J. Phys. Chem. 1988, 92, 6149-6150
6149
Dielectric Measurements of Water in the Radio and Microwave Frequencies by Time Domain Reflectometry M. Merabet and T. K. Bose* Groupe de recherche sur les dielectriques, Departement de physique, UniversitC du Quebec ci Trois- RiviPres. C.P.500, Trois-RiviZres. Quebec, Canada G9A 5H7 (Received: September 4, 1987; In Final Form: April 14, 1988)
The time domain reflectometric method is used with success to measure the dielectric properties of water from 10 MHz to 8 GHz. It is shown that special precautions must be taken into account in order to determine the dielectric properties of a substance with high dielectric constant in the microwave region.
Introduction Most of the dielectric measurements of water in the literature in the radio frequency (RF) and microwave (MW) regions have been carried out by the frequency domain Recent developments of the time domain reflectometric technique (TDR) by Cole et aL6 have opened up great possibilities for the study of dielectric materials with large dielectric constant and dielectric losses. In the past, the transmission method by Gestblom and Elmgren' and the single-reflection method by Suggett8 have been applied successfully to study the dielectric properties of aqueous solutions. The transmission method was also applied by Gestblom and Elmgren in the case of aqueous electrolyte solutions. The purpose of this article is to present the results of the dielectric properties of water between 0 and 30 OC by using the total reflection time domain technique. To the best of our knowledge, water has never been measured successfully up to 8 GHz by using only the time domain technique. Since water is the major component in microemulsions, measurement of pure watter certainly ensures the possibility of studying the critical point phenomena in microemulsions by the time domain technique. TDR is fast and covers a large range of frequencies, and as such it is particularly suitable for use in the critical region. Our successful measurement of water will likely lead to the use of the TDR technique for the investigation of dielectric properties of oil/water microemulsions, which have not been studied in the high-frequency region to the same extent as water/oil microemulsions. Although the earlier TDR results were more q u a l i t a t i ~ e ~than *'~ quantitative, recent development^"-'^ in data acquisition, processing, and reduction of system errors have improved enormously the precision of the TDR technique. Basic Principle of TDR The basic TDR system consists of a fast-rising tunnel diode pulse generator, a wide-band sampling oscilloscope, and a minicomputer for data acquisition and analysis. A step voltage pulse produced by the tunnel diode is detected at the sampler and displayed on the oscilloscope (Figure 1). In time domain methods, the voltage pulse propagates in the coaxial line until it reaches the air-dielectric interface, where a part of the pulse is reflected Hasted, J. B. Aqueous Dielectrics; Chapman and Hall: London, 1973. Kaatze, U.; Uhlendorf, V. Z . Phys. Chem. (Munich)1981,1265, 151. Van Loon, R.; Finsy, R. Reu. Sci. Instrum. 1973, 44, 1204. Gestblom, B. J . Phys. E. 1982, IS, 87. (5) Cole, R. H. IEEE Trans. Instrum. Meas. 1983, IM-32, 42. (6) Cole, R. H.; Mashimo, S.;Winsor IV, P. J . Phys. Chem. 1980,84,786. (7) Gestblom, B.; Elmgren, H. Chem. Phys. Lett. 1982, 90, 412. (8) Suggett, A.; Clark, A. H. J . Solution Chem. 1976, 5, 1. (9) Fellner-Feldegg, H.; Barnett, E. F. J. Phys. Chem. 1970, 74, 1962. (10) Springett, B. E.; Bose, T. K. Can. J . Phys. 1974, 52, 2463. (11) Gans, W. L.; Andrews, J. R. Tech. Note 672, NBS, Boulder, CO, 1975. (12) Elliott, B. J. IEEE Trans. Instrum. Meas. 1976, IM-25, 376. (13) Cole, R. H. Annu. Reo. Phys. Chem. 1977, 28, 283. (14) Chahine, R.; Bose, T. K. Reu. Sci. Insrrum. 1983, 54, 1243. (1) (2) (3) (4)
0022-3654188 12092-6149$01.50/0 , I
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and the rest is transmitted into the body of the dielectric sample. It is possible to determine the dielectric properties of a substance by only analyzing the first refle~ti0n.l~As pointed out by Arcone,16 the single-reflection method is not very precise in the presence of Ohmic conductivity. However, the total reflection from a thin sample with an open or 5 0 4 termination is better suited for precision dielectric measurements over a wide frequency range.6-17 Transmission methods developed by Gestblom and collaborators18have been shown to give very precise values of the real and imaginary parts of the dielectric constant up to very high frequencies.
Experimental Method Although the experimental measurements in a TDR system are obtained in the time domain, the analysis of the results is simpler, more precise and broad band if they are converted to the frequency d~main.'~.~~ For a sample terminating dielectric, the complex dielectric constant e* is given by6vZo
where e * = e' - jt", e' and e'' are respectively the real and imaginary parts of e*, and Vo(s)and R(s) are respectively the Laplace transforms of the incident signal vo(t)and the reflected signal r(t) in the time domain. The functionf(z) is a high-frequency correction factor which for a perfect coaxial line is given by f(z) = z cot z = 1 - (1/3)z2 - (1/45)z4 ..., for JzI < A (2) where z = j o d t * 1 / 2 / cc, = 1 / ( 1 5 ~ C ~ )is' / ~the speed of light, C, and L, are respectively the geometric capacitance and inductance per unit length of the line, d is the length of the dielectric sample, and o is the angular frequency. For cell designs which deviate from coaxial line,f(z) may be written asf(z) = 1 - az2- bF, where a and b are empirical coefficients. The cell used for the complex dielectric constant measurements of water is a short section of a 2-mm 50-R coaxial line. The length of the sample is determined by the electrical length of the inner conductor of the cell. The mechanical length of the cell is zero, and the equivalent electrical length which is essentially due to fringing fields is 0.154 mm. The empirical coefficients a and b for the correction factorf(z) are obtained by calibrating the sample cell with methanol over the frequency range 100 MHz to 10 GHz. The a and b values obtained are respectively 0.35 and 0.02, showing very slight deviation from coaxiality. The dielectric parameters for methanol used in our calibration are to = 33.2 and em = 5.1 for the low- and high-frequency limits of the dielectric constant, and 7 = 55.3 ps for the relaxation time at 20 OC. The
+
(15) Suggett, A. J. Phys. E 1975,8, 327. Arcone, S . J . Phys. E 1986, 19, 1067.
(16) (17) (18) (19) (20)
Chahine, R.; Bose, T. K. J. Chem. Phys. 1980, 72, 808. Gestblom, B.; Noreland, E. J . Phys. Chem. 1977, 81, 782. Cole, R. H. J . Phys. Chem. 1975, 79, 1459. Cole, R. H. J. Phys. Chem. 1975, 79, 1469.
0 1988 American Chemical Society
Merabet and Bose
6150 The Journal of Physical Chemistry, Vol. 92, No. 21, 1988
TRANSIENT RECORD
SYNCHRONIZATION
I
OSCILLOSCOPE
Figure 1. Basic diagram of a TDR system. Somoler
Pulse
Coupler
ChOnne’ A
_____Vorioble delay
Dielectric Et Termination
Figure 3. Frequency dependence of the real and imaginary part of the complex permittivity of water at 20 ‘C.
Chonnel B -Tarminotion
‘-6Ffi Trigger
Auxiliary Circuit
Signol
MiniComputer
Figure 2. TDR dual-channel configuration.
dielectric parameters for methanol used are quite similar to the literature values and have been obtained from a TDR experiment with 3.5-mm-diameter coaxial air line and longer cell lengths. Although the experimental result was limited to few gigahertz, it was carried out with the series expansion of z cot z . Since the dielectric constant of water is 80 at 20 “C, it can be easily seen from the value of z that any meaningful information on the dielectric properties in the high-frequency region can only be possible for extremely small lengths. This condition essentially ensures that the measurement on water has to be carried out with zero mechanical length and hence deviation from the coaxial configuration. In a cylindrical waveguide the cutoff frequency2’ isf, (GHz) = 1.9 X 102/mt’/2, where m is the diameter of the outer conductor in millimeters and t is the dielectric constant of the substance filling the waveguide. For our cell design with 2-mm 0.d. the cutoff frequency for water is around 10 GHz. We are, of course, assuming that the cylindrical waveguide cutoff frequency is valid in the fringing field space. The experimental result seems to justify the validity of our assumption. Since the relaxation time of water is very fast, great precautions have to be taken to reduce the timing errors to the minimum. If the signal averager (in our case the minicomputer) is triggered by repetitive scanning sweeps of the sampling oscilloscope, individual traces will be in constant time relationship, but the record will be broadened by drift of the tunnel diode pulse relative to the scanning sweep, with resultant distortion which may vary from one recording to another. This can be avoided by using a part of the incident signal to trigger the minicomputer, making possible extended periods of signal averaging without loss of high-frequency information. A 10-dB (8-12-GHz) directional coupler is inserted between the tunnel diode and channel A of the sampler, where the dielectric is placed. The modified fraction of the incident signal taken from the directional coupler is delayed by a line of variable length and applied to channel B (Figure 2). The impulse at the ouput of channel B is then used to trigger the minicomputer. In this the signal at the output of channel A could be expanded to a lower scale to include only the desired portion of r ( t ) . Results and Discussion Dielectric measurements of water at various temperatures between 0 and 30 OC have been carried out with an open terminal TDR technique. (21) Mashimo, S.; Umehara, T.; Ota, T.; Kuwabara, S.; Shinyashiki, N.; Yagthara, S. I . Mol. Li9. 1987, 36, 135.
i 66
71
E‘
76
Figure 4. Plot of ue” as a function of e’. TABLE I: Dielectric Parameters of Water as a Function of Temperature ‘0
T, O C 0 5 10 15 20 25 30
7,
PS
expt
lit. (ref 1)
expt
lit. (ref 1)
89.02 85.82 84.24 81.98 80.50 79.20 11.16
88.3
17.71 14.42 12.48 10.51 9.27 8.33 7.93
17.9
84.1 80.4 76.8
12.6 9.3 7.2
Figure 3 shows the experimental plot of e’ and d’ of water as a function of log f at 20 “ C . The presence of a single relaxation time is well-represented by the Debye equation (3)
where 7 is the relaxation time. By equating the real parts on both sides of eq 3, we get €‘
=
€0
- W€”7
(4)
Now plotting t’ vs we” (Figure 4) would give a straight line of slope -7. The relaxation time 7 = 9.27 ps at 20 O C is determined from the slope. This way of plotting the results enables one to determine the relaxation time outside the frequency range of the system. Table I gives the static dielectric constant as well as the relaxation time of water for seven different temperatures. The agreement with the literature values1p2is very good. We can therefore conclude that it is possible to extend the powerful technique of time domain measurement to the determination of high-frequency dielectric properties of substances with high dielectric constant.
Acknowledgment. This work was supported by the National Sciences and Engineering Research Council of Canada and by the MinistBre de l’agriculture, Gouvernement du Quzbec. Registry No. H,O, 7732-18-5.
