absorption and desorption currents

8/14/2019 Absorption and Desorption Currents

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ABSORPTION AND DESORPTION CURRENTSThe response of a linear system to a frequency dependent excitation can be transformedinto a time dependent response and vice-versa. This fundamental principle covers a widerange of physical phenomena and in the context of the present discussion we focus on thedielectric properties s' and e". Their frequency dependence has been discussed in theprevious chapters, and when one adopts the time domain measurements the responsethat is measured is the current as a function of time. In this chapter we discuss methodsfor transforming the time dependent current into frequency dependent e' and s".Experimental data are also included and where possible the transformed parameters inthe frequency domain are compared with the experimentally obtained data using variablefrequency instruments.The frequency domain measurements of & ' and & " in the range of 10~ 2 Hz-10 GHzrequire different techniques over specific windows of frequency spectrum though it ispossible to acquire a 'single' instrument which covers the entire range. In the past thenecessity of using several instruments fo r different frequency ranges has been anincentive to apply and develop the time domain techniques. It is also argued that thesupposed advantages of the time domain measurements is somewhat exaggeratedbecause of the commercial availability of equipments covering the range stated above1.The frequency variable instruments use bridge techniques and at any selected frequencythe measurements are carried out over many cycles centered around this selectedfrequency. These methods have the advantage that the signal to noise ratio isconsiderably improved when compared with the wide band measurements. Hence verylow loss angles of ~10 \ J L rad. can be measured with sufficient accuracy (Jonscher, 1983).The time domain measurements, by their very nature, fall into the category of wide bandmeasurements and lose the advantage of accuracy. However the same considerations of

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accuracy apply to frequencies lower than 0 .1 H z which is the lower limit of ac bridgetechniques and in this range of low frequencies, 10"6


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The integration may be carried out by a change of variable. Let p = (t-T). Then dp = -dTand equation (6.4) becomes

(6.5)dp

The physical meaning attached to the variable p is that it represents a previous event ofchange in voltage. This equation is in a convenient form for application to alternatingvoltages:v = K m a xexpL/( f l* + )] (6.6)

where 5 is an arbitrary phase angle with reference to a chosen phasor, not to be confusedwith the dissipation angle. The voltage applied to the dielectric at the previous instant pis(t-p) +S] (6.7)

Differentiation of equation (6.7) with respect top givesd v x 0 -/>) + < ? ] (6.8)dp

Substituting equation (6.8) into (6.5) we geti = jo)CFmax exp j[co(t -p) +S] (t -p) + S} = Qxp[j(o)t + S)] x exp(-y') (6.10)

Equation (6.9) may now be expressed as:/ = ^CFmaxexp|j(^+ )] (6.11)

Substituting the identity

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Qxp( - ja > p) =cos(ct)p) - j sm(cop} (6 . 12 )we get the expression for current as:

/ = C D CFmax exp[j(a>t + )]{ j cos(a>p) q > ( p ) dp+g sm(c op) (p(p) dp} (6.13)The current, called the absorption current, consists of two components: The first termis in quadrature to the applied voltage and contributes to the real part of the complexdielectric constant. Th e second term is in phase and co ntributes to the d ielectric loss.An alternating voltage applied to a capacitor with a dielectric in between th e electrodesproduces a total current consisting of three components: (1) the capacitive current Icwhich is in quadrature to the voltage. The quantity S o o determines the magnitude of thiscurrent. (2) The absorption current / given by equation (6.13), (3) the ohmic conductioncurrent Ic which is in phase with the voltage. It contributes only to the dielectric lossfactor s". Th e abso rption current given by equation (6.13) may also be expressed as

ia = jco CQvsa* =joC0v(4 - je "a ) (6 .14)Equa ting the real and imaginary parts of eqs. (6.13) and (6.14) gives:

e'a = s'-s!X> =

(6-15)

where C 0 is the vacuum capacitance of the capacitor and V 0 the applied voltage. Notethat w e have replaced/? by the variable t without loss of generality.

(6 .16)The standard notation in the published literature for ea' is s' - Soo as shown in equation(6.15). Equations (6.15) and (6.16) are considered to be fundamental equations ofdielectric theory. They relate the absorption current as a function of time to the dielectricconstant and loss factor at constant voltage.To show the generality of equations (6.15) and (6.16) we consider the exponential decayfunction

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' / T (6 .17)where T is a constant, independent of t and A is a constant that also includes the appliedvoltage V . Substituting this equa tion in equations (6.15) and (6.16) we have

e~tTcos(Dtdt (6.18)-f/r_:_,^j. (6.19)

W e use the standard integrals:-DX e ' p + q-DX Pe F cosqx--~-p + q

Equations (6 . 1 8) and (6.19) then simplify to

^ ' - ^ 0 0 =-T ~ ^ 21 + Q ) T (6.21)+ 0?The factor A is a constant with the dimension of s"1 and if we equate it to

A =^-?2L (6 .22)TEquations (6 .20) and (6.2 1 ) become

g' = g c o + (g'"*00) (6-23)

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These are Debye equations (3.28) and (3.29) which we have analyzed earlier inconsiderable detail. Recovering Debye equations this way implies that the absorptioncurrents in a material exhibiting a single relaxation time decay exponentially, inaccordance with equation (6.17). As seen in chapter 5, there are very few materialswhich exhibit a pure Debye relaxation.T he transformation from th e time domain to frequency domain using relationships (6. 1 8)and (6.19) also proves that th e inverse process of transformation from th e frequencydomain to time domain is legitimate. This latter transformation is carried out usingequations

2= J o ('-x)cosa>tda> (6.25)n2= -^e"(6))smcotdcQ (6.26)n

Substituting equations (6.20) and (6.21) in these and using the standard integrals

*+* 2ax sin m x _n -ma2 2 ~ ~ ^ r ex +a 2

equation (6.17) is recovered.A large number of dielectrics exhibit absorption currents that follow a power lawaccording to

7(0 = Kt~" (6.27)where K is a constant to be determined from experiments. Carrying out thetransformation according to equations (6.15) and (6.16) we get

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(6.28)0


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In this range we have to use a different version of the solution of eq. (6.14) and (6.15)~pn;rsec

xp~lsinaxdx =2apr(l-p)

f ncosec pnx p l cos axdx = 2apT(l-p)

1.E-02 1 .E+GO 1.E+02

Log ( , rad/s)Fig. 6.2 Loss factor as a function of frequency at various values of 0.5 < n < 2 . 0 . s" is constantat n = 1. There is also a change of slope from negative to positive at n >1 . 0 .The slope of the loss factor curve depends on the value of n in the range 1 < n < 2 ispositive, in contrast with the range 0 < n < 1. The loss factor decreases, remains constantor increases according as n is lower, equal to or greater than one, respectively. Thecalculated values do no t show a peak in contrast w ith the m easurements in a majority ofpolar d ielectrics and one o f the reasons is that the theory expects the current to be infinite

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at the instant of application of the voltage. Further the current cannot decrease with timeaccording to the power law because if it does, it implies that the charge is infinite. Theloss factor should be expressed as a combination of at least two power laws. Jonscher(1983) suggests an alternative to the power law, according to/ ( f )oc--- (6.3^According to this equation a plot of / versus log I yields tw o straight lines. The largerslope at shorter times is - n and the smaller slope at longer time s is -1 - m. The changeover from one index to the other in the time domain occurs corresponding to the losspeak in the frequency domain. An experimental observation of such behavior is given bySussi and Raju . The change of slope is probably associated with different processes ofrelaxation in contrast with the exponential decay, equation (6.17) for the Debye

relaxation.Jonscher1 suggests that the absorption currents should be measured for an extendedduration till the change of slope in the time domain is observed. This requirement isthought to neutralize th e advantage of the time dom ain technique.Combining equations (6.15) and (6.16) we can express the complex permittivity as

**-*= Kt) (6.31)where Co is the vacuum capacitance and V is the height of the voltage pulse. / is thesymbol fo r Lap lace transform defined as

6.2 HAMON'S APPROXIMATIONLet us consider equation (6.27) and its transform given by (6.29). If we add thecompo nent of the loss factor due to condu ctivity then the latter equation becomes

s " = -+ Kcon~ l (F(l - n} sin[(l - n)n 12 ]} (6.32)

Hamon7 suggested the substitution

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cot, ={r(l-n)sm[(l-n)7r/2]Yl/n (6.33)noting that the right side of this equation is almost independent of n in the range 0.3 < n< 1 .2 . This leads to the expression

(6.34)

The equation is accurate to within 5% for the stated range of n, but it also has theadvantage that there is no need to measure odc.

6.3 DISTRIBUTION OF RELAXATION TIME AND DIELECTRIC FUNCTION

It is useful to recapitulate from chapter 3 the brief discussion of the distribution ofrelaxation times in materials that exhibit a relaxation phenomenon which is muchbroader than the Debye relaxation. Analytical expressions are available for thecalculation of the distribution of the relaxation functions G(i) considered there. Toprovide continuity we summarize the equations, recalling that a, (3 and y are the fittingparameters.

6.3.1 C O L E - C O L E F U N C T I O N

(3.94)Incosh[(l - a} ln(r / r0 ) - cos an6.3.2. D A V I D S O N - C O L E F U N C T I O N

n(3.95)

= 0 T> TO (3.96)6.3.3 Fuoss-KiRKWOQD FUNCT I O N

The distribution function for this relaxation is given as:

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71o cosh( ) co sh ( t>s )G(T) = -- 2 . ;*= l o g ( f i > / *> ) (3.97);r cos (;r / 2) + smh (Js)

6.3.4 H A V R IL IA K -N E GA M I FUN C TION B

(6.36)

[(r /rF)2(1-a) +2(r / rff )' cos ;r(l - a) + I '2

9 = arctanacosQ;r/2)

It is necessary to visualize these functions before we attempt to correlate the results thatare obtained by applying them to specific materials. We use the variable log(i / T O ) .Fig.(6.3) shows calculated distributions for the range of parameters as shown. For the Cole-Cole distribution th e relaxation times are symmetrically distributed on either side oflog(i / T O ) = 1 while th e Davidson-Cole distribution (fig. 6.4) is not only asymmetrical butG ( T ) = O a t l o g ( T / T 0 ) = l .The Fuoss-Kirkwood distribution (fig. 6.5) is again symmetrical but it should be notedthat th e ordinate here is Log ( o o / o O p ) and hence the height of the distribution increases inth r reverse order of the parameter (3. The Havriliak-Negami distribution involves tw oparameters, a and (3, and to discern the trend in change of distribution function wecompute the relaxation times at various values of a with P = 0.5, and at various values ofP with a = 0.5.Fig. 6.6 shows the results which may be summarized as follows: (1) As ais increased th e distribution become more narrow and the function G(T) will attain ahigher value at log (T/TH) = 1 though there appears to be a change of characteristics as achanges from 0 . 2 to 0.4. (2) As P increases th e peak height increases and thedistributions become broader.Fig. 6.6 shows the application of equation of (6.35) to poly(vinyl acetate) PVAc whichhas been studied by a number of authors. W e recall that PVAc is an amorphous polarpolymer with T G ~30C. Nozaki and Mashimo9 have measured the absorption currentsand evaluated the Havriliak parameters through the numerical Laplace transformationtechnique. The present author has used their data to calculate G(T)4. The abrupt change ofdistribution function at TG is evident. Whether this is true for other polymers needs to beexamined.

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o

-2 -1 0LOG(t/t0)

Fig. 6.3 Distribution of relaxation times according to Cole-Cole function (3.93). Th e distributionbecomes broader with increasing value of a.

1.0

0.8

0.6

0.4

0.20.0

-3

Fig. 6.4 Distribution of relaxation times according to Davidso n-C ole function. G(t) = 0 for t / to >1 .

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Fig. 6.5 Distribution of relaxation times according to Fuoss-Kirkwood relaxation, eq. (3.96). Thedistribution is symmetrical about c o = cop.

e

-2

Fig. 6.6 Relaxation time in P VAc calculated by using the experimental results of Noz aki andMashimo and values of a and P (1987). At T g there is an abrupt change of G(t).

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6.4 THE WILLIAMS-WATTS FUNCTIONThe detour to the distribution function of relaxation times was necessary due to the factthat it is an intermediate step in the numerical methods of transforming the time domainabsorption currents to frequency domain dielectric loss factor. We shall, however, pursuethe analytical methods a little further to indicate the potential and limitations of makingaproximations to facilitate the transformation. We have already seen that an exponentialdecay function for absorption current results in a Debye relaxation. A power law exhibitsa peak, not necessarily Debye relaxation, in the dielectric loss factor. Williams and Wattsuggested a decay function1 0 , usually called the stretched exponential,

(6.37)

where 0 < P < 1 and T is an effective relaxation time. Sometimes this equation is calledKohlraush-Williams-Watts (KWW) equation because Kohlrausch used the sameexpression in 1863 to express the mechanical creep in glassy fibres (Alvarez et. al.,1991). For the time being we can drop the subscript for the relaxation time as confusionis not likely to occur. The complex dielectric constant is related to the decay functionaccording to eq. (6.31)

(6.38)d t

If the decay function is exponential according to Equation (6.37), or y = 1 in equation(6.37), a reference to the Table of Laplace transforms gives

1< - 1 + JCOT (3.31)

which is the familiar Debye equation. Substitution of equation (6.37) in (6.38) yields

= (6.39)

The analytical expression for the Laplace transform of the right side is quite complicated.So we first substitute y = 0.5 for which an analytical evaluation of the transform can beobtained as follows. For this special case equation (6.39) becomes

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. 1 / 2nTQ

exp- 2k(f}1 /2 (6 .40)

where k = (io)"1 /2 . This has the standard form found in tabulated Laplace transforms1 1giving1 f n }l/2 1 .k\ f k e x p ( ) e r f c -^ I I / \i / Z r __J / (6 .41)

Substituting

th e transform (6.41 ) becomes

7 (6 .42)

The error function for complex arguments, w(z), has been tabulated in reference1 2 . Asample calculation is provided h ere. Let (coio) = 0.1, (e s - S0 0)=2.0, s ro = 2 .25 . Then

z = 1.12 + j\.12; w(z) = OA957(Abramowitz and Stegun,1972);

0.894O *J* =s' - js" = 3.03 - y l .47; *' = 3.03, s " = 1.47The function w(z) may also be calculated for values close to 0.5 by interpolation becausevalues for y = 1 may also be calculated using th e Debye expression. William-Watts havecomputed equation (6 .42 ) fo r several polymers including PVAc and obtained reasonableagreement between the measured and calculated values of s"/e"max and (s'-Soo) / (cs - S o o )versus log ( G O /coma x).

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The Laplace transform of equation (6.40) fo r other values of 0 < y < 1 has been given asa summation by Williams et. al.13

/ 2)] (6.43)

The range of parameters fo r which convergence is obtained is also worked out as0.25 < p < 1.0; - l < l o g f f i > r 0 < + 4

Outside this range0


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The curves become narrower fo r higher values of y, the Debye relaxation being obtainedfor y = 1, satisfying equation (3.31).

1.0

Fig. 6.7 Transformationfrom time domain to frequency domain using Laplace transform forodd values of the exponent, y. The dispersion ratio, defined by the quantity shown on theordinate (y-axis) is almost linear fo r small value of y = 0 . 1 . For higher values the familiarinverted S shape is generated (Williams et. al. 1971. with permission of the Faraday Soc.)

The half width of the absorption ratio curve may be used to determine the dielectricdecrement (ss - S o o ) using th e relation

(6.44)

where Aw is the half width of the absorption ratio curve and k(y) is a constant. For y = 1k(y) = 1.75, and this corresponds to Fuoss-Kirkwood distribution.The three parameters that appear in the Will iams-Watt equation (6.31) are y, T and (ss -S o o ) . Numerical techniques are required to find th e Laplace transform or evaluate th eintegrals. As an example, th e function (|)(t) is approximated to a series of exponentials14and the Fourier transform of the resulting series is evaluated. The parameters y and T arefitted in terms of the peak and half-width of the dielectric loss function. The

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approximations fo r each value of y require determination of a large numbe r of parameters(about 30) by curve fitting techniques. A simpler method of calculating these parametersone by one has been proposed by Weiss et. al.15 in contrast with other methods thatinvolve simultaneous evaluations of all the three parameters. The method is demonstratedto apply in P V Ac yielding results that com pare favorably.

0Log ( < O T O )

Fig 6.8 Transformation from time d omain to frequency domain using Laplace transform for oddvalues of the exponent, y. The absorption ratio, defined by the quantity shown on the ordinate (y -axis) is almost flat for small value of y = 0 .1 . Fo r higher values the familiar peak is observed(W illiams et. al. 1971, with permission of Faraday Soc).

A summary for the function within square brackets in equation (6.38) is appropriate atthis juncture from the point of view of using Laplace transforms for evaluating thedispersion ratios and abso rption ratios. Let(6.45)d t

Then the relaxation formulas yield the following time dependent functions:A . D E B Y E FUNCTION

-T (6.46)

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B . C O L E - C O L E FUNCTIONf \~ a- ; for(-)l (6.47)

< I )Txf/(0=l - /or (-)>>! (6.48)Fa T T

C . DAVIDSON-COLE FUNCTION

(6.49)

D . WILLIAMS A N D WA TT S FU N C TION( * Yexp (6 .50)v T )

E . GE N E R A L FU N C TI ON3 0 ^= fG( r ) e xp ( }d t (6.51)i ^ ~

This expression assumes a superposition of Debye processes for an incrementalrelaxation time G ( T ) d(r) and follows directly from equation (6.46). The physicalsignificance of the superposition in the time domain and the frequency domain isexplained by Fig. 6.9.Fig. 6.10 shows the function ^(t) as a function of (t/i) for a constant value of theparameters a, P, y = 0.5. For the purpose of comparison the exponential decay and thepower law decay are also shown. For short times (t/i1) which correspond to highfrequencies, the three functions, namely Cole-Cole, Debye-Cole and William- Watts,have the same dependence on time for chosen parameters. However for large values oft/i th e Davidson-Cole function drops off rapidly due to the exponential term while th edecay acco rding to William-Watt function is less sharp.

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tog ittl

log"

logw

Fig. 6 .9 Schematic illustration of superposition of (a) currents in the time domain, (b ) lossfactor in the frequency domain (Jonscher, 1983, with permission of Chelsea Dielectric Press).

6.5 THE Gd) FUNCTION FOR WILLIAM-WATT CURRENT DECAYCurrent measurements in the time domain give a single curve for a particular set ofexperimental conditions such as temperature and electric field. From these measurementsthe fitting parameters and T are obtained using equations (6.46)-(6.50). From theseparameters G(i) is evaluated with the help of analytical equations (eqs. 3.94- 3.101 ) .Using these values of i, e' and s" may be determined with the help of equations (3.89)and (3.90). However, it is not possible to express the G(i) function analytically for theWilliam-Watt decay function except for the particular value of y = 0.5. In this case theexpression fo r G(t) is (Alvarez, 1991)

(6.52)

where x = T / T W W . For other values of y the expression for G(i) is (Alvarez, 1991)

k \ (6.53)

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l .E+OQ

1.E-01 -

I* 1.E-02.a

1 .E - 0 3

1 . E - G 4

coie-coi Davidson-ColeWiiharra et. al

*Oebye

power iawi-0 81

1 .E - 0 3 1.E-01 1.E+01Iog(t / r0)

Fig. 6.10 C alculated current versus time for several relaxations, according to equations (6.46) -(6.50). 1 -Cole-Cole, 2 -Davidson-Cole, 3-Williams et. al.,, 4-Debye, 5-t-0 .8.

Though expression (6.53) is analytical its evaluation is beset with difficulties such aswide ranging alternating terms and large trignometric functions. Hence, numericaltechniques involving inversion of Laplace transform have been developed16'17.Alvarez et. al. (1991) have also developed another important equation relating theW illiam-W att time dom ain parameter y to the frequency domain parameters of Havriliak-Negami parameters, according to

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aj3 = /23 (6.54)

= 2.6(l-7)05exp(-37) (6.55)n T Hww

where TH and tww are the relaxation times according to Havriliak-Negami and William-Watt functions, respectively. Fig. 6.11 shows these calculated relationships. Theusefulness of Fig. 6-11 lies in the fact that the values of (a, P ) for a given value of y isnot material specific. Hence they have general validity and the transformation from th etime domain to the frequency domain is accomplished as long as the value of y is known.W e now summarize the procedure to be adopted fo r evaluating the dispersion ratio andth e absorption ratio from th e time-domain current measurements. For an arbitrarydependence of the current on time we use equation (6.38) and numerical Laplacetransform yields the quantities e'- E O O and s". If the transient current is an analyticalfunction of time, then G(i) may be evaluated (section 6.3) and hence the quantity ( s*- 8ro)/(es - S o o ) from equations given in section (3.16).In chapter 5enough experimental data are presented to bring out the fact that the a- andP- relaxations are not always distinct processes that occur in specific temperature ranges.As the temperature of some polymers is lowered towards T G there is overlap of the tworelaxations, (see Fig. 5.8 fo r three different possible ways of merging). For example inPMMA the cc-relaxation merges with the p- relaxation at some temperature which is a

IQfew degrees above T G . The P- relaxation occurs at higher frequencies than the a-relaxation, and it also has a weaker temperature dependency. It is present on either sideof TG.In the merging region, tw o different ansatzes have been proposed for the analysis ofdielectric relaxation. One is the simple superposition principle according to

0(0 = (0+ (0 (6-56)where < j ) a and ( J ) p are the normalized decay functions for the a- and P- relaxationsrespectively.

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1 .0 tm a x ceases to contribute to e' and s"22. The lowest frequency at which e' and 8" areevaluated depends on the longest duration of the measurement and the current magnitudeat that instant.In the data acquisition system the measured current is converted from analog to digitaland Shannon's sampling theorem states that a band limited function can be completelyspecified by equi-spaced data with two or more points per cycle of the highest frequency.The high frequency limit fn is given by the time interval between successive readings orth e sampling rate according to 2 f n = I/At. For example a sampling rate of 2 s"1 results ina high frequency cut off at 1 H z. At high sampling rates a block averaging technique isrequired to obtain a smooth variation of current with time.For low frequency data th e current should be measured for extended periods and for ahigher fn th e time interval should be smaller. These requirements result in voluminousdata but the problem is somewhat simplified by the fact that the currents are greater atthe beginning of the measurement.

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Iog 10 t (s)Fig. 6.12 The distribution of relaxation times at selected temperatures in PMMA derived fromloss factor measurements. H -N function is used to calculate inversion from frequency domaindata. The relaxation times decrease and the distributions become narrower as the temperature isincreased (Bergman et. al., 1998, with permission of A. Inst. P hys.)

10' 1 0 * t o * t o *Time (s)Fig. 6.13 Time domain current functions calculated using data shown in fig. 6.12. Thetemperature increases in steps of 10K. The inset shows the temperature dependence of the KW Wstretching parameters for the a- and (3- relaxations in PM M A (B ergman et. al. 1 998, withpermission of A. I. P.)

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Fig 6.14 Schematic diagram fo r measuring absorption and desorption current. 1-variable DCsupply, 2-High voltage electrode, 3-Measuring electrode, 4-Guard Electrode, 5-Sample, 6-Thermocouple, 7-Environmental chamber, 8-Electrometer, 9-Amplifier, 10-Recorder/computer.Inset shows details at the electrode (Sussi and Raju, 1994, with permission of Sampe journal).The current may be obtained at greater intervals at later periods and the intermediatevalues obtained by interpolation. Quadratic schemes of interpolation have been found tobe satisfactory except at short durations where the current decay is the steepest. In thisrange the log I(t) versus log(t) data are usually used fo r quadratic interpolation.The sample under study is charged using a DC power source for the required time, asensitive electrometer and recording equipment. The latter consists of a variable gainamplifier and A/D converter connected to a personal computer. The data acquisitionsystem is capable of handling 16 channels of input with individual control for eachchannel. The frequency of sampling can be adjusted up to 4 kHz though after a fewminutes of starting the experiment a much lower sampling rate is adequate to limit thevolume of data. A block averaging scheme is employed with the num ber of readings in ablock being related to the sampling rate.

6.6.1 POLY(VINYL AC ETATE)As already mentioned in the previous chapter PVAc is one of the most extensivelyinvestigated amorphous polymers, both in the frequency domain and the time domainproviding an opportunity to compare the relative merits of the two methods overoverlapping ranges of measurements.Table 6.1 lists selected publications.

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Table 6.1Dielectric Constant and Dielectric Loss Data in PVAc.

Authors & ref.No.

Mead and Fuoss23Broens and Muller24Thurn and Wolf2 5Ishida et. al26Saito27Block, et. al.

'JOMashimo, et. al.Weiss, et. al.Nozaki and Mashimo29Nozaki and Mashimo30Shioya and Mashimo31Rendell et. al32Murthy33Dionisio et. al.34

Year

19411955195619621963197219821985198619871987198719901993

Method

F-DF-DF-DF-D

F-D and T-DT-D

T-D and F-DT

T-D & F-DF-DTTT

F-D

Temp, rangeC

50-10020-65

-100 to 150-61 to 15038.1-78.7-

23.2-28.2566.7C28-9326-8526-8526-8526-8545-70

Frequencyrange(Hz)

60- 1 0 46-1072xl065 - 1 0 60.1-1061 0 "4-110'6-106502-71941 0 " 3to 1 0 7i o - 6 - i o 66x lO" 5-1 0 610'7tol0310'6-10616- 1 0 5F - D = Frequency Dom ain, T - D = Time Domain, T = Theory

Other studies are due to Veselovski and Slutsker'Sasabe and Moynihan38, Funt and Sutherland39.

35 Hikichi and Furuichi36, Patterson37,

Fig. 6.15shows the complex dielectric constant in PVAc due to Nozaki and Mashimo(1986). The Havriliak-Negami distribution function is found to apply and the evaluatedparameters a and P are shown in Table 6.2. In a subsequent publication Shioya andMashimo (1987) have found that Cole-Cole distribution also applies well for the data.The measured data have also been analyzed according to William-Watson function;while the agreement between the H-N parameters and the W-W parameter is good atlower frequencies, some differences prevail at the higher frequencies. This is attributedto a secondary relaxation process at higher frequencies, in addition to the main relaxationprocess. The secondary relaxation is thought to be due to side group rotation, the localmode relaxation of the main chain or co-operative relaxation of both motions.

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2 0log f(Hz)

-6 -4 - 2 0 2log f(Hz)Fig. 6.15 Dispersion (a) and absorption (b ) curves in PVAc at various temperatures: 1-26.8, 2 -28 . 0 1 , 3-29.41, 4-30.98, 5-34.95, 6-37.38, 7-40.20, 8-41.34, 9-42.3, 10-45.02, 11-50.20, 1 2 -54.85, 13-58.80, 14-58.98, 15-61.23, 16-61.7, 17-63.70, 18-66.07, 19-68.91,20-71.53,21-77.62,22-84.77 (C).Solid equation (Nozaki and Mashimo, 1987). (with permission of the A. Inst.Phy.)

W e recall that the W LF equation holds true for many polymers at T > TG. For PVAc thisequation is found to hold true above 45C (Nozaki and Mashimo, 1987), and theconstants are given by900.5

T- 245.3 (6.59)

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At lower temperatures the WL F equation fails and the constants in this region arelog/m =-3.01x!04-+96.1 (6.60)

This deviation from W L F equation is not observed if one evaluates th e frequency atmaximum loss according to W-W function (Mashimo et. al., 1982). The reasons are notquite clear if we suppose that the W-W function is a mathematical tool to specify th eshape of the absorption currents. Equation (6.60) is Arrhenius law with an activationenergy of 6 eV. At T G the distribution of relaxation time shows an abrupt change asshown inFig. 6.6.

Table 6.2Dielectric Relaxation Parameters in PVAc [Nozaki and Mashimo, 1986].T(K)

301.25301.71302.90303.60305.80309.163 1 1 . 1 5315.65326.64326 . 25330.47335.05339.30343.75349.80365.95

T O ( S )5.70 x 10 32.94 x 1 0 39.25 x 1 0 24.90 x 10 25.64 x 1 01.86 x 10

3.261.207.16 x 10"31 . 3 8 x 10"24.15 x 10"21.01 x 10'33.26 x 10"41 . 0 3 x 10'43.02 x 10'53.23 x 10"6

Ac7.9228 .0178.0537.9608.0407.2797.4446.8006.2206.0045.9095.5825.2895.1974.8004.170

Coo

3.1393.1383.1353.1333.1263.1173 . 1113.0983.0683.0683.0563.0433.0313.0183 . 0012.955

a0.3900.3880.4020.4140 .4210.4500.4450.4670.5010.5130.5000.5260.5300 .5510.5600.590

P0.8210.8130.8190.8220.8230.8600.8440.8640.8800.8560.8590.8680.8720.8690.8700.873

H ( D )2.4582.4902.5152.4982.5492.3672.4402.2982.2312 . 1 6 12.1742.1102.0502.0601.9721.865

Ac = Ss - Soo; (With permission of J. Chem. Phys.)The last column in Table 6.2 is the dipole moment of the monomer according toOnsager's equation and the method of calculating it has been demonstrated in section2.8.

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Murthy (1990) has discussed the dielectric relaxation in PVAc and suggest that thedeviation from the W LF equation may be removed if a power law is assumed instead ofthe WLFequation (5.19),

log/. =4-];T>Tk (6.61)The large value of s ( 12 . 1 for PVAc) combined with the introduction of another arbitrarytemperature Tk and lack of physical basis limits th e usefulness of this equation.

6.7 COMMERCIAL DIELECTRICSAbsorption currents are measured, as described, during the application of a step voltagelasting from a few minutes to several hours. The recorded current is the sum of thepolarization currents and conduction currents due to the electric field impressed. Theabsorption current is reversible; if the applied electric field is removed and the twoelectrodes are short circuited the current flows in the opposite direction and itsmagnitude is exactly equal to the absorption current. The conduction current, however, isnot reversible, since th e electric field is removed. To remove th e effect of applied fieldon the absorption current the discharging current is preferably measured using the samesetup shown in fig. 6.14. The discharging current is also called desorption current,though we use the common term "absorption current" to denote both charging anddischarging conditions.There are a number of reasons for the generation of absorption currents. These have beenvariously identified as40:

(1 ) electrode polarization,(2) dipolar orientation,(3) charge injection leading to space charge effects,(4) hopping of charge carriers between trapping sites, and(5) tunneling of charge carriers from the electrodes into the traps.

An attempt to identify the mechanism involves a study of the absorption currents byvarying several parameters such as the electric field applied during charging, thetemperature, sample thickness and electrode materials. Wintle41 has reviewed theseprocesses (except the hopping process) and discussed their application to polymers.Table 6 .3 summarizes th e characteristics responsible for the transient currents.

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The mechanisms of generation of absorption current in specific materials is still an activearea of study and there is only agreement in the broad generality. Measurements todiscriminate the various processes should include a systematic study of the influence ofparameters such as field strength, electrode material, sample thickness, water content,temperature, and thermal history42. Indicators, which are looked for in experimentalresults, may be summarized as below:Table 6.3Summary of the characteristics of the mechanism responsible fo r transientcurrents. L is the thickness and F is the electric strength.

Process

Electrodepolarization

Dipole orienta-tion, uniformlydistributed inbulkCharge injectionforming trappedspace charge

TunnelingHopping

field depen-dence ofisochronalcurrento c Fp wherep = l

o c Fp wherep = l

related tomechanismcontrol-ling chargeinjectiono c Fp wherep = lo c Fp wherep=l

thicknessdependenceof isochronalcurrent atconstant fieldnot specified

independent

independent

L-1independent

electrode materi-al dependence

strongly depen-dent throughblockingparameterindependent

related tomechanism con-trolling chargeinjectionstronglydependentindependent

Temperaturedependence

thermallyactivated

thermallyactivated

related tomechanism con-trolling chargeinjectionindependentthermallyactivated

time depen-dence / oc t~n

initiallyn = 0.5followed byn>lO^n < 2

0 < n s ? l

O s n < 20 < n ^ 2

relation be-tween chargingan d dischargetransientsmirror images

mirror images

dissimilar

mirror imagesmirror images

(Das-Gupta, 1997, with permission of IEEE).1 . Identical absorption and desorption currents, except in sign, implies that chargecarriers are generated in the bulk and not injected from th e electrodes. If dipolarprocesses are present, polymers as a rule exhibit a distribution of relaxation times withseveral overlapping processes. This is reflected as a change of slope of the power lawindex over short segments of time.2 . Oxidation of a polymer introduces additional dipoles of a different kind and these arelikely to reside closer to the surface. The transient currents will then be inverselyproportional to the thickness.3. Charging current independent of the electrode material provides additional support forth e absence of charge injection from th e electrodes. Tw o different electrode materialswith a large difference in work function may be employed in four differentconfigurations to check th e injection of charge carriers from th e electrodes.

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4. Independence of current on electrode material indicates lack of electrode polarizationand the power law index for this process is close to 0.5.5. Currents that are independent of thickness and temperature suggest that tunneling is atleast not the principal mechanism for the transient current.6. Hopping of ionic or electronic charge generally shows a transient decay of thedischarging current divided in two successive domains having n~l and n~0, respectively.7. Presence of moisture generally reduces the activation energy for dipolar motionsshifting the a-peak to lower temperatures. This is reflected in the absorption currentsincreasing at room temperature.Lowell43 has provided an analysis of the absorption and conduction currents bydeveloping a model of the behavior of charges trapped on a polymer network. The theoryis claimed to agree with experimentally observed current-time relationship in polymers.

6.7.1 ARAM ID PAPERAramid Paper which is an aromatic polyamide is a high temperature insulating materialand finds increasing applications in electrical equipment such as motors, generators,transformers and other high energy devices. There are a growing number of uses for thismaterial in nuclear and space applications due to its excellent capability of withstandingintense levels of radiation such as beta, gamma and X-rays. The dielectric properties ofthis material are of obvious interest, and Sussi and Raju (1992) have measured theabsorption currents covering a wide range of parameters.To ensure repeatable results the samples were conditioned by heating for several hours(~ 5 hours) at high temperatures (~260C). Two different electrode materials, aluminumand silver of 50 nm thickness, were vacuum deposited on the samples at reducedpressure of 0 .1 mPa. A guard ring which was also vapor deposited was employed toreduce effects due to stray fields. Before each measurement a blank run was performedin order to free th e sample from extraneous charges. This ru n consisted of raising th etemperature of the sample at a constant rate to 200C and short circuiting the electrodesfor twelve hours. The influences of several factors on the absorption currents aredescribed below.

A. TIMEDEPENDENCEFig. 6.16 shows typical charging currents in a 127 urn thick sample at a temperature of200C and various poling electric fields for a time duration of 10 4 s. The chargingcurrents decay very slowly and no polarity dependence is observed. There is noappreciable dependence of the current on the sample thickness or electrode material.

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A similar slow time dependence of charging current has been reported fo r otherinsulating materials such as SiO 2 and SiO with aluminum electrodes44 , solid n-octadecane45 and polycarbonate above its glass transition temperature of 150C 46 .Lindmeyer suggests that a slow filling of traps by charge carriers in conjunction with aspace charge effect is responsible for the slow decay of current with respect to time. Ithas also been suggested that that a high electric field can induce traps in a polym er w iththe trap creation rate being proportional to the applied field. This theory, also, possiblyexplains the reason for not being able to observe a steady current under certainexperimental conditions, even after as long as 1 0 Ds.

O)o

8

9

Nonwx 410127 pin, T=200 'CE (M Vm ')78 S . 9 Q 0 0 0 0 0 0 0 0 0 00000000003 . 9 A A^A A AAAAA A A A A AAA

2.0

0 .80 . 6

A A A A A AAA A A A AAA A

o . 4 3 3333333

0.2

0.08, a a a a a a a a a a a a a o an2 3

Log t (s)

Fig. 6.16 Charging currents at various electric fields at 200C in aromatic polyamide with silverelectrodes. The current decays very slowly with time. This is just one of several types of decayof current in polymers (Sussi and Raju, 1992, with permission of Sampe journal).

Fig. 6.17 shows the influence of temperature on the absorption currents at a constantelectric field. The decay of current is more rapid in the lower temperature range of 70 -170C. Such behavior has also been observed in linear polyamides in which theconduction currents in the range of 2 5 - 80C and 1 20 - 155C decay more rapidly thanin the intermediate range of 80 - 110C. Obviously there is a change of conductionmechanism above 150C in aramid paper.

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As stated earlier a comparison of charging and discharging currents perm its reliableconclusions to be drawn with regard to the origin of charging current.Fig. 6.17 shows thedischarging currents at 2 0 0 C after poling at various fields in the range of 0.08 - 8MVm" 1 . The current decays fast for a duration of about 10 3 s where the power law holdswith a value of n -1.0. For times longer than 10 3 s the current decay is much lesspronounced and the decay co nstant in this range of time is between 0.1- 0.4.The data in Figs. 6.17and 6.18 clearly show that the charging and the dischargingcurrents are dissimilar for times up to 1 0 4s. Such dissimilarities have been observed inother polyme rs as well and there is no agreement on the processes responsible for them.

200%190

0 a n o o n< 8

Ol

1 9

ISO

70 0

a a o a a a o a a a

1Q(_ Nona 410IUP 76 inn, EM U M Vm '

11

00

iCo

1 2 3 4Log t (s)

Fig. 6.1 7 C harging currents in arom atic polyamide at various temperatures and constant electricfield of 6.6 MVm-1 with silver electrodes. A change of mechanism is clearly seen above 1 50 C.(Sussi and Raju, with permission of Sampe Journal).

Lindmeyer (1965) discusses these dissimilarities with the aid of energy diagrams andattributed the initial transient during charging to empty traps. The current can be as largeas allowed by the injecting barrier. As the traps become filled, the current reduces to thespace charge limited current with traps. In the discharging period, on the other hand, thetrapped carriers will be emptied towards both electrodes, the external circuit registering asmaller current.

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Das Gupta and Brockley47, on the other hand, concluded that the dissimilarity betweenth e charging and discharging currents is due to the onset of quasi steady state conductioncurrent which is superimposed on the transient current. If this reasoning holds good thenth e dissimilarity should increase with electric field or temperature.It is relatively easy to interpret the results of fig. 6.18 in terms of the universal relaxationtheory of Jonscher(chapter 4) who suggests that the time domain response of the currenthas two slopes or two values for the power law index , equation (6.30). An increase inth e value of index at longer periods is attributed to a dipolar mechanism whereas adecrease at longer periods is attributed to interfacial polarization. O n this basis it issuggested that the relaxation changes from dipolar to interfacial as the temperature isincreased from 90C to 200C.

L o g t (s)Fig. 6.18 Discharging currents at various temperatures and constant electric field of 6.6 MVm-1with silver electrodes. Charging and discharging currents are dissimilar indicating that dipolarprocesses are not dominant (Sussi and Raju, 1992, with permission of Sampe Journal).

Aramid paper is manufactured out of aromatic polyamide in the form of fibers which areamorphous and floes (like corn flakes) which are crystalline, the finished product havinga crystallinity of approximately 50%. The material abounds with boundaries between thecrystalline and amorphous regions. Both the number of distributed trap levels and themobility of charge carriers are likely to be higher resulting in a higher conductivity in

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amorphous regions in comparison with the crystalline parts. This difference inconductivity is possibly the origin of the observed interfacial polarization.B . TEMPERATURE DEPENDENCE

From Figs. 6.17 and 6.18 it is inferred that the isochronal discharging currents increasewith temperature at the same electric field in the temperature range of 90 - 2 0 0 C . Thecurrents show no significant peaks and from the slopes an approximate activation energyof 0.21 0 .05 eV may be evaluated.C . FIELD DEPENDENCE

The isochronal charging current shows a linear relationship with the electric field at eachtemperature according toI(t} = k( t)E p (6.62)

where k is a constant independent of E and p is approximately equal to one. Das-Guptaand Joyner48 and Das-Gupta et. al.49 have observed similar relationship in poly(ethyleneterephthalate) (P ET) at 173 K . The m agnitude of isochronal current is observed to dependupon the time lapsed in contrast with aramid paper which shows only a marginaldependence on time. This difference is attributed to the fact that the charging currentsvary slowly with time(Fig. 6.16) .D . EFFECT O F ELECTRODE MATERIAL

The charging currents are found to be dependent on the electrode material, silver yieldinghigher currents than aluminum. The power law index n is also higher fo r silver. Howeverthe discharging currents do not show any significant dependence on the electrodematerial.E. Low FREQUENCY DIELECTRICL.OSS F ACTOR ( E " )

The theoretical basis for transforming the time domain data into e"-log 00 has been dealtwith in considerable detail in earlier sections of this chapter. The Hamon approximationgiven by equation(6.34)

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is easy to apply to the measured data.Fig. 6.19 shows the calculated values of s" at lowfrequencies in the temperature range of 100-190C. Since dc conductivity contributes tothe loss factor appropriate correction has been applied. Two peaks are observed at eachtemperature and they correspond to relaxation times of 1.6 x 103s and 2 x 104s. Thesevalues are comparable to the relaxation time of 1 x 1 0 4s at 150C obtained by GovindaRaju50 from studies of thermally stimulated discharge currents in aramid paper. Thisstudy has shown that dipolar mechanism is responsible for the TSD currents in thetemperature range cited and the observed peak at the higher frequency in fig. 6.19 (thefirst peak) is attributed to the dipolar mechanism. W e also note that the low frequencypeak (second peak) is more dominant at higher temperatures.Sussi and Raju (1992) also examine whether the low frequency peak is possibly due tointerfacial polarization according to the Maxwell Wagner equation, considering that theamorphous and crystalline regions have different conductivity. The parameters evaluatedare shown in Table 6.3. Note the large increase of the loss factor at 200C which ispossibly due to an intensified Maxwell-Wagner effect. Higher temperatures increase theconductivity in amorphous regions more rapidly than in crystalline region. It isinteresting to compare Fig. 6.19 with Fig. 5.17 which also demonstrates a large increasein s" as the temperature is increased.

6.7.2 COMPOSITE POLYAMIDEThe demand for electrically insulating polymers has increased over the years due to theirhigh levels of electrical, chemical and mechanical integrity, making them ideally suitedfor several applications. Some organic polymers are capable of withstandingtemperatures higher than 180C and also have capability for withstanding radiation. Themechanical strength of aramid paper can be increased substantially for industrialapplications by using it in conjunction with other polymers and such materials arereferred to here as composites. For example an industrially available material iscomprised of aramid-polyester-aramid (A-P-A) with three layers of equal thickness.The isochronal currents measured by Sussi and Raju51 in A-P-A are shown infig. 6.20with regard to the influence of temperature on discharging currents. For each curve thecurrent increases with temperature and shows a well defined peak in the range of 80-100C. The position of this peak tends to shift to a higher temperature with decreasingtimes, suggesting the presence of thermal activation processes (Sussi and Raju, 1992). Acomparison withfig. 6.16 shows that the observed peaks are due to the intermediate layerof polyster which has TG in the vicinity of the observed peak. The dipolar contribution issuggested as a possible mechansm. Above 130C the isochronal currents increase almost

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monotonically with time and the increase is attributed to interfacial polarization asexplained in the previous section.

40

2 0 -

100

** .V--* . c

a B ^ s o ^ oj

\X4ff '

I I . 1 .5 -4 -3

543210

Fig. 6.19 Frequency dependence of dielectricloss factor s" at various temperatures. Pre-applied field of 6.6 MV m-1; charging time 5h;electrodes made of silver (Sussi and Raju, 1992).(with permission of Sampe Journal.)

Fig. 6.20 Temperature dependence of theisochronal discharge currents in compositearamid-polyester-aramid polymer atprescribed times. The electric field is 6.56MVm" 1 , 1 54 jam sample with aluminumelectrodes. (Sussi and Raju, 1994: Withpermission of Sampe Journal)

-12 1 1 1 I I I I I I80 100 120 140 160 180 200 220

T ( -C)

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6.7.3 POLYfETHYLENE TEREPHTHALATE)The absorption currents in PET have been measured by a number of authors. Das-Guptaand Joyner (1976) distinguish two temperature regions, 83-273K and 273-43OK. In bothregions the isochronal currents (I-T plots) are proportional to the electric field. Nothickness dependence or electrode material dependence is observed. The charging anddischarging transients are mirror images and two thermally activated energies of 0.65 and1 .6 eV are measured in the regions mentioned above respectively. The lower energy isidentified with the p-relaxation and the higher energy with the a-relaxation.

Table 6.4Loss factor in aramid paper (Sussi and Raju, 1992)T(C) s" a (Qm)'1 i(s)

(a) High frequency peak1 1 01 301 401 501 601 1 0140160

0.761 .141.312.142 . 0 3

(b)0.891.702.27

1 . 7 5 x 10'155.48 x 10'151 . 2 0 x 10'142.62 x 10 "1 44.38 x 10'14L ow frequency peak7.0 x 1 0 '1 62.3 x 10'154.0 x 10'15

1 . 6 x 10 31 .1 x 1 0 36.3 x 10 25.5 x 10 23.1 x 10 25.6 x 1 0 35.0x 10 34.0x 10 3

(with permission from SAMPE journal)i52More recent results are due to Thielen et.al who discovered that the traditional use of

three electrodes fo r measurement of current is likely to damage th e ultra-thin films (


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whereas Schottky mechanism is found to hold true for the steady state current. Watercontent increases the steady state current mo re than it increases the transient current.

-20-22-24-26-28

0 2 4 6 8 10In t(sec)

Fig. 6.2 1 Transient conductivity observed in Mylar films of various thicknesses (12, 6 , 3and 1.5 urn) for times ranging from 1 to 10 4 s T=23C, RH=52%, E p=4xl 0 7 V/m (Thielen et.al., 1994, with permission of American Inst. Phys).There are two relaxation peaks observed in PET. The high temperature measurementscovered by the Neagu et. al.54 falls into the range 20-110C where both a- and p-relaxations are detected. In fact the (3- relaxation is centered about -110C55 andattributed to local motions of glycol methylene and carboxylic groups. It is characterizedby a broad distribution. The dipoles corresponding to this relaxation are oriented in ashort time < Is and not measured by Thielen et. al (1994). The a-process is due to themotion of main chain and the distribution of relaxation times extends down to roomtemperature.The measured current is probably due to this relaxation and the relaxation time in therange of 1 0 3-1 0 4 s agrees with the expected va lue for this relaxation.

6.7.4 FLUOROPOLYMERThe fluoropolymer, commercially known as Tefzel (E. I. Dupont De Nemours & Co.Inc) is a copolymer of tetrafluoro-ethylene and ethylene considered as a high temperatureinsulating material. As far as the author is aware there is only one study carried out on

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th e dielectric properties ( & ' and e") and absorption currents in this high temperaturedielectric56 and these data are presented below.

A . DIELECTRIC CONSTANT A N D Loss FACTORFig. 6.22 shows & ' and s" as a function of temperature and frequency. The dielectricconstant decreases with increasing temperature at each frequency. Further the loss factorshows a well defined peak in the region 100-120C, the loss peak shifting to higherfrequencies as the temperature is increased. These are characteristics of dipolar relaxationand plots of the molar polarizability versus 1 /T (eq. 2.53) do not yield a straight line. It isevidence for the non-applicability of Debye theory for dipolar relaxation on the basis of asingle relaxation time. This result is not surprising since most polymers are characterizedby a distribution of relaxation times.

B . ABSORPTION CURRENTSThe absorption currents are measured at 5 < E < 40 M V m" 1 and 50 < T < 200C by W uand Raju (1965).Fig. 6-23shows the time dependence of isothermal charging currents at2 0 MVm"1 poling field and various temperatures. Results at other temperatures andelectric fields shows a similar trend, suggesting that th e mechanism fo r absorptioncurrent remains the same over these ranges of parameters. The material exhibitsabsorption currents that increase with t ime during the initial 1 - 100 s and then decreaseslowly. Similar behavior is found in PE in which th e conduction mechanism is identifiedmainly as space charge enhanced Schottky injection57.The absorption current also depends significantly on the electrode material, aluminumelectrodes yielding higher currents than silver electrodes. This dependence suggestspossible injection of charge carriers from the cathode and the accepted mechanism isfield assisted thermionic emission. The current dependence on the electrode material canthen be explained by the work function differences. Others have also reported similarresults in PET and polyimide.Injection of electrons from the cathode due to field assisted thermionic emissiongenerally leads to a space charge layer within the thickness of the material andaccumulation of charge carriers can also occur in the metal/polymer interface. The spacecharge, if sufficiently intense, will reduce the electric field at the cathode resulting in adecreased electron injection. The observed increase in current with time can only beexplained on the basis of accumulation of positive charges within the polymer and closeto the cathode. The ions originate within the bulk of the material or are injected from theanode and drift towards the cathode, thereby increasing the cathode field.

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150 .10 04100T , C 50 f, Hz

s0. 03

0.020.03

f, H zFig. 6.22 Real and imaginary part of the complex dielectric constant in Tefzel (author'smeasurements).

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Fig. 6.23 Isothermal charging currents at 20 MV/m at various temperatures (electrodes silver) influoropolymer (W u and Raju, 1995, with permission of IEEE).There are several sources for ion generation. One of them is impurity ions, the currentdue to which tends to decrease if successive measurements are repeated on the samesample within a short interval of time. A second source of ions is due to autoionization ofone of the atoms in the monomer, as demonstrated by Raju (1971) in aromaticpolyamide. There is no experimental evidence in the literature to support this mechanismin this copolymer, and it is shown in chapter 7 that ionic conduction is not a feasiblemechanism in this material.Poole-Frenkel mechanism (Ch. 7) suggests that the electric field lowers the barrier fortransition of charge carriers in the localized states. The mechanism is not specific tocharge carriers of a specific type and is similar fo r donors (creating electrons) oracceptors (generating holes). Holes generated within th e bulk drift towards th e cathodeand contribute to the intensification of the field.

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An additional mechanism, which has been suggested to account for an increase in currentwith time, is the increase of the number of traps in the bulk due to the influence of theelectric field. The new traps that are created permit more injection resulting in anincrease of the current. Such an increase in the number of traps is likely to saturate after acertain time of application of voltage and the surface potential of the polymer is expectedto vary according to fig. (6.23).At longer time duration the current decreases with time and this is the more commonbehavior for many polymers. Lindmeyer [1965] suggests that gradual filling of traps bycharge carriers leads to this type of behavior. Further the polymer-metal interface issensitive to structural features such as the density of localized molecular states in thepolymer and tunneling range of the delocalized electrons in the metal electrode. Unlessth e charge is moved away from th e site of injection, further injection is reduced. Thiseffect added to the bulk trapping causes decay of charging currents at longer periods.Isothermal absorption currents are also measured by interchanging the polarity of theelectrodes. The currents were approximately the same as the current under positivepolarity fields (fig. 6.23). This is considered to be reasonable because both electrodes arevacuum deposited with the same type of metal under identical conditions and the changein surface conditions is not significant enough to cause a change in emission.

6.7.5 POLYIMIDEso

Chohan et. al. have measured the absorption and desorption currents in Kapton and thetwo currents were dissimilar in selected temperatures in the range 39-204C. Fig. 6-24shows the low frequency loss factors versus th e frequency, which exhibit peaks that shiftto lower frequencies as the temperature decreases. This observation is attributed to theinterfacial polarization either at the electrode-dielectric interface or in boundariesbetween crystalline and amorphous regions. There is need fo r systematic study oftransient currents in this important polymer over a wider range of temperature andelectric field.

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-1C JO5O

-2 -

-3

10-6Hz 10'5Hzi- J-.. 7-8x10- 10'2Hz-AA-

-6 -5 -4 -3 -2log frequency( H z )

-1

Fig. 6.24 Hamon transform results from desorption currents. Tem perature (C ) as shown. Thepeak at 39C is outside the frequency window (Chohan et. al. 1995). (with permission ofChapman and Hall).

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6.8 REFERENCES1 A. K. Jonscher, "Dielectric Relaxation in Solids", Chelsea Dielectrics Press, London

19832 E. Von Schweidler, Ann. Der. Physik, 24 (1907) 711.3 M. F. Manning and M. E. Bell, "Electrical conduction and related phenomena in soliddielectrics", Reviews of Modern Physics, 12 (1940) 215-256,4 G. R . Govinda Raju , Annual Report, Conference on Electrical Insulation and DischargePhenomena, 2001.5 I. S. Gradshteyn, I. M. Ryzhik, Yu. V. Geronimus and M. Yu. Tseytlin, "Tables ofintegrals, series, and products", Academic Press, New York, 1965.6 M A. Sussi and G. R. Govinda Raju , SAMPE Journal, 28 (1992) 29-36.7 B. V. Hamon, Proc. IEE, Proc. IEE (London), 99 (1952) 1 5 1 .8 F. Alvarez, A. Alegria and J. Colmenero,, Phys. Rev., B, Vol. 44, No. 14 (1991) 7306-7312.9 R. Nozaki and S. Mashimo, J. Chem. Phys. 87(1987) 2271.1 0 G . Williams and D. C. Watts, Trans. Farad. Soc., Vol. 66 (1970) 80 .1 1 M . Abramowitz and L. Stegun, Handbook of Mathematical Functions, N.B.S., formula118 .1 9 M. Abramowitz and L. Stegun, Handbook of Mathematical Functions, N.B.S., page325.13 G. Williams, D.C. Watts, S. B. Dev and A. M. North, Trans. Farad. Soc., 67 (1972)pp. 1323-1335.14 C. T. Moynihan, L. P. Boesch and N. L. Laberge, Phys. Chem. Glasses, 14 (1973)1 2 2 .1 5 G. H. Weiss, J. T. Bendler and M. Dishon, J. Chem. Phys., 83 (1985) 1424-1427.16 S. W . Provencher, Comput . Phys. Commun., 27(1982) 213.17 J. -U. Hagenah, G. Meier, G. Fytas and E. W. Fischer, Poly. J. 19 (1987) p. 441 .18 R. Bergman, F. Alvarez, A. Alegria and J. Colmenero, J. Chem. Phys., 109 (1998)7546-7555.19 F. I. Mopsik, IEEE Trans. Elec. Insul., EI-20 (1985) 957-963.2 0 Y. Imanishi, K. Adachi and T. Kotaka, J. Chem. Phys., 89 (1988) 7593.2 1 M. A. Sussi and G. R. Govinda Raju , Sampe Journal, 2 8, (1992) 2 9.2 2 H. Block, R. Groves, P. W. Lord and S. M. Walker, J. Chem. Soc., Farad. Trans. II,

68(1972) 1890.2 3 D. J. Mead and R. M. Fuoss, J. Am. Chem. Soc., vol. 63 (1941) 2832.2 4 O. Broens and F. M. Muller, Kolloid Z., Vol. 141 (1955) 20.25 H. Thurn and K. Wolf , Kolloid Z., Vol. 148 (1956) 16.2 6 Y. Ishida, M. Matsuo and K. Yamafuji , Kolloid Z., vol. 180 (1962) 108 .

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