A. Dielectric Permittivity of Free Water as a Function of Frequency and Temperature
By Guy Serbin
The complex dielectric constant of water as a function of frequency f may be described using
the Cole-Cole [28] model [22, 29]:
ε *H 2O ( f ,T ) = ε ∞ +
ε s (T) − ε ∞
⎛ f ⎞
⎟⎟
1 + j ⎜⎜
⎝ f rel (T) ⎠
1− β
−
jσ dc ( T )
2πfε 0
(4)
where εs(T) is the static dielectric constant of water at a temperature in degrees Celsius T(°C),
ε∞ the dielectric constant at infinite frequency and equals 4.25 for the 0-60°C temperature range,
frel(T) the relaxation frequency of water at a given T(°C) in Hz, β=0.0125 and is a factor which
accounts for the possible spread in relaxation frequencies, σdc(T) the DC conductivity of the
medium at a given temperature and ε0 the permittivity of free space which is 8.854 · 10-12
Farads/m. εs(T), frel (T) and σdc(T) may be calculated via [16, 30, 31]:
εs[T(°C)] = 87.74 – 0.4008 T(°C) + 9.398·10-4 T(°C)2 –1.410·10-6 T(°C)3
(5)
frel[T(°C)] = (1.1109·10-10 – 3.824·10-12 T(°C) + 6.938·10-14 T(°C)2 – 5.096·10-16 T(°C)3)-1
(Hz)
(6)
σdc[T(°C)] = σdc(25°C) exp [-ΔT (2.033·10-2 + 1.266·10-4 ΔT + 2.464·10-4 ΔT2)]
≈ σdc(25°C) [1 - 0.02 ΔT]
(dS/m)
(7)
where ΔT = [25- T(°C)], and σdc(25°C) the DC conductivity of the medium at 25°C.
In accordance with the Debye [21] model, the relaxation frequency (frel) is dependent on the
viscosity of the polar liquid (among other things), such that more viscous liquids relax at lower
frequencies than less viscous ones. The viscosity of a liquid is temperature dependent, such that
as the temperature of the liquid increases the viscosity will decrease as the molecules are more
energetic.
Rotational relaxation effects for all temperatures are manifested with increasing
frequency, and peak at the relaxation frequency, characterized by a sharp decrease in ε'
accompanied by an increase in ε".
This characteristic behavior is due to the decrease in
polarizability of the medium (the inversion of the electromagnetic field becomes so rapid that
time interval is insufficient for water molecule to align themselves with the induced field) and an
associated increase in rotational losses. In summary, the relaxation frequency is the frequency at
which ε" attains a maximum value, above that frequency both ε' and ε" decrease with increasing
frequency.
B. References
[16]
D. Or and J. M. Wraith, “Temperature effects on soil bulk dielectric permittivity
measured by time domain reflectometry: A physical model,” Water Resources Research,
vol. 35, pp. 371-383, 1999.
[21]
P. Debye, Polar Molecules. Dover, Mineola, N.Y., 1929.
[22]
J. B. Hasted, Aqueous Dielectrics. London: Chapman and Hall, 1973.
[28]
K. S. Cole and H. R. Cole, J. Chim. Phys., vol. 9, pp. 341, 1941.
[29]
T. J. Heimovaara, “Frequency Domain Analysis of Time Domain Reflectometry
Waveforms 1. Measurement of the Complex Dielectric Permittivity,” Water Resources
Research, vol. 30, pp. 189-199, 1994.
[30]
A. Stogryn, “Equations for calculating the dielectric constant of saline water,” IEEE
Transactions on Microwave Theory and Techniques, vol. MIT-19, pp. 733-736, 1971.
[31]
F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing: Active and
Passive, vol. III: From Theory to Applications. Dedham, MA: Artech House, Inc., 1986.