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Dielectric Constant and Polarization

V Background information and definitions
V Polarization and polarizability
V Refractive index
link Water and microwaves
link Complex dielectric permittivity
link Dielectric spectroscopy

Background information and definitions

The electric dipole moment (μ) of a molecule is directed from the center of negative charge (-q) to the center of positive charge (+q) distance r away. The units are usually given in Debye (= 3.336 ˣ 10-30 C ˣ m).

 

μ = qr

 

In liquid water, molecules possess a distribution of dipole moments (range ~1.9 - 3.1 D) due to the variety of hydrogen bonded environments.

 

If two charges q1 and q2 are separated by distance r, the (Coulomb) potential energy is V (joule)

 

V = (q1 x q2)/(4 x pi x vacuum permittivity x r)

 

where ε0 is the permittivity of a vacuum (= 8.854 x 10-12 C2 J-1 m-1; the ability of a material to store electrostatic energy).

 

In a medium it is lower

 

where ε is the medium's permittivity.

V = (q1 x q2)/(4 x pi x medium permittivity x r)

 

The dielectric constant (εr) of the medium (also known as the relative permittivity) is defined as

 

Dielectric constant=medium permittivity/vacuum permittivity

 

and clearly approaches unity in the dilute gas state. In liquid water, it is proportional to the mean-square fluctuation in the total dipole moment. In liquid water, the dielectric constant is high and there is a linear correlation between it and the number of hydrogen bonds [239]. [Back to Top to top of page]

 

The dipole moment (μ) and the dielectric constant (ε) of liquid water are related by the relationship,

relationship

where ε0 , ε , g, μ , k, T, NA and Vmol are measured dielectric constant, the dielectric constant at a frequency sufficiently low for atomic and electronic polarization but sufficiently high for intermolecular relaxation processes, the parameter of local ordering degree related to the probability of the central water molecule instantaneously hydrogen bonding to surrounding molecules, the dipole moment in the vapor phase, the Boltzmann constant, the temperature in K, Avagadro's number and molar volume, respectively [2727].

Polarization and polarizability

The polarization (P) of a substance is its electric dipole moment density (see also). The charge density vector (D) is the sum of the effect of the applied field and the polarization.

D = ε0E + P

 

But as

 

D = εE

P = (εr - 1)ε0E

 

The relative permittivity (dielectric constant) (εr) is related to the molar polarization of the medium (Pm) using the Debye equation

 

(Dielectric constant -1)/(Dielectric constant +2) = mass density x molar polarization/molar mass

 

where ρ is the mass density (kg m-3), M is the molar mass (kg). At high relative permittivity (dielectric constant), such as water, the left hand side of the above equation approximates to unity and the molar polarization (calculated from equation(1) below = 181.5x10-6 m3 at 25 °C) should approximate to the molar volume (18.0685x10-6 m3 at 25 °C) but it clearly does not in the case of water. The molar polarization of the medium (Pm) is defined as

 

Molar polarization = (Avogadro number/(3 x vacuum permittivity)) x (polarizability +(dipole moment squared/(3 x Boltzman constant x temperature))            (1)

 

where α is the polarizability of the molecules, a which is the proportionality constant between the induced dipole moment μ* and the field strength E (μ* = αE), NA is the Avogadro number, Mk is the Boltzmann constant (=R/NA), T is the absolute temperature and μ is permanent dipole moment. Unfortunately, in line with many other anomalies of water, this equation is not a good predictor for the behavior of water, which shows a minimum molar polarization at about 15 °C. The term in (εr + 2) comes from the relationship between the local field (E') and the applied field (E). 

 

E' = (E/3)(εr + 2) 

 

The polarizability (α) may be given as the polarizability volume (α´) where

 

polarizability volume = polarizability/(4 x pi x vacuum permittivity)

 

The second term in equation (1) is due to the contribution from the permanent dipole moment, which is negligible when the medium is non-polar or when the frequency of the applied field is sufficiently high that the molecules do not have time to change orientation. In this case the Clausius-Mossotti equation holds (but again not for water):

 

(Dielectric constant -1)/(Dielectric constant +2) = mass density x Avogadro number x polarization/(3 x molar mass x vacuum permittivity)

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Refractive index of water, from [2573]

Refractive index of water, from [2573]

Refractive index

The refractive index (ηr) in the visible and ultraviolet [2573] is the ratio of the speed of light in a vacuum (c) to that in the medium (c´); ηr = c/c´. It is also related to the relative permittivity (εr), the absorption coefficient (α) and wavelength (λ) [177].

 

(refractive index)squared equals relative permittivity + (wavelength times absorption coefficient)/(4 times pi)

 

This reduces to approximately ε = ηr2 where ε is the relative permittivity at visible frequencies (400 THz - 800 THz, ηr ~ 1.34). ε represents the total polarizability (atomic + electronic) whereas ηr2 represents the electronic polarizability only. In most liquids, the atomic polarizability amounts only to about 10% of the electronic polarizability whereas in water ε ~ 2 ˣ ηr2 , an anomalously high value as the atomic and electronic polarizability are approximately equal [2884].

 

Broken spectrer

Broken spectre, By Brocken Inaglory, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=15799557

 

The change in refractive index with wavelength within the range of visible light gives rise to rainbows and Brocken spectres.

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Footnotes

a The polarizability describes the change in dipole moment in the presence of an electric field. [Back]

 

 

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This page was established in 2001 and last updated by Martin Chaplin on 16 October, 2017


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