Simulation of contact electrification of polymers by an analogue model

Journal of Electrostatics, 6 (1979) 15--27

15

© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

SIMULATION OF CONTACT E L E C T R I F I C A T I O N OF POLYMERS BY AN A N A L O G U E MODEL

M. HENNECKE, R. HOFMANN and J. FUHRMANN Universitilt Kaiserslautern, Fachbereich Chemie, Postfach 3049, D-6750 Kaiserslautern (G.F.R.) (Received September 16, 1977; in revised form April 7, 1978)

Summary A simple analogue model of the electrification o f polymers by intermittent contact is proposed. The parameters of the model can be interpreted in terms of solid-state physics. The time-dependent densities of excess electrons on the polymer surface and in the polymer bulk are computed. Statements are made about the influence of polymer parameters (for example, energy level of the polymer band-gap states and barrier heights between these levels), whilst experimental parameters (such as work function of the contacting metal, contact ratio and contact frequency on transient and steady-state contact electrification) are accounted for.

1. I n t r o d u c t i o n Contact electrification of p o l y m e r solids depends on the mechanisms of charge injection and of charge storage on the surface and in the bulk. Charge storage and charge transport in polymers are discussed in terms of an electron energy level system [ 1--3] with localized band-gap levels, i.e. localized traps for electrons or holes. Within this concept, the density and the energetic situation of localized levels both on the surface and in the bulk are responsible for the electrostatic charging of p o l y m e r solids. Contact electrification due to metal--polymer contact has the advantage of a fixed b o u n d a r y condition, the well defined Fermi energy of the contacting metal [4]. Contact electrification can be investigated by the following different types of experimental technique: (1) transient contact electrification associated with charge rate effects, (2) steady-state contact electrification (for example, due to sequential intermittent contact), and (3) equilibrium contact electrification due to thermal, mechanical and electrochemical equilibrium at the interface [ 5]. In most of the experiments to investigate this effect, steady-state contact electrification due to sequential intermittent contact has been used [3,6--8]. The results of these experimental techniques depend on the steady-state parameters, which include contact ratio and contact frequency.

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The correlation between steady-state electrification and the experimental parameters can be shown by an analogue computer using a simple kinetic model. The experimental parameters investigated here are contact ratio, contact frequency and work function of the contacting metal. Band bending is neglected in this approach, because we do not consider hole traps and, hence, the build-up of the consequent double-layer space charges which may affect the transition rates is ignored. 2. Specification and range of validity of the kinetic model The computer simulation of contact electrification is based on the following model (Fig. 1). The model system describes three different spatial regions: the contacting metal, the boundary layer of the polymer with surface states (S), and bulk states (B = B1, B2 . . . . ) in the adjacent polymer bulk. Excess electrons can exist in these polymer regions. One gets particularly simple expressions for the density of electron traps on the surface and in the bulk of the polymer if one assumes a single energy level system for each polymer region. It has been shown experimentally that a multi energy level system with an inherent occupation probability is adequate for polymers [ 5,9--11 ]. Nevertheless, the simplification to a single energy level system does not essentially affect the statements about the influence of surface and bulk on the total charge. We presume that prior to the first metal contact, no excess electrons are present in the polymer. Intrinsic charge carriers are neglected. ._. _I_

1

~_~

S

_L

B1

,

B2

B: BI+ B2 2

L,.

m

F'

I

I_. I__ 1 1;~2 F' BI~--~B2 F 1;-T~ S ~

Fig. 1. M o d e l o f e l e c t r o n i n j e c t i o n f r o m m e t a l t o p o l y m e r o n t h e basis o f a single e n e r g y level s y s t e m o f e l e c t r o n s o n t h e surface a n d in t h e b u l k o f t h e dielectric. F, F ' are F e r m i e n e r g y o f t h e c o n t a c t i n g m e t a l a n d c o u n t e r e l e c t r o d e . S, B1, B2 (B1 = B 2 ) are surface a n d b u l k s t a t e levels. T h e t i m e c o n s t a n t s for e l e c t r o n t r a n s f e r are: ~'F f r o m m e t a l t o p o l y m e r s u r f a c e ; 7 s f r o m p o l y m e r surface t o b u l k ; rB1 = rB2' between the bulk states ( b a n d b e n d ing is n e g l e c t e d ) ; rB1, f r o m p o l y m e r b u l k t o s u r f a c e ; ~s, f r o m p o l y m e r surface t o gas p h a s e o r t o dissipative surface levels; 7"B2 f r o m b u l k t o c o u n t e r e l e c t r o d e .

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An electron injection from metal to the p o l y m e r can occur if vacant localized energy levels in the surface are situated energetically lower than the Fermi energy of the contacting metal. Due to the higher number of free electrons in the metal than in the polymer, the electron density in the metal will n o t change during contact. If one denotes the density o f available states before any excess electron is present in the surface by Nssat and the density of excess surface electrons b y Ns, then (Nssat - Ns) is proportional to the driving force of the excess electron flow from metal to polymer, i.e. dNs/dt. If reverse flow from p o l y m e r to metal is neglected, the following linear dependence can be assumed:

dN~/dt

= (Nssat

-

(la)

Ns)/r F

The rate constant 1/T F represents the reciprocal relaxation time for electron transfer from metal to surface. Electrons can flow from the surface states (S) to adjacent bulk states (B1). This transfer can take place if excess electrons are present in S and if electron levels (B1) are available:

dNbs/dt = - N s ( N B I s a

(lb)

t - NB1)/r s

Analogous to this, one has to account for a reverse flow of excess electrons from B1 to S:

dN~/dt = Nm(Nssat - Ns)/TBI'

(lc)

Last, b u t n o t least, flow from S is considered wherein the charge can be dissipated from S to the surroundings (gas phase, surface conductivity) at a rate proportional to Ns:

dNd/dt = - N s / r s'

(ld)

Balancing these mechanisms, one gets the rate of build-up of excess electron density in S: dNs

dt

1

TF(NSsat Ns) +

1 -NBI(Nssat TBI'

Ns) - -

1 --gs(gBlsatTs

1 NB1) -- --Ns T S'

(2) In the region at level B1 the charge density is reduced by bulk conduction wherein electrons leaving level B1 go to B2. This transfer is suggested in order n o t to reduce the total charge in the region under consideration. In Figs. 3--8 the mean value of NB1 and NB2 is plotted. We introduce this transfer because we want to simulate a continuous spatial charge distribution in the bulk region adjacent to the p o l y m e r surface. Excess electrons leaving level B2 go to the subsequent counterelectrode by p o l y m e r bulk conduction. Therefore, the balance equations for levels B1 and B2 give: 1 1 1 dNBI_ -~rs(NBlsat -- N m ) + - - N B 2 - - NBI(Nssat - Ns) - - - N m

dt

rs

TB2'

TBI'

TBI

(3)

18

dNs2 dt

1 -

1 NB1 --

TBZ

-

-

TB2

1 NB2 -- - -

TB2'

NB2

(4)

etc. Calculations of the time-dependent electron densities of states S, B1 a n d B2 from the linear differential equations (2)--(4) are executed by an analogue computer. The analoguecircuit diagram is shown in Fig. 2. The contact frequency f of the contacting metal and the contact ratio r (that is, the ratio of contact time to time of discharge) are generated by adjusting the frequency and d.c. offset of a triangular function generator. The function generator controls the comparator K which is arranged as a switch in the current path of the contacting metal. We omit the normalization parameters and scaling factors of excess electron density and time in Figs. 3--8 because we only need relative values for the statements given in the discussion.

K

TI

AT

~,-q(~

Z~

a 2 Z~2'

Fig. 2. Analogue computer circuit used for simulation of transient, equilibrium and steadystate contact electrification. The symbols are explained in the text. 3. E x a m p l e s a n d d i s c u s s i o n

The analogue model allows one to vary the parameters given in Table 1 so that one can study the influence of the physical properties correlated with these parameters on transient and steady-state contact electrification.

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f

51

/

511

f 4 BI

BII 3

01 L_ tu

2

~. time (arbitrary units) Fig. 3. T r a n s i e n t d e n s i t y o f excess e l e c t r o n s o n t h e s u r f a c e S a n d in t h e b u l k B a c c o r d i n g to the model (see Fig. 1) w i t h 1/T F = 0.5 a n d 1/T F = 1 ( b r o k e n lines). PS a n d PB are halfv a l u e times at w h i c h h a l f t h e v a l u e o f the equilibrium excess electron density is r e a c h e d o n t h e surface S a n d in t h e b u l k B.

In a given polymer, the parameters in Table l(a) are largely fixed because of chemical, mechanical and thermal pretreatment of the sample. The experimental parameters {Table l(b)) 1/TF, f and r for the electrification experiments determine the magnitude of the electrostatic charge of the polymer sample characterized by the parameters in Table l(a). The polymer parameters listed in Table l(a) are kept constant throughout this paper. The absolute and relative values of these parameters have been estimated with regard to the conditions underlying the experimental data shown in Fig. 4(b). The problem we shall consider first is that of transient electrification, i.e. the rate effect during the time immediately after a permanent metal--polymer contact is established which leads to equilibrium contact electrification. Secondly, we will consider steady-state contact electrification. 4. Transient contact electrification

The time-dependent contact electrification {polymer parameters shown in Table l(a)) during the period subsequent to permanent metal--polymer contact is shown in Fig. 3 wherein the computed surface and bulk charge densities are given separately. The curves are characterized by the half-value time p, the time at which half the value of the equilibrium excess electron density is reached. As indicated by Ps and PB, the equilibrium value of the charge den-

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TABLE 1 Varied and fixed parameters of the model. The absolute and relative values of the parameters have been estimated with regard to the conditions underlying the experimental data shown in Fig. 4(b). Parameter

Correlated with physical properties

Relative values used within the examples

(a)

rs

time constant of charge carrier kinetics from the surface state (S) to the bulk state near the surfact (B1); band bending

0.2

r s,

time constant of the kinetics of discharge from the surface state to the gas phase; surface conductivity

0.1

Nssat

saturation density of charge carriers in the surface state

1

rB1

bulk conductivity; space-charge limitation

0.1

rB1,

time constant of the kinetic of charge carriers going over from the bulk state near the surface (B1) to the surface state; band bending

0.3

(b)

NB1Sat saturation density of charge carriers in the bulk level near the surface (B1)

0.75

r B2

bulk conductivity; geometry of the probe

0.04

rB2,

bulk conductivity

0.1

~F

Fermi energy (work function) of the metal; barrier height

varied (0.5 and 1)

r

contact ratio

varied (1/2--1/80)

f

contact frequency

varied with reference to p

s i t y is a p p r o a c h e d e a r l i e r o n t h e s u r f a c e t h a n in t h e b u l k (Fig. 3). T h e c h a r g e d e n s i t y o n t h e p o l y m e r s u r f a c e is h i g h e r t h a n t h a t in t h e p o l y m e r b u l k in b o t h transient and equilibrium contact electrification. T h e t i m e c o n s t a n t s Ps a n d PB d e p e n d o n t h e F e r m i e n e r g y via 1/rF a n d t h e p o l y m e r p a r a m e t e r s l i s t e d in T a b l e l ( a ) . A n a l o g o u s t o this, t h e t i m e a t w h i c h t h e s t e a d y s t a t e is r e a c h e d in an i n t e r m i t t e n t c o n t a c t e x p e r i m e n t d e p e n d s o n the same parameters without any significant influence of contact ratio r and c o n t a c t f r e q u e n c y f. 5. S t e a d y - s t a t e c o n t a c t e l e c t r i f i c a t i o n

5.1 Effect of work function of contacting metals on steady-state electrification The influence of the Fermi energy of the metal on contact electrification a t c o n s t a n t c o n t a c t r a t i o ( h e r e r = 1 / 1 7 ) a n d c o n s t a n t c o n t a c t f r e q u e n c y is

21

shown in Fig. 4(a). The calculated time dependence of the charge density on the surface and in the bulk is plotted for different values of the parameter that describes contact with different metals. The shape of the calculated c u r v e s agrees with a measured curve [12] of a s~ady-state experiment (Fig. 4(b}). ~

S

2J

~5

~

3

~

2

::::::::::::::::::::: .......................................................

fo

fs

x S II

2o

J, time (arbitrary units)

(b)

xxxxx~xx~xxXXxxxxxxxxxxxxxxxxxxx× xxXxx

xx U

xxx

x xx x x

i

i

,t'ime ~ h

Fig. 4. (a) Steady-state density of excess electrons on the surface S and in the bulk B o f polymer with different work function o f the contacting metals I and II. The workfunctions are adjusted by the parameter 1/r F (I: 1; II: 0.5). The ratio of contact period to the halfvalue time PS is unity, the contact ratio is r = 1/17. For reference, the transient electrification curves S and B are plotted only to half the scale time using workfunction II. (b) Steady-state steel contact electrification of polystyrene (f = 10 h-~, r = 1/17); the charge is measured half-way between t w o contacts.

22

T h e c o n t a c t f r e q u e n c y is c h o s e n in these calculations (with regard t o the halfvalue t i m e Ps) t o give a value o f u n i t y f o r t h e r a t i o b e t w e e n t h e c o n t a c t p e r i o d ( t h a t is, the reciprocal f r e q u e n c y ) and Ps. As s o o n as steady-state values are r e a c h e d in the calculation, the p a r a m e t e r 1/rv is c h a n g e d and the acquisition o f a n e w steady-state value is simulated. If the r e m a i n i n g p a r a m e t e r s are k e p t fixed, the m o d e l yields a specific s t e a d y state o f charging for each value o f the p a r a m e t e r 1/rF with n o d e p e n d e n c e o n previous values (Fig. 5).

EL. 0 ..Q

U~ E 7O O~ r0 c" U

~ilil I

"

5

10 t a

•

-. . . . .

B

17;~:::EE:L,~ S

15

20

time (arbitrary units)

Fig. 5. Steady-state density of excess electrons on the surface S and in the bulk B of polymer; the contact metals were varied at time ta. The work functions of metals I and II are the same as in Fig. 4(a); r = 1/17, 1/fp s = 1 for metal I.

5.2 Effect o f the contact ratio r on steady-state electrification T h e c o n t a c t ratio r is t h e r e l a t i o n b e t w e e n charging and discharging times. T h e e f f e c t o f r o n t h e t o t a l charge at c o n s t a n t f r e q u e n c y f d e p e n d s primarily on t h e injection time. C o m p a r e d with this, t h e t i m e laws o f charging and discharging kinetics are o f m i n o r i m p o r t a n c e . Fig. 6 shows the excess e l e c t r o n d e n s i t y o f surface and b u l k states f o r c o n t a c t ratios 1/2, 1/6, 1 / 1 5 and 1/80. T h e s t e a d y - s t a t e charge values are c o r r e l a t e d with t h e c o n t a c t ratios. F o r the t i m e laws c h o s e n in o u r m o d e l , the c o n t a c t ratio has a significant i n f l u e n c e on t h e c o n t r i b u t i o n o f surface and b u l k t o t h e m e a s u r e d t o t a l charge. This i n f l u e n c e can be e s t i m a t e d f r o m T a b l e 2. It can be seen t h a t t h e i n f l u e n c e o f the surface charge o n t h e t o t a l charge decreases with decreasing c o n t a c t ratio. C o n s e q u e n t l y , t h e c h o i c e o f the c o n t a c t ratio influences t h e c o n t r i b u t i o n o f the surface as well as t h e m e a s u r e d t o t a l charge. This is t h e r e a s o n w h y diffi-

23

ECI >, 4

S

.5

(1)

C

(2)

"(3 Oh ..C {9

(3) 1

l

(4)

1'0 •

1%

2"o

t i m e ( a r b i t r a r y units)

Fig. 6. Time-dependent density of excess electrons on the surface S and in the bulk B for four contact ratios: (1) rl = 1/2; (2) r~ = 1/6; (3) r3 = 1/15; (4) r4 = 1/80. The contact frequency (f = 1/ps) and the w o r k function (metal II) are constant; steady-state contact electrification. TABLE 2 Ratio of surface to bulk excess electron density calculated from the steady-state values in Fig. 6 Contact ratio

1/2

1/6

1/15

1/80

Surface/bulk ratio

1.22

1.02

0.95

0.89

culties c a n arise w h e n c o m p a r i n g e x p e r i m e n t a l results o b t a i n e d w i t h d i f f e r i n g c o n t a c t ratios.

5. 3 Effect o f contact frequency f on steady-state electrification As s h o w n in Fig. 7, t h e c o n t a c t f r e q u e n c y in o u r m o d e l d o e s n o t i n f l u e n c e t h e t i m e o f r e a c h i n g s t e a d y - s t a t e a n d t h e r a t i o b e t w e e n surface a n d b u l k charge density. To interpret the effect of frequency on the steady-state charge d e n s i t y , we distinguish t h r e e d i f f e r e n t f r e q u e n c y regions: a l o w e r region w i t h t h e l i m i t i n g f r e q u e n c y f = 0, a higher region w i t h t h e l i m i t f = oo, a n d an interm e d i a t e r e g i o n g o v e r n e d b y t h e p o l y m e r p a r a m e t e r s leading t o t h e t i m e c o n s t a n t s Ps a n d PB-

24

L_

3

c

2

-----5(I)

c~ L-

~S(2)

i •

1'o 1~ 2"o time (arbitrary units)

Fig. 7. T i m e - d e p e n d e n t d e n s i t y o f excess e l e c t r o n s o n t h e surface S a n d in t h e b u l k B for t w o d i f f e r e n t f r e q u e n c i e s : (1) i'1 = 0 . 1 / p s ; (2) f~ = 2/p s. T h e w o r k f u n c t i o n is c o n s t a n t ( m e t a l II); r = 1 / 1 7 ; steady-state contact e l e c t r i f i c a t i o n . F o r h i g h e r frequencies, the b u l k curves c o i n c i d e w i t h (2); the surface curves lie in the centre o f (2).

In the low-frequency region, the steady-state charge density approaches a limiting value which depends simply on the equilibrium values Nseq and NBeq and the contact ratio r: r

lim Ns Nse q ; f-* 0 r+1

r

lim NB = f-~ 0 r + 1 NBeq

(5)

This results because, in the low-frequency limit, the intermittent contact experiment is a sequence of transient state experiments. In the high-frequency region, at high contact frequencies compared with the values 1 / p s or 1 / p B, the analogue-computed steady-state value reaches a high-frequency limit which depends on the time laws of charging and discharging kinetics (i.e. the polymer parameters in Table l(a) and 1/rF) as well as on the contact ratio r. This limit consists of superimposed contributions, wherein an oscillating excess electron density varies with decreasing amplitude during increasing contact frequency (see Fig. 8). The integral mean value Nsos of this oscillating contribution to the steady-state limit can be calculated in the following manner: 1

2

i mdt~ t

Nsos = ~ m c t c + ~ tc + td

where mc and md are the slopes of the linearized charging and discharging curves and tc and td are the charging and discharging times. This equation reduces to

(6)

25

pp.v

l'

e.c.d

0 0bl

011

i

10 f

P, Fig. 8. Dependence of the peak-to-peak value (p.p.v) of the alternating excess electron density referred to the equilibrium charge density (e.c.d) on the ratio of contact frequency f t o t h e half-value t i m e Ps ; s t e a d y - s t a t e c o n t a c t e l e c t r i f i c a t i o n ; r =

Nsos

~1

~mctc

--1

~mc

r

r+lf

ps/pds.

1

(7)

because tactc = mdt d and r = tc/td. Eqn. 7 indicates that the high-frequency limit includes at least two frequency-dependent contributions irrespectiveof the overall frequencyindependent electrification. In real steady-state electrificationexperiments, electrificationis measured at a given discharging time. The time at which electrificationis measured can be correlated with contact frequency in different ways depending on the experimental set-up. The type of correlation can determine the measured electrificationwhich is an apparent steady-statevalue. Apparent steady-state values m a y depend on the frequency region selected by the experimenter rather than on the steady-state value. In the intermediate region, the steady-state value depends on frequency. The frequency range of this region is determined by the time constants Ps and PB, and two additional constants pd and pd, and markedly depends on the contact ratio r*. If r = ps/pd, the intermediate-frequency region, in which the steady-state value depends on frequency, is contracted to a narrow range. * T h e t i m e c o n s t a n t s pd a n d pd are i n t r o d u c e d f o r t h e t r a n s i e n t discharging p r o c e s s in a n a n a l o g o u s m a n n e r as are Ps a n d PB f o r t h e c h a r g i n g process. T h e initial p a r t o f t h e c h a r g e a n d discharge c u r v e s o f t h e excess e l e c t r o n d e n s i t y in t h e surface c a n b e r e g a r d e d as l i n e a r u p t o Ps o r p sd.

26

5.4 Effect o f measurement techniques on steady-state electrification As shown in the previous t w o sections, to interpret results of contact electrification one needs to know the experimental conditions (r,f) exactly. Comparing experimental results obtained from different experimental techniques, differences in the techniques of observing the gauging depth and the surveyed volume have to be accounted for; that is, the dependence of the experimental signal on contact ratio and on contact frequency is influenced by weighting factors, assigned to surface and bulk regions of the measured probe by the experimental technique (Kelvin--Zissman [ 13], CPD [ 14], probe scanning techniques). Additionally, the t y p e of weighting affects the time at which the steady-state value is approached. In our model, we can simulate different gauging depth and surveyed volume by applying appropriate chargedensity weighting factors to the regions S, B1 and B2. 6. Conclusion The contact electrification of polymers can be simulated by an analogue computer using a simple kinetic model for charge injection, charge storage and dissipation on the surface and in the bulk of polymers. Transient contact experiments leading to equilibrium electrification and steady-state contact electrification b y intermittent contact are simulated. The analogue-computed excess electron densities may be converted to total charge b y volume integration, i.e. through multiplication by appropriate weighting factors. The weighting factors for the surface and bulk regions must be adapted to the experimental conditions. Under the assumption of energetically different energy levels for excess electrons or holes on the surface and in the bulk of polymers, the dependence of contact electrification on different values of the work function of the contacting metal, contact frequency and contact ratio is investigated. Within our model, the contact electrification of polymers shows the expected dependence on the work function of the contacting metal and on the contact ratio. The existence of different time laws for the charging and discharging kinetics leads to a remarkable dependence of the steady-state charge density levels on b o t h contact ratio and contact frequency. Additionally, by using small contact ratios, it is possible to intensify the contribution of surface or bulk charge to the total charge of the polymers. The steady state depends on frequency in range outlined b y the polymer and experimental parameters. At sufficiently low and high frequencies compared with this range, the steady-state values approach their limits. The low-frequency limit can be calculated from the equilibrium value in a simple manner, whereas the highfrequency limit cannot be given by an analytical expression b u t can be created b y the analogue computer. The dependence of the steady-state level on contact frequency f is reduced to a minimum frequency range if the contact ratio r equals the ratio of the time constants of charging and discharging curves. In our model, the contact frequency influences neither the time of reaching steady state nor the relation between surface and bulk charge.

27

Acknowledgements Financial s u p p o r t f o r this w o r k has been p r o v i d e d b y A r b e i t s g e m e i n s c h a f t Industrieller F o r s c h u n g s e i n r i c h u n g e n e.V. References 1 F.A. Vick, Theory of contact electrification, Br. J. Appl. Phys. Suppl., 4 (1953) 1. 2 H. Krupp, Physical models of the static electrification of solids, Static Electrification, 1971, Inst. Phys. Conf. Ser. No. 11, 1971, p. 1. 3 J. Fuhrmann, Elektrische Eigenschaften polymerer FestkSrper in starken elektrischen Feldern, Coll. Polymer Sci., 254 (1976) 129. 4 D.K. Davies, Charge generation on solids, Adv. Stat. Electr. (Brussels Auxilia), 1 (1969) 10. 5 J. Fuhrmann, Contact electrification of dielectric solids, J. Electrostat., 4 (1977/1978) 109--118. 6 A. W~ihlin and G. B~ickstr~m, Sliding electrification of Teflon by metal, J. Appl. Phys., 45 (1974) 2058. 7 J. Lowell, The electrification of polymers by metals, J. Phys. D: Appl. Phys., 9 (1976) 1571. 8 U. Benninghoff, J. Fuhrmann and G. Rehage, Measurements of electrostatic charging and discharging of polystyrene, Adv. Stat. Electr. (Brussels Auxilia), 1 (1969) 96. 9 J. Fuhrmann, R. Hofmann and J. Ki~rschner, Excess charges in amorphous organic polymers, Physics of non-crystalline Solids, Fourth International Conf., ClausthalZellerfeld, Sept. 1976, in G.H. Frischat (Ed.), Non-crystalline Solids, Trans. Tech. Publications, CH-4711 Aedermannsdorf, 1976, p. 291. 10 J. Fuhrmann, R. Lamour and G. Rehage, Gleichstromleitf'~igkeit und PolarisationsstrSme des Polystyrols, Coll. Polymer Sci., 62 (1977) 131. 11 D.M. Taylor, Electron-beam charging of polyethylene terephthalate films, J. Phys. D: Appl. Phys., 9 (1976) 2269. 12 J. Kiirschner and J. Fuhrmann, unpublished. 13 W.A. Zisman, Rev. Sci. Instrum., 3 (1932) 367. 14 I.G. Garton, Charge transfer from metal to dielectric by contact potential, J. Phys. D: Appl. Phys., 7 (1974) 1814.