“Artificial” Dielectrics

An interesting set of results is reported in work [1], obtained in the study, in the centimeter-wave region, of the electrical properties of compounds consisting of paraffin wax and small metallic particles immersed in it.

The authors of [1] call such materials “artificial” dielectrics, thereby emphasizing that the electrical properties of such a compound differ sharply from the properties of the basic (binding) dielectric—paraffin. In essence, these “artificial” dielectrics are a “macroscopic reproduction” of ordinary dielectrics. (It is known that one of the fundamental formulas of the physics of dielectrics—the Clausius–Mossotti formula—was originally obtained by considering a model of a dielectric consisting of conducting spheres separated from one another by an insulating medium.)

To measure the indicated materials at a frequency of 9364 MHz ($\lambda = 3.2$ cm), a waveguide method [2] was used, which consists, in its essential features, of the following. The specimen for measurement was made in the form of a plate and placed in a rectangular waveguide, where its input impedance was determined (by measuring the standing-wave coefficient $k$ and the shift in the position of the standing-wave minimum $x_0$) for two positions of the specimen:

1) short-circuited—$z_{\text{k.z}}$ and 2) “open-circuited,” when the specimen is located at a distance $\frac{\lambda}{4}$ from the short-circuited end of the line—$z_{\text{raz}}$. Further, having found $z_{\text{k.z}}$ and $z_{\text{raz}}$, they determined the propagation constant $\gamma$ and the characteristic impedance of the specimen—$z_0$, and from these two quantities calculated the dielectric constant, as well as the magnetic permeability (their real—$\varepsilon'$, $\mu'$—and imaginary—$\varepsilon''$, $\mu''$—parts).

Measurements were carried out on specimens of paraffin with powders of metals: zinc, copper, aluminum, carbonyl iron, magnetite ($\mathrm{Fe_3O_4}$), and also carbon; in addition, specimens were prepared and measured,

paraffin with powder of the dielectric rutile, which has a very high dielectric constant \((\varepsilon \simeq 100)\).

Some of the results reported by the authors\(^1\) are given in the table.

Samples of paraffin wax (without any inclusions) had, at \(\lambda = 3.2\) cm, the following electrical parameters:

\[ \varepsilon = 2.25,\quad \operatorname{tg}\delta < 0.0002. \]

An investigation of the dependence of the dielectric constant of these materials on the concentration of metallic powder \((\theta)\) showed that, with increasing concentration, \(\varepsilon\) increases strongly (Figs. 1, 2). (Fig. 1 refers to inclusions of Zn, Fig. 2 to Al.)

Fig. 1.

Fig. 2.

Compounds with aluminum powder possess the highest dielectric constant (at equal concentrations); \(\varepsilon\) of samples with zinc powder is greater than that of samples with copper powder, but less than with aluminum. This is probably explained by the fact that, as the authors’ microscopic investigations showed, the aluminum particles had the form of elongated chains; the copper powder consisted of lumps of irregular shape, while the zinc particles had a shape close to spherical.

The polarizability of the first (for equal transverse dimensions) is greater than that of the second, and that of the second is greater than that of the third.

The experimental results were compared with Levin’s theoretical formulas^3 for \(\varepsilon\) and \(\mu\) of a heterogeneous system consisting of a base material \((\varepsilon_1,\mu_1)\), in which spherical inclusions with electrical characteristics \(\varepsilon_2,\mu_2\) are arranged symmetrically at the vertices of cubes (\(\theta\) is the volume concentration of the inclusions):

\[ \varepsilon=\varepsilon_1\left[1+\frac{3\theta}{\dfrac{\varepsilon_{\mathrm{eff}}+2\varepsilon_1}{\varepsilon_{\mathrm{eff}}-\varepsilon_1}-\theta}\right], \tag{1} \]

\[ \mu=\mu_1\left[1+\frac{3\theta}{\dfrac{\mu_{\mathrm{eff}}+2\mu_1}{\mu_{\mathrm{eff}}-\mu_1}-\theta}\right]. \tag{2} \]

In formulas (1) and (2), \(\varepsilon_{\mathrm{eff}}\) and \(\mu_{\mathrm{eff}}\) are, respectively, the effective dielectric constant and effective magnetic permeability of the spherical inclusions. For ultrahigh-frequency fields, \(\varepsilon_{\mathrm{eff}}\) and \(\mu_{\mathrm{eff}}\) differ from \(\varepsilon_2\) and \(\mu_2\):

\[ \frac{\varepsilon_{\mathrm{eff}}}{\varepsilon_2} = \frac{\mu_{\mathrm{eff}}}{\mu_2} = \frac{2\sin\xi-\xi\cos\xi}{(\xi^2-1)\sin\xi+\xi\cos\xi}, \tag{3} \]

where

\[ \xi=\frac{2\pi a}{\lambda}\sqrt{\varepsilon_2\mu_2}, \]

\(a\) is the particle radius, and \(\lambda\) is the wavelength in free space.

It is interesting to note that formula (1) coincides exactly—only written somewhat differently—with the well-known formula of Odelevskii,^4 obtained by an entirely different method for the dielectric permittivity of a heterogeneous system:

\[ \varepsilon=\varepsilon_1\left(1+\frac{\theta}{\dfrac{1-\theta}{3}+\dfrac{\varepsilon_1}{\varepsilon_2-\varepsilon_1}}\right) \tag{4} \]

when \(\varepsilon_{\mathrm{eff}}=\varepsilon_2\), i.e., when \(\xi\to0\), or for very large \(\lambda\).

The theoretical and experimental analysis carried out in work^1 shows that the losses in such “artificial” dielectrics, formed by well-conducting particles, are predominantly magnetic in character, while the dielectric losses are very small.

The paper under review shows that it is possible to obtain “artificial” dielectrics which, in the centimeter-wave region, possess very high values of dielectric permittivity (\(\varepsilon\simeq60\), Fig. 2) and small dielectric losses, provided that the powder-like metallic particles are sufficiently small and have high electrical polarizability.

V. Sarafanov

References Cited

1. K e l l y, S t e n o i e n, I s b e l l, J. Appl. Phys. 24, No. 3, 258 (1953).
2. B i r k s, Proc. Phys. Soc. 60, No. 339, 282 (1948).
3. L e v i n, J. I. E. E. 94, Pt. III, No. 27, p. 65 (1947).
4. O d e l e v s k i i, ZhTF, vol. XXI, issue 6, 667 (1951).