A study of potentiostat noise

J. Electroanal. Chem., 93 (1978) 155--161 © Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands

155

Preliminary n o t e A STUDY OF POTENTIOSTAT NOISE*

I. EPELBOIN, C. GABRIELLI, M. KEDDAM, L. RAILLON

Groupe de Recherche No. 4 du C.N.R.S., Physique des Liquides et Electrochimie associJ l'Universit~ Pierre et Mar:ie Curie, 4, Place Jussieu, 75230 Paris Cedex 05 (France) (Received 29th June 1978)

INTRODUCTION

It has been shown that analysis of electrochemical noise is of great value [1]. However, the measurement of the quantities characterizing the random signal generated by this noise is technically very demanding since this signal may often be of the same order of magnitude as the spurious noises originating in the experimental apparatus. The accuracy of the measurement is limited b y these spurious noises mainly generated by the electronic instrumentation and particularly by potentiostats when included. By judicious use of two identical, b u t uncorrelated, measurement channels, noise arising in the amplifiers can be eliminated. Yet, at present, there is no means by which the potentiostat noise can be obviated. In this note we shall show how the potentiostat noise can be evaluated in the determination of the accuracy of such electrochemical noise measurements. These results are used to criticize previous results in which a correlation function was chosen for the characterization of the data. THE POTENTIOSTAT NOISE Theoretical

derivation

The various noise sources of possible significance to voltage regulation are shown in Fig.1. The Operational amplifier noise is assumed to be represented by a single source, Vs, at its output, equivalent to a source, v e = v s / A v , across the two inputs to this device, i.e. the current noise sources to each input are neglected consistent with the use of FET input stages. Each quantity is represented as the sum of t w o parts, the mean value denoted by { } and, the fluctuating c o m p o n e n t denoted by a small letter, in such a way that the potential of the working electrode, for example, reads:

Vwe(t) = ( V w e ( t ) } + V~ve(t)

*This paper is a part of L. Raillon's thesis

(1)

(docteur ing~nieur).

156

=VA +' D'~

iR

÷

i

~Vc +( P~CE ~CE

?

? + 3R°f

)

+\ VB

II=+L


Fig.1. S c h e m a o f a p o t e n t i o s t a t i c r e g u l a t i o n , s h o w i n g t h e noise sources as s h a d o w e d circles, a n d m e a n s t a t i o n a r y voltages as o p e n circles. Vp = < Vp > + vp : r e f e r e n c e voltage source. V A : o u t p u t voltage o f a voltage gain A v o p e r a t i o n a l amplifier. V c : c o u n t e r - e l e c t r o d e voltage. V B : voltage m e a s u r e d t h r o u g h r e f e r e n c e electrode. W o r k i n g e l e c t r o d e is g r o u n d e d .

T h e m e a n value < VWE > can be e q u a t e d with the m e a n c u r r e n t , { I >, flowing t h r o u g h t h e i n t e r f a c e b y the usual non-linear c u r r e n t - p o t e n t i a l relationship:

<% e >= f(< I>)

(2)

Hence, each interface can be defined by its small-signal impedanceat each frequency f: (3)

z (f) = v (f)/i(f)

and by an electrochemicalnoise source vWE about each mean polarization point studied. Assuming that the current flowingthrough the reference electrode is negligible, the equations describing the system can be written as:

VB = <%e >+ zi + Vwe

(4)

V C = V s +< Yce >+Zcei+Vce

(5)

VA = V c + R l + v R

(6)

V A = A v ( V + - V_) + vs

(7)

V+- V_ = (( Vp >+Vp)- (VB- -vRef)

(8)

where the various parameters are defined in the legend of Fig.1. On equating (6) and (7) and by using eqns.(5) and (8) we have: Av(< Vp > + Vp - V B + ( V R e f > + VRef) + v s = V B + < Vce > + Zcei + Vce +

RI

+ vr

157 The m e a n of the t w o sides of the equality (9) lead to the steady-state relation:

Av(< Vp >-< V B > +< VRef> ) = < V B > +< V e > + R =

(i0)

-< Vce > - R +Av(< Vp > +< VRef>)

(11)

1 +A v Since the gain, Av, of the operational amplifier is very large (Av (f=0 Hz) > 50000, A v (f up to 1 kHz) > 1000) e q n . ( l l ) can be simplified as:

( YB >= ( Yp >+

(12)

or, alternatively, as the potential of the working electrode (< VB >= ) is measured versus the reference electrode:

< Vwe > - < VRef> = ( Vp >

(13)

Similarly the low amplitude fluctuatingterms are related by: Av(V p -

zi -

Vwe + V,Ref) +

v s = zi +

Vwe +

Z cei +

Vce +

Ri

+ vr

(14)

Substituting A w e ! for vs and z/WE for vwE w h e r e / W E is the equivalent current noise source related to vWE : i=_

(A v + 1)z

Av •

(A v + 1)z + Zce + R lwe

+

(A v + 1)z + Zce

+ R (Vp + ve + VRef) +

Vce + VR

(15)

(A v + 1)z +Zce + R so t h a t in the f r e q u e n c y range such t h a t A v >> 1: Vp + ve + VRef i = - iwe +

(16) z

E q u a t i o n (16) can be+interpreted as the whole fluctuating c u r r e n t being the sum of b o t h the noise of electrochemical interest and a parasitic term. The current is normally estimated by measuring the voltage, VAC , across t h e resistor R, and VAC

(17)

= R i + vR

R VAC = - R/we + - - (Vp + ve + VRef) + v R

(18)

Z

By assuming t h a t each t e r m is uncorrelated with a n y other, the variance of VAC is given by: R2

=R2+ (++)+ lz] ~

(19)

158

The four parasitic sources of noise are then: (1) Vp : the reference voltage source has a certain drift but is quite regular for short times and the noise has been shown not to exceed 1 nV/x/Hz in a 1--104 Hz bandwidth for the nickel-cadmium battery. (2) VR, for which the thermal noise given by e n = 4VrRka n V / V ~ z , is also generally negligible within the usual range of possible values of R (~< 100~2). (3) Vref, for which the thermal noise should be negligible if a good reference electrode having a low o u t p u t resistance is used (a few 100 ~2). (4) Ve, which is for an actual low noise operational amplifier a b o u t 6--10 nV/x/] in the frequency-independent range (1 kHz < f < 100 kHz) and even more when f < 1 kHz. Then this last noise is generally the dominant term in eq.(18), so that: ( V A c 2 ) -~ R 2 ( i w e 2 ) + R 2 (

Ve2 ) / I z 12

(20)

Thus the effect of the potentiostat noise is inversely related to the modulus of the impedance. A real potentiostat departs from the ideal by imposing a fluctuating voltage on the cell in addition to the d.c. level:

v B = zi + Vwe

(21)

VB = z ( - i w e

(22)

+ Ve[Z ) + Vwe

i.e. VB = Ve

(23)

This potentiostat noise then largely arises in the operational amplifier which therefore has to be chosen for low noise.

Measurement o f the potentiostat noise By using a d u m m y cell consisting of two resistors R and z, and an Analog Device 45J operational amplifier for the potentiostat, the previous approximations are valid. As predicted in (23) ( VB2 ) depended neither on z nor on R as ascertained by varYing these resistors between 10 and 104 ~2, and also did not depend on the current ( I ) flowing through the d u m m y cell. Including the thermal noise, vR, of the resistor R (usually insignificant with respect to the electrochemical noise) the relation: R2 ( VA B )2 = ~

Iz 12

( Ve2 ) + ( Vz2 ) + ( VR2 )

(24)

was verified to an accuracy of 5% for the same range of resistors. A Rockland 852 (8 pole Butterworth) filter was used to delineate a very narrow spectral window in the determination of the potentiostat noise as a function of frequency. As the equivalent bandwidth of a low-pass filter of this kind is: oo

Beq=

f o

¢o

]H(f)[2df=

f o

dx

l +x 16 - 1 , 0 0 6 4 5 4 5 4

159

fmax - fmin will be a good approximation for such a band-pass configuration. The voltage noise power e 2 (fi, fi+l ) in the (fi, fi+l ) frequency range is related to the voltage power spectral density ¢Ve %(f) by: e2 (fi'fi+ 1) ~VeVe(fmi) =

fi+l

(26)

-- fi

and was arbitrarily attributed to the mean frequency, fm = f~'f~+l (the geometrmal mean has been chosen because that is castor to plot on a loganthmm scale). Note that V~v v (f) = 12 nV/x/-H-z from Fig.2, for 2 kHz ~< f ~< 50 kHz as previously assumede In the low frequency range the usual l f f noise was found. •

:

.

.

i

•

•

~~'u"e "U"eZd B :

--X----..,..X,

.i-3C X% Xq,X

~x-x-x-xo x°x"x"x.,xb~

+20

1

+10

0dB 1Hz

[/Hz 10Hz

100Hz

IK

10K

100K

Fig.2. Spectral density of the noise voltage ~ vAv^ (f) of the p o t e n t i o s t a t versus frequency f. The reference vahle of 0 dB was chosen to be 1 n ~ ~ ~Iz-1, so that 10 log [~Pveve(f)/nV 2 Hz -1 ] in dB is plotted on the vertical axis.

I N F L U E N C E OF THE P O T E N T I O S T A T NOISE IN THE E L E C T R O C H E M I C A L NOISE MEASUREMENTS

Equation (20) expresses ( vAC2 ) as a current noise measured across the resistor R. The potentiostat noise can also be interpreted as a current noise. Thus, i = Ve/Z with corresponding power spectral density: ~i i (fmi) = ¢VeVe(f)= ~VeVe(f) = ¢%ve(f) PP z z z(f)z*(f) Iz(f) l2

(27)

160

(~E.pLp/dB ~X~)~ ,)(,w)41 ,~ x~x'

÷I(

-X~xJ-, 0 dB j,x "~

~.xBX--X'x-~ X

L/

,~x~,x_x.x,.x.,X-X--)'~)'C,=X~),

,,xf×

-10 = ')(.

-20

-30L -

,,,,,,,,~,,~

/

/

/,k

I

~

I

10Hz

/

.oo,L3

I

I

100Hz

~1K

/Is

.,/o

,o4

x,,,~ "~l o

401'

'IH:

k,ooT-

600 ~

~

1oo 2oo I

f/Hz I

10 K

100K

Fig.3. Calculated spectral density of the noise current, ff//plp(f), induced by the p o t e n t i o s t a t across two electrochemical interfaces: (A) anodic dissolution of iron at a c u r r e n t of 4 m A (current density 20 m A cm-2). (B) Diffusion limited r e d u c t i o n o f ferricyanide ion (V ffi 0). Reference value of 0 dB is 1 nA2/Hz. Inset: the electrochemical impedances z of (A) and (B), including electrolyte resistance.

The total power

Pii(f) = f

f

Pipip(F)in the bandwidth

@/p/p(f)df = ~

o

~VeVe(fmi)

d.c. to F is given by:

(fi+l - fi )

(28)

i [ Z ( f m i ) [2

P i i ( F ) = ~ i e:(fi'fi+l) • I z(fmi) I:

(29)

Since only the sum @~(f) = @/e I/e'(f) + @/p/p(f)

(30)

can be measured, the electrochemical noise can be estimated accurately only in a frequency range for which

~iw e wi e (f) >> ~ip pi (f)

(31)

The following examples illustrate that when low values of the electrochemical impedance z (f) are encountered, electrochemical noise measurements are inaccurate despite the use of potentiostats with a good noise performance. Firstly the measurement of the noise generated by the interface Fe--1M H2SO4

161

will be considered as an example of a small value of electrochemical impedance, see Fig.3(A ). For each frequency bandwidth (fi,fi+ 1 ) the current power spectral density due to the potentiostat has been calculated using eqn.(29) and plotted. Secondly, the measurement of the noise generated by the diffusion of ferricyanide in KC1 aqueous solution (diffusion limited system) is by contrast characterized by a high value of electrochemical impedance, especially at low frequencies. The current power spectral density due to the potentiostat has been obtained in the same way as for the previous example and plotted in Fig.3(B). The current background noise is n o w fairly low for f < 100 Hz. DISCUSSION

It is n o w possible to assess the accuracy of the correlation function studies previously published [2--4]. For the Fe--H2SO4 system: (1) the valUe of the peak observed for r = 0 corresponding to the noise power in the frequency range up to 1 kHz is 15,000 nA 2 ; (2) the tall of the correlation function is indicative of a noise power o f 3000 nA 2 in the frequency range up to some tens of Hz. As these figures have to be compared with (1} 14,000 nA 2 and (2} 2000 nA 2 found above Fig.3(A), it seems that the potentiostatic m e t h o d leads to a poor accuracy in the low frequency range and even worse for high frequencies. The measured correlation function for the diffusion example provides a value for the noise power of 250 nA 2 for a frequency range up to 10 Hz compared with a s p u r i o u s noise power of 2.6 nA 2 obtained from Fig.3(B). This good accuracy, at least in the low frequency range, is made possible b y the high value of impedance. These spurious noises originating in the potentiostat have been shown to hinder the measurement of electrochemical noise to an extent inversely proportional to the electrochemical impedance. It can be seen that, firstly an analysis of the electrochemical noise spectral density is a necessary control. The accuracy of the measurement is itself frequency-dependent. Secondly, potentiostatic studies are not suitable for interfaces with low electrochemical impedances -- galvanostatic regulation is then recommended.

REFERENCES 1 2 3 4

G. G. G. G.

Blanc, Blanc, Blanc, Blanc,

I. Epelboin, C. GabrieUi and M. Keddam, J. Electroanal. Chem., 62 (1975) 59. I. Epelboin, C. Gabrielli and M. Keddam. J. Electroanal. Chem., 75 (1977) 97. C. GabrieUi and M. Keddam, C.R.A.S., 283 (1977) S~r. C, p.107. Doctoral Thesis, Univ. P i e c e et Marie Curie, Paris, Nov. 1976.