THEORETICAL AND EXPERIMENTAL STUDY OF ICE IN THE PRESENCE OF A SPACE CHARGE

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THEORETICAL AND EXPERIMENTAL STUDY OF ICE IN THE PRESENCE OF A SPACE CHARGE

A. Zaretskii, V. Petrenko, I. Ryzhkin, A. Trukhanov

To cite this version:

A. Zaretskii, V. Petrenko, I. Ryzhkin, A. Trukhanov. THEORETICAL AND EXPERIMENTAL

STUDY OF ICE IN THE PRESENCE OF A SPACE CHARGE. Journal de Physique Colloques,

1987, 48 (C1), pp.C1-93-C1-98. �10.1051/jphyscol:1987113�. �jpa-00226257�

THEORETICAL AND EXPERIMENTAL STUDY OF ICE IN THE PRESENCE OF A SPACE CHARGE

A.V. ZARETSKII, V.F. PETRENKO, I.A. RYZHKIN and A.V. TRUKHANOV Institute of Solid State Physics, The USSR Academy of Sciences, Chernogolovka 142432, USSR, Moscow District

R6sum6 : Une 6tude th6orique et experimentale de la glace par la m6thode des Qlectrodes bloquantes est propos6e dans ce travail. L1analyse effectu6e permet de comprendre les processus physiques provoquant une dispersion de charge d'espace (SC-dispersion) basse fr6quence et de dhterminer la mobilit6 et la concentration des porteurs de charge dans la glace. La valeur de la mobilit6 de H30+ a ht6 obtenue & partir des r6sultats exp6rimentaux.

Abstract : Ice, with ideally-blocking electrodes, has been described theoretically and studied experimentally. The analysis performed makes it possible both to understand the physical processes causing the low-frequency space charge dispersion (SC-dispersion) and to develop a method for determining mobility and concentration of proton charge carriers in ice. The value of the H3O+ -defect mobility has been obtained from the experimental data.

1. INTRODUCTION

When investigating the dielectric properties of ice in a low-frequency range, one can often observe along with the principal Debye dispersion (D-dispersion)(l), an additional dispersion (2,3). There may be up to three dispersion **stepsH (3).

This dispersion is caused by accumulation of charges in the vicinity of the electrode region due to the limited ability of electrodes for charge-exchange, by a space charge. We shall call this dispersion the SC-dispersion. Though its physical nature is clear, SC-dispersion has not been studied practically so far.

The type and character of the SC-dispersion are determined by the ability of electrodes to exchange charges with ice. The SC-dispersion is not observed in the samples of ice with ohmic electrodes but it is maximal in the samples containing ideally blocking electrodes.

The experimental studies of the electrical properties of ice with the electrodes close to ideally blocking ones, performed on the basis of the SC-dispersion theory, allow determination of many parameters of the proton-ice subsystem carriers.

2. THEORY OF THE SC-DISPERSION IN ICE Let us consider a plane capacitor, fig. 1.

t ^{METAL F i g . 1} A n i c e s a m p l e with blo -

X INSULATOR c k i n g e l e c t r o d e s , w h e r e 1 a n d

I ^METAL S a r e t h i c k n e s s a n d a r e a of t h e s a m p l e .

I we assume that the capacity of the insulating layer (which excludes the exchange of protons between ice and electrodes) is infinitely large (the insulator being extremely thin), we can speak of ideally blocking electrodes.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987113

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Let the surface charge on metal plates vary with time as per the harmonic law. For determining the effective dielectric permittivity of the sandwich, one should obtain the amplitude of the potential difference v(w) between metal plates, v(w ) can be obtained from the closed system of equations for all the quantities connected with the electroc field in ice. The number of these quantities is ten : E (x,t) and

fl (x,t) are x-components of the electric field strength and of the configuration vector, respectively ; ni(x,t) are the concentrations of defects of the proton subsystem (i = 1,2,3,4) ; ji(x,t) are x-components of the defect flux density. The values of i = 1,2,3,& correspond to ionic H 0 and OH- defects and to orientational D and L defects : 1 : If+", 2 : "-", 3 : !ID", ? ^{: flL".}

The corresponding equations are written and solved in the general form in ( 4 ) . In the case where only two types of defects are responsible for the conductivity of ice, it is possible to get an analytical expression for the dielectric permittivity of ice. Assuming that only L defects and H30+ ions participate in the conductivity, we can write :

1, m, kl and k2>0

where ei, nio, Di, mi are the charge, concentration, diffusion coefficient and the mobility of the i defect, respectively (e+ - ^eL = e) ; e is the proton charge ; Eo = the dielectric permittivity in vacuum, kg = the Boltzmann constant ;goa = the high frequency dielectric permittivity of ice ( E m = 3.2 (1) ,Es=static dielectric permittivity ; gf = 3.85 kg T r , , (5) ; ro,

the distance between oxygen atoms (rm

= 0.276 nm) ; T is the temperature and w i s the angular frequency.

Fig. 2 shows frequency dependences of real &'(w) and imaginary &If(w) parts of the total comljlex dielectric permittivity, calculated according to Eq.(l).

These graphs and the corresponding Argand diagrams depicted in fig. 2 resemble a

superposition of classical dependences with (w) described by Debye ( 6 ) .

tion of L and ~ ~ defects. 0 '

- 1 0 0 % ~ ( c ) and ( D ) a r e Argand dia- g r a m s corresponding to curves 500 ( 1 ) and ( 2). In the calculation

the following parameters a r e u s e d : _{% ( I ) = 10} 22 -3 _m . _,

21m-3 .

% ( 2 ) = 10

-8 2

- 2 0 2 _ , 4 0 500 1000 m = 1 . 7 1 0 m v-f.secl

tg w ( s e c ) f' L

-6 2 -1 -1 m =8.2 1 0 m v s e c

- ³

n + ( 2 ) = I O ~ ; ~ T= ~ 263 - ~ K ; 1 = 10 m . n : ( l ) = l ~ ~ ~ r n - ~ ;

The fitting of the curves in fig. 2 (or similar ones to Eq. 2b shows that interpolation of the theoretical dependence (Eq. 1) by simple formula (Eq. 2b) can be done to a high accuracy, the values of a1 and a2 being close or even equal to unity. This makes it possible to hope that with a wide range of the parameters employed one can obtain mathematically simple and physically clear interpolation formulas, describing the behaviour of ice with ideally blocking electrodes.

3. INTERPOLATION FORMULAS DESCRIBING SC - DISPERSION

According to (Eq. l), we find families of dependences &' ^{( W} , ^{n ~ , n , , m ~ ,} m+T, 1) by varying one of the parameters. The curves t t ( w ) obtained are fitted to (Eq. 2b).

Through optimization all the seven parameters involved in (Eq. 2b) can be obtained.

Fig.3 shows an example of the corresponding dependencies of this parameters on nL.

Fig. 3 D e p e n d e n c e s of strength and ap- propriate times of relaxation of the D -

d i s p e r s i o n ( ES1- eog , ^tl) and the SC- d i s p e r s i o n ( S. s2- t 2 ) o n the con- centration of L defects y,. T h e upper part of the figure f e a t u r e s polydispersity,

a of the D -dispersion and a 2 , of the S C -dispersion. T h e parameters u s e d

in c a l c u l a t i o n a r e ;

n = 6 . 1 0 l ~ m - ~ ; m = 8.2 1 0 - ~ m ~ ~ - ~ s e c - ~ ; m+-1.7 10-8m%-'sec-1, T=263K,

L- -3 1 =1.10 m.

Analogous calculation has been done upon varying all the latter parameters in (Eq.

1). Thus procedure allows determining to a high accuracy the analytical dependence

of seven parameters in (Eq. 2b) on the quantities involved in (Eq. 1).

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In this case we obtain the following parameters, characterizing the SC-dispersion :

When the inequalitites are weaker than (Eq. 3) and "crossover" approaches (mL nL = m+ n+), the interpolation relations become more complicated. This problem is considered in detail elsewhere. The interpolation formulas derived not only elucidate the physics of the processes responsible for the Sc-dispersion, but also may serve as a good first approximation for describing the experimental data obtained by (Eq. 1 ) .

4. EXPERIMENTAL RESULTS

To study experimentally the SC-dispersion in ice and apply the developed theory, one must have electrodes whose properties are close to those of blocking ones. Let us formulate the subsequent requirements for real systems :

1. Charge exchange between ice and electrode should be excluded.

2. The capacitance of the insulating interlayer CB should be much greater than the effective diffuse boundary layer capacitance Cdl.

We used mostly electrodes of highly pure aluminium coated with sprayed sapphire (A1203). The sapphire film thickness varied from 0.05 to 0.1 microns. The results we believe to be the most interesting are presented in this paper ; the experimental techniques and other results would be described elsewhere.

The problem of the type or character of SC-dispersion is most important. Fig. 4 shows the low-frequency part of Cole-Cole plots of the SC-dispersion in ice.

The SC-dispersion of ice is described by the relation similar to (Eq. 2b), which does not differ much from the conclusions of the theoretical analysis.

600 1 ^{SC -} dispersion d

Fig.4. Low-frequency part of Cole-Cole plots for some tem- peratures. 1 = 1.2 mm. T h e i c e i s not doped purposely, but i s

not chemically p u r e either.

The most significant feature of SC-dispersion is the dependence of its strength

( - sl) and it relaxation time t2 on the dimensions of the sample. But, at

present, we have insufficient data to define such dependences unambiguously. We hope

to do it in the near future.

nature of change of the SC-dispersion while approaohing and withdrawing from the wcrossoverll. A certain analog of figt 3 can be experimentally obtained by changing the concentration of L defects during the change of temperature (T) of the ice. For this purpose we must have crystals doped with an impurity delivering number of T-independent H O+ ions to the ice.

We have succedea in growing such ice crystals and fig. 5 shows the variation of the parameters of the D and SC-dispersion in the region of "crossoverw, as a function of temperature.

It should be noted in conclusion that the many experiments the SC-dispersion was a handicap to gain to required data. Assuming that the frequency dependence of the dispersion is determined by (Eq. 2), we succeded in subtracting the contribution Of the SC-dispersion from the quantities under measurement and in determining more exactly the D-dispersion parameters. In particular, we managed to show that the procedure used earlier for data handling are really valid. Furthermore, as is shown below, the study of the SC-dispersion offers more important informations on the proton-ice subsystem.

Fig. 5. Dependence of strength ( 1) , ( 3 ) and relation time ( 2 ) ,(4) of the D- and SC-dispersi~ns~respectively. T h e ice i s doped with ~ ~ 0 + = d o n o r s .

1 = 1.2 mm.

I I

3.0 5.0 60

107 TEMPERATURE, K1

5. MOBILITY OF ION DEFECTS

As it follows from eqs. (1) and (4) the type of the SC-dispersion is directly connected with defect parameters of the proton subsystem such as the concentration and mobility of charge carriers. From Equations (11, (3) and (4), far from the

ncrossoverll, for the minority carriers one can write :

While approaching the flcrossover", the formulas become more complicated, but it is

still possible to determine the parameters. From the same data, previously used in

C 1-98 JOURNAL DE PHYSIQUE

fig. 5, we managed to obtain the value of mobility of minority carriers at some temperatures ( T < T C r w 200 K) - see e.g. fig. 6. We believe them to be H30+ ions in our case. Determination of mobility at T > ^{240 K} is difficult because of the possible influence of the surface conductivity (even in the measurements with guarded electrodes).

K-S

Fig.6. Mobility of minority charge carriers

( ~~0~ ions ). T h e c u r v e s calculated from theo- r i e s of Pintar and G o s a r ( P - ^G ) / 8 , 9 / and of Kim and Schmidt ( K - ^{S ) /lo/} a r e a l s o in-

It is difficult to judge the temperature of the H30+ ion mobility from the studies made so far. Here, we may only state, that the values obtained are very large and closer to the theoretical (8-10).

ACKNOWLEDGEMENTS

We are grateful to Dr. A.I. Evtushenko for growing crystals, to Mr. E.O.Potapov for technical assistance and to Mr. P.N. Okolev for preparing most of the diagrams.

REFERENCES

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(3) Von Hippel A., Knoll D.B. and Westphal W.B., J. Chem. Phys., 54, (1971), 134.

(4) Petrenko V.F. and Ryzhkin I.A., Phys. Stat. Sol. b, 121, (19841, 421.

(5) Hubman M., 2. Phys., B32, (19791, 127.

(6) Debye P., Polar Molecules, N.Y., Chem. Catalogue Company, ( 1929).

(7) Cole K.S. and Cole R.H., J. Chem. Phys., 9, (1941), 341.

(8) Gosar P., Nuovo Cimento, 30, (1963), 931.

(9) Gosar P. and Pintar M., Phys. Stat. Sol., 4, (19641, 675.

(10) Kim D.Y. and Schmidt V.H., Canad. J. Phys., 45, (1967), 1507.