Dielectric Losses in Copper Sulfate at High Frequency

M. P. Tonkonogov

Statement of the Problem

At a constant voltage, heating of a dielectric is determined by the small residual current; in an alternating field, however, because of the very considerable time required for polarization to become established, the initial (or nearly initial) value of the electric current may prove decisive. It is therefore clear that the power dissipated in the dielectric, \(W = C V^2 \omega \operatorname{tg}\delta\) (where \(C\) is capacitance, \(V\) is voltage, \(\omega = 2\pi f\) is angular frequency, and \(\delta\) is the loss angle), for high-frequency current may be thousands of times greater than for direct current.

One must distinguish conduction losses, associated with the through displacement of charges between the electrodes, and relaxation losses, caused by the orientation of dipoles in an alternating electric field or by thermal ionic polarization.

For relaxation losses, the presence of a maximum in the dependence of \(\operatorname{tg}\delta\) on frequency and temperature is characteristic, this maximum shifting with increasing temperature toward higher frequencies.

The condition for the maximum of \(\operatorname{tg}\delta\) is described by the equation \((\omega \tau)_m \simeq 1\), where \(\tau\) is the relaxation time.

At present, the existence of relaxation dielectric losses in liquids and amorphous bodies has been demonstrated on the basis of a vast amount of experimental material [1]. The nature of dielectric losses in ionic crystals is still not entirely clear.

The first to observe a relaxation maximum in the dependence \(\operatorname{tg}\delta = f(T)\) in phlogopite mica, in 1947, was K. A. Vodop’yanov [6].

At low frequency, the dielectric properties of crystalline hydrates and micas were studied by A. P. Izergin [9]. Nevertheless, even in the comparatively recent monograph by Skanavi [7] (published in 1949), conductivity is considered the principal source of losses in inorganic crystals.

The absence of a developed theory of dielectric losses creates great difficulties for the systematization and interpretation of experimental material. At the same time, the insufficient amount of experimental data, in turn, is a hindrance to the development of the theory.

Thus, the study of dielectric losses in crystals containing polar molecules or radicals is a topical scientific problem.

In electrical and radio engineering, crystals with polar molecules have found wide application as insulators possessing high electrical and mechanical qualities; therefore, the study of the dielectric properties of such substances is of urgent interest for technology.

The present work studies the mechanism of dielectric losses in the comparatively simple crystals of copper sulfate¹.

Experiment and Its Discussion

Measurements of $\operatorname{tg}\delta$ and $\varepsilon$ were carried out by the resistance-variation method, developed by Karandeev [10] and improved by Vodop’yanov [11], [12]. The maximum relative error of $\operatorname{tg}\delta$ did not exceed 30%. Copper sulfate $\mathrm{CuSO_4\cdot 5H_2O}$ was studied, as well as $\mathrm{CuSO_4\cdot H_2O}$ and $\mathrm{CuSO_4}$ (anhydrous salt). The measurements were performed in the radio-frequency range $2\cdot 10^5$–$7.5\cdot 10^7$ cps.

Fig. 1. Dependence of $\operatorname{tg}\delta$ and $\varepsilon$ of copper sulfate on frequency. The specimen was prepared by double recrystallization of technical copper sulfate. $a$ is the specimen thickness.

The principal method of investigation was the study of the dependence of $\operatorname{tg}\delta$ and $\varepsilon$ on frequency, temperature, and the content of water of crystallization, which can provide valuable material for elucidating the conditions of polar groups in the crystal and the mechanism of dielectric losses.

The frequency dependence of $\operatorname{tg}\delta$ in $\mathrm{CuSO_4\cdot 5H_2O}$ has three maxima (Fig. 1). The structural formula of copper sulfate is

        H
        |
H2O     O—H
  \    /   \
   Cu++     O        H····O−      O−
  /   \    / \      /      \    /
H2O    O—H   O — H····O−  S++ 
        |                 /    \
        H              O−      O−

indicates that there are three groups of crystallization-water molecules with different conditions of binding [13]. According to Menshutkin [14], dehydration of copper sulfate also occurs in three stages.

We put forward the hypothesis that the maxima in the frequency dependence of the loss angle in copper sulfate are due to the orientation of weakly bound crystallization-water molecules with different values of activation energy.

Fig. 2. Dependence of \(\operatorname{tg}\delta\) and \(\varepsilon\) of copper sulfate on frequency. Double recrystallization of the reagent “chemically pure for analysis.”

Purification of the crystal by repeated recrystallization does not change the character of the curves \(\operatorname{tg}\delta = f(\omega)\); however, the absolute magnitude of \(\operatorname{tg}\delta\) decreases very sharply (Figs. 2 and 3). From the standpoint of the proposed hypothesis, one may think that impurities, loosening the lattice of copper sulfate, promote the formation of weakly bound dipoles participating in the process of dielec-

Fig. 3. Curves 1—dependence of \(\operatorname{tg}\delta\) and \(\varepsilon\) of \(\mathrm{CuSO_4 \cdot 5H_2O}\) on frequency. Triple recrystallization of the reagent “chemically pure for analysis.”
Curves 2 and 3—dependence of \(\operatorname{tg}\delta\) of the monohydrate \(\mathrm{CuSO_4 \cdot H_2O}\) on frequency at temperatures \(66^\circ\mathrm{C}\) (curve 2) and \(115^\circ\mathrm{C}\) (curve 3).

tric relaxation. The absolute magnitude of the maxima of \(\operatorname{tg}\delta=f(\omega)\) apparently depends on the number of weakly bound dipoles. Another confirmation of the hypothesis was provided by experiments with \(\mathrm{CuSO_4\cdot H_2O}\) (Fig. 3, curves 2 and 3). In copper sulfate calcined to the monohydrate, the losses increased noticeably. Although only \(20\%\) of the \(\mathrm{H_2O}\) molecules remained in the crystal after calcination, the number of weakly bound dipoles increased, which, we believe, explains the overall increase in losses. The increase in the number of weakly bound dipoles in this case may be associated with loosening of the crystal structure owing to the formation of vacant sites as a result of the partial removal of water of crystallization after calcination.

Fig. 4. Temperature dependence of \(\operatorname{tg}\delta\) for \(\mathrm{CuSO_4\cdot 5H_2O}\).
Frequencies \(f_1=2\cdot10^6\) Hz; \(f_2=10^7\) Hz.

Distortion of the crystal lattice by impurities and the formation of vacant sites promote the appearance of weakly bound dipoles in crystals; however, this alone does not exhaust the mechanism of the phenomenon, in which, apparently, an important role belongs to structural features.

The maximum of \(\operatorname{tg}\delta=f(\omega)\) in \(\mathrm{CuSO_4\cdot H_2O}\) shifts with increasing temperature toward higher frequencies, which indicates the relaxation character of the process. The experimental curves (Fig. 3) can be used to calculate the activation energy by the formula proposed by Skanavi and Demeshina [15] and first applied to crystal hydrates by Vodopyanov [16]:

\[ U=\frac{kT_1T_2}{T_2-T_1}(\ln f_2-\ln f_1);\qquad u=0.115\ eV. \]

\(f_1\) and \(T_1\) are the frequency and temperature in \({}^{\circ}K\) at which \(\operatorname{tg}\delta\) has a maximum (curve 2 in Fig. 3).

\(f_2\) and \(T_2\) correspond to curve 3 in Fig. 3.

The process of dielectric relaxation is possible only under the condition that the activation energy of the process is less than the activation energy for the conduction process.

We obtained \(U\) approximately an order of magnitude smaller than \(U\) calculated from conductivity, which is indirect confirmation of the correctness of the assumption concerning the role of weakly bound dipoles in explaining the phenomenon of dielectric losses in crystals.

The relaxation character of the dielectric losses in copper sulfate is clearly manifested in the temperature–frequency dependence of \(\operatorname{tg}\delta\). Indeed, the temperature dependence \(\operatorname{tg}\delta=f(T)\) has two maxima (Fig. 4), of which the low-temperature one is double. With increasing frequency the maxima shift toward higher temperatures, which also testifies to the relaxation character of the process. Apparently, the low-temperature maximum corresponds to less firmly bound \(\mathrm{H_2O}\) molecules, while the high-temperature maximum is due to more firmly bound dipoles.

Fig. 5. Temperature dependence of \(\varepsilon\) in \(\mathrm{CuSO_4\cdot 5H_2O}\).
Frequencies \(f_1=2\cdot10^5\) Hz; \(f_2=10^7\) Hz; \(d=1.21\) mm.

In analyzing the temperature dependence of \(\operatorname{tg}\delta\) of copper sulfate, it must be borne in mind that, as the temperature rises, the crystal gives off water of crystallization. Therefore there is a superposition of two processes—dielectric relaxation and dehydration. Thus, in the region of the low-temperature maximum, \(\varepsilon\) has two drops, while in the region of the high-temperature maximum there is one drop (Fig. 5).

Fig. 6. Temperature dependence of \(\operatorname{tg}\delta\) in \(\mathrm{CuSO_4}\).

It remains to verify that the observed effects are not associated with the relaxation of any ions. Indeed, the dependence \(\operatorname{tg}\delta=f(T)\) in \(\mathrm{CuSO_4}\) is characteristic of conductivity losses (Fig. 6). We consider the proposed hypothesis proven.

We do not undertake to explain many details, for example, the relative magnitude of the maxima in the frequency dependence of \(\operatorname{tg}\delta\), but for us this is not the main point. What is important is that the relaxation character of dielectric losses in crystals has been established.

Conclusions

We have obtained new experimental data confirming Vodopyanov’s [17] point of view on the possibility of orientation of polar molecules in crystals with different values of activation energy.

1. Three frequency maxima of the loss angle have been found in copper sulfate. The maxima of $\operatorname{tg}\delta$ are accompanied by a change in $\varepsilon$, although this change is not always sharply pronounced.

2. The absolute magnitude of the dielectric-loss angle in a crystal of $\mathrm{CuSO_4\cdot 5H_2O}$ depends noticeably on the degree of purification of the crystal.

3. A frequency maximum of the loss angle has been found in the monohydrate $\mathrm{CuSO_4\cdot H_2O}$; with increasing temperature the maximum shifts toward higher frequencies.

4. The temperature dependence of $\operatorname{tg}\delta$ of copper sulfate $\mathrm{CuSO_4\cdot 5H_2O}$ has two maxima. The low-temperature maximum is double and corresponds to less firmly bound groups of molecules of water of crystallization. The high-temperature maximum is associated with more firmly bound $\mathrm{H_2O}$ molecules. Both maxima shift with frequency toward higher temperatures.

On the other hand, the dielectric constant in the region of the maxima of $\operatorname{tg}\delta$ decreases sharply. Three stages of decrease in the dielectric constant with increasing temperature are observed; two of them are located in the region of the low-temperature maximum of $\operatorname{tg}\delta$, and the last (small) one in the region of the high-temperature maximum.

Thus, the temperature dependence of the loss angle reflects two processes occurring in copper sulfate at high frequency with increasing temperature—dielectric relaxation and dehydration.

1. The molecules of the water of crystallization that are part of copper sulfate can orient themselves in an external field with three different values of the activation energy. This conclusion is supported by the following additional considerations:

1) according to Menshutkin, the dehydration of copper sulfate takes place in three stages;

2) the structural formula of copper sulfate indicates that there are three groups of $\mathrm{H_2O}$ dipoles, bound in different ways in the complex molecule $\mathrm{CuSO_4\cdot 5H_2O}$;

3) the observed effects cannot be associated with the relaxation of any ions, since in the anhydrous salt $\mathrm{CuSO_4}$ there are no such effects.

I express my gratitude to K. A. Vodop’yanov, under whose guidance and at whose suggestion the present work was carried out.

REFERENCES

1. P. P. Kobeko. The Amorphous State. Publishing House of the Academy of Sciences of the USSR, 1952.

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3. F. M. Gol’tsman. Author’s abstract of Candidate dissertation. Leningrad University, 1952.

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5. Ya. I. Frenkel. Acta Physicochim. USSR, 3, 23, 1935; ZhETF, 6, 902, 1936.

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7. G. I. Skanavi. Physics of Dielectrics. Gostekhizdat, 1949.

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11. K. A. Vodop’yanov. Proceedings of the 1st Regional Conference of Physicists of Western Siberia, issue 3, p. 1, Tomsk, 1934.

12. K. A. Vodop’yanov and V. F. Ivlev. ZhTF, 8, 1521, 1938.

13. Ya. K. Syrkin and Dyatkina. The Chemical Bond and the Structure of Molecules. Goskhimizdat, 1946.

14. B. N. Menshutkin. Course of General Chemistry. Goskhimizdat, 1933.

15. G. I. Skanavi and A. I. Demeshina. ZhETF, 19, 3, 1949.

16. K. A. Vodop’yanov. DAN SSSR, 84, 919, 1952.

17. K. A. Vodop’yanov. ZhTF, 19, 1067, 1949.

Karaganda,
Mining Institute

1. At low and audio frequencies the dielectric permittivity of copper sulfate was studied by Errera [8], and the dependence of the loss angle and dielectric permittivity on temperature at technical frequency was first investigated by A. P. Izergin [9]. At high frequency, as far as we know, copper sulfate has not been studied. ↩