Remarks on the hamon approximation

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Advances in Molecular Relaxation Processes, 7 (1975) 13-20 @ Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

REMARKS ON THE HAMON APPROXIMATION

YAW0 KITA AND NAOKAZU KOIZUMI

Institute for Chemical Research, Kyoto University, Uji, Kyoto-fu 611 (Japan)

(Received 14 August 1974)

CONTENTS

I. Introduction . . . . . II. Hamon approximation .

III. Dielectric loss . . . . . IV. Relaxation intensity . . V. Relaxation frequency . .

Summary . . . . . . References . . . . . . . . .

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13 14 14 17 18 19 20

I. INTRODUCTION

Hamon proposed an approximation method1 to evaluate dielectric loss by measuring the absorption current or the transient charging current which flows into a dielectric medium when a step voltage is applied to it. The approximation has been used extensively’ for the analysis of dielectric relaxation processes as exhibited by polymeric solids in a very low frequency region where a conventional bridge technique is not applicable. Recently, Williams and Watts3*4 discussed the limits of applicability of the Hamon approximation in connection with a new empirical formula for the response function.

It would be worthwhile to examine the accuracy of the Hamon approximation in analysis of the relaxation processes which are found for polar liquids and solids. Most polar dielectrics show the Cole-Cole’, Davidson-Cole6, or Williams- Watts3 type of dielectric relaxation. The knowledge of the response functions and the dielectric loss as a function of frequency for these types of relaxation makes it possible to investigate the accuracy of the Hamon approximation. The purpose of this work is to examine the accuracy and errors when the Hamon approximation is applied to evaluation of the dielectric loss and the relaxation time from transient charging current data.

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II. THE HAMON APPROXIMATION

The relationship between the electric displacement o(t) and the electric

field E(t) may be expressed by the response function 4(t) at time t after application

of the electric field E(0) at t = 07.

D(t) = @‘(r)+(~,-c,)/’ E(u)d(r-u)dzr (1) -CC

where E,, and E, are the instantaneous and equilibrium values of dielectricconstant,

respectively.

The Hamon approximation referred to here is the conversion method of

the response function 4(t), which is defined in eqn. (1) for a given electric field,

into the dielectric loss. The absorption or charging current density i(t) for an

applied step electric field V is related to the response function 4(t) as i(t)/ V = 4(t).

By taking time t after application of a step electric field as a/o, the normalized

dielectric loss E”(w)~~,,,,,~ is given by

= E”(O)

co -E’x

=

where w is the angular frequency and c(( = 0.63) is a numerical constant.

(2)

III. DIELECTRIC LOSS

The response functions for the Cole-Cole, Davidson-Cole, and Williams-

Watts types of relaxation are written as’, 9* 3

4(t) = ‘, nz,(- V+‘(~/r)“V-(nB) Cole-Cole (3)

4(f) = t (t/~)~- ’ exp (- t/r)/r(fl) Davidson-Cole (4)

4(t) = s (t/~)~-r exp [ -(t/~)~] Williams-Watts (5)

where z is the generalized relaxation time for respective types of relaxation, /I is

the distribution parameter of relaxation times and r(B) is gamma function. Thus,

the dielectric loss for the particular type of relaxation to be expected by the

Hamon approximation eqn. (2) can be calculated by substituting the response

functions of eqns. (3)-(5) into eqn. (2).

On the other hand, the true values of the normalized dielectric loss E”(w)~,,,

are given by a Fourier transformation of the response function as

a,

s”(m)tr”e = s

sin ot4(t)dt (6) 0

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The dielectric loss E”(O),,,, as a function of frequency is given by using eqns.

(3)-(6) in the following expressions 5, 6B 4.

G4tr”e = co8 8 sin /%I; tan 61 = wz Davidson-Cole (8)

Cole-Cole (7)

E”(~L = c O” (-l>“-‘r(ng+ ‘) sin nB n ( 1 PI= 1 (m>“V(n + 1) 2 Williams-Watts (9

Values of E”(w)~~~~~ calculated by eqn. (2) and E”(w)~~“~ as a function of

frequency are compared in Figs. 1, 3 and 5. The ratio of E”(o)~_,~ to E”(W),,,,

0.0

0.6-

-i -i 6 i i log (w/U$

1

Fig. 1. Comparison of E”(w)~~,,,~~ with E”(U) Lrue for the Cole-Cole type of distribution as a function of the reduced frequency, i.e. frequency divided by relaxation frequency. The distribution parameter B is indicated in the figure.

Fig. 2. Plots of the ratio of E”(o)~~,,,~~ to E”(o)~~~~ for the Cole-Cole type of distribution against the reduced frequency. The distribution parameter @ is indicated in the figure.

Fig. 3. The same as in Fig. 1 for the Davidson-Cole type of distribution. The distribution parameter @ is indicated in the figure.

Fig. 4. The same as in Fig. 2 for the Davidson-Cole type of distribution. The distribution parameter ,!I is indicated in the figure.

E 0.9

E

c 0.4

ti

‘2 0.2

0.0

log (W/WR)

Fig. 5. The same as m Fig. 1 for the Williams-Watts type of distribution. The distribution parameter B is indicated in the figure.

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Fig. 6. The same as in Fig. 2 for the Williams-Watts type of distribution. The distribution parameter p is indicated in the figure. (Also see ref. 3.)

is plotted against frequency in Figs. 2, 4 and 6, to demonstrate relative errors for

three types of relaxation. As seen from Figs. 1-6, the error in E”(o)~~~,,,, is small

at frequencies higher than the relaxation frequency oR( = l/r), but significant

errors are involved at lower frequencies. With the Cole-Cole and Williams-Watts

types, the greater the value of fi, namely the narrower the distribution of relaxation

times, the larger becomes the error in evaluating the dielectric loss at lower fre-

quencies. In the case of the Davidson-Cole type, large errors are introduced even

for a small value of /?, and the Hamon approximation is useless in evaluation of

E”(W) at frequencies lower than about one tenth of the relaxation frequency. It was

noted that the approximation can be used in relaxation processes of the Cole-Cole

type with a value of fi < 0.5.

IV. RELAXATION INTENSITY

The relaxation intensity or the magnitude of dielectric dispersion (se - e,)namon

in the Hamon approximation may be expressed by the Kramers-KrBnig relation

!Eg_-dhlon = 2 m E”(W)Hamon dw

(~o-4true s 7.c 0 w

Substitution of eqn. (2) into eqn. (10) by using t = 0.63,‘~ gives

t&o -&mon = 1 a, d@*63/0) dw

(co -4tr”e s n 0 o.+

= o&-~o~~(t)df E 1.01

(10)

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In the above equation, the following condition for response function is used:

s -d(t)dt = 1 (12)

0

The response functions given by eqns. (3)-(5) satisfy eqn. (12) (ref. IO). Therefore

the relaxation intensity may be obtained within an accuracy of 1 % by the Hamon

approximation, regardless of the type of relaxation.

V. RELAXATION FREQUENCY

The error introduced by the use of the Hamon approximation for the

relaxation frequency, which is an important parameter in the relaxation process,

was estimated. The relaxation frequency on by the Hamon approximation is the

value of w which maximizes s”(~)n~,,,,,~. Then

[““‘(cd)5mon]w=mm = 0 (13)

The ratio of wn to mR, the reciprocal of the generalized relaxation time character-

istic of the particular type of relaxation, can be taken to estimate the error in wH.

By putting oR/o = x, eqn. (13) is rewritten as

[d”“(cd)~-on]o=~” = - !!$ [““‘(~>]X~~R,w. = 0 (14)

In the Davidson-Cole type of relaxation6, .s”(x)namon is given as

s”(x)“amcNl cc xaexp(-ax)

Differentiation of .s”(x)uamO,, with respect to x gives

dx cc xB-l(/3-cIx) exp (-MX)

Using eqn. (14) one obtains

1 wH a 0.63 -=--__-_= X *R P P

Similarly, wn/oR in the Williams-Watts type is derived as follows.

s”(x)“amon a xB exp ( - c?x8)

dE”(X)Hamon a pxD- I(1 _ -/,$J) exp (_ apxfl)

dx

(15)

1 OH -=-= CI = 0.63 x WR

(16)

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TABLE 1

COMPARISON OF 0” RELAXATION FREQUENCY EVALUATED BY THE HAMON APPROXIMATION WITH WR

TRUE RELAXATION FREQUENCY

Distribution oH/wR parameter -__-

B CC” DCb WW’

I.0 0.63 0.63 0.63

0.9 0.69 0.70 0.63

0.8 0.74 0.79 0.63 0.7 0.81 0.90 0.63 0.6 0.87 1.05 0.63 0.5 0.93 1.26 0.63 0.4 0.99 1.58 0.63 0.3 1 .os 2.10 0.63 0.2 1.09 3.15 0.63 0.1 1.11 6.30 0.63

i( Cole-Cole distribution.

’ Davidson-Cole distribution. ’ Williams-Watts distribution.

Since no explicit expression for the ratio ~~~~~ in the Cole-Cole type can

be obtained, numerical calculations were carried out to estimate on/c+. The

values of wn/e+ for three types of relaxation are listed in the Table 1. The relaxa-

tion frequency wH by the Hamon approximation is fairly close to the reciprocal

of the generalized relaxation time in a range of B of 0.3 < p < 0.5 for the Cole-

Cole type distribution. But the approximation is not applicable to determining

OR in the Davidson-Cole and Williams-Watts types of relaxation, as shown in

Table 1.

It would be useful to examine how close or, is to op, the peak frequency

where E”(o),~“~ is a maximum. For the Cole-Cole type of relaxation, wp is identical

to wR. In the Davidson-Cole type, the following relation is derived easily from

eqns. (8) and (15),

WH 0.63 -= ---

0~ P tan [74(2P f 31 V)

In the Williams-Watts type, the ratio oH/oP was examined by numerical cal-

culations. The ratios of wu to cup for three types are shown in Table 2. Thus, the

Hamon approximation is applicable to the determination of op in Cole-Cole and

Williams-Watts types of relaxation with broad distributions of relaxation times.

but it is not valid for the Davidson-Cole type.

All computations were carried out with a Facom 230-75 Computer at the

Data Processing Centre, Kyoto University.

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TABLE 2

COMPARISON OF WH RELAXATION FREQUENCY EVALUATED BY THE HAMON APPROXIMATIONWITH Op

PEAK FREQUENCY=

Distribution wH/wp parameter ~

B cc DC ww

1.0 0.63 0.63 0.63 0.9 0.69 0.64 0.67 0.8 0.74 0.66 0.71 0.7 0.81 0.68 0.76 0.6 0.87 0.70 0.81 0.5 0.93 0.73 0.85 0.4 0.99 0.76 0.91 0.3 1.05 0.80 0.95 0.2 1.09 0.84 1.02 0.1 1.11 0.91 1.07

a The frequency at which the maximum of dielectric loss takes place.

SUMMARY

The accuracy of the Hamon approximation for evaluating the dielectric loss, the relaxation time, and the relaxation intensity from transient absorption current data was discussed with regard to dielectric relaxations of the Cole-Cole, David- son-Cole and Williams-Watts type. It was found that the Hamon approximation enables us to evaluate the relaxation intensity and the dielectric loss, at frequencies higher than the relaxation frequency, with good accuracy, regardless of the type of relaxation, but it introduces significant errors in calculations of the dielectric loss at lower frequencies and in the relaxation time, particularly in relaxations of the Davidson-Cole type and other types with a narrow distribution of relaxation times. The approximationis applicable, with fairly good accuracy, to the Cole-Cole type of relaxation with a broad distribution of relaxation times.

REFERENCES

1 B. V. HAMON, Proc. Inst. Elec. Engrs. (London), Monograph No. 27, 99 (1952) 151. 2 See for instance G. WILLIAMS, Polymer, 4 (1963) 27. 3 G. WILLIAMS AND D. C. WATTS, Trans. Faraday SOL, 66 (1970) 80. 4 G. WILLIAMS, D.C. WATTS,S.B.DEVANDA. M.NORTH, Trans.Furuday Soc.,67 (1971) 1323. 5 K. S. COLE AND R. H. COLE, J. Chem. Phys., 9 (1941) 341. 6 D. W. DAVIDSON AND R. H. COLE, J. Chem. Phys., 18 (1950) 1417. 7 H. FRGHLICH, Theory of Dielectrics, 2nd edn., Oxford University Press, London, 1958. 8 K. S. COLE AND R. H. COLE, J. Chem. Phys., 10 (1942) 98;

S. TAKAHASHI AND H. NISHINO, Denki Shikensho Zho, 14 (1950) 70. 9 S. TAKAHASHI, Denki Shikensho Zho, 15 (1951) 222.

10 N. KOIZLJMI AND Y. KITA, Bull. Inst. Chem. Res., Kyoto Univ., 50 (1972) 499.

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Remarks on the hamon approximation