3.5 Protonic Transport as a Fundamental Mechanism of the Dielectric…
125
sidered model, n i is obtained assuming electrostatic interaction among ions. Thus,
the pH is a measure of proton activity (but not of the concentration of excess protons)
at the static limit only. The majority of these excess protons sit in the potential and
conduct only at high frequencies.
The high-frequency ion diffusion coefficients,
D ∞ =
σ D1 k B T
n i q 2 ,
(3.25)
are equal 1.1·10
−14 for ice and 7·10
−9 m
2 /s for water. They exceed the self-diffusion
coefficient of H 2 O molecules by 10 and 350% for ice and water, respectively. This
means that the excess protons (or holes) in liquid water do not remain bound to
water molecules and can spontaneously change hosts by proton exchange. As a
consequence, the ensemble of excess protons forms a kind of gas, if we switch to
the frame of reference of a neutral water molecule and a crystal-like structure in a
laboratory frame of reference. Note that the charge carrier here is not equivalent to
the hydrogen proton, as a water molecule has two hydrogen protons and either of
them can participate in the charge transfer.
It should be noted that we do not explicitly discuss the hydrogen bonds between
the H 2 O molecules in this model. However, the manifold of the excess-proton migration and the corresponding dissolution of the excess protons among neutral water
molecules on large time and space scales can be considered as an analogue of hydrogen bonding (see Sect. 4.1).
The spectra of water and ice in terms of the real,
, and imaginary,
, parts of
the complex dielectric function
∗
(ν) =
(ν)+i
(ν) have fundamental similarities
discussed in Sect. 3.4. The general difference between the two spectra is a frequency
shift shown by the arrow in Fig. 3.9. The static dielectric constant ((0)) is shown
in Fig. 3.12 as a function of temperature. The experimental curve (0, T ) looks like
a single line with a small gap near the melting point substantiated, presumably, by
the 8% density change between ice and water.
9 Because (0) is proportional to the
integral of
(ν), considering the similar mechanism for ice and water dielectric
functions, we expect similar mechanisms for their dielectric constants.
From (3.23) and (3.24), one can obtain for the static dielectric constant:
ε(0) =
σ dc τ c
ε 0
.
(3.26)
This relation links (0) with DC conductivity σ dc and is proportional to the transition time τ c .
10 The DC conductivity and τ c are both functions of temperature:
9 Note that the difference between the concentrations of excess protons in ice and water in Table 3.5
is also about 8%.
10 Note that this formula is different from the classic Debye formula (0) = σ D1 τ D1 / 0 , which
connects high-frequency conductivity σ D1 and dielectric relaxation time τ D1 with the static dielectric
constant.
125
sidered model, n i is obtained assuming electrostatic interaction among ions. Thus,
the pH is a measure of proton activity (but not of the concentration of excess protons)
at the static limit only. The majority of these excess protons sit in the potential and
conduct only at high frequencies.
The high-frequency ion diffusion coefficients,
D ∞ =
σ D1 k B T
n i q 2 ,
(3.25)
are equal 1.1·10
−14 for ice and 7·10
−9 m
2 /s for water. They exceed the self-diffusion
coefficient of H 2 O molecules by 10 and 350% for ice and water, respectively. This
means that the excess protons (or holes) in liquid water do not remain bound to
water molecules and can spontaneously change hosts by proton exchange. As a
consequence, the ensemble of excess protons forms a kind of gas, if we switch to
the frame of reference of a neutral water molecule and a crystal-like structure in a
laboratory frame of reference. Note that the charge carrier here is not equivalent to
the hydrogen proton, as a water molecule has two hydrogen protons and either of
them can participate in the charge transfer.
It should be noted that we do not explicitly discuss the hydrogen bonds between
the H 2 O molecules in this model. However, the manifold of the excess-proton migration and the corresponding dissolution of the excess protons among neutral water
molecules on large time and space scales can be considered as an analogue of hydrogen bonding (see Sect. 4.1).
The spectra of water and ice in terms of the real,
, and imaginary,
, parts of
the complex dielectric function
∗
(ν) =
(ν)+i
(ν) have fundamental similarities
discussed in Sect. 3.4. The general difference between the two spectra is a frequency
shift shown by the arrow in Fig. 3.9. The static dielectric constant ((0)) is shown
in Fig. 3.12 as a function of temperature. The experimental curve (0, T ) looks like
a single line with a small gap near the melting point substantiated, presumably, by
the 8% density change between ice and water.
9 Because (0) is proportional to the
integral of
(ν), considering the similar mechanism for ice and water dielectric
functions, we expect similar mechanisms for their dielectric constants.
From (3.23) and (3.24), one can obtain for the static dielectric constant:
ε(0) =
σ dc τ c
ε 0
.
(3.26)
This relation links (0) with DC conductivity σ dc and is proportional to the transition time τ c .
10 The DC conductivity and τ c are both functions of temperature:
9 Note that the difference between the concentrations of excess protons in ice and water in Table 3.5
is also about 8%.
10 Note that this formula is different from the classic Debye formula (0) = σ D1 τ D1 / 0 , which
connects high-frequency conductivity σ D1 and dielectric relaxation time τ D1 with the static dielectric
constant.