ELSEVIER Physica B 222 (1996) 123 130

Dynamic transport on humid silica surface II

Altaf Husain 1, A.K. Jonscher* Royal Holloway, Universi(v qf London, Egham, Surrey. TW2OOEX. UK

Received 19 June 1995; revised 11 September 1995 and 12 January 1996

Abstract

A recent study of dynamic transport on humid silica surfaces provides data on the frequency dependence of the complex capacitance of humid silica rods. This is now supplemented by data on the time dependence of charging and discharging currents on humid silica rods with the relative humidity, rod diameter, rod length, electrode material and the applied step voltage amplitude as the principal parameters. The results confirm the "fractal" nature of charge transport and reveal important complementary information on the non-linearity of the charging and discharging processes.

1. Introduction

It is known that transport of charge on humid insulating surfaces exhibits very interesting phe- nomena which point to a large storage of charge in this process, with strong low-frequency dispersion (LFD) in the frequency domain (FD) characteristics [1--4]. Recent data [1], referred to as 1, relating to humid silica rods of a range of diameters d and spacings s between ring electrodes provide a de- tailed picture of the processes involved, as well as giving a list of the relevant literature. The present paper gives complementary information on the time-dependence (TD) of the charging and dis- charging currents, it(t) and id(t), under step-func- tion excitation. While the T D behaviour of linear systems should be the Fourier transform of the corresponding FD response, this is manifestly not true of transport on humid insulators which ex- hibits significant non-linearities in the form of it(t) and id(t) having very different magnitudes and dif-

* Corresponding author. 1Permanent address: Physics Department, University of

Karachi, Karachi, 75270, Pakistan.

ferent time dependences. Thus the TD information is essential for the correct interpretation of the dynamic behaviour of surface transport and any conclusions based solely on FD behaviour are li- able to be suspect.

It is important to point out that despite the evident non-linearity, certain broad features of the FD and T D information remain approximately consistent between themselves, the most notable of these being the time dependence which follows the general trend

i(t) ~ t - " (1)

with the exponent n close to zero, which is the Fourier transform of the FD L F D response which at 'low" frequencies goes as co "-1 . The presence of discharging current is a clear indication of the im- portance of charge accumulation in the system and the present results will give detailed information on this process. The fact that the FD and TD re- sponses are approximately compatible with one another may be understood in terms of the fact that any FD measurement, whether by a bridge or a fre- quency response analyser as in Ref. [1], is necessar- ily measuring the response to the fundamental

0921-4526./96/$15.00 (C 1996 Elsevier Science B.V. All rights reserved PII S092 1 -4526(96)0002 1-X

124 A, Husain, A.K. Jonscher /Phys ica B 222 (1996) 123 130

frequency only, rejecting rigourously all higher har- monics, thereby linearising the system's response.

2. Experimental details

The samples of silica rods used in the present study and their treatment were the same as those described in Ref. [1] and will not be repeated here. The principle of TD measurement is well known and may be found in standard references [ l-Y], and will be only briefly outlined here. The instrument used is the Chelsea time-domain system represent- ed schematically in Fig. 1 consisting of a voltage source for energising the sample and a separate source providing the reference current for a differ- ential logarithmic amplifier which gives a voltage output proportional to the logarithm of the ratio of the measured current to the reference current. There is a logarithmic or linear timebase for plot- ting the data on the computer display or on the plotter.

The diameters of the samples measured were d = 2, 5, 8, 14, 20, 28 mm, the spacings between elec- trodes were s = 2, 4, 8, 16, 32 mm. Measurements were taken for most of these combinations, exactly as in the case of FD measurements, but we present here in detail only the summary of typical data.

3. Voltage dependence

We begin our presentation with the dependence of the TD response on the amplitude V of the step voltage. Fig. 2(a) gives the set of data for &(t) with s = 2 m m and d = 2 mm and with the V range 0.2-14 V. The general time dependence of &(t) and &(t) is "flat" with some tendency to fall-off at longer times and lower voltages and rapid fall-off above 6 V. There is also a clear tendency to a reduction of the amplitude of the response beyond 8 V, with a shortening of the time to the fall-off. At the highest V in some samples there is even a rise of &(t) before the final drop. The corresponding set of data for id(t) is shown in Fig. 2(b) with broadly similar features, including in particular a drop of ampli- tude at the higher voltages with a more pronounced fall-off at longer times below 2 V.

Fig. 1. Schematic diagram of the Chelsea t ime-domain instru- ment used in the present experiments.

log[ i:( t)/ A ]

-7.0 -

-7.5 -

-8.0-

-8.5- [-~

I

-4

. . . . . . . . . . . . . . . . i -2-2-2: . . . . . . . . . . . ~ --- :-z: : . . ---- . i ---- . -- . : -- : - - - t . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . ! . . . . . 4 . . . . . . . . . . . . . . i . . . . ~ ,! ! q , o ' :

. . . . . . . . . . . . . . . . . i

~ i i i ! i i

I I I I I I I I -3 -2 - 1 0 l 2 Iogt/s 4

- 8 . 0 - log[id(t)/A]

- 8 . 5 -

- 9 . 0 -

- 9 . 5 -

- 1 0 . 0 -

i . . . . . . . . . ' - . . . . . . . . .

i i l '~ i i ~ ~ ~ -- "'~'-"e~& .4

- 3 - 2 -1 0 1 2 3

Fig. 2. The time-dependence of &(t) and &(t) for a range of values of the step voltage V for a rod with s = 2 m m and d = 2 m m a t 9 7 % RH.

Fig. 3(a) and (b) give similar data for a longer sample with a much larger diameter, s = 32 mm, d = 28 mm. The contrast with Fig. 2 is that the fall-off ia(t) at the lowest V is much more pro- nounced and occurs at the very early stages, t = 10 -z s, approximately. The recovery of the "flat" time dependence takes place gradually at

A. Husain , A .K. J o n s c h e r / Phys i ca B 222 (I 996) 1 2 3 - 1 3 0 125

log(i/A) i i ~: i - ~ . s - ..~ .......... ~: ............................... ~"-'.-'-" :*. '-'..=..'-~: ...... .*...z ,-.i

• • • "~ . . . . . . ~ . . . . . . . . . . ~ - - • " f - - 1 0 [

. . . . . . ~ . ~ . ~ . . . . i . . . . . . . . . . ~

- s. o . . . . . . . . . . . . . ~ .............. ~ * ' = ' ~ ' ~ ' = ' ~ ..... 5= . . . . . . . 4 = = * = * - ~ - 0 . 2 - - - - . i i i i i lo~ds i I I I

- 4 - 3 - 2 - 1 2 3 4

.. : ~ i i i i i - o- ~ . . . . . . . . . . . . . . i 4 i

log(i~/ A)

- 9 - i i i i i v~ i i

- 1 0 ~- .............................................................................................

L

! o . 4 ~ , , , . 2 , i i , i . . , ! i ~ i r ~ ! toglvs) i I I t I t t

4 - 3 - 2 -1 0 1 2 3 4

Fig. 3. The t ime-dependence of &(t) and id(t) for a range of values of the step vol tage V for a rod wi th s = 3 2 m m and d = 28 m m at 9 7 % RH.

l og[ i c /A] at l s "0

- 7 . 5 . . . . . . . . . . ; : : ~ : : : . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . : y ' 5 .

i i .................................................................. l o g ( v / v ) I I

- o . 5 o . o 0 . 5 1 . o

Fig. 4. The dependence of i~ at t = 1 s on the cha rg ing vol tage

V for a n u m b e r of rods wi th s = 2 m m and different d as shown

in mm. The line & ,~ V is shown for compar i son .

log[i -6.0 -

-6.5 -

-7.0 -

-7.5 -

-8.0 -

J A ] at I s i i i a )

.......... ~ ..................... i .......................... i ..................... ~-~i---~--t ....... i oOO~i .. . - ' ~ ' ~

.......... i ...................... i ......................... i--.-;-~--.- ~:~: .......... ~o

. . . . . . i ......................... i ........... ............. ! _ , z V ~ 28 i md~ / " ] *.---2 i 12o ~ ~ - - i i

d=:2 ~...~ _5~_.~.. L ,,--4.5 :: " . ' ~ ..... . ..................... i ............... 1 3 . . . . . . . . . . . . . ! ..........

4 . ~ ;i " ' ~ 0 ~ 8 1ogl V/V ) ......... i ................ 1 ........................ i ..................... l ~ .............. 1 ...........

-1.0 -0,5 0.0 0.5 1.0

higher V and goes over into a slowly decaying form at 12 and 14 V, accompany ing the general reduc- tion of ampli tude described above. In this sample, there is no fall of ampli tude with rising V and the decay of i d ( t ) continues to much higher voltages.

Part , but not all the differences between the re- sponses in Figs. 2 and 3 may be due to the fact that the longer sample has a much l o w e r f i e l d for a given voltage, but this is not sufficient to account fully for the differences.

To study in more detail in ampli tude dependence of V, we have taken the values ofic(1 s) at t = 1 s for samples with s = 2 m m and variable d and plot ted them against V in Fig. 4 showing that at low voltages both currents are propor t iona l to V and the peak of ampli tude is reached for values of V increasing with s. A more complete picture is obtained from the normal isat ion of the data for both ic and id in Fig. 5 which brings out the broadly c o m m o n nature of the voltage dependence of the currents, a l though it should be unders tood that the displacements in the two cases are not the same and

log( -7.2 -

-7.6-

-8.0-

-8.4 -

/A) at Is

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

F ........... ~ - X ........ T- .................... i .................... ~ ................... i " ~ " i i i il°8(v/v)

-1.5 -1.0 -0.5 0.0 0.5 1.0

Fig. 5. (a) N o r m a l i s a t i o n of the da ta in Fig. 4 for i~, to show

c o m m o n trends, The d i a m o n d s wi th number s relate to the

relat ive d i sp lacements of the var ious vol tage sets in the nor-

mal i sa t ion ; (b) gives the co r r e spond ing no rma l i s a t i on of da t a re la t ing of ia. The relat ive d i sp lacement of ic and id are no t the same. The s imi la r i ty of the shapes of the normal i sed p lots shou ld

be noted.

neither do the voltage at the peak of the behaviour cor respond to one another.

We may define the conductance G ( s , d) a function of s and d as the ratio ic(1 s ) / V . The dependence of G ( s , d ) as function of d with s = 2 m m is shown in

126 A. Husain , A .K. J o n s c h e r / P h y s i c a B 222 (1996) 1 2 3 - 1 3 0

-6.6-~ log[G(2,d)/S] ............... T .............. T ................ ~ " . : ~

-6"8- i .................. i .................. i ................. i ............... ~ ~ . ~ ................. i

-7 .0 - ! FRA ~ - c x p ( ~ 3 0 ) ~ . ; ' ~ ' i ................. i ................. i

................. ............ i ................ i ................. i . . . . . . . . . ............ i ................. i ................................... i

-7.6 . . . . . [3....~:..'.i ............ i.... TD 6: ~ exp(d,'25) ....." ................ i 5" ' i ! i ! i i

-7.8 .................. i .................. i ................. ~ ................. ~ .................. d / m m I I I ! I

0 5 10 15 20 25 30

Fig. 6. T h e d e p e n d e n c e o f t he c o n d u c t a n c e G(2, d) for a r o d w i t h

s = 2 m m a n d v a r i a b l e d g i v i n g t i m e - d o m a i n d a t a ( T D ) a n d a l so

for c o m p a r i s o n f r e q u e n c y - d o m a i n ( F D ) d a t a , b o t h g i v i n g ex -

ponen t ia l d e p e n d e n c e o f d.

log[G(s,28)/S] i i i -6.6 ................................. i .................. i .................................... ! ................................... i

-68- i< ........................... i ................. .................. i ..................................... i

% 0 .............. .......................... i . . . . . . . . . . . . . . . . .

-7.2 - ' . . . . . i i i

i I I I l I

0.4 0.6 0.8 1.0 1.2 1.4 1.6 log(s/ram)

Fig. 7. The dependence of the conductance G(s,28) for a rod with d = 28 mm and variable s giving time-domain data (TD) and also for comparison frequency-domain (FD) data, both giving power-law dependence of s- o.T0.

Fig. 6 giving a clear exponential dependence on d

G(s = 2, d) oc exp(d/e), (2)

where ~ is approximately 30 and is in agreement with the corresponding data reported in Ref. I-1] for the FD response which are also reproduced here.

By contrast, the dependence of G(s, d) on s with constant d = 28 mm shown in Fig. 7 gives a frac- tional power law dependence

G(s,d = 28) oc s -°'7°, (3)

and again good agreement is seen with the FD results reported in Ref. [1].

4. Relative magnitudes of it(t) and id(t)

The relative magnitudes of i¢(V) and id(V) were the subject of various studies in the past dealing with mica, sand and glass which were listed in Ref. [1] where the principal form of behaviour was that i¢(V)oc V - Vo, with Vo being a "built-in" voltage, while id(V)= VoG = const., where G is the conductance defined above. In our experi- ments we have found consistently a different behaviour, whereby both i~(V) and id(V) were volt- age-dependent, while the ratio ic(V)/id(V ) re- mained relatively constant. The varmtion of it(V) and id(V) taken at the time t = l s was already shown in Fig, 5 where it was clear that these cur- rents show a remarkable degree of "parallelism" in their voltage dependence.

This difference of behaviour may be connected with the fact that, unlike mica, sand and glass, silica rods do not have any significant dry conductivity. However, this is just a guess at the resent time. The conclusion here is that in the present case there is no "threshold" for the saturation of id(V).

5. Dependence on electrode material

The precise role played by the material of the electrodes is not well understood at present. While our results show clearly that there is no significant voltage drop on the electrodes, ruling out blocking action, the material of the electrodes does influence the magnitude of it(V) and id(V) as shown in Fig. 8. At lower voltages both currents increase propor- tionally to V with a tendency to saturation and eventually fall, especially for id(V). Au has the high- est, Ag the lowest currents with approximately a factor of 10 between them.

The electrodes will be found to influence parti- cularly the total charge Qd described below.

6. The total charge

It is apparent that there is a visible tendency to fall-off of both it(t) and id(t) at longer times and this is particularly pronounced for long samples, while being fairly independent of d. Of the two quantities,

A. Husain, A.K. Jonscher /Phys ica B 222 (1996) 123-130 127

- 7 . 0 - 1 ........... 7 ....................... ~ .......................... ~ .................. . . ~ . ' " log(//A) ] i ~ i . - i

- 7 . s . . . . . . . . . . . . . i ....................... i ~ v . . . . . . . . i . L . . : 2 . . ' .__

- 8 . 0 . . . . . . . . . . . . . . . . . . .

. 9 . ~ ~~~:::::: - i o . o + ~ - i ........

-~o,~-I¢'~,-.. ,, - ...,.....:..- ~...._~gLv...~...

i i - 0 . 5 0 .0 0 .5 1.0

Fig. 8. The voltage dependence of ic (open symbols) and id (solid symbols) for several electrode materials.

the total charge Q¢ involved in the charging process and Qa involved in the discharging process, the latter is more important because it relates to the charge s t o r e d in the system which must of necessity be finite, while Q~ tends to diverge since it involves the throughput of charge going on indefinitely.

It is now possible to estimate the value of the integral

;o Q a = i a ( t ) d t = In 10 10X+rdX (4) ~3

where Y = logia(t) and X = logt. The dominant contribution to this integral comes from the highest values of X + Y as illustrated in Fig. 9, where the element of integration indicated by the dotted verti- cal lines corresponds to 10 r dX. The integration is not affected significantly by the exact functional shape ofi(t) away from the highest values ofX + Y. Using this procedure we have estimated the total charge corresponding to the various charging con- ditions. Allowing for the inevitable scatter inherent in the present method of evaluation, the results of this calculation for the series of rods with d = 28 mm and a range of s are shown in Fig. 10 as Qa vs V showing that Q a ( V ) is almost constant at low voltages and then grows rapidly by some six orders of magnitude over a change of V by a factor of 2 approximately beyond a threshold voltage Vxh. Fig. 11 shows the normalisation of the data of Fig. 10 which follow a broadly common pattern with varying s and confirm the impression of an exponential dependence for V > VTh. The satura-

Y = log~Ix) . . _ _ l",,, N x + r = - o . 4 3 4 3 - - - ~ i

0 0 i " ' - . . . . . . . . . . . . . . . . . ! . . . . . . . . ~..

-o,si l~i~) ~ ......................... , '- ................... i--

-1.0! ...................... i ..................... ! ........... :. """'T

- 1 . s i .......................... i ......................... ' .......... '-.." ......... ! ......... t--

. i . 5 0 ~ t-2.0 i .......................... i ] i i .......................... i .......... : . . . . . . . . . . . . . . !

- 3 - 2 - 1 0 1 X = logx

Fig. 9. The determination of the integral of a function f ( x ) in logarithmic coordinates, where the lines with slopes - 1 corres- pond to constant values of logf(x) + log x. The area under the curve is 1 and the extreme tangent of slope - 1 corresponds to a value 10 .0.43`* = 1/e = 0.3679, so that the integral in Eqn. (3) works out as 0.8471. The successively lower lines each contribute one-tenth less than the previous one. The points along the line moving away in either direction from the extreme tangent con- tribute values indicated by the numbers in boxes which have rapidly diminishing values. Thus the integral is given ap 9roxim- ately by In 10 x 10 x÷r.

Q a / C .~ . I .~ I'i'l d = 2 8 rnm

I o - 4 .......... - O ' - ' L t : : : : : : : : : . ~ . . . . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . .

o-, i.o: ........... i ......... .............. i .... e 1 - ~ . . . . -., ~. . .T . . . . . . . . i . . . . . . . . . > ' : ~

, o - ..i ..................... .................... h-"-:h ......... -8 i ".',i / i ..°"i.." i

1 0 - ~. . . . . . . . . . . . . . . . . . . . . . ÷ 4 - ; , . . . . . . . . . . . . . . . . . . . 4 . . . - . . , = e ............ %, . . . . . . . . . . . . . . . . . . . . . i . . . . .

1 0 - 9 - "- .................. / : ~ .............. ,~-'-9- ........... :,:2:: " * " s= 2 m m

-,o v ~ ' , ~ ; , ....'" v~ ..'" .-~- , = , r a m 1 0 - --+ ........... *'- ...... -/- .................. ;,'~ ............... 0"" s = 8 m m

- ~ ~ "..'._'.'_T.'..'.~.~. 1 0 . . . . . . . . . . . . . . . . . . . . . . . s =32 m m

i i i i i 0 1 2 3 V / V 4

Fig. 10. The estimated charge Qd(V) in the discharge process of five rods of different s and the same d = 28 ram. The onset of rapid growth of Qd(V) is visible at 0.5 V for s = 4 and 8 mm, 1 V for 16 m m and 2 V for 32 mm. The s = 2 m m data appear to be saturated at all voltages. Vrh represents threshold voltage for the onset of rapid growth of charge.

tion shown in Fig. 1 1 may be misleading the charge for s = 2 mm is indeed almost independent of V but its position on the normalised plot is arbitrary.

1 2 8 A. Husain, A.K. Jonscher / Physica B 222 (1996) 123-130

/ q / t " ~d~" "~" s = 2 m m AV--4.0V i / ~ T"*I

. ' - ' & " s = 4 n u n A V = 3 . 0 V . . . . . . . . . i . l C r 3 U J . - . - _ - . - . - [ , , j 1 0 " - 4 , . - s = 8 m m A V = 2 . 0 V l ~ l z " ~ i

- ~ - ,=16ram a v = l . 0 v . . .A~42L"I~ i

~o"°" --[ ................. ! ~xp(sv)~~-!-! ................ i . . . . . . . . . . . . . . . . " . . . . . . j z _ . . . . . . . . i . . . . . . . . . . . . . . . } ..~ ................. ~ ................ + ............. i. ~.4.......~. .............. + ................ i

i i i :'" i i i - 9 i ~ i *~ . ~ ~ ~ i

1 0 '~ ............... ~ ................ r . . . . . . ; ' ~ ; ' r ",." ........ ~ .............. T ................

1 0 . 1 o i i ~ .-" : i J i i i --i .............................. +--.'------t-i- '- ........... i ............... + ................ i

1 o : ~.,,.."~- Tv'z.~ ............ i V + A V / V i i t I I 0 1 2 4 5 6

Fig. 11. The normalisation of data of Fig. 10 in semi-logarith- mic presentation, the shift in voltage AV with respect to the s = 32 mm data being 1 V for each of the subsequent rods with s = 16, 8, 4 and 2 ram, suggesting that the charge increases expo- nentially with V and decreases exponentially with the length s.

The determination of the total charge Qc in the charging process is less certain since the saturation of i¢(t) is much less pronounced. For s = 4 and d = 28 m m the value of Q¢ for 8 and 10 V is 10 -4C which is between 10 and 100 times higher than the highest values of Qd in Fig. I0. There is no doubt, however, that much higher values of Q~ may be reached at other voltages and with different geometries.

Both the magnitude and the voltage dependence of the stored charge are critically determined by the nature of the metallic electrodes, as is shown in Fig. 12. Au and NiCr give very similar results and In gives results smaller by a half-a-decade, in all three cases the voltage dependence falls between V °'s and V. By contrast, Ag is in a class by itself with much stronger voltage dependence and much higher absolute levels being observed.

- 7 1 0 ": .................................. ; " - i ............. d = 5 m m , s = 8 r a m .....

~ / ~ , , " 0 .... i O ° .

] • i - - . - ° " O

1 o ................ ~ ............. ~ : : - " ~ , - - • 4 . . , % P / i "

~i ," i , - , . . . '~ i ~..~.~'0< i _ ~ . - - - . . ~ ~ . 1 < - -o - ^ , ~]i(~ / ............ " ~-'" .-Av- Ag b + ...... , - " ' ~ ,,o., - - ~ - N i C r

1 09 4 ~ / -""" ~Zd,~ v .-0.-. In //V i | , i , i , , I ' ' • • • ' ' ' 1 2 3 4 5 6 2 3 4 5 6

1 1 0

Fig. 12. The values of Qd(V) for identical rods with different electrodes. The data for Ag are in a class of themselves and only a fraction of them is shown, with values as high as 1 0 - 6 at 2V. The two lines with slopes 0.5 and 1 indicate the broad range of voltage dependence which is seen.

The data in Fig. 10 show that the longer the sample the higher the voltage for the onset of rapid steep rise of Qa. This implies that a critical field is needed to cause a rapid growth of stored charge.

7. Energy stored in the system

The presence of stored charges Q~ implies a stor- age of energy which may be described in terms of the product

Ws = Qs Vs, (5)

where V~ is a "storage voltage" retained in the system, which is likely to be comparable to the "built-in" voltage Vo discussed earlier in our pre- sentation. A similar concept may be applied to the energy dissipated in the system during the dis- charge process,

Wa = Qc Vc (6)

and similar observations apply to this as to Ws. We note here that not only is Q¢ much larger than Q~ but also the relevant voltages are very different, v~>> Vs.

8. General features

The results shown in the present paper extend very considerably the picture obtained in Ref. [1] on the basis of FD measurements. The following broad features are evident.

A. Husain, A.K. Jonscher/Physica B 222 (1996) 123-130 129

(a) it(t) shows the general trend to almost time- independent behaviour consistent with LFD re- sponse established as a common feature in Ref. [1]. ia(t) gives a more complicated picture, with a tend- ency to "exhaustion" at progressively shorter times with falling V, and that almost independently of s, but the very existence of id(t) confirms the presence of charge storage in the surface conduc- tion process.

(b) Both ic(t) and id(t) are linear in V for lower step voltages, subsequently peaking and decreasing for higher V. The process scales with both s and d but does not follow any simple laws, although approximate scaling with V/s is apparent for shor- ter rods. There is no evidence of saturation of ia(V) nor of the existence of a "built-in battery voltage" Vo.

(c) There is good evidence for "fractar ' nature of transport, whereby the conductance is not a linear function of d/s as would be expected for uniform surface flow and follows instead of fractional power law dependence on d and an exponential depend- ence on s. This strongly suggests the importance of filamentary flow on the surface, which is hardly surprising in view of the likely non-uniformity of condensation on the surface.

(d) The charge Qd(V) recovered in the dis- charge process is very low at low V and then rises rapidly by six decades, with apparent saturation at high V.

9. Interpretation of results

The existence of discharge current id(t) provides the most direct demonstration of the essential dif- ference between LFD and direct current (DC) con- duction, since the latter could not possibly show any discharge.

The ratio of the recovered charge Qa(V) to the injected charge Qc(V) and the corresponding ratio of the energy stored to the energy lost in the pro- cess, is clearly brought out by our measurements. It is evident that ic(t) at constant V and Qc(V) at constant t are dominated by irreversible charge transport between electrodes, and only a fraction of the order of one-tenth to one-hundredth is re- covered. This means that the process of charge

storage accounts only for a small fraction of the total charge and the question of its precise mechanism has to be further elucidated, especially in relation to the role of the material of the elec- trodes.

Very illuminating in this respect is the evidence of initial growth and subsequent fall with V of both it(V) and id(V). This shows the finite ability of charge to respond to the driving electric field, and the most likely cause of this is the injection, rather than the collection, process. According to this no- tion, injection cannot supply charge at an unlimited rate and leads to a lowering of current. This then explains also a corresponding reduction of the stor- age process, which appears to maintain a fairly constant ratio to injection. One possible specific mechanism is the exhaustion of the available hu- midity at short times where the currents are rela- tively high and the supply from the ambient does not become effective until later. Our results suggest that a small fraction, 1-10% of the total charge transported through the system is in fact stored.

Little is understood at present of the mechanisms of charge storage on humid surfaces, except that it seems to be related to the electric field in the inter- electrode space, rather than to the voltage on the sample. The apparent saturation of Qa with V pro- vides an indication of the restriction on the density of the species that can be accommodated on the surface. The presence of memory effects, whereby the charge stored can be preserved even after dry- ing out of the sample and its subsequent re-humid- ification, indicates likewise that storage represents a fairly long-lived process.

Saturation of id with the charging voltage V ob- served in earlier measurements may be seen as a corresponding limitation of the stored charge Qd and stored energy Ws in those samples. By contrast, in the present experiments both Qa and W~ increase with V.

Consider now the theoretical interpretation of the strongly dispersive frequency dependence of the LFD processes observed in our results relating to transport on humid surface. Since both it(t) and id(t) are slowly falling with time, common sense suggests a strongly dispersive Fourier transform giving a frequency dependence c~"-1 with a small

130 A. Husain, A.K. Jonscher / Physica B 222 (1996) 123-130

value ofn. However, the necessary condition for the validity of the fractional power law dependence is the independence of frequency of the ratio of the energy stored to energy lost per cycle. This has been discussed earlier [3, 5] and relates the values of n to the ratio of Wd/Ws which is very large in our situ- ations, leading to small values on n.

References

[1] A.K. Jonscher and A. Husain, Physica B 217 (1996) 29. [2] A.K. Jonscher, Dielectric Relaxation in Solids (Chelsea

Dielectric Press Ltd, London, 1983). [3] A.K. Jonscher, Universal Relaxation Law (Chelsea Dielec-

trics Press Ltd, London. 1996). [4] A.K. Jonscher, J. Mater. Sci. 26 (1991) 1618. [5] A.K. Jonscher, J. Mater. Sci. 30 (1995) 2491.