72
2 The Interaction of Electromagnetic Waves with Water
The first approach follows Debye’s idea of the molecular dipole reorientation
mechanism [5, 30, 31]. The starting assumption here is that the dielectric constant
arises from the ordering of polar H 2 O molecules in an external field E, which acts
against thermal disordering. Assuming that water molecules are rigid dipoles, μ 0 ,
and that the applied E and the locally acting E
fields are equivalent, one obtains
ε(0) − ε ∞ =
nμ
2
0
3 0 k B T
,
(2.48)
which is Debye’s equation for the static dielectric constant [5]. Equation (2.48) leads
to a static dielectric constant which is much smaller than the experimental value. For
example, at μ 0 =1.85 D and room temperature it gives (0) = 13.
A more successful calculation of the local field in polar liquids was made by
Onsager [32]. He treated the molecule as a real cavity in a statistical continuum of
the uniform dielectric constant equal to that of the liquid in bulk. On the basis of
this model, the electrostatic theory of continuous media leads at once to a simple
expression for the local field and the average torque effective in orienting a dipole
molecule relative to the field E. The molecular dipole moment in Onsager’s approach
contains a permanent part and a part induced by the local field E
, and derived the
following formula:
(ε(0) − ε ∞ )(2ε(0) + ε ∞ )
ε(0)(ε ∞ + 2)
2
=
nμ
2
0
9ε 0 k B T
,
(2.49)
which at (0) ∞ gives
ε(0) − ε ∞ =
nμ
2
0
3ε 0 k B T
(ε ∞ + 2)
2
6
.
(2.50)
For room temperature, μ 0 =1.85 D, and ∞ ≈ 2, (2.50) gives the value (0) ≈ 30,
which is higher than those obtained by Eq. 2.49, but still several times lower than the
experimental one.
Kirkwood [33] suggested an extension of Onsager’s theory and assumed that
Bernal–Fowler tetrahedrally coordinated molecule (see Sect. 1.2) orients in the
dielectric continuum together with its first coordination shell. The field of the center dipole μ 0 causes a favored average orientation of neighboring dipoles and thus
an increased moment μ = g · μ 0 of the tetrahedral group of five H 2 O molecules. If
we assume a modified Bernal–Fowler structure for water and introduce a g-factor of
about 2.7, the experimental value, (0) ≈ 78, of the dielectric constant of liquid water
at 25
◦ C is obtained from (2.50) with a dipole moment in the liquid 26% greater than
that in the vapor.
Although the theory described above reduces the problem relating to the dielectric
polarization of polar liquids to the problem of the calculation of the g-factor and the
sum of the dipole moments of overpolarized molecules in its immediate environment
in water, there are still several points which do not allow one to fully accept this theory:
2 The Interaction of Electromagnetic Waves with Water
The first approach follows Debye’s idea of the molecular dipole reorientation
mechanism [5, 30, 31]. The starting assumption here is that the dielectric constant
arises from the ordering of polar H 2 O molecules in an external field E, which acts
against thermal disordering. Assuming that water molecules are rigid dipoles, μ 0 ,
and that the applied E and the locally acting E
fields are equivalent, one obtains
ε(0) − ε ∞ =
nμ
2
0
3 0 k B T
,
(2.48)
which is Debye’s equation for the static dielectric constant [5]. Equation (2.48) leads
to a static dielectric constant which is much smaller than the experimental value. For
example, at μ 0 =1.85 D and room temperature it gives (0) = 13.
A more successful calculation of the local field in polar liquids was made by
Onsager [32]. He treated the molecule as a real cavity in a statistical continuum of
the uniform dielectric constant equal to that of the liquid in bulk. On the basis of
this model, the electrostatic theory of continuous media leads at once to a simple
expression for the local field and the average torque effective in orienting a dipole
molecule relative to the field E. The molecular dipole moment in Onsager’s approach
contains a permanent part and a part induced by the local field E
, and derived the
following formula:
(ε(0) − ε ∞ )(2ε(0) + ε ∞ )
ε(0)(ε ∞ + 2)
2
=
nμ
2
0
9ε 0 k B T
,
(2.49)
which at (0) ∞ gives
ε(0) − ε ∞ =
nμ
2
0
3ε 0 k B T
(ε ∞ + 2)
2
6
.
(2.50)
For room temperature, μ 0 =1.85 D, and ∞ ≈ 2, (2.50) gives the value (0) ≈ 30,
which is higher than those obtained by Eq. 2.49, but still several times lower than the
experimental one.
Kirkwood [33] suggested an extension of Onsager’s theory and assumed that
Bernal–Fowler tetrahedrally coordinated molecule (see Sect. 1.2) orients in the
dielectric continuum together with its first coordination shell. The field of the center dipole μ 0 causes a favored average orientation of neighboring dipoles and thus
an increased moment μ = g · μ 0 of the tetrahedral group of five H 2 O molecules. If
we assume a modified Bernal–Fowler structure for water and introduce a g-factor of
about 2.7, the experimental value, (0) ≈ 78, of the dielectric constant of liquid water
at 25
◦ C is obtained from (2.50) with a dipole moment in the liquid 26% greater than
that in the vapor.
Although the theory described above reduces the problem relating to the dielectric
polarization of polar liquids to the problem of the calculation of the g-factor and the
sum of the dipole moments of overpolarized molecules in its immediate environment
in water, there are still several points which do not allow one to fully accept this theory: