.1 o r less. E a c h h a r m o n i c a m p l i t u d e is t h e n p r e d i c t e d to be 0.5r times the p r e v i o u s one. T h e results p r e s e n t e d in the t a b l e a r e b a s e d o n the simple m o d e l , i.e., o n the o b s e r v a t i o n t h a t V= coefficients for m a g n i t u d e , real a n d i m a g i n a r y p a r t o f Y, a r e i d e n t i c a l . T h e c a l c u l a t i o n b e c o m e s m o r e c o m p l e x if this is n o t the case.

TABLE A1. Harmonics of simple nonlinear model. Exact DC Fundamental 2nd 3rd 4th 5th

0.5r(1 + 0.75r 2 + 0.625r 4 + ...) (1 + 0 . 7 5 r 2 + O. 1 2 5 r 4 + . . . )sin - O. 1 8 8 r s cos O.5r( 1 + r 2 - 0 . 5 6 r 4 + . . . )cos 0.25r2(1 - 1.25r2)sin 0.125r3(1 - 1.5r2)cos - O . 0 6 r 4 sin

Approx.

O.5r 1 . 0 sin 0 . 5 r cos 0 . 2 5 r 2 sin 0 . 1 2 5 r 3 cos - O . 0 6 r 4 sin

9 is equal to V/V= or i! V= Yo for v o l t a g e or c u r r e n t c l a m p r e s p e c t i v e l y , sin and cos a r g u m e n t s are t h o s e o f c o r r e s p o n d i n g harmonics.