3.1 Dielectric-Terahertz Spectrum of Ice
109
Whalley and Klug [14] argue that the strong infrared absorption in the O–H
stretching region is already an indicator of LO–TO splitting, which occurs in the
vibrational spectrum of ice, as well as in liquid water, due to the unconditional
similarity of the spectra. This similarity hints that liquid water should also have
collective vibrations that propagate in a similar way to those in ice and that are caused
by long-range Coulomb interactions. Thus, the IR spectrum of ice and water can be
fully understood only considering the phononic modes, which are usually missing
in conventional molecular-dynamic simulations. Note that at terahertz frequencies
water behaves more like a moderately rigid, isotropic, elastic solid as highlighted by
Walrafen [11], while ice looks more like a crystal.
3.2 The Temperature Dependence of Spectral Parameters
Figure 3.3 shows the temperature dependencies of the main microscopic transport
parameters of water and ice: the static conductivity, σ dc , the high-frequency conductivity, σ D1 , the relaxation frequency, ν D1 , and the self-diffusion coefficient, D sel f .
Curves are collated on the basis of data from [2, 5, 15–18] and represented as
Arrhenius plots.
2 The perfect linear behavior of all quantities indicates that in the
given temperature intervals, i.e., 0–100
◦ C for water and −60 − 0
◦ C for ice, both
obey Arrhenius law: A = A 0 exp(E a /k B T ), where A is the transport parameter, A 0
is the pre-exponential factor, and E a is the activation energy. The corresponding
best-fit parameters are given in Table 3.2. The figure and table show that parameters
D sel f , ν D1 and σ D1 have approximately the same activation energy E a ≈ 0.16 eV for
water, and E a ≈ 0.6 eV for ice. Additionally, at the phase transition, all quantities
change synchronously by about 6–7 orders of magnitude, following the spectral shift
between ice and water, except the static conductivity σ dc , the activation energy of
which is 1.2 and 2.5 times higher than that for other quantities for ice and water,
respectively. Interestingly, σ dc changes by one order of magnitude only, and in the
megahertz frequency region the conductivity of ice exceeds the conductivity of water
by a factor of 35 (see Fig. 3.1). Thus, ice is an order of magnitude more conductive
in the megahertz region than water.
Figure 3.4 compares the temperature dependence of H 2 O molecule mobility,
μ H 2 O = D sel f /k B T , with the low frequency, σ dc , and the high-frequency, σ D1 , conductivities, which are plotted in dimensionless units. One can see that μ H 2 O (T) coincides with σ D1 (T) but differs from σ dc (T). This means that the high-frequency conductivity mechanism is diffusion controlled, or, in other words, the energy barrier of
conduction is the same as for diffusion. Although the static electrical conductivity
σ dc is also expected to be diffusion controlled, the higher activation energy than that
for H 2 O mobility indicates an interaction between charge carriers (H 3 O
+ and OH
−
ions). This fact is missing in the account of the concentration of hydrogen ions in
the concept of pH (see Sect. 1.3.3).
2 The plot displays the logarithm of a quantity plotted against reciprocal temperature.
109
Whalley and Klug [14] argue that the strong infrared absorption in the O–H
stretching region is already an indicator of LO–TO splitting, which occurs in the
vibrational spectrum of ice, as well as in liquid water, due to the unconditional
similarity of the spectra. This similarity hints that liquid water should also have
collective vibrations that propagate in a similar way to those in ice and that are caused
by long-range Coulomb interactions. Thus, the IR spectrum of ice and water can be
fully understood only considering the phononic modes, which are usually missing
in conventional molecular-dynamic simulations. Note that at terahertz frequencies
water behaves more like a moderately rigid, isotropic, elastic solid as highlighted by
Walrafen [11], while ice looks more like a crystal.
3.2 The Temperature Dependence of Spectral Parameters
Figure 3.3 shows the temperature dependencies of the main microscopic transport
parameters of water and ice: the static conductivity, σ dc , the high-frequency conductivity, σ D1 , the relaxation frequency, ν D1 , and the self-diffusion coefficient, D sel f .
Curves are collated on the basis of data from [2, 5, 15–18] and represented as
Arrhenius plots.
2 The perfect linear behavior of all quantities indicates that in the
given temperature intervals, i.e., 0–100
◦ C for water and −60 − 0
◦ C for ice, both
obey Arrhenius law: A = A 0 exp(E a /k B T ), where A is the transport parameter, A 0
is the pre-exponential factor, and E a is the activation energy. The corresponding
best-fit parameters are given in Table 3.2. The figure and table show that parameters
D sel f , ν D1 and σ D1 have approximately the same activation energy E a ≈ 0.16 eV for
water, and E a ≈ 0.6 eV for ice. Additionally, at the phase transition, all quantities
change synchronously by about 6–7 orders of magnitude, following the spectral shift
between ice and water, except the static conductivity σ dc , the activation energy of
which is 1.2 and 2.5 times higher than that for other quantities for ice and water,
respectively. Interestingly, σ dc changes by one order of magnitude only, and in the
megahertz frequency region the conductivity of ice exceeds the conductivity of water
by a factor of 35 (see Fig. 3.1). Thus, ice is an order of magnitude more conductive
in the megahertz region than water.
Figure 3.4 compares the temperature dependence of H 2 O molecule mobility,
μ H 2 O = D sel f /k B T , with the low frequency, σ dc , and the high-frequency, σ D1 , conductivities, which are plotted in dimensionless units. One can see that μ H 2 O (T) coincides with σ D1 (T) but differs from σ dc (T). This means that the high-frequency conductivity mechanism is diffusion controlled, or, in other words, the energy barrier of
conduction is the same as for diffusion. Although the static electrical conductivity
σ dc is also expected to be diffusion controlled, the higher activation energy than that
for H 2 O mobility indicates an interaction between charge carriers (H 3 O
+ and OH
−
ions). This fact is missing in the account of the concentration of hydrogen ions in
the concept of pH (see Sect. 1.3.3).
2 The plot displays the logarithm of a quantity plotted against reciprocal temperature.