4.2 A Phenomenological Model for the Broadband Dielectric Response
139
molecules and intrinsic ions, thus accounting for the nuclear (protonic) quantum
effects. We assume that each molecule/ion obeys rapid oscillatory dynamics around
the equilibrium position, being simultaneously in Brownian diffusion motion (see
Sect. 4.2.1). The water is considered electrically neutral, containing an equal number
of positive and negative ions (the electroneutrality principle), so that in equilibrium
we have
n w
t w
=
n ±
t ±
,
(4.1)
where n w and n ± are concentrations of H 2 O molecules and ions, respectively. The
concentration of H 2 O molecules can be found as n w = n 0 − n ± ≈ n 0 , where n 0 =
55.5 mol/l is the full concentration of all species in water.
Figure 4.3 represents the details of atomic-molecular transport in water, relevant
to its electrodynamic properties. For simplicity, the ions and H 2 O molecules are
represented by gray and white spheres, respectively, and the only one type of ions
(H 3 O
+ ) is considered. Protons and proton holes are depicted by small solid black and
open circles, respectively, and oxygen atoms are assumed to be in the center of the
each sphere. The numbers indicate the polarization relaxations that have a reflection
in the spectrum.
Let us assume that at the initial moment, the H 3 O
+ ion oscillates with a characteristic frequency ν s inside the hydration shell (event 1 in the figure). This state we
call a “dressed”-charge state. When the observation time increases, the excess proton
(charge) escapes the hydration shell and becomes “naked.” The transformation goes
by the charge transfer between the molecular species. The excess proton changes the
host molecule, being occasionally squeezed between two neighboring species (event
2). The charge hops the distance of 2.8 Å of two molecular radii, while the proton
as a particle just sticks from one molecular species to another (i.e., does not hop at
all). The time spent by the ion in the vibrational state determines the half-width ν s
of the ν s mode. The subsequent relaxation of the naked state to the dressed state
is characterized by the relaxation time t ± , coinciding with the secondary relaxation
time 1/ν D2 . Due to continuous diffusion, the naked charge becomes dressed again
at a distance l from the initial position, equal to
√
D 1 t ± , where D 1 is a diffusion
coefficient of the solvated charge.
Further, the charge, separated from the parent molecule, diffuses independently,
repeating a series of Zundel-to-Eigen cation transformations (event 3). The molecule
abandoned by the charge moves like a neutral particle (event 4) until it again encounters another (or the same) charge and turns back to the ionic state (event 5). The
time interval between the events of charge separation and ion reconstruction is the
lifetime t w of the molecule H 2 O.
11 The time between events 1 and 5 is also the time
of the main (Debye) dielectric relaxation [23]. Within the time t w , an excess charge
(the polarization configuration of molecules) passes a distance L in the laboratory
frame of reference.
11 Here, the lifetime is defined as the period between the transformations of a molecule into an ion
by adding/removing one proton.
139
molecules and intrinsic ions, thus accounting for the nuclear (protonic) quantum
effects. We assume that each molecule/ion obeys rapid oscillatory dynamics around
the equilibrium position, being simultaneously in Brownian diffusion motion (see
Sect. 4.2.1). The water is considered electrically neutral, containing an equal number
of positive and negative ions (the electroneutrality principle), so that in equilibrium
we have
n w
t w
=
n ±
t ±
,
(4.1)
where n w and n ± are concentrations of H 2 O molecules and ions, respectively. The
concentration of H 2 O molecules can be found as n w = n 0 − n ± ≈ n 0 , where n 0 =
55.5 mol/l is the full concentration of all species in water.
Figure 4.3 represents the details of atomic-molecular transport in water, relevant
to its electrodynamic properties. For simplicity, the ions and H 2 O molecules are
represented by gray and white spheres, respectively, and the only one type of ions
(H 3 O
+ ) is considered. Protons and proton holes are depicted by small solid black and
open circles, respectively, and oxygen atoms are assumed to be in the center of the
each sphere. The numbers indicate the polarization relaxations that have a reflection
in the spectrum.
Let us assume that at the initial moment, the H 3 O
+ ion oscillates with a characteristic frequency ν s inside the hydration shell (event 1 in the figure). This state we
call a “dressed”-charge state. When the observation time increases, the excess proton
(charge) escapes the hydration shell and becomes “naked.” The transformation goes
by the charge transfer between the molecular species. The excess proton changes the
host molecule, being occasionally squeezed between two neighboring species (event
2). The charge hops the distance of 2.8 Å of two molecular radii, while the proton
as a particle just sticks from one molecular species to another (i.e., does not hop at
all). The time spent by the ion in the vibrational state determines the half-width ν s
of the ν s mode. The subsequent relaxation of the naked state to the dressed state
is characterized by the relaxation time t ± , coinciding with the secondary relaxation
time 1/ν D2 . Due to continuous diffusion, the naked charge becomes dressed again
at a distance l from the initial position, equal to
√
D 1 t ± , where D 1 is a diffusion
coefficient of the solvated charge.
Further, the charge, separated from the parent molecule, diffuses independently,
repeating a series of Zundel-to-Eigen cation transformations (event 3). The molecule
abandoned by the charge moves like a neutral particle (event 4) until it again encounters another (or the same) charge and turns back to the ionic state (event 5). The
time interval between the events of charge separation and ion reconstruction is the
lifetime t w of the molecule H 2 O.
11 The time between events 1 and 5 is also the time
of the main (Debye) dielectric relaxation [23]. Within the time t w , an excess charge
(the polarization configuration of molecules) passes a distance L in the laboratory
frame of reference.
11 Here, the lifetime is defined as the period between the transformations of a molecule into an ion
by adding/removing one proton.