METHODFOR DETERMINING DIELECTRIC RELAXATION T~MEB
Acknowledgment. The authors wish to thank Professor 1. Ragawa of Nagoya University for his contiming interest and encouragement, and also Pro-
4197
fessor L. Kotin of University of Illinois (Chicago) and Professor Y. Kobatake of Osaka University for critically reading our manuscript.
A Method for Determining Dielectric Relaxation Times
by Abhai Mansingh and Pradip Kumar Phusics Departmend, Allahabad University, Allahabad, India
(Received June 81,1966)
A graphical method has been proposed to solve Frohlich’s expression for complex dielectric constant at microwave frequencies in order to evaluate the minimum and maximum dielectric relaxation times r1 and T Z . Calculations have been made at three frequencies for ethyl bromide and butyl bromide a t 25” and dibenzyl ether at 20,40,and 60”. It has been found that the results are affected considerably by the inaccuracy of measurements and by the nonapplicability of Frohlich’s distribution function.
Introduction Dielectric relaxation is the lag in dipole orientation behind an ahernatling electric fie1d.l Bergmann, Roberti, and Smyth2have given a method for analyzing the dielectric relaxation of some substances in terms of two relaxation times, one for the molecular orientation and the other for the intramolecular orientation process. Recent experimental observations3have been made showing very clearly the existence of the two processes. When the dielectric relaxation occurs by two or more mechanisms giving more or less overlapping dispersion regions, the analysis of the corresponding dielectric constant and loss values and the calculation of the relaxation times are often difficult and approximate. Fong and Smyth4 have described a method for such cases in terms of the Debye theory. Frohlich6considered that a distribution in relaxation time might arise from a distribution of activation energies in a given system. This is consistent with the model that each rotating segment is surrounded by a different field of force, but the lengths of the relaxing segments are equal throughout the system. Frohlich6 assumed that each process obeyed the Arrhenius relation r = ro exp(v/kZ“), H to H v being the range of
+
activation energy, R being the minimum potential barrier. To explain the circular arc plot for a large number of molecules Cole and Colee empirically introduced a parameter a: in Debye’s expression for complex dielectric constant e*, and called a the distribution parameter for the relaxation time. Imtead of using a, Frohlich5 assumed minimum and maximum relaxation times, r1 and r2,and used a distribution function of the form f(T) =
AT
f(r) = 0
< T < 72) (71 > ra < r ) (TI
T,
(1)
where A = v / k T . (1) P. Debye, “Polar Molecules,” Chemical Catalog, New York, N. Y., 1929. (2) K.Bergmann, D.M. Roberti, and C. P. Smyth, J. Phys. Chem., 64, 665 (1960). (3) W. P. Purcell, K. Fish, and C. P. Smyth, J . Am. Chem. Soc., 82,6299 (1960). (4)F. K.Fong and C. P. Smyth, J. Phys. Chem., 67, 226 (1963). (5) H.Fr6hIich, “Theory of Dielectrics,” Oxford University Press, London, 1949,pp. 93-95. (6) K. S. Cole and R. H. Cole, J. Chem. Phys., 9, 341 (1941).
Volume 69, Number 18 December 1066
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ABHAIMANSINGH AND PRADIP KUMAR
With such a distribution function, he derived expressions for E‘ and e“, the dieIectric constant and the dielectric loss, respectively. Assuming Frohlich’s distribution function, H i g a ~ i ?gave , ~ a method for determining r1and 7 2 . It is observed that different relaxation mechanisms are not equally probable. Smaller values of T indicate more probable mechanisms. Matsumoto and Higasi,g therefore, modified the eq. 1as
f(7)
=0
(0< n
TI
-
e,
T,
Theory Taking into consideration the distribution of relaxation time, Frohlich’s6 expression for the complex dielectric constant is
where €0 and c, are the static and optical dielectric constants, respectively, r is the relaxation time, w is the angular frequency, f(7) is the distribution function for the relaxation time given by eq. 1, and j = 2/zri. Only the case where n = 1 will be considered. Using eq. 1,eq. 2 can be written as
€0
d‘ 3
€I
In some substances, Davidson and Cole’O showed that as the temperature of measurement is lowered, skewed arcs are obtained. These were beautifully explained by Matsumoto and Higasig by putting n = a/z, 1, 2/3, l/2, and ]/a. When the experimental values of Davidson-Cole parameter p are very near to the theo1, 2/8, 1/2, retical values obtained by putting n = and l/ a, the method described here may also be applied. For intermediate values of 0 it may be difficult to choose a proper value of n and evaluate the integral. The purpose of the present paper is to determine 71 and T~ using Frohlich’s equation and making no approximation. Since an algebraic solution was impossible, a graphical approach has been followed. The results are considerably affected by the inaccuracy of measurements.
E*
Dividing eq. 4b by eq. 4a and evaluating the integrals -
< a, < T < 72) (71 > 7 2 < 7)
f(r) = 1 / A P
and
-
e,
d7
1 T(1
E,
- jw.)
(3)
where 71 and T~ are the minimum and maximum relaxation times. Multiplying the numerator and denominator of the right-hand side of this equation by (1 j w ~ and ) equating the real and imaginary parts of both sides and using E* = e’ j t ” , the following equations are obtained
+
+
The Journal of Physical Chemistry
Introducing two variables, w1 = 1 / and ~ ~ w2 = and rearranging the terms one gets
1/72,
or
where
The factor €‘‘/e’ - E,, for any substance at a constant temperature and a given angular frequency w , is a constant. Now a graphical solution to eq. 5 is sought. Experimentally observed values of E’, e”, and em at a given temperature, and the angular frequency (a) are substituted into eq. 5. A curve of log wh vs. f (om)is traced by varying wk independently. A typical curve (dibenzyl ether) is shown in Figure la. An infinite number of lines parallel to the abscissa can be drawn to cut the curve at two points which correspond to equal values of f((Jk). For every such line the condition f(wJ = f(oJ is satisfied. The infinite number of solutions clearly show that observation a t one single frequency is not sufficient to determine the correct values of T~ and 7 2 . Therefore, another curve must be traced (Figure lb) with the help of observed data for a different value of w . To find the relaxation times 7 1 and T ~ two , curves between log w1 and log w2, as ob(7) K.. Higasi, “Dielectrio Relaxation and Molecular Structure,” Hakkaido University, Sapporo, Japan, 1961, Chapters VI and VlI. (8) K. Higasi, K. Bergmann, and C. P. Smyth, J. Phys. Chem., 64, 880 (1960). (9) A. Matsumoto and K. Higasi, J. Chem. Phys., 36, 1776 (1962). (10) D. W. Davidson and R. H. Cole, ibid., 12, 1484 (1951).
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METHOD FOR DETERMINING DIELECTRIC RELAXATION TIMES
4
UI
Figure 2. Plots of log W, and log w2. Curves a, b, and c here correspond to curves a, b, and c, respectively, of Figure 1.
Table I: Values of the Relaxation Times 71 and for Ethyl Bromide and Butyl Bromide Authors' values 71
Substances
Ethyl bromide at 25" n-Butyl bromide a t 25"
A B
tained from different lines parallel to the abscissa for the two curves a and b in Figure 1, are traced. The point of intersection of the two curves thus drawn, Figure 2, gives the relaxation times. If the accuracy of measurement is very good, observations a t two frequencies only will suffice and give good values of r1 and TZ. With anything less than very good accuracy observations a t more frequencies will be needed. Curves corresponding to different observation frequencies o may not intersect a t a common point; thus several values of 71 and 72 may be obtained. If the spread of the calculated values is not large, an average may be taken and better values of relaxation times r1 and r2 may be obtained. If the inaccuracies are large, the lines may not intersect or the spread in the values may be large. The relaxation times for a few substances have been determined by the above method and the results are compared with other methods. The data were easily available from the Princeton group.11J2 The results of the calculations are given in Tables I and 11. The points of intersection of the lines for observations a t wave lengths 10 and 3.22 cm., 10 and 1.27 cm., and
A B C
a
T12
x
l O l * sec. lola sec.
C Figure 1. Plots of f(wk) %rg. log wb for dibenzyl ether a t 60' for three wave lengths: (a) 1.25 cm.; (b) 3.22 cm.; and (c) 10.0 em.
x
TZ
Higasi's valuesa Tl
x
1012 sec.
72
1012
x sec.
0.5 1.2 1.8
16.6 11.0 7.3
1.8
8.0
8.5 3.4 3.0
10.1 20.2 28.2
3.1
24.6
See ref. 7.
3.22 and 1.27 cm. are given in Table IA, B, and C, respectively. Table IIA, B, and C lists the points of intersection of the lines for observations a t wave lengths 10 and 3.22 cm., 10 and 1.25 cm., and 3.22 and 1.25 cm., respectively.
Discussion I n the case of ethyl bromide and butyl bromide, the spread in the calculated values is very large and hence these values cannot be averaged to given correct values for r1 and rz. The large discrepancies in the values of r1 and 7 2 are apparently due to experimental inaccuracy. This may also be due to the nonapplicability of the Frohlich distribution function. If observations a t a larger number of frequencies are available, the average (11)W.M.Heston, Jr., E. J. Hennelly, and C. P. Smyth, J . Am. Chem. Soc., 70, 4093 (1948); H. L. Laquer and C. P. Smyth, ibid., 70, 4097. (1948); E. J. Hennelly, W. M. Heston, Jr., and C. P. Smyth, sbid., 70, 4102 (1948); F. H. Branin, Jr., and C. P. Smyth, J. Chern. Phys., 20, 1121 (1952). (12) D.M.Roberti, 0. F. Kalman, and C. P. Smyth, J . Am. Chem. SOC.,82, 3523 (1960).
Volume 69, Number 12 December 1966
ABHAIMANBINC-H AND PRADIP KUMAR
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Table I1 : Values of the Relaxation Times 11 and 12 for Dibenzyl Ether Bergmann’a valuea by 71
Substance
Dibenzyl ether
at zoo Dibenayl ether a t 40’ Dibenzyl ether &t 60’
A B C A B
x
Authors’ values 10‘0 ra x 1012
71
x
Bergmann’u values by his method”
Higasi’s method‘
1oia
I 2
x
1012
I1
x
1012
I*
x
lOlZ see.
sea.
seo.
ma.
880.
12.6 11.2 10.2
64.5 72.5 127.4
11.0
79.0
3.9
33.0
8.1
51.5 50.4
8.0
42.0
4.2
25.0
8.0
19.0
4.2
17.0
c
8.4 8.8
43.1
A B
6.5 6.7
31.8 30.9
C
6.8
29.8
seo.
See ref. 2,
of the calculated n and r2 would give a better efltirnation of the relaxation times. The accuracy of measurements for dibenzyl ether is definitely greater than that for the alkyl bromides. In the case of dibenzyl ether, it is seen that as the temperature increases, the spread in the values of 71 and r2 decreases. Assuming the accuracy of measurements to remain the same, it seems possible that Frohlich’s distribution function becomes more probable as the temperature increases. The methods of Bergmann, Roberti, and Smyth2 and Fong and Smyth4 are approximate. The former method is good for the cases where two widely separated relaxation times occur, while the latter one is
The Journal of Physical Chemistrg
applicable to the cases where the distribution function is completely unknown. Higa~i’s’~~ method may be called a semiempirical method. It is based on a comparison of the Cole-Cole arc plot and the pseudoelliptic plot. The results, therefore, depend on the empirical parameter a. Since the method suggested here depends on the solution of Frohlich’s equation without any approximations, it is expected to give better results, provided Frohlich’s distribution function is applicable.
Acknowledgments. Thanks are due to Professor Krishnaji for supervision and to the C.S.I.R. (India) for financial assistance.
