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Water Complex Dielectric and Polarization

link Microwave introduction
V The complex dielectric permittivity
V Polarization
link Relative permittivity (dielectric constant) and polarization
link 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, c

complex permittivity = optical permittivity + ((static permittivity - intermediate permittivity)/(1 +  i times angular frequency x first relaxation time) + ((intermediate permittivity - optical permittivity)/(1 +  i times angular frequency x second relaxation time))                 (1)

 

where εr* is the complex permittivity, εS is the relative permittivity at low frequencies (static region), ε2 is the intermediate relative permittivity, εis the relative permittivity at high frequencies (optical permittivity), ω is the angular frequency in radians.second-1, τD and τ2 are relaxation times and i =  i. τD 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. τ2 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].

 

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:

simplifying the complex expression

 

gathering real and imaginary terms

tan(delta) = loss factor/real permittivity

 

 

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

 

complex permittivity = optical permittivity + ((static permittivity - intermediate permittivity)/(1 +  angular frequency squared x first relaxation time squared)) + ((intermediate permittivity - optical permittivity)/(1 +  angular frequency squared x second relaxation time squared))+i times (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

 

The real part corresponds to the relative permittivity (dielectric constant):

 

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

 

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 (εS - ε2) >> (εS - ε) the permittivity may be approximated to within the accuracy of current instrumentation by:

 

relative permittivity = optical permittivity + ((static permittivity - optical permittivity)/(1 +  angular frequency squared x relaxation time squared))                 (2)

 

As τD >> τ2 and (εS - ε2) >> (εS - ε) the permittivity may be approximated by: 

 

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

 

which shows small deviations between about 100 - 1000 GHz which reduce with temperature increase. [Back to Top to top of page]

Polarization

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

 

P = E εr0

 

As       E = Emax{cos(ωt) - i.sin(ωt)}        and     εr* = εr´ - i.Lf 

 

P = Emax0r´ - i.Lf){cos(ωt) - i.sin(ω t)}

 

Therefore, taking only the real part:

P = Emax0{(εr´cos(ωt) - Lf sin(ωt)}

 

where εr´ varies with frequency as equation (2) above. This equation is equivalent to:

 

P = Pmax.cos(ωt - δ)

 

where  δ = atan(Lfr´) and Pmax increases by a factor  secant(δ). [Back to Top to top of page]


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 τD, τ2 and τ3 at 25 °C of 8.26 ps (19.3 GHz, corresponding to cooperative relaxation of long range H-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;  εS = 78.4, ε2 = 5.85, ε3 = 3.65, ε = 2.4 (compared with the bimodal relaxation times τD and τ2 at 25 °C of 8.21 ps (19.3 GHz, corresponding to cooperative relaxation of long range H-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;  εS = 78.4, ε2 = 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 τ1 and τ2 [2423]. [Back]

Full dielectric spectrum, after ref. 1497

 

 

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 (VS) and intermolecular librations (VL), may be added [1497]. When the intermolecular stretching vibration is included, the following equation has been used [1563]

 

complex permittivity = optical permittivity + ((static permittivity - intermediate permittivity)/(1 +  i times angular frequency x first relaxation time) + ((intermediate permittivity - optical permittivity)/(1 +  i times angular frequency x second relaxation time)) + stretch term

with the following values determined [1563]

  AS, THz2 ωS, THz λS, THz ε
H2O 1386 33.3 33.9 2.34
D2O 1248 33.7 31.8 2.29
H218O 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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This page was established in 2001 and last updated by Martin Chaplin on 26 March, 2017


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