G. A. Brehm and W. H. Stockmayer
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Rapid Evaluation of Dielectric Relaxation Parameters from Time-Domain Reflection Data G. A. Brehm and W. H. Stockmayer” Department of Chemistry, Dartmouth College, Hanover, New Hampshire 03755 (Received January 8, 1973) Publication costs assisted by the National Science Foundation
A rapid approximate method is presented for obtaining the Davidson-Cole dielectric response parameters from time-domain reflection measurements. The method is restricted to systems of moderate polarity and negligible dc conductivity.
Time-domain reflectometry (TDR) has excited current i n t e r e ~ t l -as~ a convenient method for the study of relatively fast dielectric relaxation, corresponding roughly to the interval 107-1011 Hz in the frequency domain. Because of the quadratic relation between dielectric constant and refractive index, a calculation of the dielectric correlation function5 from the observed time dependence of the reflection coefficient is not simple, even for Debye dielecA method of obtaining the dielectric response function in the general case has been described by Suggett, Loeb, and c0workers,3~~ who perform a Fourier inversion of the reflection curve by applying a sampling of theorem of Shannon6 as adapted by Samulon.7 The computation is lengthy, inviting expensive data-acquisition equipment, and requires measurements of fairly high precision. For many purposes, it is quite sufficient to express the relaxation behavior of a dielectric material in terms of just two parameters, typically an average response time and a breadth parameter; the familiar Cole-Cole* and Davidson-Cole9 functions are two examples of this level of treatment. Recently van Gemert and deGraan10s1l have presented sample calculations of reflection curves for Cole-Cole and Davidson-Cole dielectrics, and in particular have stressed the limitations imposed by high dc conductivity. However, their results also do not lend themselves to easy evaluation of TDR measurements. In the present pap& we offer a simple approximate method for obtaining two dielectric relaxation parameters, essentially those of the Davidson-Cole function, from TDR data for systems of moderate polarity and negligible dc conductivity. In the frequency domain the reflection coefficient p * ( w ) is related to the complex dielectric constant E*(w) as follows t*(o) = [l
- P”(W)l2/[1 + P*(O)l’
(1)
Clearly, if the dependence of e* on frequency exactly obeyed a Davidson-Cole function the reflection coefficient could not do so; however, for moderate values of the ratio of static to high-frequency dielectric constants, E O / E , , the reflection coefficient could still be well approximated by a Davidson-Cole function, of course with adjusted parameters. This remark is the basis of our method. Observing also that the Davidson-Cole function is essentially empirical (though not without theoretical sense), we therefore write P*(W>
= pm
+ (Po - Pa)(l +
iW7)-‘
The Journal of Physical Chemistry, Vol. 77, No. 11, 1973
(2)
in which po and p,, are clearly the values of the reflection coefficient at zero and infinite frequency, respectively. We then assert that the dielectric constant can be approximately written as €*(a)E t, -k (€o - &)(I iWTo)-’ (3)
+
and that the parameters 70 and of the latter expression can be related to those, 7 and a , of the former through eq 1. Our procedure for doing this was the following. For various values of po, p,,, and a , the complex dielectric constant was evaluated as a function of UT by means of eq 1 and 2, and Cole-Cole plots were constructed. The two characteristic features of €*(LO)were taken to be the frequency Wm (in units of 7-1) for which the loss factor has its maximum value Em”, and the reduced magnitude tm”/(tO - e m ) of this maximum. These are related for Davidson-Cole dielectrics to the parameters 70 and p by the expressions12 WmTo = t a n $m 4m = n / ( 2 2p) (4a)
+
- E,)
sin p4m
(4b) Thus for each set of values of EO, E , , T, and a we can find related values of the Davidson-Cole parameters 70 and p. By starting with eq 2, we have ensured a simple transformation to the time domain. Thus, the time variation of the voltage step reflected from the surface of the dielectric is P m -k (Po - Pm)(t/T)‘’Y*(a, t/T) P(t) Em”/(to
= (COS$m)fl
in which Y * ( c x , ~ / T= )
( t / ~ ) ~‘’‘yo-1e-: -~ dy/l?(a)
(5)
is the incomplete y function.13 By means of eq 5 we can (1) H. Fellner-Feldegg,J. Phys. Chem., 73, 616.(1969). (2) H. Fellner-Feldegg and E. F. Barnett, J. Phys. Chem., 74, 1962 (1970). (3) A. Suggett, P. A. Mackness, M. J. Tait, H. W. Loeb, and G. M. Young, Nature (London), 228, 456 (1970). (4) H. W. Loeb. G. M. Young, P. A. Quickenden, and A. Suggett, Ber. der Bunsenges. Phys. Chem., 75,1155 (1971). (5) G. Williams, Chem. Rev., 72, 55 (1972). (6) C. Shannon, Proc. IRE, 37, 10 (1949). (7) H. A. Samulon, Proc. IRE. 39, 175 (1951). (8) K. S. Cole and R. H. Cole, J. Chem. Phys., 9 , 341 (1949). (9) D. W. Davidson and R. H. Cole, J. Chem. Phys., 18, 1417 (1951). (10) M. J. C. van Gemert, J. Phys. Chem., 75, 1323 (1971). (11) M. J. C. van Gemert and J. G. deGraan, Appi. Sci. Res., 26, 1 (1972). (12) N. G . McCrum, B. E. Read, and G. Williams, “Anelastic and Dielectric Effects in Polymeric Solids,” Wiley, New York, N. Y., 1967. (13) M. Abramowitz and J. A. Stegun, “Handbook of Mathematical Functions,” Dover Publications. New York, N. Y., 1968.
Rapid Evaluatiori of Dielectric Relaxation Parameters I
I
I
I
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I .o
I
0.4 0.8
0.3 0.6 n
B
0.2
0A
3 0.I
0.2
0.0
0.2
0.4 0.0
/ tZl3 Figure 1. Curves for the determination of the frequency of maximum dielectric loss factor, a m , from time-domain reflection measurements. From top to bottom, the curves correspond to the following values of the ratio EO/E, : 1, 2, 3, and 4.
0.2
0.4
tllS
calculate reflection curves for various chosen values of the parameters and then attempt to characterize these by their appropriate principal features. In addition t o the initial and final reflection coefficients p , and po, we choose to designate the times t i 1 3 and t 2 / 3 a t which p ( t ) has fallen one-third and two-thirds, respectively, of the way from its initial to its final value. The practical results of the calculations outlined above are shown in Figures 1 and 2. We have preferred to replace the time constant T O of eq 3 by the circular frequency of maximum loss, am,as given by eq 4b. Thus, for various values of the dielectric constant ratio Q / E , , we plot the dimensionless quantities w m t l l 3 and p against the ratio t 1 / 3 / t 2 / 3 . From these graphs, therefore, it is possible from a measured reflection directly to ob.tain Wm and curve in the time domain. Although in principle both po and p,, (and thence eo and em,) can be read from such
tll3 / t2l3 Figure 2. Curves for the determination of the breadth parameter from time-domain reflection measurements. From top to boitom, the curves correspond to the following values of the ratilo C O / E ~ :4 , 3, 2, and 1.
curves, in practice it is usually advisable to get eo by separate low-frequency bridge measurements. A time-marker system4 aids in obtaining reproducible values of p m . We have used Figures 1 and 2 routinely for rapid evaluation of the reflection curves from numerous polymer solutions, finding the method quite adequate for the precision of our measurements. I t is not intended as a replacement for more precise methods.4
Acknowledgments. This investigation was supported in part by a National Institutes of Health Fellowship (No. F 0 1 GM43303-03) to G. A. B. from the General Medical Sciences division, and in part by the National Science Foundation under Grant No. GP-30343X. The authors thank Dr. F. I. Mopsik of the National Bureau of Standards for valuable criticisms and suggestions.
The Journal of Physical Chemistry, Vol. 77,No. 7 1 , 7973
