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M. E. BAIRD
Relationships between the Two Types of Frequency-Temperature Representation of Dielectric Relaxation Data by M. E. Baird Department of Applied Physics, Institute of Science and Techmlogy, University of Wales, Cardif, Wales Accepted and Transmitted by The Faraday Society (June 6,1867)
Relationships are derived between the plots of log (frequency of maximum dielectric loss) a t a given temperature TOK against 1/T and of log (frequency) against the reciprocal of absolute temperature (T,) of maximum loss for materials conforming to a Fuoss-Kirkwood distribution of relaxation times. The theory is applied to experimental data for polyoxymethylene where the width of the distribution of relaxation times changed comparatively rapidly as the temperature was varied and a considerable discrepancy between the loci occurred. Good agreement between calculated and observed differences in the loci was obtained. The activation energy should be obtained from the slope of the plot of log (frequency of maximum loss) against 1/T since the slope of the other plot may be considerably in error.
When a material conforms with the Arrhenius relation r = K1 exp(Q/RT), where K1 is a constant, R is the gas constant per mole, T is the temperature on the absolute Kelvin scale, and r is the most probable relaxation time, the activation energy Q is usually obtained from the slope of the curve of log (frequency of maximum dielectric loss), log fm, at a given temperature T, against 1/T. However, measurements are often made as a function of temperature at constant frequency, f, and it is of interest to derive relationships between the loci of log fm against 1/T and log f against 1/Tm, where T , is the temperature of maximum loss at a given frequency. Many materials conform approximately to a Cole-Cole1 or Fuoss-Kirkwood2 distribution of relaxation times. These distributions are reasonably similar3 and so relationships derived for a Fuoss-Kirkwood distribution should hold for a wide variety of materials. Relationships are derived in this paper, and although an understanding of their physical significance is limited by the empirical nature of the Fuoss-Kirkwood distribution, the results of the analysis are considered to have some use and show that considerable discrepancies may occur between the loci.
value of E” occurs at an angular frequency wm4 given by W,T equal to unity. When the temperature is varied at constant w , the maximum value of E” would occur for a relaxation time T~ given by a r m equal to unity if eo - E , and /? were independent of temperature. In either case the maximum value of E” is given by e l t , = P ( e 0 - 4 / 2 . However, both EO - ern and may vary with temperature. For dipolar relaxation, eo - e, will show anomalous behavior near a transition, but otherwise for a liquid or disordered solids eo - E , is approximately equal to Kz/T, where K2 is another constant. Materials Conforming to the Arrhenius Relation with Both p and eo - E , Being Functions of Temperature. Differentiating with respect to temperature gives the condition for maximum e” as (wrm)2@Tm=
where
Theory For a Fuoss-Kirkwood2 distribution, the loss factor d l (imaginary part of complex dielectric constant) is given by
where w is the angular frequency, eo - E , is the dielectric constant increment, and p is the parameter determining the width of the distribution (0 < /3 I 1 and as /3 increases the distribution becomes narrower). When w is varied at constant temperature, the maximum The Journal of Physieal Chemistry
d In
A=
[
(EO
d(:)
-
E.,,)
ITm
(1) K.S.Cole and R. H. Cole., J. Chem. Pfiys.,9,341,(1941). (2) R. M. Fuoss and J. G. Kirkwood, J . Amer. Chem. Soc., 6 3 , 385, (1941). (3) W. Kauzmann, Rev. Mod. Phys., 14,12,(1942). (4) C. J. F.Bottcher, “The Theory of Electric Polarization,” Elsevier Publishing Co., Amsterdam, The Netherlands, 1952,Chapter 10. (6) H. Frohlich, “Theory of Dielectrics,” Clarendon Press, Oxford, England, 1958,Chapter 2.
1463
RELATIONSHIPS BETWEEN LOCIFOR DIELECTRIC RELAXATION BL 5 -
4 ! i
!a-
E $ 2 -
8
cl
1 \
0 -
4.0
4.5
',
5.0
1000 X 1 / T .
Figure 1. Comparison of loci for dielectric relaxation of polyoxymethylene: curve 1, observed log f m against 1 / T plot; curve 2, observed logf against 1/Tm plot; curve 3, calculated logf against l / T m plot.
Distributions usually become narrower as the temperature is raised and d@/dTis then positive. If wln is found at a particular temperature, To (and relaxation time 70) and w is held at this value while T is varied, the value of the relaxation time, T,, giving maximum E." is now different from 70, so that Tm and To are also different. Simple mathematical manipulation gives
When p is independent of temperature (fixed distribution of relaxation times) and eo - ern = KZ/T, A = T, and eq 2 reduces to (WTm)2'
=
(EL +
l)/(%
- 1)-
1
+2R T, @Q
when RT,/@Q
