Electrical conduction in biaxially-oriented polypropylene

Journal of Electrostatics, 31 (1993) 51-63 Elsevier

51

Electrical conduction in biaxially-oriented polypropylene P. K a r a n j a 1 a n d R. N a t h Department of Physics, University of Roorkee, Roorkee 247 667, India (Received November 9, 1992; accepted in revised form April 13, 1993)

Summary The electrical conduction in biaxially-oriented polypropylene (BOPP) has been studied. Space charge limited current (SCLC), Schottky and Poole-Frenkel effects and ionic conduction processes are discussed. Thermally stimulated polarization current (TSPC) studies showed a decrease in conductivity from 4.3 × 10-~6 to 3.0 x 10-17 (ohm cm)-1 at 80 °C after hexane treatment of BOPP. The decrease can be attributed to the removal of impurities by hexane treatment. The conduction is extrinsic in nature. Thermally stimulated current investigations show the presence of impurity traps in BOPP. A good fit of the ionic theory to be experimental data of the steady state currents in the temperature range 50-100°C suggests ion conduction to be predominant. Bulk electronic conduction also plays important role at high temperatures.

1. Introduction A d e t a i l e d u n d e r s t a n d i n g of e l e c t r i c a l c o n d u c t i o n in p o l y m e r s is useful in p r a c t i c a l a p p l i c a t i o n s . T h e e l e c t r i c a l c o n d u c t i o n in p o l y m e r s is g o v e r n e d by g e n e r a t i o n a n d t r a n s p o r t of c h a r g e carriers. T h e c h a r g e c a r r i e r s could be ionic i m p u r i t i e s a n d / o r e l e c t r o n i c carriers. Since c o n d u c t i o n is influenced by polym e r i c s t r u c t u r e , for this r e a s o n it h a s b e e n studied in v a r i o u s p o l y m e r s in r e l a t i o n to t h e i r s t r u c t u r e s [1-4]. T h o u g h t h e e l e c t r i c a l c o n d u c t i o n in p o l y p r o p y l e n e (PP) h a s b e e n studied by s e v e r a l a u t h o r s [5-8], no a t t e m p t h a s b e e n m a d e to i n v e s t i g a t e in detail t h e c o n d u c t i o n m e c h a n i s m s in b i a x i a l l y - o r i e n t e d p o l y p r o p y l e n e (BOPP). C o m m e r c i a l p o l y p r o p y l e n e films h a v e b e e n r e p o r t e d to c o n t a i n v a r i o u s i m p u r i t i e s s u c h as a n t i o x i d a n t s , r e s i d u a l c a t a l y s t s , a n t i s t a t i c a g e n t s a n d some i m p u r i t i e s i n t r o d u c e d d u r i n g m a n u f a c t u r i n g [9, 10]. T h e p r e s e n c e of t h e s e i m p u r i t i e s a d v e r s e l y effect t h e e l e c t r i c a l p r o p e r t i e s of p o l y m e r s . T h e h y d r o c a r b o n s s u c h as h e x a n e h a v e b e e n r e p o r t e d to r e m o v e a r o m a t i c m o l e c u l e s s u c h as

1Permanent Address: Kenya Polytechnic, Nairobi, Kenya 0304-3886/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.

52

P. Karanja, R. Nath/ Electrical conduction in polypropylene

anthracene, benzoic acid, simple derivatives of phenanthrene and antioxidants from the polymers [9, 11, 12]. In this paper the results of field dependence of current in BOPP are reported between temperatures 50 and 100 °C and field upto 6 x l0 s V/cm. In addition, we have also carried out thermally stimulated polarization current (TSPC) and thermally stimulated current measurements in the virgin and the hexane treated BOPP. The results were analysed using various conduction models: space charge limited conduction (SCLC) [13] Schottky and Poole-Frenkel (PF) effect [14, 15] some modified PF models [16-18] and ionic conduction [19]. 2. E x p e r i m e n t A roll of 35 ~m thickness of BOPP was obtained from COSMO Ltd., New Delhi. Circular samples of diameter 2 cm were cut from the roll of the film. Samples of 10, 15, 20 ~m thickness were also obtained from COSMO Ltd. The samples were cleaned ultrasonically with acetone for 10 min and then dried. Aluminium electrodes of thickness 500 A and area 2.54 cm 2 were vacuum deposited on both sides of the samples. The samples were then conditioned by heating to 120°C with electrode shorted for one hour in order to remove memory effects and stray charges. Hexane treatment was carried out by keeping the samples in hexane for 24 hours. After removal from hexane the samples were thoroughly washed with distilled water and dried. Steady-state current-voltage measurements at elevated temperatures were taken with standard apparatus. Currents were measured one hour after successive step increase in voltage. The X-ray scan (figure not shown) was obtained with a Philips diffractometer PW 1710, using Ni filter CuK~ radiation of wavelength 1.54/~. Weight crystallinity (Xw) value of 66.5% and the average crystallite size 130/~ were obtained for BOPP [20]. The open circuit thermally stimulated current experiment was performed in the temperature range 20-170 °C at a constant heating rate of 3 °C/min. The samples were negatively corona charged at room temperature. The measurements of the absorption currents were carried out by applying a constant electric field (2 x l0 s V/cm) to the sample at elevated temperatures (50 and 70°C). The charging current was subsequently registered through Keithey 610C electrometer for 1 hr. The field was then removed and the discharge current of the short circuited sample was recorded. 3. R e s u l t s and d i s c u s s i o n 3.1. Thermally stimulated polarization current (TSPC) We have carried out TSPC experiment on the virgin and the hexane treated BOPP films in order to check the effect of impurities on electrical conductivity. TSPCs were measured in virgin (Fig. l:a) and hexane treated (Fig. l:b) BOPP.

P. Karanja, R. Nath/Electrical conduction in polypropylene

53

12o

ioo

8C

o

~6o

b

40

20

0 50

60

70 80 90 TEMPERATURE (°C)

I00

I I10

Fig. 1. TSPCs for virgin (curve a) and hexane treated (curve b) electric field= 105 V/cm. The sample was first heated to 120 °C under applied field of 105 V/cm to release any previously stored charge. The temperature was then lowered to room temperature and the sample reheated under field at a constant rate of 2 °C/min to obtain the desired TSPC. From the initial rise method, an activation energy of 1.40 eV and 1.36 eV were obtained for virgin and hexane treated BOPP samples respectively. At 80 °C conductivities of 4.3 x 10- ' 6 and 3.0 x 10-1T (ohm cm)-1 for virgin and hexane treated sample were obtained. According to Rogers [21] the hydrocarbons show greatest permeation rate through polyethylene. Therefore hexane can be assumed to penetrate the bulk of polypropylene also. The decrease in conductivity can be attributed to the removal of impurities by hexane. Therefore the conduction in BOPP can be extrinsic in nature. As far as electrical properties are concerned hexane treatment can probably cause a reduction in the concentration of ionizable centres and/or mobile ions and also a reduction in the concentration of traps. The hexane treatment has also resulted in reduction of electrical conductivity in polythylene [12, 22]. The removal of impurities by hexane certified by decrease in electrical conductivity in BOPP motivated us to carry out the trapping and the conduction studies in BOPP. 3.2. Nature of traps Thermally stimulated current experiment was performed to know about the nature of traps in BOPP. Figure 2 shows a comparison of open circuit TSC of the virgin (curve a) and the hexane treated (curve b) BOPP negatively corona charged to initial surface potential of 1000 V. The TSC of virgin BOPP

P. Karanja, R. Nath/ Electrical conduction in polypropylene

54

(curve a) exhibits three peaks at temperatures around 45, 78 and 120 °C and of hexane treated BOPP shows a broad peak centred around 130 °C. The lower temperature peaks (Fig. 2, curve a) may originate from impurity traps in the amorphous region. It is known that hexane can penetrate the amorphous region and leach out impurities there in and/or from crystalline amorphous boundaries [9, 11]. It thus follows that the absence of lower temperature peaks in the hexane treated sample is due to the removal of impurity traps from the sample. The higher temperature peak is attributed to the charge trapping of crystalline-amorphous boundaries. The crystallineamorphous boundaries provide deep traps, hence the peak occurs at high temperature. 3.3. Charge and discharge current transients The time dependence of absorption current in 35 ~tm BOOP was investigated over a period of 1000 s at elevated temperatures 50 and 70 °C and at field 2 × 105 V/cm. Figure 3 shows the plots of charge and discharge current transients. The charge and discharge currents were observed to be mirror images of one another. The presence of symmetry in the charge and discharge current transients rules out the possibility of the existence of space charge limited currents [23]. 3.4. Steady-state currents Figure 4 shows the experimental current-field characteristics of BOPP at elevated temperatures (50-100°C). Currents were measured after successive step increases in voltage, one hour apart. The results were analysed using various standard conduction theories as follows:

0.6 0-5 ~ 0-4

e

~ -0.3 z

b

~0.2-u 0.1 o

I

20

40

60

80 I00 120 14.0 TEMPERATURE (°C)

160

t ISO

Fig. 2. Open circuit TSCs of Virgin (curve a) and hexane treated (curve b) BOPP. Samples were negatively corona charged to initial surface potenial of 1000 V at room temperature. Arrows indicate peak temperatures.

P. Karanja, R. Nath/ Electrical conduction in polypropylene

55

lo-gho~O°C

16'°--

1(512

1

~~o~" "

I I Itllll

1o

I

I00 TiME (s)

I I IlllLI I000

Fig. 3. Charge and discharge current transient plots of BOPP at 50 and 70°C and F= 2 x 105 V/cm.

3.4.1. P o w e r law

Power-law dependence of the steady state c u r r e n t on electric field (I ocF~) was investigated by plotting log I v s log F (Figure not shown). An exponent n varied between 3.8 to 2.3 in the temperature range 50-100 °C for fields varying from 1 to 5 x l0 s V c m - 1. The decrease in exponent n with t em perat ure has been reported in LDPE [24]. Values of n greater t han 2 have been observed in polyethylene (PE) by others [24-27]. Some authors [28] have invoked the presence of trap distribution to account for the higher exponent. The decrease in n with rise in t em pe r at ur e supports presence of space charge limited currents [29]. Figure 5 shows the dependence of cur r ent on thickness of the sample. Absence of the d- 3 dependence of the c u r r e n t eliminates the possibility of SCLC mechanism. 3.4.2. Schottky effect

The S c h o t t k y effect is due to thermoionic emission over a field - - lowered b a rr ier at the metal - - polymer interface, arising from the image force interaction between the carrier and the electrode. The current is given by I = A T 2 exp [flsF1/~ - ¢ ) / k T ]

(1)

P. Karanja, R. Nath/ Electrical conduction in polypropylene

56

4o-9 lO0°C 90°C

/

i() I0

,

j,ooc ~

60oc

[0-II

10-12

IO 13 0

I .I

I .2

I .3

I .4

F(MVcm

-I )

I

I

.5

.6

.T

Fig. 4. Steady state current-field plots of BOPP at elevated temperatures. where A is the Richardson constant, ¢ is the Schottky barrier and fls= (e3/(4 ~eeo)) in is Schottky coefficient. Figure 6 shows typical schottky plots (log I vs F 1t2) of the experimental current-field data for BOPP. The slope of curve at 80 °C gave a relative permittivity e = 1.63 compared to s = 2.36 obtained from the dielectric measurements. Schottky dependence was f ur t her examined by measuring the c u r r e n t dependence on electrode material. Figure 7 shows current-field measurements on BOPP at 100 °C, with aluminium and silver electrodes on opposite sides of the film. The c u r r e n t magnitudes were unaltered when the polarity of the voltage was reversed on silver electrodes. This clearly demonstrates t hat the steady state currents do not depend on the work function of the electrodes, but r a t h e r are due to bulk effects, eliminating the Sc hott ky effect as a conduction mechanism in BOPP.

3.4.3. Poole-Frenkel effect and modifications P o o l e - F r e n k e l conduction is a bulk limited process caused by thermal excitation of carriers over a field lowered coulombic potential barrier surrounding charge donor sites. The conductivity in this case is given by [30]. a--- ao exp (flp~ F1/2/kT)

(2)

P. Karanja, R. Nath/ Electrical conduction in polypropylene

--T H E O C R A L E T I 0

I (~i5

/

EXPERIMENTAL

57

/

¢

I(~17

I

i0 7

I

i

I I llTl

i0 8 d"3 ( cm-3 }

I

I

I

I i iii[

109

Fig. 5. Thickness dependence of conduction currents in BOPP. where a0 exp ( - W J ( k T ) ) is the low field conductivity, Wt is the barrier energy and tier = (e3/~eo) 1/2 is the Poole-Frenkel coefficient. Figure 8 shows typical Poole-Frenkel plots of l o g a v s F 1/2. These yield a value of dielectric constant 5=17.5 at 90°C compared with the value of e = 2.36 for polypropylene. Various arguments have been put forward to reconcile the high value of e obtained from Poole-techniques. Ieda [16] proposed modification in the Poole-Frenkel model by considering an electron to be free while attempting to escape in the opposite direction of the applied field and when separated from its parent atom by a certain critical distance. This yields 2 k T in the denominator of the argument of the exponential term of eqn. (2). e is reduced by a factor of four to 4.4 which is still high compared with 2.36. Thus the Poole-Frenkel mechanism in the above forms is not applicable to the conduction in BOPP. Poole-Frenkel mechanism has also been modified by a number of other workers [16, 17, 31, 32]. In one of these models the conduction current in SiO films [17] was explained to arise from the electron jumps over the Coulomb potential well from the occupied P o o l e - F r e n k e l (PF) sites to the empty ones. The PF sites are positively charged when empty and neutral when occupied by an electron. By neglecting the contribution of electrons moving against the field, the model gave the steady state current as [17]. I = Cexp [ ( - 1/2kT) [¢o -2(1.25a) 1/2 (2.43~/~ 2 + eF) in]

(3)

P. Karanja, R. Nath/ Electrical conduction in polypropylene

58

(~9

o

.~./~ O0oc

_

j/ ooc /

16~o

50°C

iO-It

I0 -13

I 200

I

I 400

I

1 600

I

t 800

FI/2( VI/2 cm-I/2]

Fig. 6. Typical Schottky plots (log I vs F 1/2) of the experimental steady-state current-field data of BOPP. where the pre-exponential factor C=(Ae,~7/2) (NN~''2, A is the area of cross section, I is the current, 7 is the attempt to escape frequency, ~ is the mean free path, N is the density of PF sites, N¢ is the effective density of states in the conduction band, T is the absolute temperature, ~b0 is the depth of the PF sites from the bottom of conduction band, ~ = e:/4~eeo and F is the applied field. Figure 9 shows a quantitative comparison of our experimental results at 50°C with those which would be predicted by eqn. (3), with C = 1.78 x 10-TA, = 9.75 × 10-27 cmC2F, ~bo= 1 eV, e = 2.36. There seems to be some agreement at high fields but at low field this model fails to give satisfactory agreement. This model also gave similar behaviour at 100 °C. This model has been shown to predict reasonable agreement for PE [33] but does not seem to fit well for polypropylene. Considering anot he r model proposed by Calderwood [18] based on the modified P o o l e - F r e n k e l mechanism in which electrons are liberated from the PF sites in three-dimensions, assuming mobility to be field independent the above model resulted in the following relative conductivity expression:

a/ao ~- 2/A 2 (1 + A sinh A - cosh A)

(4)

where A = flev(F1/2/2kT), flee is the P o o l e - F r e n k e l coefficient and ao is the low field conductivity.

P. Karanja, R. Nath/ Electrical conduction in polypropylene

59

Id 6

x

o

id T

_

o

E id 8

I 0

-I

I

I

.2

I

I

-4

'5

.5

F ( M V c m -I )

Fig. 7. E l e c t r o d e

effect on current-field

j

io-15

~ io-,6

in B O P P .

lO0°C

J

~

characteristics

8°°C 5ooc

/"

"{,d,~ I018

I 200

I o/

t 400

I

I 600

I

I 800

F I/2 ( v l / 2 cm- I ~ )

Fig. 8. Typical Poole-Frenkel plots (log avs F 1/2) of the steady-state conductivity-field data of BOPP.

60

P. Karanja, R. Nath/ Electrical conduction in polypropylene

Figure 10 shows a comparison of relative conductivity a/aovs F 1/2 plots obtained by theory (solid lines) and the experiment (open circles). The theory completely disagrees with the conductivity data at 50 °C while at temperatures 80 and 100 °C shows good agreement with the experiment. This may suggest the bulk electronic conduction in BOPP can occur at high temperature. 3.4.4. Ionic conduction For ion hopping, the current is given by [19]

I = 2Anq~5 exp ( - ¢lkT) sinh (,~qF/2kT)

(5)

i°e.

I = a sinh (bF)

(6)

where I is the steady state current, A is the area, n is the carrier concentration, q is the carrier charge, ~ is the hopping distance, 5 is the attempt to escape frequency, ~b is the barrier height, k is the Boltzmann's constant, F is the electric field, and T is the absolute temperature. Equation (6) was used to fit the

-I0 IO

03

50°C

0 °

o o

o ° °

o

/

50°C

0

i0-II

,oOO

o

//8ooc

0 0 / ~ 0 0 ~ 0 0 i i 0 °C

o

1612 _

THEORETICAL 0 EXPERIMENTAL

o

I

Id '3

300

0 EXPERIMENTAL -- CALCULATED

- -

I

l

400 500 600 FI/2( VI/2cm-Ifz }

I

I

700

800

Fig. 9. Comparison of experimental results with theory predicted by eqn. (3). C= 1.78 x 10-7 A, a=9.75 x 10-27 cmC2F, ~bo=1 ev, e=2.36 and T=50°C.

]

200

]

[

400 600 800 FI/2 ( Vl/2cm-I/2 )

]

I000

of theoretical (eqn. (4)) and experimental relative conductivity a/ao vs F In plots for BOPP. F i g . 10 C o m p a r i s o n

61

P. Karanja, R. Nath/ Electrical conduction in polypropylene

experimental data by adjusting the two parameters a and b at different temperatures. Figure 11 s h o w s the typical theoretical fit (solid lines) to the data for BOPP in the temperature range 50-100 °C. The good fit suggest ion h o p p i n g as a d o m i n a n t m e c h a n i s m of c o n d u c t i o n in BOPP. C o n d u c t i o n by ion h o p p i n g has been found in PP and other polymers [6, 34-36]. The ions are of impurity origin however, at this stage the nature of ion impurities is uncertain. The h o p p i n g distance approximately 61 ~ at 50 °C was calculated from parameter b in BOPP. The average repeat u n i t l e n g t h of PP is about 6.5 A [37]. This implies t h a t an ion jumps over about 10 repeat units in the amorphous region.

.48

.44

.4C

- 36

.52

io-4 .28 O

A ~-24

-2C

10- 5 o

.16

o

E

o

N

.12

I.I e V

o 10- 6 08

.04

0

10- 7

0

.2

-4

.6

F ( M V crt~-I }

Fig. 11. Typical steady state current (Z) vs field (F) for BOPP at different temperatures solid lines are theoretical plots of eqn. (4)

with parameters a (Amp) and b (cm/V) as follows; 100 °C: 2.36 x 10- ~~, 8.4 x 10- 6; 90 °C; 1.54×10 -H, 8.2×10-6; 80°C: 9.1x10 -~2, 7.8x10 -6 , 70°C; 2.07 x10-12, 8.9x10-6; 60 °C; 1.58 x 10-12, 8.8 x 10-6; 50 °C: 2.31x10 -13, 1.1x10 -s. Dot points and circles represent experimental data.

I

2-5

2.6

I

I

I

I

I

2.7

2.8

2.9

3.0

3.1

3.2

I0 3 / T { K "l)

Fig. 12. Arrhenius plot of a/~. (eqns. (5) and

(6)) vs I/T for BOPP.

62

P. Karanja, R. Nath/ Electrical conduction in polypropylene

This m a y a p p e a r large but is not u n c o m m o n in polymers [6, 33, 38]. It m a y be i n t e r e s t i n g to k n o w a b o u t the size of a m o r p h o u s l a y e r in BOPP. This c a n be c a l c u l a t e d from the r e l a t i o n [39]. Xv=l¢/(l¢+l~)

(7)

w h e r e Xv = (p/pc)Xw = 64.5%, is the v o l u m e crystallinity, p = 0.91 is the density of BOPP, Pc =0.936 is density of the crystalline phase, lc (130/~) is a size of c r y s t a l l i t e and la is the size of a m o r p h o u s layer. F r o m eqn. (7) la comes out to be 71.5 A, which perhaps makes the ion jump (61/~) possible in the a m o r p h o u s region. Using the c a l c u l a t e d values of ,~ from p a r a m e t e r b, values of a/~. (proportional to c a r r i e r density n) vs 1 / T are plotted on a semilog scale. F i g u r e 12 shows a typical A r r h e n i u s plot of l n a vs 1/T. An a c t i v a t i o n e n e r g y of 1.1 eV was o b t a i n e d from the slope of the line. The electric field can lower the p o t e n t i a l b a r r i e r and h e n c e at high electric field the a c t i v a t i o n e n e r g y will d e c r e a s e [24]. This could be the possible e x p l a n a t i o n for the difference in a c t i v a t i o n e n e r g y o b t a i n e d form TSPC plots (Fig. 8).

4. Conclusions F r o m the p r e s e n t studies the following conclusions can be drawn: (i) The c o n d u c t i o n in B O P P is extrinsic in n a t u r e as r e v e a l e d by the TSPC and the TSC e x p e r i m e n t s on h e x a n e t r e a t e d samples. (ii) Ion c o n d u c t i o n t h e o r y provides the best fit to the e x p e r i m e n t a l results in the t e m p e r a t u r e r a n g e 50-100 °C. (iii) A p p l i c a t i o n of the modified P o o l e - F r e n k e l model proposed by C l a d e r w o o d [17] shows t h a t bulk e l e c t r o n i c c o n d u c t i o n also becomes i m p o r t a n t at high temperatures.

Acknowledgement One of the a u t h o r s (P.K.) would like to t h a n k C o m m o n w e a l t h Fellowship plan, India and K e n y a G o v e r n m e n t s for the g e n e r o u s grant.

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M. Ieda, IEEE Trans. Elect. Insul., E1-19 (1984) 169. K. Iida, J.S. Kim, S. Nakamura and G. Sawa, IEE Trans. Electr. Insul., 27 (1992) 391. S. Mita and K. Tahagi, Jpn. J. Appl. Phys., 14 (1975) 197. C.A. Hogarth and T. Iqbal, Phys. Stat. Sol. (a), 65 (1981) 11. D.K. Das-Gupta and K. Joyner, J. Phys. D, 9 (1976) 2041. R.A. Foss and W. Dannhauser, J. Appl. Polym. Sci., 7 (1963) 1015. M. Matsui and N. Murasaki, Electrets, Charge Storage and Transport in Dielectrics, ed. M.M. Perlman, Electrochem. Soc. Inc., Princeton, 1972. [8] M. Tavakoli and J. Hirch, J. Phys. D, 21 (1988) 454. [9] T. Umemura, T. Suzuki and T. Kashiwazaki, IEEE Trans. Electr. Insul., EI-17 (1982) 300.

P. Karanja, R. Nath/ Electrical conduction in polypropylene

63

[10] L.Y. Gao, D.M.Tu, S.C. Zhou and Z.L. Zhang, IEEE Trans. Electr. Insul., EI-25 (1990) 535. [11] I. Boustead and A. Charlesby, Proc. Roy. Soc. Lond., A-316 (1970) 291. [12] R.H. Partridge, Polym. Lett., 5 (1967) 205. [13] M.A. Lampert and P. Mark, Current Injection in Solids, Academic Press, NY, 1970, Chap. 4 and 5. [14] P.N. Murgotroyd, J. Phys. D, 3 (1970) 151. [15] R.I. Frank and J.G. Simmons, J. Appl. Phys., 38 (1967) 832. [16] M. Ieda, G. Sawa and S. Kato, J. Appl. Phys., 42 (1971) 3737. [17] H. Adachi, Y. Shibata and S. Ono, J. Phys. D, 4 (1971) 988. [18] J.H. Calderwood, in Charge Storage, Charge Transport and Electrostatics, Ed. Y. Wada, M.M. Perlman and H. Kokado, Elsevier, Amsterdam, 1979, p. 359. [19] N.F. Mott and R.W. Gurney, Electronic Process in Ionic Crystals, Oxford Univ. Press, London, 1940, p. 40. [20] P. Karanja and R. Nath, IEEE Trans. Electr. Insul., 28 (1993) 294. [21] C.E. Rogers, Physics & Chemistry of Organic Solid States, Vol. 2, Interscience, New York, Chapter 6, 1965, p. 510. [22] D.K. Das Gupta, J. S. Dully and D. E. Cooper, J. Electrostat., 14 (1983) 99. [23] D.K. Das Gupta and K. Joyner, J. Phys. D, 9 (1976) 829. [24] S. Pelisson, H. St-Onge and M.R. Wertheimer, IEEE Trans. Electr. Insul., 23 (1988) 325. [25] P. Fischer, J. Electrostat., 4 (1978) 149. [26] L. Brehmer, M. Pinnow, M. Kornelson, H. Von Berlepsch and J. Hanspach, J. Electrostat., 14 (1983) 19. [27] R. Nath, T. Kaura and M.M. Perlman, IEEE Trans. Electr. Insul., 26 (1990) 419. [28] J.L. Hartke, Phys. Rev., 125 (1962) 1177. [29] S. Nespurek and J. Sworakowski, J. Appl. Phys., 51 (1980) 2098. [30] J.G. Simmons, Phys. Rev., 155 (1967) 657. [31] A.K. Jonscher, Thin Solid Films, 1 (1967) 213. [32] J.L. Hartke, J. Appl. Phys., 39 (1968) 4871. [33] D.K. Das Gupta and M.K. Barbarez, J. Phys. D, 6 (1973) 867. [34] K. Ikezaki, T. Kaneko and T. Sakakibara, Jpn. J. Appl. Phys., 20 (1981) 609. [35] M. Kosaki, K. Sugiyama and M. Ieda, J. Appl. Phys., 42 (1971) 3388. [36] T. Umemura and K. Akiyama, IEEE Trans. Electr. Insul., E-21 (1986) 137. [37] B. Wunderlich, Macromolecular Physics, Vol. 1, Acad. Press, N.Y., 1973, Ch. 2. [38] R. Nath and M.M. Perlman, J. Electrostat. 24 (1990) 283. [39] J. Schultz, Polymer Materials Science, Prentice Hall Inc. N.J., 1974, p. 176.