Evaluation of
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Dielectric Behavior by TDS
Appendix A The integral correction I for finite rise time Tr in eq 20 is
I = (2c/d) JTrdt’ 0
[
We assume that R(t’) and R ( t t’ in the interval 0 < t’ < Tr:
NOTE ADDED IN PROOF: After the manuscript went to the printer, a referee has pointed out to me a difficulty in the expansion of e x p ( - ~ c * ’/*S/C)
- t’) both vary linearly with
R(t’) = R ( T r ) t ’ / T r
R(t - t’) = R(t) - t’iz(t) which gives
-
given by the last of eq 34 if s2 +(s) does not remain finite in the limit s m, as in the cases of skewed arc or circular arc relaxation functions. In such cases, the expansion of the exponential should not be made, but the approximation to given by the second of eq 34 can be used in the exponential. The predicted reflection starting a t time T , which permits an estimate of T , has been confirmed by explicit calculations and observations. References a n d Notes (1) For a review, see M. J. C. van Gemert, Philips Res. Rep., 28, 530
For reflected pulses R ( t ) of the form shown in Figure 2c, the term ljTrR(t)/R(t)is much less than unity for times t > Tr - T,, and to a sufficient approximation
(1973). (2)H. Fellner-Feldegg, J. Phys. Chem., 76, 21 12 (1972). (3)A. H. Clark, P. A. Quickenden, and A. Suggett, J. Chem. SOC.,Faraday Trans. 2,11, 1847(1974). (4) M. J. C. van Gemert, J. Chem. Phys., 60,3963(1974). (5) A. Suggett in “Dlelectric and Related Molecular Processes”, Vol. I, Chemical Society, London, 1972,p 100. (6)M. F. lskander and S. S. Stuchly, I€€€ Trans. lnsfrum. Meas., IM-21,
which is the result used to obtain eq 21. An alternative approximation is to evaluate the integral I by the trapezoidal rule using values of the integrand at t’ - 0 and t’ = Tr, which gives the same result eq A3.
425 (1972). (7)R. H. Cole, J. Phys. Chem., 78, 1440 (1974). (8)A. Lebrun, Cah. Phys., 60, 11 (1955);Ann. Phys., I O , 16(1955). (9)D. W. Davidson and R. H. Cole, J. Chem. Phys., 19, 1484 (1951). (IO) Convenient numerical tables of Q(t) for /3 in intervals of 0.02from 0.98 to 0.30 have been given by N. Koizumi and Y. Kita, Bull. lnst. Chem. Res., Kyoto Univ., 50, 499 (1972).
Evaluation of Dielectric Behavior by Time Domain Spectroscopy. II. Complex Permittivity Robert H. Cole Chemistry Department, Brown University, Providence, Rhode lsland 029 12 (Received February 2 1, 1975) Publication costs assisted by the Materials Science Program, Brown University, with support from the National Science Foundation
Simple explicit formulas are derived for evaluation of permittivity t * ( ~ of) a dielectric sample in a coaxial line from Fourier transforms of incident and reflected voltage pulses. These take exact account of finite reflected wave amplitude and provide good approximations to propagation effects for wavelengths greater than one-sixth the sample length. Use of the sample as termination of the line is shown to have several advantages over the more common method of inserting it in a matched line. Simple numerical and analytical procedures for evaluation of the Fourier transforms are given, together with a discussion of errors.
I. Introduction A variety of methods has been developed1 for evaluating the steady state permittivity e* from observations of the voltage pulses transmitted through or reflected from a dielectric sample in a coaxial line as a function of time after an incident pulse arrives at its front surface. The analyses have usually been based on explicit solutions for frequency components of the pulse wave forms as obtained by Laplace transformation of the basic propagation equations and current-voltage relation characterizing the dielectric. The scattering and transmission coefficients so derived
represent the combined effects of the relation of observed voltages to the sample voltage and current at the front surface, propagation effects in the sample, and boundary conditions at the back surface. The usual expressions are better suited for calculation of the expected behavior given the dielectric properties than for evaluation of permittivity from observed voltage waveforms, as they involve reflection coefficients p* = (t*1/2 1)/(e*l/* 1) and propagation functions exp(-iwde*l/2/c), where d is the sample length and c = 0.300 mm psec-l the speed of propagation in vacuo. The result is that explicit
+
The Journal of Physical Chemistry, Vol. 79, No. 14, 1975
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solutions for t* cannot be obtained even for the simple case of Debye relaxation, and the effects of the several complicating factors are so to speak scrambled together. In the preceding companion paper1 (hereafter referred to as part I), a quite different approach was developed of solving the basic propagation equations without recourse to Laplace transformations by expanding the voltage and current in the dielectric in Taylor series as a function of sample length and voltage-current relations at the surface. In first order, these results confirm Fellner-Feldegg’s “thin sample” result2 and show simple relations of observed reflection signal to the dielectric response function in this limit of small sample length and reflected pulse amplitude. In the second approximation, one finds that the principal difference except a t short times is the consequence of the fact that the voltage at front surface of the dielectric sample is the sum of incident and reflected pulse voltages, not just the incident voltage as assumed in the first-order approximation, and that the resultant distortion of the simple relations is significant at all times for which dielectric response to the incident pulse is incomplete. Although the second analysis provides simple approximate corrections which give results of useful accuracy for considerably greater sample lengths and reflection signals, it too becomes inaccurate as these increase and no account can be simply taken of effects of finite speed of propagation in the sample. In this paper, we formulate the problem in terms of frequency components of voltage and current at the sample surfaces as related to dielectric properties and the boundary conditions. This leads quite directly to convenient relations which take exact account of finite reflection signal amplitude and approximate propagation effects by a series expansion in powers of (ud/c)%*. From these relations, simple explicit solutions for t* in terms of incident and reflected voltages are obtained which are correct to terms of ) ~ (wd/c)6 if necessary and give results of order ( ~ d / cand useful accuracy up to frequencies for which the sample length is an appreciable fraction of the wavelength in the sample. Two experimental arrangements are considered: insertion of a dielectric sample in a coaxial line terminated in its characteristic conductance, and termination of the line by a finite length of sample. The latter is shown to have advantages of larger reflected signals for comparable accuracy, simpler analysis, and simpler cell designs. In both cases, there are no intrinsic low-frequency limits and evaluation of properties of dielectrics with appreciable ohmic conductance is possible. 11. Basic Analysis
We cclnsider a section of uniform transmission h e of length d filled with dielectric of complex relative permitti.vity 6%. Regarded as a symmetrical four terminal network, this section has an input admittance yin given by3
where yo is the open circuit admittance of the section, 2, its short circuit impedance, and yd the input admittance of the line or network used to terminate the section. If the geometric capacitance and inductance per unit length of line are C, and L,, then from transmission line theory yo and 2, are given by3 The Journal of Physical Chemistry, Vol. 79, No. 14, 1975
Robert H. Cole
y o = iwCcdE*
(‘”,” ”>
Z, = iwLCd(---) tanh x
where x = iw(LcCc)1/2t*1/2d= i(wd/c)t*lI2, as the speed of propagation c in vacuo is c = (LcCc)-1’2. (In these expressions for 2, and x, a nonmagnetic sample with relative permeability p* = 1is assumed.) If the component of incident pulse voltage Vo(t) of frequency w/2* is denoted by uo(iw), i.e., uo(io) = LVo(t) where L is the Laplace transform, and the component of reflected pulse voltage - R ( t ) a t the input of the dielectric section by -r(iw), the input voltage and current are given r(iw)], by u(iw) = uo(iw) - r(iw) and i(iw) = G,[V,(iw) where G, = (C,/L,)1/2 is the characteristic conductance of the coaxial line, assumed loss free, connecting the sample section to the pulse generator. The input admittance of the dielectric section and termination is then related to uo and r by
+
Combining eq 1,2, and 3 gives on rearrangement and using GJC, = c
(x coth x + i w L c y d d ) (4) & (5) + In this equation, the relation of voltages uo and r, calculable from observed incident and reflected pulses, to the input voltage and current of the sample is accounted for by the ratio ( u g r)/(uo - r ) . The result is not an explicit solution for t*, as € * appears i n ’ t h e argument x = i(wd/ ~ ) t * l /but ~ , the key to approximate solutions is that x coth x can be expanded as a series in powers of x 2 valid for 1x1
> 1/X.
+
IV. Examples of Use of t h e Analysis The measurement system and cell used to obtain the results presented here are described in part I. Numerical evaluations of Fourier transforms were made using the Brown University IBM 360/70 computer and a Fortran program written by T. G. Copeland. Results from a 6.0-mm sample of 1-butanol at 24’ in a matched 50-ohm line are shown as dispersion 6’ and absorption E’’ curves against logarithm of frequency in Figure 2, and as complex plane plots in Figure 3. Open circles connected by the dashed curves are values from the thin sample formula e* - 1 = (2c/d)r/iwuo; the filled circles connected by solid curves result from use of eq 8-10. Although the results from the thin sample approximation can be fitted below 1 GHz by a semicircle and hence a Debye relaxation function, the high-frequency limit a t e l = 1.5 and fre-
quency fc = 0.253 GHz of maximum E” are both much too small. The corrected values are fitted to 3 GHz by a Debye function with €1 = 3.0 and fc = 0.328 GHz (relaxation time 7 = 484 psec), in good agreement with other steady state and TDS results. Above 3 GHz, the corrected values show indications of a higher frequency relaxation process, but are increasingly unreliable because of timing errors and approximations in evaluations of the transforms. Results from the sample termination method have been given for 1-propanol in a previous communication;ll another example for a 11.3-mm sample of 1-octanol a t 24’ is shown in the complex plane locus of Figure 4. The indicated Debye relaxation at frequencies below 1 GHz has parameters €1 = 1.4 and fc = 0.105 GHz from the thin sample approximation, corrected to t l = 2.5 and fc = 0.123 GHz by eq 14-16. The corrected locus for 1-octanol gives definite evidence a t frequencies above 0.6 GHz of a second faster relaxation process. A convenient analysis to show this more clearly is , shown in Figure 5 . For a single by a plot of E’ against f ~ ”as Debye relaxation the relation of E’ to f t ” is a straight line of shape l/fC,l2 with transition to a second straight line of slope l/f; if a second Debye process with higher relaxation frequency f;is also present. Within their accuracy, the data have this behavior with fc‘ = 2.2 GHz. For two Debye relaxations described by E * - E,
= (EO
1
-
+
~
E,) W
+
T
€1
~ 1
-
+
E, iwT2
The Journal of Physical Chemistry, Vol. 79,
No. 14, 7975
1474
Robert H. Cole
the relaxation times 71 and 7 2 are related to the indicated values T~ and T b from f c and f d by
+
For Tb 1, which can be taken as an approximate limit beyond which the formulas should be used with increasing caution. (We may note that the first zero of x coth x for E* real occurs a t 1x1 = a/2
The Journal of Physical Chemlstry, Vol. 79, No. 14, 1975
= 1.57, the condition for quarter wave resonance of the dielectric sample.) This limit corresponds to a frequency f = c/2adl~l1/2, which for a sample with d = 3 mm, I tl = 3.2 is 8.8 GHz. At higher frequencies, the formulas given here may still be useful, as by providing starting values of E* for numerical solutions of eq 6 or 12 by a Newton-Raphson or other iterative meth0ds.l’ Our examples of the use of TDR, and most of those reported to date, have been for quite strongly polar substances with relatively large values of static permittivity EO falling to much smaller values in the frequency or time range of the measurements. The evaluation of smaller changes, as for polar solutes at low concentrations, will present greater difficulties, but we believe that the present methods a t least considerably extend the range of possibilities. Finally, it should be remarked that most of the discussion has been for nonconducting dielectrics, although the possibilities for study of samples with finite conductance have been considered, and it has also been assumed that the system of interest is nonmagnetic. If the relative permeability p* is appreciably different from one, our formulas for e* are valid only a t relatively low frequencies. This is because the propagation function x is given by x = i ( w d / ~ ) ( t * p * ) land / ~ the line inductance by L , = iwLcdp* ((tanh x ) / x ) . Evaluation of E* if 1x1 is not much less than one then requires a knowledge of p* and vice versa, and simultaneous evaluation of the two requires further information, as .obtained by the simultaneous reflection-transmission method of Nicolson and Ross18 for example. An alternative is to measure the reflection from a sample terminated in a short circuit, which together with the open circuit measurement can be analyzed by our methods to give both E* and u*.
Acknowledgments. This work was supported by the Brown University Materials Science Program and the National Science Foundation. I thank Drs. T. A. C. M. Claasen and M. J. C. van Gemert for a copy of their paper16 prior to publication, and Dr. A. M. Nicolson for helpful discussion. References and Notes For reviews, see ref 1 and 5 of part I (preceding paper in this issue). H. Fellner-Feldegg,J. Phys. Chem., 76, 21 16 (1972). See, for example, W. C. Johnson, “Transmission Lines and Networks”, McGraw-Hill, New York, N.Y., 1950. From specifications for General Radio Type 900-WO precision open circuit termination, General Radio Catalog U, 1970. H. Levine and C. Papas, J. Appl. Phys., 22, 29 (1951). E. 0. Tuck, Math. Comput., 21, 239 (1967). C. Shannon, Proc. lnst. Radio Eng., 37, 10 (1949). H. A. Samulon, Proc. lnst. RadioEng., 39, 175 (1951). A. M. Nicolson, Electron. Lett., 9, 317 (1973). A. M. Nicolson. WESCON Technical Paper, No. 13, 22 (1969). See also ref 18. R. H. Cole, J. Phys. Chem., 79, 93 (1975) R. H. Cole, J. Chem. Phys., 23, 493 (1955). A. Lebrun, Cah. Phys., 60, 11 (1955); Ann. Phys., I O , 16 (1955). See, for example, P. Girard and P. Abadie, Trans. Faraday SOC., 42A, 40 (1946). D. W. Davidson and R. H. Cole, J. Chem. Phys., 19, 1484 (1951). T. A. C. M. Claasen and M. J. C. van Gemert, submitted for publication in J. Chem. Phys. S. 0 . Dev, A. M. North, and R. A. Pethrick, Adv. Mol. Relax Processes, 4, 159 (1972). A. M. Nicolson and G. F. Ross, E € € Trans. lnstrum. Meas., IM-19, 377 (1970).
