148
4 The Dielectric Properties and Dynamic Structure of Water and Ice
time-resolved spectroscopy, which are two orders of magnitude slower [46]. Bai and
Herzfeld used [47] a force field for subatomic particles based on a semi-classical
treatment of explicit valence electron pairs with heuristic potentials trained on the
structural and thermodynamic properties of water monomers and dimers in all of
their protonational states. The authors found that about 1 mol/l of simulated water
molecules have an anomalously low O–O distance of less than 2.5 Å, which assumes
no barrier for proton transfer and indicates the formation of ionic pairs. This concentration of ionic pairs is in remarkable agreement with the concentration of short-lived
species in the ionic model. The authors also observed no spontaneous ionization in
over 10 ns, which means that ion pairs annihilate in sub-nanosecond timescales.
This is also consistent with the ionic model, which assumes a picosecond lifetime
of ionic species. The special role of the ultrashort interactions, and the formation of
the corresponding short-lived ionic species, is consistent with the elusive rise of the
dissociation constant K w of water with increasing pressure. Thus, the results by Bai
and Herzfeld provide more satisfactory agreement with experiment than DFT does
with different functionals and indirectly support the ionic model.
Summarizing, the comparison of the hundred-year-old Bernal–Fowler model with
the ionic model shows the tremendous advantages of the latter in the description of
ultrashort and nanoscale dynamics, and the corresponding experimental observations. The experiments of the past decade reveal the significant spatial-time heterogeneity of water, which can be neglected at large scales and relatively long observation times, but which cannot at picosecond or at nanometer scales. As Bernal–Fowler
water does not account for the heterogeneity of water and nuclear quantum effects,
the experimental data described above can be explained by the ionic model only.
Thus, the ionic model has wider application and validity range and allows one to
account for more dynamic properties of water than models based on the concept of
a hydrogen-bound network of equivalent molecular species.
4.4 Instantaneous Structure of Water and Ice
Figure 4.6 shows the I-structure of water and ice according to the ionic model. The
inset shows the effective potential for the central ion (excess proton). The potential
has two characteristic periods and differs for water and ice. The height of the potential
peaks and the length of periods are in accordance with the barriers (activation energies) of conductivity (see Table 3.2) and the diffusion lengths shown in Table 4.2. The
spatial period of a smaller (by period) potential associated with diffusion length l ≈
3.4 Å, which corresponds to the elementary step of the translational diffusion of the
ion. The period of this potential is approximately the same for water and ice, but the
amplitude differs by about 0.4 eV. The larger-by-period potential (see the enveloping
dashed black line) associated with the rearrangement of the ionic atmosphere of the
central ion, and associated with the second characteristic diffusion step L ≈ 0.84 nm,
which is half the unit step of DC conductivity. The height of the enveloping potential
is determined by the difference between the activation energies of σ dc and σ D1 (see
4 The Dielectric Properties and Dynamic Structure of Water and Ice
time-resolved spectroscopy, which are two orders of magnitude slower [46]. Bai and
Herzfeld used [47] a force field for subatomic particles based on a semi-classical
treatment of explicit valence electron pairs with heuristic potentials trained on the
structural and thermodynamic properties of water monomers and dimers in all of
their protonational states. The authors found that about 1 mol/l of simulated water
molecules have an anomalously low O–O distance of less than 2.5 Å, which assumes
no barrier for proton transfer and indicates the formation of ionic pairs. This concentration of ionic pairs is in remarkable agreement with the concentration of short-lived
species in the ionic model. The authors also observed no spontaneous ionization in
over 10 ns, which means that ion pairs annihilate in sub-nanosecond timescales.
This is also consistent with the ionic model, which assumes a picosecond lifetime
of ionic species. The special role of the ultrashort interactions, and the formation of
the corresponding short-lived ionic species, is consistent with the elusive rise of the
dissociation constant K w of water with increasing pressure. Thus, the results by Bai
and Herzfeld provide more satisfactory agreement with experiment than DFT does
with different functionals and indirectly support the ionic model.
Summarizing, the comparison of the hundred-year-old Bernal–Fowler model with
the ionic model shows the tremendous advantages of the latter in the description of
ultrashort and nanoscale dynamics, and the corresponding experimental observations. The experiments of the past decade reveal the significant spatial-time heterogeneity of water, which can be neglected at large scales and relatively long observation times, but which cannot at picosecond or at nanometer scales. As Bernal–Fowler
water does not account for the heterogeneity of water and nuclear quantum effects,
the experimental data described above can be explained by the ionic model only.
Thus, the ionic model has wider application and validity range and allows one to
account for more dynamic properties of water than models based on the concept of
a hydrogen-bound network of equivalent molecular species.
4.4 Instantaneous Structure of Water and Ice
Figure 4.6 shows the I-structure of water and ice according to the ionic model. The
inset shows the effective potential for the central ion (excess proton). The potential
has two characteristic periods and differs for water and ice. The height of the potential
peaks and the length of periods are in accordance with the barriers (activation energies) of conductivity (see Table 3.2) and the diffusion lengths shown in Table 4.2. The
spatial period of a smaller (by period) potential associated with diffusion length l ≈
3.4 Å, which corresponds to the elementary step of the translational diffusion of the
ion. The period of this potential is approximately the same for water and ice, but the
amplitude differs by about 0.4 eV. The larger-by-period potential (see the enveloping
dashed black line) associated with the rearrangement of the ionic atmosphere of the
central ion, and associated with the second characteristic diffusion step L ≈ 0.84 nm,
which is half the unit step of DC conductivity. The height of the enveloping potential
is determined by the difference between the activation energies of σ dc and σ D1 (see