Water Complex Dielectric and Polarization

Microwave introduction
The complex dielectric permittivity
Polarization
Dielectric constant and polarization
Dielectric spectroscopy

The complex dielectric permittivity

It has been suggested that a bimodal relaxation time expression is the most appropriate description of the dielectric properties of water [135].^b

(1)

where ε_r* is the complex permittivity, ε_S is the relative permittivity at low frequencies (static region), ε₂ 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 τ₂ 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 as hydrogen bonds are weakened and broken. τ₂is small (~1 ps [135] 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].

Variation of the dielectric parameters with temperature

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:

tan(delta) = loss factor/real permittivity

and the complex permittivity rearranged to give real permittivity and imaginary (the loss factor) parts:

The real part corresponds to the dielectric constant:

and the imaginary part corresponds to the loss factor (Lf):

Loss factor = (static permittivity - intermediate permittivity) x angular frequency x first relaxation time /(1 + angular frequency squared x first relaxation time squared] +(intermediate permittivity - optical permittivity) x angular frequency x second relaxation time /(1 + angular frequency squared x second relaxation time squared

As (ε_S - ε₂) >> (ε_S - ε_∞) the permittivity may be approximated to within the accuracy of current instrumentation by:

(2)

As τ_D >> τ₂ and (ε_S - ε₂) >> (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 ]

Polarization

The polarization (P) of a substance is its electric dipole moment density (see also). It varies with the applied field (E = E_maxe^-i^ωt) and the permittivity. It is given by the real part of the expression:

P = ε_r*ε₀

As E = E_max{cos(ωt) - i.sin(ω t)} and ε_r* = ε_r´ - i.Lf

P = E_max.ε₀(ε_r´ - i.Lf){cos(ωt) - i.sin(ω t)}

Therefore, taking only the real part:

P = E_max.ε₀{(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 ]

Footnotes

^a It has been shown that the different values for τ₂ correspond to different frequency ranges and the most appropriate relaxation time expression is trimodal [1247]. This analysis gives relaxation times τ_D, τ₂ and τ₃ at 25°C of 8.26 ps, 1.05 ps and 0.135 ps respectively; ε_S = 78.4, ε₂ = 5.85, ε₃ = 3.65, ε_∞ = 2.4 (compared with the bimodal relaxation times τ_D and τ₂ at 25°C of 8.21 ps and 0.392 ps respectively; ε_S = 78.4, ε₂ = 5.54, ε_∞ = 3.04) [1247]. The fast relaxation time was thought possibly associated with the free rotation of water molecules having broken hydrogen bonds. [Back]

Full dielectric spectrum, after ref. 1497

^b For use at higher frequencies up to 100 THz (that is, into the far infra-red) two extra terms, representing the intermolecular stretch (V_S) and intermolecular librations (V_L), may be added [1497]. [Back]