Microwave introduction
The complex dielectric permittivity
Polarization
Relative permittivity (dielectric constant) and polarization
Dielectric spectroscopy
It has been suggested that a bimodal relaxation time expression is the most appropriate description of the dielectric properties of water [135].^{ b, c}
where ε_{r}* is the complex permittivity, ε_{S} is the relative permittivity at low frequencies (static region), ε_{2} is the intermediate relative permittivity, ε_{∞ }is the relative permittivity at high frequencies (optical permittivity), ω is the angular frequency in radians.second^{-1}, τ_{D} and τ_{2} are relaxation times and i = . τ_{D} is relatively long (18 ps at 0 °C [135]), due primarily to the rotational relaxation within a hydrogen-bonded cluster, but reduces considerably with temperature (8 ps at 25 °C [2503]) as hydrogen bonds are weakened and broken. τ_{2 }is small (~1 ps [135], [2503] or 0.2 ps [343])^{ a} and less temperature dependent being determined primarily by the translational vibrations (near 200 cm^{-1}) within the hydrogen-bonded cluster [240].
Plotted opposite are equations derived for pure water over the range for -20 °C ~ +40 °C [683], extrapolated (dashed lines) to indicate trends; relaxation times are in ps. Further data has been published [1185].
Equation (1) may be simplified:
and the complex permittivity rearranged to give real permittivity and imaginary (the loss factor) parts:
The real part corresponds to the relative permittivity (dielectric constant):
and the imaginary part corresponds to the loss factor (Lf):
As (ε_{S} - ε_{2}) >> (ε_{S} - ε_{∞}) the permittivity may be approximated to within the accuracy of current instrumentation by:
As τ_{D} >> τ_{2} and (ε_{S} - ε_{2}) >> (ε_{S} - ε_{∞}) the permittivity may be approximated by:
which shows small deviations between about 100 - 1000 GHz which reduce with temperature increase. [Back to Top ]
The polarization (P) of a substance is its electric dipole moment density (see also). It varies with the applied field (E = E_{max}e^{-i}^{ωt}) and the permittivity. It is given by the real part of the expression:
P = E ε_{r}*ε_{0}
As E = E_{max}{cos(ωt) - i.sin(ωt)} and ε_{r}* = ε_{r}´ - i.Lf
P = E_{max}.ε_{0}(ε_{r}´ - i.Lf){cos(ωt) - i.sin(ω t)}
Therefore, taking only the real part:
P = E_{max}.ε_{0}{(ε_{r}´cos(ωt) - Lf sin(ωt)}
where ε_{r}´ varies with frequency as equation (2) above. This equation is equivalent to:
P = P_{max}.cos(ωt - δ)
where δ = atan(Lf/ε_{r}´) and P_{max} increases by a factor secant(δ). [Back to Top ]
^{ a} It has been shown that the different values for τ_{2} correspond to different frequency ranges and the most appropriate relaxation time expression is trimodal [1247]. This analysis gives relaxation times τ_{D}, τ_{2} and τ_{3} at 25 °C of 8.26 ps (19.3 GHz, corresponding to cooperative relaxation of long-range hydrogen-bond-mediated dipole-dipole interactions), 1.05 ps (150 GHz, possibly associated with dipole-dipole interactions due to the free rotation of water molecules having no more than one hydrogen bond) and 0.135 ps (1.18 THz, possibly associated with dipole-dipole interactions due to the free rotation of water molecules having no hydrogen bonds) respectively; ε_{S} = 78.4, ε_{2} = 5.85, ε_{3} = 3.65, ε_{∞} = 2.4 (compared with the bimodal relaxation times τ_{D} and τ_{2} at 25 °C of 8.21 ps (19.3 GHz, corresponding to cooperative relaxation of long-range hydrogen-bond-mediated dipole-dipole interactions) and 0.392 ps (406 GHz, possibly associated with dipole-dipole interactions due to the free rotation of water molecules having broken hydrogen bonds) respectively; ε_{S} = 78.4, ε_{2} = 5.54, ε_{∞} = 3.04) [1247]. A strong case has been proposed that the ionization of water can lead to the rearrangement of the water clusters and the values of both τ_{1} and τ_{2} [2423]. [Back]
^{ b} For use at higher frequencies up to 100 THz (that is, through the terahertz into the far infrared) two extra terms, representing the intermolecular stretch (V_{S}) and intermolecular librations (V_{L}), may be added [1497]. When the intermolecular stretching vibration is included, the following equation has been used [1563]
with the following values determined [1563]
A_{S}, THz^{2} | ω_{S}, THz | λ_{S}, THz | ε_{∞} | |
H_{2}O | 1386 | 33.3 | 33.9 | 2.34 |
D_{2}O | 1248 | 33.7 | 31.8 | 2.29 |
H_{2}^{18}O | 1184 | 31.1 | 26.7 | 2.28 |
[Back]
^{ c} A recent model shows improved behavior in the supercooled region for use in atmospheric cloud measurements [2262]. [Back]
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