142
4 The Dielectric Properties and Dynamic Structure of Water and Ice
The diffusion coefficient D of excess protons differs at each plateau (σ dc , σ D1 or
σ D2 ) and is associated with the conductivity by the Nernst–Einstein relation:
σ i =
q
2
k B T
n ± D i ,
(4.3)
where subscript i means dc, D1 or D2, q is the elementary charge, k B is the Boltzmann constant, n ± is the concentration of charge carriers. The conductivity plateaus
σ D1 and σ D2 are connected with the dielectric contributions D1 and D2 (see
Table 4.1) of the corresponding relaxation bands ν D1 and ν D2 by the Debye formula:
σ i = ε 0 ε i 2πν i .
(4.4)
Further, the average distance between the charges is determined by the concentration n ± :
L ≈
3
4π n ±
1/3
.
(4.5)
The system of (4.1)–(4.5), using experimental data from Table 4.1, allows one
to determine the parameters of the atomic-molecular dynamics of H 2 O molecules
and H 3 O
+ and OH
− ions in water. The parameters calculated at room temperature
are given in Table 4.2, which constitutes the core parameters of the ionic model of
water. The table shows that, within the ionic model, a microscopic description of the
conductivity spectrum is achieved with the concentration of short-lived intrinsic ions
of n ± = 1 mol/l on the average distance of 0.8 nm from each other. This concentration
is 2% of the all the molecular species in water, but due to the ultrashort (2.3 ps at room
temperature) lifetime, most of them disappear faster than they make a contribution
to the static conductivity σ dc . Nevertheless, any “snapshot” of water is represented
by “swarms” of about 50 H 2 O molecules per an H 3 O
+ or OH
− ion.
Figure 4.5 shows the instantaneous water structure according to the abovecalculated parameters of the ionic model (compare with Bernal–Fowler water in
Fig. 1.5). It consists of electrostatically interacting molecular and ionic species. The
electrostatic attraction of molecular dipoles by the ionic species increases the local
density. The density between the ions of different signs is lower, as the forces on
the molecular dipole, placed between the negative and positive charges, are balanced, and the center of the molecular dipole is in indifferent equilibrium. Note that
the concept of water made of a linear combination of two components was introduced by Röntgen [25]. The idea of LDW and HDW was used for the explanation
of water’s thermodynamic properties [26], and recently verified by the neutron and
X-ray diffraction techniques [27]. However, the structure of the areas of high and low
density still lacks clarity on the microscopic level. The ionic model provides a microscopic picture, substantiating the existence of high- and low-density regions in water,
and explaining the fast density fluctuations observed in scattering experiments by the
4 The Dielectric Properties and Dynamic Structure of Water and Ice
The diffusion coefficient D of excess protons differs at each plateau (σ dc , σ D1 or
σ D2 ) and is associated with the conductivity by the Nernst–Einstein relation:
σ i =
q
2
k B T
n ± D i ,
(4.3)
where subscript i means dc, D1 or D2, q is the elementary charge, k B is the Boltzmann constant, n ± is the concentration of charge carriers. The conductivity plateaus
σ D1 and σ D2 are connected with the dielectric contributions D1 and D2 (see
Table 4.1) of the corresponding relaxation bands ν D1 and ν D2 by the Debye formula:
σ i = ε 0 ε i 2πν i .
(4.4)
Further, the average distance between the charges is determined by the concentration n ± :
L ≈
3
4π n ±
1/3
.
(4.5)
The system of (4.1)–(4.5), using experimental data from Table 4.1, allows one
to determine the parameters of the atomic-molecular dynamics of H 2 O molecules
and H 3 O
+ and OH
− ions in water. The parameters calculated at room temperature
are given in Table 4.2, which constitutes the core parameters of the ionic model of
water. The table shows that, within the ionic model, a microscopic description of the
conductivity spectrum is achieved with the concentration of short-lived intrinsic ions
of n ± = 1 mol/l on the average distance of 0.8 nm from each other. This concentration
is 2% of the all the molecular species in water, but due to the ultrashort (2.3 ps at room
temperature) lifetime, most of them disappear faster than they make a contribution
to the static conductivity σ dc . Nevertheless, any “snapshot” of water is represented
by “swarms” of about 50 H 2 O molecules per an H 3 O
+ or OH
− ion.
Figure 4.5 shows the instantaneous water structure according to the abovecalculated parameters of the ionic model (compare with Bernal–Fowler water in
Fig. 1.5). It consists of electrostatically interacting molecular and ionic species. The
electrostatic attraction of molecular dipoles by the ionic species increases the local
density. The density between the ions of different signs is lower, as the forces on
the molecular dipole, placed between the negative and positive charges, are balanced, and the center of the molecular dipole is in indifferent equilibrium. Note that
the concept of water made of a linear combination of two components was introduced by Röntgen [25]. The idea of LDW and HDW was used for the explanation
of water’s thermodynamic properties [26], and recently verified by the neutron and
X-ray diffraction techniques [27]. However, the structure of the areas of high and low
density still lacks clarity on the microscopic level. The ionic model provides a microscopic picture, substantiating the existence of high- and low-density regions in water,
and explaining the fast density fluctuations observed in scattering experiments by the