Theoretical description of dielectric relaxation of ice with low concentration impurities
Introduction
In nature, impurities are always present in water, and the question of their influence on the electrical properties of ice is not completely resolved.
The process of dielectric relaxation of ice in pure water has been studied in detail both experimentally [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12] and theoretically [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. The main peak of the dielectric loss of ice is symmetrically broadened below a temperature of 240 K and is well described by the Cole-Cole expression for complex dielectric permittivity (CDP). Above a temperature of 240 K, the main peak of the imaginary part of the CDP has a Debye shape. The main intrigue in the analysis of experimental results is related to the temperature behavior of the relaxation time τ (see Fig. 1). The Fig. 1 shows the temperature dependences of the dielectric relaxation time τ of ice Ih from different studies [1], [4], [6], [7], [8], [9], [11], [12]. We note the characteristic features of the presented temperature dependence of the relaxation time. If in early studies (see [1]) it was shown that the characteristic ice relaxation time follows the Arrhenius law with an activation energy of approximately 0.575 eV above 200 K [2], [3], [4], [5], [6], [7], then the results of later works [5], [6], [7], [8] demonstrate a deviation from Arrhenius behavior in the temperature range 210–245 K and a change in the slope of the relaxation time on a logarithmic scale with a decrease in the activation energy down to 0.2 eV (see Fig. 1). It was demonstrated [9] that this transition or high-temperature crossover depends on a method of ice samples preparation. Detailed measurements of the CDP of ice samples prepared by various methods [9] showed that samples prepared with stirring at water freezing (in order to avoid rapid ice crystallization) did not exhibit the high-temperature crossover, in contrast to the samples with regular freezing (without stirring). Detailed experiments performed in a wide temperature range [4], [5], [7] demonstrate also a second low-temperature crossover below the temperature of ~170 K, where activation energy rises up to 0.48 eV [5]. Furthermore, there is the discrepancy in the reproducibility of the low-temperature crossover.
Theoretical studies are mainly based on the “wait-and-switch” model [23], [24], [25], [26], [27]. According to this model, the presence of a network of hydrogen bonds limits the rotational diffusion of dipoles of water molecules and, therefore, the reorientation of dipole moments does not occur freely, as in a gas. Crystalline ice is a well-ordered structure in terms of oxygen sites and, at the same time, it has certain disordering in terms of hydrogens position. Ice is a proton semiconductor and charge carriers in it are disordered protons. Proton hopping create defects of two types in the ice structure: ionic and orientation. In the first case, the proton jumps along the hydrogen bond from one H2O molecule to another [15], resulting in the formation of a pair of ionic defects H3O + and OH–, and in the second - on the adjacent hydrogen bond in one H2O molecule, resulting in a pair of orientation Bierrum defects, called L- and d-defects [16], [17], [18]. Formally, such a jump can be considered as a rotation of the H2O molecule. These defects can diffuse over the ice crystal lattice. In accordance with the “wait and switch” model [23], [24], [25], [26], [27], the reorientation of the dipole moment of a water molecule is the result of an intermittent change in the direction of the dipole moment. In this case, the reorientation of water molecule dipole is possible only when it encounters a corresponding defect in the hydrogen bonds network, otherwise the water molecule remains in a waiting mode. Based on this idea, in [19] it was possible to obtain a relationship between the CDP and the mean square displacement (MSD) of defects
Here are the density, the effective charge, the conductivity, and the Laplace image of the MSD of the defect α, respectively, and T is the temperature in the energy units.
It was shown in [19] that orientation defects obey normal diffusion law (<r2(t)> = 6DLDt), while ionic defects demonstrates the anomalous diffusion behavior (<r2(t)> = 6D±tα±). Currently, there are large number of approaches, where anomalous diffusion motion might be derived. The most reasonable for the ice is based on the effect of ionic defects blockage created by orientation defects. For example, the H3O+ ionic defect jump can be blocked by orientation d-defects [12] (see Fig. 2), then its further migration is impossible until at least one of d-defects moves away. Blocking of ionic defect jumps also occurs, when an H3O+ ion has passed through a certain fragment of the H-bonded network, and, as a result, the next ion would not be able to move along the same path. The release of the path is possible when a d-defect passes through it. Same is fair for the OH– and L defects. Thus, in the presence of defects in the ice structure, so-called “traps” arise, falling into which protons cannot continue further migration for a certain time and are localized.
Based on these ideas, a phenomenological model was proposed in [19] to justify a high-temperature crossover, which was then expanded in [12], [22] to justify a low-temperature crossover. According to this model, a change in the activation energy near a temperature of 240 K is explained by the transition from the predominant motion of Bierrum orientational defects at high temperatures to the predominant motion of ionic defects at low temperatures. At low temperatures, a strongly correlated movement of ionic and orientational defects occurs [12], [22], which explains the change in the activation energy of ice near a temperature of 150 K. A microscopic approach based on the multiple-trapping model (MT) [28], [29], to describe the relaxation behavior of ice at low temperatures, was developed in [20], [21]. According to this approach, a proton moves along delocalized states with energies above the mobility edge. This motion is interrupted by trapping into localized states formed by traps from orientational defects located below the mobility edge, with subsequent activation of the proton back to the conducting state at the mobility edge. In this case, localized states are widely distributed over energies. The microscopic description made it possible to find a relation between the parameters of the dielectric response of ice and the parameters of the microscopic structure of the medium. The deviation from the Arrhenius behavior of the ice relaxation time at low temperatures and the slowdown of the relaxation process in the framework of the microscopic description are explained by the predominance of the processes of proton capture in traps in comparison with the processes of release from them.
The aim of this work is an applying the microscopic approach to describe the dielectric relaxation of ice in the presence of impurities of low concentrations using the example of a partially crystallized gelatin-water mixture with low gelatin concentrations. The study of the dynamics of water around biomolecules is an actual problem in solving the problem of the role of water in the functioning of living organisms. The need to study the dynamics of water in partially crystallized protein-water mixtures is due to their importance in the modern food industry.
Section snippets
The two-level model (single trap)
In this section, we propose the simplest model for describing the non-Arrhenius behavior of the temperature dependence of the relaxation time of ice in a wide temperature range, including both temperature crossovers.
As mentioned above, orientation defects obey normal diffusion motion with MSD < r2(t)> = 6DLDt, where DLD is the diffusion coefficient of orientation defects, and ionic defects obey an anomalous diffusion motion with MSD < r2(t)> = 6D±tα±, where D± is the diffusion coefficient of
Multiple trapping model (mt model)
In the previous section, we formulated the simplest version of the model with single localized energy level. However, this is a rather rough approximation, since in reality localized states are widely distributed in energies. Therefore, it becomes necessary to extend the model by introducing the energy distribution of localized states with a some distribution density ρ(E). Then, assuming that protons in localized states are distributed in accordance with the Fermi-Dirac statistics, we write the
Description of the dielectric relaxation of ice in water-gelatin mixture of low concentration
In this section, we will demonstrate the consistency of the model proposed in the previous section by describing the experimental temperature dependences of the dielectric relaxation time of ice in partially crystallized water-gelatin mixtures with a low concentration of gelatin obtained in [30]. In this article, the authors performed the detailed study of the process of dielectric relaxation of ice in a water-gelatin mixture with gelatin concentrations of 1–5 wt% by the method of broadband
Conclusion
In this work, the authors have developed a simple model of dielectric relaxation of ice with impurities, which was successfully tested by describing the temperature dependence of the dielectric relaxation time of ice in partially crystallized water-gelatin mixtures of low concentration. The model is based on the concept of proton hopping, which is controlled by traps from orientation defects generated by impurity molecules. The non-Arrhenius behavior of the ice relaxation time at high and low
Data availability statement
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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