Electrical noise in aqueous 1-1 electrolytes

Ekcrrochimico

Acm, 1974, Vol. 19, pp.iSl-186.

Pergamon

Press. Printed in Great Britain.

ELECTRICAL

NOISE

IN AQUEOUS

l-1 ELECTROLYTES

D. VASILESCLJ,M. TEBOUL, H. KRANCK and F. GUTMANN* Laboratoire

de Blophysique, U.E.R.D.M. Universitt 06034 Nice Cedex. France

de Nice,

(Receioed 20 October 1973)

Abstract--An equivalent

noise conductivity A, for aqueous uni-univalent electrolytk is defined on the basis of the Langevin equation. Apparatus for the recording of noise spectra and their evaluation by means of correlation methods are described. It is shown that A, of NaCl and of KC1 solutions parallels that obtained by conventional bridge methods though no external electric field is applied in the case of noise measurements. 4” for both these electrolytes is shown to obey the Debye-Hueckel-Onsager limiting law in its concentration dependence and to exhibit the usual Arrhenius form of temperature dependence. The activation energy obtained is close to that obtained from conventional bridge methods applying an external field, and closely approximates that of the viscosity of the solvent, viz. water. The noise generated by the electrolyte is shown to be white noise, being independent of frequency without any llffrequency component being present. There is no contact noise

1. INTRODUCTION 1.1 Thermal noise in resistive circuit elements: gevin

tistics as V(t) for any Z. This may be summarized by the relations

the Lan-

(0) =o

Equations

this introduction we shall briefly review some modern ideas on the connexion between the Langevin Equations and thermal noise and specially its power spectrum, the noise being produced in a resistance forming part of an electric circuit. We shall, in the main, follow the argument put forward by Papoulis[ l] and also that proposed by Friedmann[2]. Consider an electrically neutral particle of mass m, which is free to move in one linear dimension only in a liquid in which it is subject to Brownian motion. Let its macroscopic position as a function of time t be x(t). The Brownian motion is governed by the Langevin Stochastic differential equation: In

Here,fis a friction coefficient, and F(t) a random fimction of time which includes the effect of all collisional processes between the particle and the solvent; it gives Newton’s Equation its stochastic form. v(t) = dx/dt is the velocity of the particle. The random variable u(t) has a mean value (c)} and an autocorrelation function C(7)

=


+

7)@)>.

In this notation[l], a Fourier Transform of R,(t) yields the power spectrum S,(w) of the velocity. +,. Thus: R,(T) emior dz S”(W) = (4) s -1 where w = 2~v, v being the frequency of a noise component j = (~ 1)‘j2. The ‘&tocorrelation function ,R,.(T) is ‘the inverse Fourier Transform of the power spectrum +-z R,(r) = (2~)-l S,(w) ejor do. (5) s -Ix‘ The behaviour of the function F(t) may then be closely approximated by comparing it with the whitenoise produced by a generator having an internal, ideal and noisefree resistance R as shown in the equivalent circuit of Fig. 1. The spectral density, (or power spectrum) of the thermal noise produced by such a circuit is constant, i.e. frequency independent, and only a function of temperature T: S,Jw)

symbol < > will be employed to denote mean values. The stochastic process here considered is stationary, that is to say that v(t) and v(t + T) follow the same sta-

(e’(t)>

(6)

= R,(O)

= (27r- ’ s_+I S,(w) dw

University of Nice. Permanent

address: Physical Chemistry Department, Sydney, Sydney, NSW, Australia 2006. EA Vol. 19 No. S---A

= 2kTR.

This is known as the Nyquist-Johnson Theorem[3]. k is Soltzmann’s Constant. The mean value , where e(t) is the instantaneous terminal noise voltage appearing across the resistance R producing thermal noise, is given by

(2)

The

* Visiting Professor,

(3)

R”(O)= (aZ> 1

The University

of

2S,(w) dw. 181

(7)

D. VASILESUJ,

582

M. TEROLJL,H. KRANCK

and

F. GUTMANN

and finally I&(T) =

e(t)

Fig. 1. Series equivalent circuit of a noise generating resistance.

‘$ exp

-TV

.

(3

(16)

If the relaxation time t $ (f/m)- I, then the acceleration term dv/dt in equation (9) is negligible and the displacement of the particle is given by

f x(t) = 2 n(c)dt. f sD

(17)

This last expression has a limiting value if the frequencies are not too high (the Planck condition hv + kT requires u + 1Ol3 Hz). With this restriction, a good approximation of (e’(t)> is given by: . A detailed discussion of this result is given by Lange[4]. The Nyquist-Johnson Theorem is usually quoted in the form of equation (8). The Langevin equation (1) can also be written as dv z f pu = n(t)

This is a case of a Wiener--Levy process[ 11 from which the mean square displacement of the particle position follows as

where /3= f/m. It is seen that F(t)/m has been replaced by the noise function n(t) = n,(t). This means that

1.2.1 The mechanical mobility. Consider the motion of an uncharged particle, we can then define its mechanical mobility u by:


=o

S”,(o) = c( = a constant. 1

(10)

D=PoL leads to the well known equation (x2> G 2Dt. 1.2 Application

to electrolytes

u =J.-‘.

The autocorrelation function of the velocity components then immediately follows as

s +m

2 -CC 0 which yields upon integration

o( +B

eJwrdru

To obtain a/2A we note that, at equilibrium, the mean value of the kinetic energy per degree of freedom is kT/2: Thus = ;<@)@)>

= ;m R,(O) = ;kT

(21)

e = electronic charge in MKS units

/*”= eu.

R,(r) = (2n)-l

(20)

(‘3)

resu 1ting in

;m

m2

(12)

(11)

The power spectrum of the velocity components S”(w) is related to that of the thermal noise, S’,(w) according to S”(W) = S,(w)lHO’wN2

Writing for the diffusion coefficient D

The coefficient of friction f appeared already in equation (1). u has the dimension time/masse: t/m. FriedmannL21 shows that the quantity mu,’ which has the dimension of a time, is of the order of I O- ’ set for a colloidal particle 1OOOa in diameter, in water for comparison, a Nat ion in water has an mu value of the order of lo- I4 sec. 1.2.2 The electrical mobility. Considering now a monovalent ion in Brownian Motion: we can introduce an electrical mobility ,u, of dimension m2 V-’ set-I:

Equation (9) can be solved using the methods of the general theory of linear stochastic differential equations (see eg Papoulis[ I]): The characteristic function of equation (9) is Ei(jw) = 0’0 + /3-l.


(15)

(22)

This “noise mobility” is the equivalent of the conventional mobility derived from the motion of an ion under the influence of an external electric field. For an Nat ion in aqueous solution, mu = that u= 10-‘4sec, m = 3.82 x 10m2bkg so 2.6 x 10” set/kg and p,, = 4.2 x lo- ’ mz V- 1 set- f which compares with the experimental value of 5.06 x 10-8m2 V-l see-‘. 1.2.3 The equivalent noise conductivity. For a system of N identical monovalent ions, an equivalent noise conductivity o,, may be defined by CT,= Nepc,.

(23)

This treatment assumes the ions to be acted on only by random forces of thermal origin, mn(t). which also cause the noise phenomena; coulombic ion-ion inter-

Electrical noise in aqueous actions are supposed to be negligible. In effect, this means that the ions behave like a dilute Lifschitz-Landau[5] plasma, ie, like an assembly of perfect gasmolecules, which have a kinetic energy determined by thermal agitation only: their electrostatic energy of interaction, by comparison, is negligible. For two monovalent ions separated by a mean distance (d), this condition may be expressed as

1 I electrolytes

very small compared to the permanent dipole polarisation which enters h; moreover in such systems EL 9 1 so that one may write, as a rough approximation, ELN h/T which is the required result. For water, it then follows that the plasma approximation holds for a concentration range of about 10m2 to 10e4 molar, and over a temperature interval of about l(r7O”C. For a real uni-univalent electrolyte which contains ions of both polarities,

(24) using MKSA units. Here. E,, stands for the permittivity of free space. E; for the relative permittivity of the solvent (.for water at 25”C, EL = 78.3) and T for the absolute temperature. For N ions in a volume V, the mean distance between two neighbouring ions is approximately (d)

so that the plasma

-v

criterion,

(;y3 equation

183’

aPI = Ne@:

+ PL,)

so that O, = 2Ne(p,,).

(24) becomes

2.

As has been pointed out[9], the quantity EAT varies but little over a limited temperature interval. This has been proposed as an entirely empirical relation which in fact is being obeyed: E;T for water varies from 2.2 x 104at 70°C to 2-37 x 10“ at 10°C. However, the relation can be readily derived from any of several expressions for the permittivity as a function of temperature, such as eg the Onsager, Van Arkel, or Froehlich equations by expanding the-often quite complicated--expressions for ELin a power series yielding 1+e:=a+;

(27)

where a is determined by the induced polarisation. now E: is large, as in water, the induced polarisation

If is

EXPERIMENTAL

The noise voltage 4t) produced by the electrolyte is compared, at every instant of time, to a reference signal derived from a white noise generator in the form of an ohmic, metallic, resistance connected ta the input of the noise spectrometer, point “A” in Fig. 2. The alternating signal obtained from either the electrolyte or from’ the noise generator, is amplified and filtered. It may then be analyzed by: (1) Square Law detection by means of a thermocouple followed ey by a ballistic galvanometer or galvanometer recorder. This yields the spectral density of the signal at a selected frequency v; (2) By applying it to a signal correlator followed by a digital voltmeter or a potentiometric recorder. This yiklds the autocorrelation function Al,(z).

Noise voltage visuolisotion

0, rscorder

Fig. 2. Block-diagram

(28)

where the p’ now refer the cations and anions, respectively. Noise measurements by themselves do not permit the separate evaluation of ,u: and p; but yield only an average mobility,

of the noise spectrograph: (a) Autocorrelations measurements.

measurements:

(b) rms voltage

D. VASILESCU, M. TEBOUL,H.KRANCK

184

and F.GUTMANN

2.1 The noise spectrograph The noise spectrograph consists of: (a) A measuring cell in the form of a cylindrical capacitor of, in our case, 7 cm3 volume in which the electrolyte forms the dielectric. The electrodes are concentric and gold plated. The geometric (empty) capacitance of the system is about 1 pF. The cell is surrounded by a mantel through which a heat exchange liquid is circulated. Its temperature is controlled by means of a thermostat, in the present case a HAAKE type 52, the temperature range extending from - 30 to + 100°C. The entire system is kept within an evacuated vessel which also contains the resistances forming the noise generator. The switches serving to adjust their resistance values are also mounted within this evacuated enclosure. (b) A highly stable high gain (about 130dB) wide band (20 Hz-150 kHz) amplifier which is nearly completely non-microphonic. Its high input impedance and very low inherent noise level permit the detection of noise voltages produced either by the electrolyte or by the reference noise generator, ranging from 1000 to 10 Mfl equivalent noise resistance. (c)A filtering device such as a highly frequency selective, narrow band amplifier, in order to isolate a narrow frequency interval Av centered around a mean frequency v which is adjustable between 20Hz and 20 kHz. A General Radio Type 123-A amplifier having Aviv = S/l00 was found satisfactory. For higher accuracy and better resolution, this amplifier may be replaced by a wave analyzer. The L.E.A. Type AF-10s employed maintains Av = 30 Hz constant over the frequency range of 20 Hz-50 kI-Iz. (d) A correlator such as the P.A.R. Model 101-A. This receives the filtered signal E,(t) and allows the computation of the autocorrelation function S(z) with a precision of 1 per cent. The sampling period was 50,usec. The value of R,(T = 0) thus obtained has already been shown to be related to the (e:(t)> of the noise signal. RR(O) is read off a digital voltmeter such as the DANA Model 3800-A or else recorded ey by a Beckmann Potentiometric Recorder Model 100.5. To measure the noise voltage spectrum directly, the correlator may be replaced by a square law (rms) detector consisting eg of a thermocouple connected to a ballistic galvanometer, having high sensitivity and a long time constant, such as the AOIP Model G-326-C. Alternatively, a recording galvanometer amplifier such as +he SEFRAM “Graphispot” may be employed. 2.2 Materials Merck “Suprapure” chlorides were dissolved in conductivity water which had been twice re-distilled in a quartz still, to yield solutions of NaCl and of KC1 5 x lo-‘to 10-4M. 2.3 Measurement

techniques

The reference resistance R, producing the same rms noise voltage as the electrolyte under test, is deter-

Fig. 3. The equivalent conductivity A, as a function of (centre) frequency: 0 refers to 10e3 M KC1 and shows A,, values from noise measurements, A refers to the same electrolyte but gives n values obtained by conventional bridge method; x refers to lo-‘M NaCl and gives A, values obtained from noise measurements. mined by means of interpolating between 4 resistances selected so as to bracket the resulting resistance value. These resistors are kept under electrical and thermal conditions identical with those of the electrolyte. The equivalent noise resistance R, values obtained for the electrolytes in this study were all far below the input impedance of the amplifier so that R, may be equated to R,. The errors of these measurcmcnts are estimated to be below 3%. It should be pointed out that these noise measurements do not involve the application of an external voltage, or field, to the system under test. Thus, conductivity data derived from noise measurements do not suffer from electrode polarisation effects invariably present in the case of conventional methods at lower audio frequencies; see Fig. 3. 3. RESULTS

3.1 E&d

AND

DISCUSSION

ofjkequency

The noise emission spectra obtained are linear and, to the accuracy of the measurements, there is no evidence for any relaxation or resonance phenomena. There is no noise component varying with l/v. The electrolyte produces a white noise spectrum as mentioned in the introduction; see also Fig. 3. This refers to a 10m3 M KC1 solution the resistance values of which were also obtained by conventional methods using a Wayne Kerr Admittance Bridge Type B-224, but employing the same measuring cell assembly. Under the conditions of these experiments the bridge applied a voltage of about 68 V to the system, corresponding to a field of about O-6 V/cm. In that case, electrode polarisation effects are seen to appear at low frequencies of the applied electric field, though such are entirely absent in the measurements of the equivalent noise resistance. At high frequencies, platinised platinum electrodes in conjunction with a Boonton RX Bridge Model 250A were employed. Both these instruments express the

Electrical

noise in aqueous

impedance of the sample as a parallel combination of a conductance and a capacitance. The thermal noise produced by these electrolyte solutions does not vary with frequency and thus is white noise justifying the use of the Langevin Equation (9). Moreover, since the equivalent noise resistance values agree to the accuracy of the measurement with the resistance values obtained by means of conventional bridge methods, there is no excess noise arising from the electrodes. This is remarkable because in solid semiconductors the contact noise frequently exceeds the thermal noise. The absence of contact noise is probably due to the immobilisation of the charges within the rigid part of the double layer because of the high electric fields in that region; the energy of the carrier due to the field is much larger than kT. 3.2 Concentration

efSects

The following data were obtained at 25°C and with a filter center-frequency of 1-6 kHz, which is the frequency of the built-in oscillator powering the Wayne Kerr Bridge. To facilitate comparison with the conductance values obtained by means of the bridge and applying an external electric field to the sample, the results of the noise measurements are also given in terms of equivalent noise conductivity A, which are defined thus:

l-l

electrolytes

185

independent of the presence or absence of an external electric field. The equivalent noise conductivities of the electrolyte solutions studied thus obey the Debye-Onsager Limiting Law over a 20-fold concentration range, Departurcs occur at higher concentrations as to be expected. From equation (24) we can now calculate the Landau Screening LengthC7-J I (32) which for 25°C results as 7.155 A. The mean interionic separation (d) is 43.6 A for 10m2 and 203 w for 10e4M solutions. Thus, the plasma condition equation (26), (6) %- 1 is no longer obeyed for solutions of concentration greater than a few times lo-’ M. At concentrations much below about 10m4 M the residual conductivity of the water itself makes itself felt. The conductivity water used in this study had still at 25”C, of between 1.6 and a conductivity, 2.6 x 10 6 (Q-cm) 1 which thus is comparable to that of a 10m5 M KC1 solution. The departures from the Debye-Onsager relation evident below about 5 x lo- 5 M are due to this effect.

the solutions urider test all being uni-univalent. Here, C denotes the molar concentration of the electrolyte. The cell constant K was determined at 1 MHz by means of a Boonton RX Bridge Model 250-A, it had a value of 1.131 cm- I. The results are given in Fig. 4, which shows plots of the equivalent conductivities vs JC. The equivalent conductivities thus measured agreed with those obtained from noise measurements to within 3 per cent, which is the accuracy of the measurement, and over the entire frequency range; thus A, was

ZOL

1

1

1

f

1

2-8

29

30

3-I

32

yl

1 33

1 34

1

35

Fig. 5. Arrhenius plot for the noise equivalent conductivity of KC1 and NaCl and Arrhenius plot of water fluidity (4-l): x KC1 10m3 M; o NaCl 1O-3 M; A Water fluidity (q-l), values of Handbook of Chemistry and Physics[lZ]. In this diagram the activation energy value in electron-volt is computed by the relations: Fig. 4. Theequivalent conductivity of NaCl at 25°C: x Noise measurements; A Bridge (Wayne Kerr v = 1.66 kHz); o Bridge (Boonton RX mctcr Y = 1 MHz); 0 Tables and Owe$S]).

and KC1 US CL” measurements measurements datas (Harned

E,

=

0.198

,mg104

B(lO”/T“K) or:

E, = 0.198

wag,, ‘I- ‘) a( 103/T “K)

D. VASILESCU. M. TEROUI., H. KKANCK and F. GUTMANN

‘186

Table 1. A comparison of data derived from noise measurements with those listed in Refs. [8,12]. The mean noise mobility is defined in equation (29) and the equivalent noise conductivity in equation (31)

Concentration C mole .litre- 1

Salt KC1 KC1

Equivalent noise conductivity *” ~~‘cm’mole-’

10mz lo-’

A, - A

A (Table) a-‘cm’mole-’

141.24 1458

Mean mobility

All

141.27 146-9 5

10-X 8 x lo-)

7.32 7.55

Cation Anion Error mobility mobility p+ x 108 p- x 10s I<&> -&.n (Table) (Table) 7.18 7.47

7.46 7.76

2 x 1o-2 2-l x 10-Z l
.

(4”)

NaCl

10-a

124.5

123.74

6 x 10-j

6.46

5.06

7.76

10-Z

rnw 6.41

3.3

Temperature

effects

An Arrhenius type plot of logn us l/T is seen from Fig. 5 to be linear for both KC1 and NaCl solutions 10m3 M over the temperature interval of IO-90°C. The center frequency was selected as 1.6 kHz. The activation energy E, ‘v 0.14eV obtained from the slopes was the same and did not vary with concentration within the range studied. The diffusion coefficient, which is a characteristic parameter of Brownian Motion, varies with temperature in an exponential fashion. Neglecting interionic interactions, one can thus writer91 D = D,T exp ( - EJkT).

While for water a simple relationship of the form of qcc exp (--E$kT) does not hold[l I], it is still remarkable that over the temperature range of 1&8O”C the fluidity value for water given by the Natl. Bureau of Standards[12] can be fitted exactly to the Arrhenius type plot of the noise equivalent conductivities, as seen from Fig. 5. The fluidity values yield an activation energy of O-148 eV while our noise measurements result in an activation energy of 0.14 eV. AcknowledgementsOne of us (F.G.) wishes to thank the University of Nice for their hospitality during his stay.

(33)

Here D, is a constant and E, the activation energy of the process. The Einstein relation then yields the wellknown relation for the mobility ,u. The noise equivalent conductivity has been shown to obey an Arrhenius type of relation, see Fig. 5. We can thus compare its experimentally determined activation energy with that of conventionally obtained conductivity data such as those given in the tables of Gunning and Gordon[lO] : for KC1 E, is 0.156 eV and for NaC10.153 eV while our noise measurements yield E, = 0.14 eV for both cases. 3.4 Mobility The noise mean mobility
REFERENCES 1. A. Papoulis, Probability, Random Variables and Stochas-

tic Processes. McGraw-Hill, New York (1965). 2. H. Friedman, in Water and Aqueous Solutions (Edited by R. A. Horne), p. 723. Wiley, New York (1972). 3. J. B. Johnson, Phys. Rev. 32, 97 (1928); H. Nyquist, Phys. Rev. 32, 110 (I 928). 4. F. H. Lange, Correlation Techniques, p. 167. Iliffe Books, London (1967). 5. L. Landau and E. Lifchitz, Physique Statistique (Edited by Mir), p. 349. Moscow (1967). 6. D. Vasilescu, M. Teboul, H. Kranck and B. Camous, Biopolymers 12, 341 (1973). 7. H. Falkenhagen, W. Ebeling and W. D. Kraeh, in Ionic Interactions (Edited by S. Pctrucci), Vol. 1, pp. l-l 13. Academic Press, New York (1971). 8. H. S. Harned and B. B. Owen, The Physical Chemistry ofEIectrolytic Solutions. Reinhold, New York (1964). 9. I. O’M. Bockris and A. K. N. Reddy, Modern Electrochemistry, Vol. 1, p. 342. Plenum Press, New York (1970). 10. H. E. Gunning and A. R. Gordon, J. them. Phys. 10, 126 (1942). 11. J. Jarzynski, in Water and Aqueous Solutions (edited by R. A. Horne), p. 701. Wiley, New York (1972). 12. Handbook of Chemistry and Physics (Edited by R. C. Weast), p. F36. Chemical Rubber Co., Ohio (1971).