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 infra-red) 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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