LOW-FREQUENCY AND HIGH-FREQUENCY DIELECTRIC LOSSES IN AMORPHOUS POLYMERS

P. F. VESELOVSKII

The study of the dielectric properties of amorphous polymers is of great scientific and practical interest.

The study of dielectric polarization and losses makes it possible, on the one hand, to establish a connection between the electrical properties and the chemical structure of polymers, and also to draw a number of conclusions about the structure of polymers and the nature of the forces of intermolecular interaction. On the other hand, the investigation of high-frequency dielectric losses is of substantial importance in selecting material for high-frequency insulation and various radio-engineering components operating at temperatures below room temperature. The same may be said with regard to low-frequency dielectric losses, which characterize the behavior of a polymer at low frequencies and elevated temperatures.

Therefore the practical aspect of the matter apparently comes down to a detailed study of the observed phenomena, to the ability to control them, and to obtaining high-frequency materials with predetermined properties.

Investigations carried out under the direction of P. P. Kobeko and G. P. Mikhailov [1, 2, 3] established the existence of two regions of anomalous dispersion of the dielectric constant and, correspondingly, two anomalous regions of dielectric losses in amorphous polar polymers. Subsequently, dielectric losses at high frequencies or at low temperatures (below the softening temperature \(T_g\)) were associated with the thermal motion of a polar radical and were called dipole-radical losses [1, 4]. Dielectric losses observed in the region of the highly elastic state at temperatures above \(T_g\) were associated with the thermal motion of sections of the polymer chain (segments) and were called dipole-elastic losses [1, 4]. A characteristic example of the observed regularities may be the dependence of \(\varepsilon\) and \(\lg \tan \delta\) of polyvinyl acetate (p.v.a.) on temperature, obtained by us at a frequency \(f = 1.6\ \text{kc/s}\) [5], (Fig. 1).

Similar dependences were obtained for all cellulose polyesters investigated by us (diacetate, triacetate, acetobutyrate) and for all samples available to us of the homologous series of acetals of polyvinyl alcohol:

In Fig. 2 is shown the dependence of \(\varepsilon\) and \(\lg \delta\) of p.v.a. on frequency, obtained by us at room temperature [6].

The increase in \(\operatorname{tg}\delta\) for P.V.A. upon transition to low frequencies (Fig. 2) is associated with dipole-elastic losses, which is confirmed by comparison with the temperature dependence of \(\varepsilon\) and \(\operatorname{tg}\delta\) for P.V.A. (Fig. 1). Similar

Fig. 1. \(f = 1.6\ \text{kHz}\), × — \(\varepsilon\); ○ — \(\operatorname{tg}\delta\).

dependences were obtained for all the cellulose esters and polyvinyl alcohol acetals investigated¹. It should be noted that, in the case of dipole-elastic losses, \(\operatorname{tg}\delta\) at the maximum reaches values of \(\sim 10^{-1}\),

Fig. 2. \(t = 20^\circ\text{C}\).

In the case of dipole-radical losses, however, \(\operatorname{tg}\delta\)—in the region of the maximum—is more than an order of magnitude smaller.

The dielectric permittivity in the case of dipole-elastic losses in the region of the maximum changes sharply with the temperature and frequency of the alternating electric field; the change in dielectric permittivity reaches \(\Delta\varepsilon \sim 4\), whereas in the case of dipole-radical losses \(\Delta\varepsilon \sim 1\).

Fig. 3.

The dielectric losses observed in amorphous polar polymers are caused by relaxation processes, as may be judged, for example, from the character of the frequency-temperature dependences of \(\varepsilon\) and \(\operatorname{tg}\delta\), similar to those shown in Fig. 3 for cellulose triacetate.

In Fig. 4 analogous dependences are given, obtained by us for P.V.A. Dipole-radical relaxation losses were found in all the amorphous polymers studied in the present work, with the exception of P.V.A. Low-frequency, or high-temperature, so-called dipole-elastic losses are also caused by relaxation processes. As an example, Fig. 5 gives the dependence of \(\operatorname{tg}\delta\) of P.V.A. on temperature, measured by us at a number of frequencies indicated in Fig. 5.

Dielectric losses of this type were found in all the amorphous polymers studied in the present work.

The relaxation time of dipole-elastic losses depends sharply on the action of various factors, for example temperature, orientation (stretching), the introduction of various components into the polymer composition, etc. [1, 7, 8].

We have investigated the effect of composition on the relaxation time ($\tau$) in copolymers of synthetic butadiene rubber (SKB) with styrene. It turned out that, as the percentage concentration of styrene in the copolymer increases, the relaxation time changes in parallel with the change in the position of the glass-transition temperature $T_g$ of the copolymer. Fig. 6 and Table 1 give the results of these investigations [9].

Fig. 4.

Table 1

The relaxation time of dipole-elastic losses changes sharply with changes in the chemical structure of the polymer. Fig. 7 gives the results of our investigations carried out on polymers of a homologous series of acetals of polyvinyl alcohol. It should be noted that $\tau$ changes approximately in the same way as $T_g$ changes as the side radical becomes more complex (internal plasticization). The increasing dimensions of the nonpolar part of the polar radical contribute to a reduction in the forces of molecular interaction. This corresponds to greater freedom of rotation of the polar groups, a lowering of $T_g$, and a shift of the maximum of $\operatorname{tg}\delta$ into the region of lower temperatures.

The introduction of polar and nonpolar plasticizers acts in the opposite way to the formation of “crosslinks,” shifting the maximum of $\operatorname{tg}\delta$ in different directions. These factors have the same influence on $T_g$ and $\tau$. Some results of our studies are given in Fig. 5 [5].

Fig. 5.

On the basis of the data presented, one may conclude that, in the relaxation process leading to the appearance of dipole-elastic

Fig. 6. $f = 1$ kHz.

losses, considerable segments of the polymer chain participate, so that losses of this type can, to a certain extent, be “controlled.”

High-frequency dielectric losses have been little studied, and as yet there are no established views on the specific molecular mechanism of the process that gives rise to them. In the works of G. P. Mikhailov it was observed that, upon orientation (stretching) of polymers, the relaxation time of this process does not change. A more detailed study of losses of this type showed that the relaxation time does not depend on internal and external plasticization, “cross-linking” of polymer chains, etc.

Fig. 7. \(f = 1.7\) MHz.

Figure 9 presents data on the study of the dependence of \(\operatorname{tg}\delta\) on the chemical structure of the polymer—on the size of the polar group (internal—

Fig. 8. 1—p.v.a.; 2—p.v.a. plasticized with \(40\%\) naphthalene; 3—“cross-linked” p.v.a.

… plasticization). From the figures presented it is evident that increasing sizes of the radicals carrying polar groups do not affect the position of the maximum of $\operatorname{tg}\delta$ on the temperature scale or the relaxation time $\tau$. At the same time (Figs. 9a, 15, and 16), members with an even and an odd number of $\mathrm{CH}_2$ groups form, as it were, two independent families. External plasticization with a polar and a nonpolar plasticizer likewise does not affect the position of the maximum of $\operatorname{tg}\delta$ on the temperature and frequency scales, nor the relaxation time $\tau$,

Fig. 9. a) $f = 20$ kc/s; b) $f = 5$ kc/s.

as can be seen from the data presented in Fig. 10 [6]. From the curves shown in Fig. 10 it is evident that, with increasing concentration of plasticizer, the losses in the region of the maximum are obscured by low-frequency (dipole-elastic) losses; however, when the temperature is lowered they appear again.

Figure 11 presents results on the investigation of the influence of “crosslinking” on dipole-radical losses [6]. From Fig. 11 it is evident that “crosslinking” practically does not affect the position of the frequency maximum of $\operatorname{tg}\delta$, whereas the maximum of $\operatorname{tg}\delta$ for dipole-elastic losses (Fig. 8) shifted considerably into the region of higher temperatures or lower frequencies. On the basis of the results presented, one may conclude that in the relaxation process associated with dipole-radical losses the polar groups of a single monomer unit participate. Since the relaxing element is small, losses of this type cannot yet be “controlled.” Only by decreasing the concentration of polar groups per unit volume can $\operatorname{tg}\delta$ be reduced; however, $\tau$ in this case remains unchanged. Since dielectric losses of both types are caused by relaxation processes, on the basis of existing concepts and methods of treatment

of the experimental results, it is of interest to consider the dependences of \(\lg f\) on \(\frac{1}{T}\), where \(f\) and \(T\) are the frequency and temperature at which the maximum of \(\operatorname{tg}\delta\) is observed. Since, for the maximum of \(\operatorname{tg}\delta\), the approximate equality \(f \cdot \tau \cong 1\) holds, where \(\tau\) is the most probable relaxation time [10], the experimental data can also be represented in the form of the dependence of \(\lg \tau\) on \(\frac{1}{T}\).

Fig. 10.

a) Dependence of \(\operatorname{tg}\delta\) of PVA, plasticized with naphthalene, on frequency at \(t = 20^\circ\text{C}\). The numbers on the curves correspond to the percentage concentration of naphthalene.

b) Dependence of \(\operatorname{tg}\delta\) of PVA, plasticized with 30–40% naphthalene, on frequency at different temperatures.

The study of relaxation processes in polymers has until now been carried out over a very narrow frequency range and temperature interval. This circumstance made it impossible to compare and analyze the dependences of \(\lg \tau\) on \(\frac{1}{T}\) for both types of relaxation.

In Figs. 12, 13, and 14 are presented the dependences of \(\lg \tau\) on \(\frac{1}{T}\), obtained

by us for P.V.A., the homologous series of polyvinyl alcohol acetals, and cellulose ethers. From the figures presented it is evident that the dependences of \(\lg \tau\) on \(\frac{1}{T}\) for the high-frequency dielectric losses of all the amorphous polymers investigated in the present work represent a well-satisfied linear dependence, which can be described by a function of the form [6].

Fig. 11. Dependence of \(\operatorname{tg}\delta\) of “cross-linked” P.V.A. on frequency at \(t = 20^\circ\mathrm{C}\).

1 — P.V.A.; 2 — “cross-linked” P.V.A.

\[ \tau = \tau_0 e^{\frac{\Delta U}{kT}}, \tag{1} \]

where \(\tau\) is the relaxation time characteristic of the relative majority of polar groups, by whose motion the dielectric losses are determined (the time of “sedentary life” of a polar group in an equilibrium position); \(\tau_0\) is the time (period) of one natural oscillation of this group; \(\Delta U\) is the potential barrier separating equilibrium positions. The rectilinear

Fig. 12. 1. For dipole-radical losses of P.V.A. 2. For dipole-elastic losses of P.V.A. 3. For dipole-elastic losses of P.V.A. plasticized with 40% naphthalene.

the dependence of $\lg \tau$ on $\dfrac{1}{T}$ confirms the exponential character of the dependence of $\tau$ on $T$, and proves that the activation energy $\Delta U$ does not depend on temperature and can be calculated as the tangent of the angle of inclination $\Delta \lg \tau / \Delta \dfrac{1}{kT}$.

Fig. 13a. Dependence of $\lg \tau$ on $\dfrac{1}{T}$ for acetals of polyvinyl alcohol with an even number of $\mathrm{CH}_2$ groups in the radical of the polymer chain. $\times$ and $\circ$ refer to dipole-radical losses for P.V.E., P.V.B., and P.V. hexylal. The symbols Э, Б, О refer to dipole-elastic losses. (P.V.E., P.V.B., and P.V.O.).

Fig. 13b. $\times$ — obtained from studies of dielectric losses of P.V.P.; ● — P.V.F. — materials with an odd number of $\mathrm{CH}_2$ groups in the radical of the polymer chain.

For high-frequency dielectric losses, $\Delta U \simeq 8—11\ \text{kcal/mol}$ ($\Delta U$ P.V.A. $= 8.6\ \text{kcal/mol}$, $\Delta U$ P.V.E.; P.V.B. and P.V.G. $= 10\ \text{kcal/mol}$, $\Delta U$ P.V.P. $= 11\ \text{kcal/mol}$, $\Delta U$ cellulose polyesters $= 10.4\ \text{kcal/mol}$).

Fig. 14. $\times$ — obtained from studies of dipole-elastic and dipole-radical losses of cellulose triacetate, $\bigcirc$ — cellulose acetobutyrate.

If it is assumed that in expression (1) \(\tau_0\) does not depend on temperature, and this expression is extrapolated to \(T=\infty\), then \(\tau_0\) is found from the plot of the dependence of \(\lg \tau\) on \(\frac{1}{T}\) at \(\frac{1}{T}=0\). It is of interest to note that the \(\tau_0\) determined from the relaxation losses under consideration proves to be of the order of \(10^{-13}\) sec for PVA and all members of the homologous series of acetals of polyvinyl alcohol, and \(\tau_0 = 10^{-12}\) sec for cellulose polyesters.

The dependences of \(\lg \tau\) on \(\frac{1}{T}\) for low-frequency dielectric losses for all the amorphous polymers studied, as is seen from Figs. 12 and 13, are curvilinear. As the temperature increases, the radius of curvature increases, so that in the limit they tend toward the straight lines of dipole-radical losses. At a temperature considerably above the softening temperature of the polymer \(T_g\), the experimentally obtained values of \(\lg \tau\) satisfactorily fall on the straight line of dipole-radical losses. The curvilinear dependence of \(\lg \tau\) on \(\frac{1}{T}\) for dipole-elastic losses may be connected with the fact that the activation energy \(\Delta U\), in contrast to dipole-radical losses, is some function of temperature and cannot be calculated from the slope \(\Delta \lg \tau / \Delta \frac{1}{kT}\). Extrapolating the dependence of \(\lg \tau\) on \(\frac{1}{T}\) to \(\frac{1}{T}=0\), one can determine the natural frequency of oscillations of the relaxing element. For low-frequency dielectric losses it is equal to \(10^{12} - 10^{13}\) \(1/\)sec. The results obtained for the natural frequency of oscillations of the relaxing element agree satisfactorily with the value of the pre-exponential factor in the dependence of \(\tau\) on \(T\), calculated according to the theory of Frenkel or Eyring. Extrapolating the dependence of \(\lg \tau\) on \(\frac{1}{T}\) to \(\lg \tau = 0\), one can determine \(T_g\). The \(T_g\) values obtained in this way agree satisfactorily with the results of other measurements published in the literature. The curvilinear dependence of \(\lg \tau\) on \(\frac{1}{T}\) for dipole-elastic losses, it seems to us, can be explained by the fact that, with increasing temperature, the size of the relaxing element of the polymer chain decreases and tends in the limit to the size of a monomer unit. Therefore the curvilinear dependence of \(\lg \tau\) on \(\frac{1}{T}\) for dipole-elastic losses, with increasing temperature, tends toward the straight line of dipole-radical losses.

In Figs. 15 and 16 the results of our subsequent processing of the experimental data are presented. Fig. 15 shows the dependence of \(\tan \delta\) in the region of the maximum of dipole-elastic and dipole-radical losses on the number of carbon atoms in the radical. From the figure presented it is seen that, for the first members of the homologous series of acetals of polyvinyl alcohol, characteristic notches are observed. Similar dependences were also obtained at other frequencies used in the present work. Fig. 16 gives the dependence of the temperature at which the maximum of \(\tan \delta\) is observed at a frequency \(f = 1.7\) MHz on the number of carbon atoms in the radical of polyvinyl acetals. As is seen from Fig. 16, what appear to be two intersecting curves are formed. Applying S. N. Zhurkov’s theory of local nodes and the hardening of polymers [11], this result can be interpreted as follows: if, in PVF, the formation of a local node is associated with interaction in the plane of a six-membered ring consisting of 4 carbon atoms and 2 oxygen atoms, then upon transition to PVAc it is strongly com-

...the influence of the \(\mathrm{CH}_3\) group is manifested; as is known, it always leads to a sharp increase in \(T_g\). For example, there is the transition from normal compounds to iso-compounds, from polystyrene to polymethylstyrene, etc. The transition from P.V.E. to P.V.P. leads to the appearance of an uncompensated polar \(\mathrm{CH}_2\) group [4], which may be responsible for such a large value of \(\operatorname{tg}\delta\) in the region of the maximum, as compared with the other members of the homologous series of acetals of polyvinyl alcohol, and also explains the position of the \(\operatorname{tg}\delta\) maximum on the temperature scale, close to the temperature of \(\operatorname{tg}\delta_m\) for P.V.B.

Fig. 15. \(\times\) — dipole-elastic losses, \(f = 1.7\) MHz; \(\bullet\) — dipole-radical losses, \(f = 20\) kHz.

In what follows, on passing from P.V.P. to P.V.B., P.V.G., and to P.V.O., these effects are smoothed out; internal plasticization causes only a decrease in the concentration of dipoles per unit volume and the associated decrease in the relaxation time, and consequently also in heat resistance. The assumption put forward here requires additional verification. In particular, it should be taken into account that hydroxyl groups are present in all the polymers considered, and we had no information on their amount or concentration per unit volume. However, it should be borne in mind that hydroxyl groups cannot explain the phenomena that were found in the investigation of P.V.A., polyvinyl acetals, and cellulose polyesters. Moreover, infrared spectroscopy data show that, for example, in the esterification of cellulose, a large part of the hydroxyls proves not to be included in hydrogen bonding, and that the interaction is essentially reduced to interaction of the acetyl groups [12].

The study of the dielectric losses of polar polymers over a wide temperature interval and frequency range, as well as comparison of the results obtained, makes it possible to elucidate certain regularities inherent in molecular relaxation processes in polymers, to come closer to an understanding of the specific molecular mechanism responsible for the dielectric properties of a polymer, and to establish the relation between the dielectric properties of a substance and its chemical structure.

Fig. 16. Dependence of the temperature at which the maximum of $\tg\delta$ is observed on the number of carbon atoms in the side radical of the polymer chain, $f = 1.7$ MHz.

At present, work is continuing on the investigation of the dielectric polarization and losses of amorphous polymers, in particular “cross-linked” polymers, as well as on the systematization and processing of the results obtained.

REFERENCES

1. P. P. Kobeko. Amorphous Substances, Moscow–Leningrad, Publishing House of the USSR Academy of Sciences, 1952.
2. P. P. Kobeko, G. P. Mikhailov, and Z. I. Novikova. ZhTF, 29, 3, 1949.
3. A. P. Votinov, P. P. Kobeko, and G. P. Mikhailov. ZhTF, 14, 18, 1944.
4. G. P. Mikhailov. Dissertation, Leningrad, 1950.
5. P. F. Veselovskii and A. I. Slutsker. ZhTF, 25, issue 7, 1955.
6. P. F. Veselovskii and A. I. Slutsker. ZhTF, 25, issue 5, 1955.
7. G. P. Mikhailov. ZhTF, XXI, 1935, 1951.
8. G. P. Mikhailov and B. M. Feinshtein, ZhTF, XXII, 759, 1952.
9. P. F. Veselovskii. Proceedings of the Kalinin LPI, No. 181, 1955.
10. P. Debye. Polar Molecules, GTTI, 1931.
11. S. N. Zhurkov. Proceedings of the First and Second Conferences on High-Molecular Compounds, Publishing House of the USSR Academy of Sciences, 1945.
12. Nikitin. ZhFKh, issue 23, 1949.

Leningrad
Polytechnic Institute

Questions and Answers

R. L. Müller.—In the case under consideration, it is necessary, along with measurements of $\operatorname{tg}\delta$, to make measurements of heat capacities. Since the matter concerns oscillations of radicals, the heat capacities must be measured after converting them to molecular heat capacities. Zhurkov et al. measured the heat capacity in the region where torsional–oscillatory motions are switched off, and in doing so obtained a jump in heat capacity.

P. F. Veselovskii.—I agree that our measurements are insufficiently accurate. Measurements are needed not only of heat capacity, but also data from X-ray structural analysis. We are studying these questions and shall continue to do so. We are interested in the question of the statistical theory of thermal motion.

1. In P.V.A. high-frequency losses could not be detected. ↩