138
4 The Dielectric Properties and Dynamic Structure of Water and Ice
absorption spectroscopy in the form of a broad absorption band between 900 and
1500 cm
−1 , with overtones assigned to the broad absorption continuum between
1,500 and 2,700 cm
−1 . In addition, one should also expect the intermolecular spectral
activity of the excess proton, which should appear in the terahertz frequency range.
Proton dynamics are responsible for static conductivity too, thus, the dynamics of
the solvated proton is expected to contribute to the whole frequency range from the
IR down to low-frequency static conductivity σ dc .
The essence of the proton current is the continuous mutual transformations of
Eigen and Zundel cations (see Sect. 1.3). The Eigen cation is a hydronium ion
solvated by three water molecules (see Fig. 4.2a), which has a spectral activity near
the 5 THz oscillatory mode, ν s , due to the vibration of the central ion in the shell
of surrounding molecules (see Sect. 2.6.2). The Zundel cation is a transition point
between two vibrational states. The switch between two Eigen states goes by proton
tunneling, followed by the relaxation of the hydration shell (see the two circles around
Zundel cation in Fig. 4.2a). The spontaneous tunneling of the charge induces the
delayed adaptation of the nearest molecular dipoles, which presumably corresponds
to the second relaxation (see the blue area in Fig. 4.2). As the polarization of the
hydration shell is due to the reorientations of H 2 O molecules, the second relaxation is
also associated with molecular rotations. The local current induced by the migration
of naked (unsolvated) ion contributes to the plateau σ D2 .
Continuing to increase the time of observation, we switch to a series of proton
transfers, which contribute to another conductivity plateau σ D1 . This plateau can be
considered as a current made by the dressed (solvated) ions, as the charge drags its
solvation shell. That is why, in the real spectrum, this plateau is distorted by the
solvation and the broadening of the peak ν s . The lifetime of an ion in the oscillatory
state between two transitions can be found from the half-width ν s of this peak.
Finally, a decrease of the conductivity below ν D1 is due to the finite lifetime of
ionic species and due to their mutual electrostatic interactions, which, as discussed
in Sect. 3.5, effectively reduce the cumulative mobility of the charge carriers at
relatively long observation times (more than a few dozen picoseconds).
Thus, we correlated the intramolecular and intermolecular dynamics of the excess
proton with the broadband dielectric spectrum of water and identified the characteristic frequency ranges, where we can expect the spectral signatures, which correspond
to the vibrations, tunneling, translations, drift, and diffusion of excess protons and
proton holes.
4.2.3 The Ionic (Protonic) Model of Water
Now, accounting for the details of proton transport described above, we formulate the
ionic model of water. First of all, we represent water as an ensemble of H 2 O molecules
and intrinsic ions, H 3 O
+ and OH
− (see Fig. 3.11), obeying thermal collisions, which
transform to each other by the transfer of a proton (or proton hole). In other words,
we admit spontaneous autoprotolysis and periodic proton exchange between water
4 The Dielectric Properties and Dynamic Structure of Water and Ice
absorption spectroscopy in the form of a broad absorption band between 900 and
1500 cm
−1 , with overtones assigned to the broad absorption continuum between
1,500 and 2,700 cm
−1 . In addition, one should also expect the intermolecular spectral
activity of the excess proton, which should appear in the terahertz frequency range.
Proton dynamics are responsible for static conductivity too, thus, the dynamics of
the solvated proton is expected to contribute to the whole frequency range from the
IR down to low-frequency static conductivity σ dc .
The essence of the proton current is the continuous mutual transformations of
Eigen and Zundel cations (see Sect. 1.3). The Eigen cation is a hydronium ion
solvated by three water molecules (see Fig. 4.2a), which has a spectral activity near
the 5 THz oscillatory mode, ν s , due to the vibration of the central ion in the shell
of surrounding molecules (see Sect. 2.6.2). The Zundel cation is a transition point
between two vibrational states. The switch between two Eigen states goes by proton
tunneling, followed by the relaxation of the hydration shell (see the two circles around
Zundel cation in Fig. 4.2a). The spontaneous tunneling of the charge induces the
delayed adaptation of the nearest molecular dipoles, which presumably corresponds
to the second relaxation (see the blue area in Fig. 4.2). As the polarization of the
hydration shell is due to the reorientations of H 2 O molecules, the second relaxation is
also associated with molecular rotations. The local current induced by the migration
of naked (unsolvated) ion contributes to the plateau σ D2 .
Continuing to increase the time of observation, we switch to a series of proton
transfers, which contribute to another conductivity plateau σ D1 . This plateau can be
considered as a current made by the dressed (solvated) ions, as the charge drags its
solvation shell. That is why, in the real spectrum, this plateau is distorted by the
solvation and the broadening of the peak ν s . The lifetime of an ion in the oscillatory
state between two transitions can be found from the half-width ν s of this peak.
Finally, a decrease of the conductivity below ν D1 is due to the finite lifetime of
ionic species and due to their mutual electrostatic interactions, which, as discussed
in Sect. 3.5, effectively reduce the cumulative mobility of the charge carriers at
relatively long observation times (more than a few dozen picoseconds).
Thus, we correlated the intramolecular and intermolecular dynamics of the excess
proton with the broadband dielectric spectrum of water and identified the characteristic frequency ranges, where we can expect the spectral signatures, which correspond
to the vibrations, tunneling, translations, drift, and diffusion of excess protons and
proton holes.
4.2.3 The Ionic (Protonic) Model of Water
Now, accounting for the details of proton transport described above, we formulate the
ionic model of water. First of all, we represent water as an ensemble of H 2 O molecules
and intrinsic ions, H 3 O
+ and OH
− (see Fig. 3.11), obeying thermal collisions, which
transform to each other by the transfer of a proton (or proton hole). In other words,
we admit spontaneous autoprotolysis and periodic proton exchange between water