ELECTRICAL AND MECHANICAL STRENGTH AND MOLECULAR FORCES

A. F. Ioffe, Leningrad.

The electrical theory of solids developed by Born and his collaborators, as is well known, quite correctly describes a number of properties of crystalline bodies; namely, it gives the constants of density, elasticity, the dielectric constant, and also the symmetry of the crystal. However, the cohesive forces calculated from the theory exceed by hundreds of times the strength observed in practice. For example, a specimen of crystalline rock salt breaks when the tensile stress reaches \(0.4\ \mathrm{kg/mm^2}\), whereas according to the electrical theory of cohesive forces one should expect a stress equal to \(200\ \mathrm{kg/mm^2}\).

If we attempt to destroy a crystal by electrical forces, then, as soon as the field strength inside the crystal reaches approximately \(300\,000\ \mathrm{V/cm}\), breakdown occurs, whereas the electrical theory shows that only at fields of the order of \(100\,000\,000\ \mathrm{V/cm}\) will the field strength be able to overcome the cohesive force and tear ions out of the crystal lattice, which would lead to the rapid destruction of the crystal.

As regards mechanical strength, the contradiction between the enormous cohesive forces and the small values of strength can be explained by the fact that rupture does not occur simultaneously over the entire cross section of the crystal. The reason for this could be sought, presumably,

A. F. IOFFE

Griffith’s hypothesis consists in the presence of the smallest cracks in the crystal. Near the sharp edges of these cracks the stresses may exceed the mean stress hundreds of times and lead to further growth of the cracks.

The gradual growth of a crack through the entire cross-section of the crystal leads to the same consequences as the simultaneous rupture of the crystal over the whole cross-section. At the same time, it is clear that the first mechanism requires considerably smaller forces for the destruction of the crystal than the second.

In a properly formed single crystal one may expect cracks only on the outer surface. Experiment has shown that on fresh surfaces such inhomogeneities do not arise instantaneously, but with the passage of time. If experiments are conducted under conditions of continuous, sufficiently rapid removal of the surface layer of the crystal, then one may expect that the formation of surface cracks will be hindered, and thereby the cause of the reduction in mechanical strength will be partially eliminated. This idea was realized by M. A. Levitskaya and me by placing the crystal being subjected to tension into hot water. Thus, throughout the experiment its surface was continuously dissolved. It turned out that the parts of the crystal immersed in water possess considerably greater strength than the dry parts. Even in those cases when the cross-section of the dry part exceeded the cross-section of the wet part by 20 times, rupture always occurred in the dry part. Only when the wet cross-section of the specimen, by gradual dissolution, was brought to a very small fraction (from $\frac{1}{100}$ to $\frac{1}{400}$) of the dry one did rupture occur in the part of the specimen immersed in water. In this way values of tensile strength from 30 to 160 kg/mm² were attained, which are already close to the theoretical value of the cohesive force.

Against these experiments it could have been objected that by the moment of rupture the crystal had undergone considerable plastic deformation, and therefore the measured high values of strength pertain not to a single crystal, but to a polycrystalline formation that had undergone hardening.

In order to eliminate this objection, we carried out experiments under several different conditions. In doing so we avoided contact of the surface of the specimen with water, since one cannot be completely certain that water, as such, has no influence on the mechanical properties of the crystal; moreover, we used a theoretically simpler type of deformation: namely, instead of uniaxial tension we applied hydrostatic tension.

A sphere of rock salt was carefully cooled in liquid air and then suddenly immersed in boiling water or molten lead. In this case, between the outer heated shell and the inner cold core there arises a tension which, within several seconds, reaches a maximum at the center of the sphere.

This maximum value of the hydrostatic tension does not depend on the diameter of the sphere and, upon immersion in boiling water, reaches 25 kg/mm², and in molten lead—70 kg/mm². Since the surface of the sphere is free, radial stresses cannot arise at the surface of the sphere, and the behavior of the sphere under the conditions described does not depend on the condition of its surface. Thus the influence of cracks is eliminated, although the cracks themselves do exist. In addition to the radial stresses, which at the surface of the sphere, as we have seen, are equal to zero, there is also a tangential pressure, which, although it does not vanish, also cannot lead to rupture of the hot surface.

The experiments showed that sudden heating, if measures are taken so that the entire surface is heated, did not produce cracks in the crystal. Rock salt withstood, without being destroyed, a tensile force of up to 70 kg/mm² (7,000 Atm) in those cases where, owing to the corresponding distribution of stresses, the influence of surface cracks was eliminated. From this we conclude that the cohesive forces of rock salt do indeed have the large values that are obtained from the electrical theory of solids. The too small value of the usually observed strength limit can be attributed entirely to the influence of surface defects.

In an analogous way it has also been possible to explain the contradiction between the values of the theoretical and practical electrical strength. Here, too, the excessively small breakdown voltages are explained by secondary effects; when these are eliminated, it has been possible to observe the true theoretical strength, exceeding by a factor of 500 the strength observed under ordinary conditions. The causes of premature breakdown may be regarded as: 1) the heat released when current passes; 2) the formation of new ions during the motion of a small number of primary ions (impact ionization); 3) nonuniformity of the field (local overvoltages); and 4) the growth of cracks in the crystal owing to mechanical stresses caused by the electric field.

The last two causes of breakdown have been studied in our laboratory by A. F. Walter and L. D. Inge, G. Kh. Gorovits and B. Ya. Pines, and published in Russian and foreign journals.

The first interpretation of the phenomenon of breakdown belongs to K. Wagner and served as the subject of numerous disputes in technical circles. All dielectrics have a large temperature coefficient of electrical conductivity. Therefore the heat produced in a dielectric when current passes through it increases its electrical conductivity and thereby increases the current flowing through the dielectric, which entails a new release of heat and a rise in temperature. The temperature of the dielectric therefore rapidly increases until an equilibrium is established between the release of heat inside the dielectric and the loss of heat to the outside. If this equilibrium state is reached at a temperature below the melting temperature, then the result of the phenomenon described will be a certain heating of the dielectric and the consequent increase in the energy lost in the dielectric. However, if thermal equilibrium cannot be reached below the melting temperature, melting, and at times partial evaporation, of the dielectric occurs.

It is clear that, under such a conception, one may expect a very strong dependence of the breakdown voltage on temperature. The closer the temperature of the experiment is to the melting temperature,

the smaller the voltage required for breakdown. A calculation carried out at our institute by V. A. Fock and N. N. Semenov and, at the same time, by Rogowski and Karman in Aachen shows that the breakdown voltage in the interval from the melting temperature down to the very lowest temperatures must increase exponentially as the temperature is lowered, and the exponent in the temperature dependence of the breakdown voltage must be approximately equal to one half of the corresponding exponent in the temperature dependence of the electrical conductivity, i.e.

\[ \lg V_a = \frac{L}{T} + B, \tag{1} \]

\[ \lg \sigma = \frac{A}{T} + B, \tag{2} \]

where \(L \lessgtr \frac{1}{2} A\).

These conclusions, as well as other consequences of the thermal theory of breakdown, have been confirmed with great accuracy by experiments carried out by A. F. Walter, L. D. Inge, and N. N. Semenov with glass, rock salt, and porcelain in the region of high temperatures. It turned out that, upon approaching the melting temperatures, the breakdown voltage falls to 50–100 V. The totality of the experiments proves beyond doubt that breakdown of rock salt at temperatures above \(200^\circ\) C occurs entirely at the expense of the heat released in the dielectric. The dependence of the logarithm of the breakdown voltage on the reciprocal of the absolute temperature \(\frac{1}{T}\) in the interval from \(200^\circ\) to \(800^\circ\) C is expressed by a straight line whose slope is approximately equal to one half the slope of the straight line expressing the dependence of the electrical conductivity on the same quantity (Fig. 1).

At \(200^\circ\) C the slope of the straight line changes abruptly, and down to the temperature of liquid air the breakdown voltage does not depend on temperature. This circumstance indicates that here the cause of breakdown has nothing in common with the release of heat and the melting of the dielectric. In this

in the temperature interval, breakdown occurs as soon as the field reaches a certain definite magnitude, of the order of several hundred thousand volts per centimeter.

My investigations of the electrical conductivity of crystals led me to one case where fields a hundred times greater did not result in breakdown. The passage of current through calcite and certain other dielectrics changes the distribution of potential inside the dielectric in such a way that the entire potential drop falls across a layer of very small thickness near one of the electrodes. The thick-

Fig. 1.

ness of this layer was determined and proved to be from 1 to 10 $\mu$, while the potential drop reaches several thousand volts. Consequently, the field strength inside this layer reaches $10^7\ V/\mathrm{cm}$.

After we had convinced ourselves that the increase of the breakdown voltage in such layers cannot be explained by their high specific resistance, we turned our attention to the small thickness as a possible cause of the increased strength of polarization layers. Just as the rapid fall of the breakdown voltage on approaching the melting temperature serves as incontrovertible proof of the existence of thermal breakdown, so too the increase of the electrical strength in the region of very

small thicknesses may be regarded as an indication of the existence of impact ionization.

Indeed, let us consider a dielectric in which an electric field \(\mathcal{E}\) has been created, sufficient that, in the presence of a certain number of free ions, impact ionization begins. In order to accumulate the energy necessary for ionization, an ion must traverse some distance \(\lambda\) and some potential difference \(P\). Denoting by \(D\) the thickness of the dielectric and by \(V\) the applied potential difference, we may expect that the number

Fig. 2.

of ionizations by collision which an ion will produce on its path from one of the electrodes to the other will be expressed in the form

\[ Z=\frac{D}{\lambda}=\frac{V}{P}. \tag{3} \]

Each ionization doubles the number of ions. Therefore each \(N_0\) ions moving from one of the electrodes on their way produce \(N\) new ones, where

\[ N=N_0\,2^z. \tag{4} \]

A calculation for the case of free ions uniformly distributed throughout the whole volume of the dielectric in the amount \(n_0\)

ions in \(1\ \mathrm{cm}^3\), gives the mean density of \(n\)-ions in the form

\[ n=n_0\frac{P}{V}\left(e^{\frac{V}{P}}-1\right). \tag{5} \]

So long as \(z\) remains small, impact ionization has no other consequences besides an increase in the current strength and a distortion of the potential distribution. For example, for

\[ \begin{aligned} z&=5 \qquad n \simeq 20\,n_0,\\ z&=10 \qquad n \simeq 4000\,n_0,\\ z&=20 \qquad n \simeq 10^8\,n_0, \end{aligned} \]

whence it is seen that approximately at \(z=15\) the dangerous limit is reached.

The value of the path \(\lambda\), over which an ion accumulates energy sufficient for ionization, may be estimated as \(10^{-5}—10^{-6}\ \mathrm{cm}\), i.e. approximately several hundred molecular distances. Therefore, if the thickness of the dielectric is \(D=1\ \mathrm{mm}\), the number of ionizations reaches

\[ Z=\frac{V}{P}=\frac{D}{\lambda}=10^4, \]

and the number of newly formed ions

\[ n \simeq e^{10000} n_0. \]

It is perfectly clear that under such conditions impact ionization, once begun, inevitably leads to breakdown. The condition for breakdown in dielectric layers whose thickness is large in comparison with \(\lambda\) is the attainment in the dielectric of a field sufficient for the occurrence of impact ionization.

The matter is quite different in layers whose thickness is not too large in comparison with \(\lambda\). Here everything is determined by the value of \(n\). As the condition for breakdown we put forward the attainment of some definite mean density \(n_x\) of ions within the dielectric. Formula (5) shows that breakdown must occur at a definite \(\frac{V}{P}\).

Assuming at first that \(P\) depends neither on \(V\) nor on \(\mathcal{E}\), we see that the breakdown voltage should not depend

...on the thickness \(D\), and the mean field strength \(\mathfrak{E}=\dfrac{V}{D}\) must be inversely proportional to it.

Experiments carried out by K. D. Sinelnikov, I. V. Kurchatov, and myself with thin layers of glass, mica, rosin, sealing wax, oil, and benzol gave the results shown in Figs. 3 and 4.

Fig. 3.

According to these results, we must regard layers with thickness greater than \(5\,\mu\) as thick in comparison with the length \(\lambda\), whereas for layers of thickness less than \(5\,\mu\), formula (5) is applicable. However, the breakdown voltage is not

Fig. 4.

directly connected with impact ionization, the fundamental premise of which is the exponential dependence of the number

newly formed ions on \(\dfrac{V}{P}\) or \(\dfrac{D}{\lambda}\). The number of ions is determined directly from the strength of the current flowing through the dielectric. Therefore we turned to measuring the current \(J\) as a function of the field strength \(\mathfrak{E}\) and the thickness \(D\). Experiments showed that only a few seconds are sufficient for the electrical conductivity of the dielectric to decrease noticeably owing to the heat evolved in the dielectric. Therefore the voltage was applied for only 0.1 sec., with intervals of 2 sec. The current curves, with stepwise raising of the voltage, were taken with a string galvanometer with photographic recording. The results are given in Fig. 5.

Fig. 5.

Before the onset of impact ionization, i.e. approximately up to fields of the order of \(3 \times 10^{5}\ \mathrm{V/cm}\) (for glass), Ohm’s law holds. Above this limit the increase of current with field is the more rapid, the greater the thickness, as a consequence of the relation:

\[ \lambda = \frac{V}{P} = \frac{\mathfrak{E}D}{P}. \tag{6} \]

Each point of the curves given makes it possible to calculate both \(P\) and \(\lambda\). The curve corresponding to the initial number of ions \(n_0\) would have to run, after the onset of impact ionization, approximately horizontally. The ratio of the current \(J\), corresponding to a definite field strength \(\mathcal{E}\), to the current \(J_0\) that would flow in the dielectric if the number of ions remained equal to \(n_0\), gives the relative increase in ion density.

\[ \frac{J}{J_0}=\frac{n}{n_0}. \tag{7} \]

After this it remains to compute \(P\) from formula (5). First, according to Fig. 6, \(P\) turns out to be a function of \(\mathfrak{E}\), which, incidentally, was to be expected. The increase in the kinetic energy of the ion occurs along the path \(\lambda\), which contains a large number of atomic distances. Therefore the motion of the ion cannot fail to be accompanied by a loss of energy. The higher the field \(\mathfrak{E}\), the shorter the path and, consequently, the smaller the loss of energy. Therefore a rather small potential difference \(P\) is sufficient for the ion to acquire the energy \(eP_0\) needed for ionization. At very large fields \(P\) approaches \(P_0\), while the path \(\lambda\) reaches a length of only a few atomic distances.

Fig. 6.

In order to substantiate a theory that is correct not only qualitatively but also quantitatively, we would have to take into account also the distribution of the volume charges arising as a result of impact ionization and causing a redistribution of the potential inside the dielectric.

In the region of impact ionization, the mobility of the ions likewise does not remain constant, but is a function of the field strength; however, all these corrections, as well as more

detailed description of the mechanism of impact ionization will serve as the subject of a more extensive communication.

Here we shall consider only one question: what maximum values of the electric field can be attained in going over to ever thinner layers of a dielectric? We have succeeded in obtaining glass films with a thickness down to \(0.014\,\mu\) and mica sheets down to \(0.05\,\mu\). At the same time, however, it turned out that already at a thickness of \(0.2\,\mu\) it is possible to attain a field of \(150\,000\,000\ \mathrm{V/cm}\), and with a further decrease

Fig. 7.

of the thickness by a factor of 15 the breakdown gradient remains constant. This is especially clearly seen in logarithmic coordinates (Fig. 7).

Using the most varied electrodes (brass, mercury, soot, India ink, water, air ions), we became convinced that the material of the electrodes has no significance whatsoever and that the cause of breakdown lies within the dielectric.

We assume that in this new region, where the breakdown voltage does not depend on the thickness, the true electrical strength of the dielectric is manifested. It indeed has the magnitude predicted by theory, about \(10^8\ \mathrm{V/cm}\).

The elastic energy of an ion in a field of \(1.5 \times 10^8\ \mathrm{V/cm}\) reaches \(10^{-12}\) erg, which corresponds to the kinetic energy of a gas molecule at \(1600^\circ_{\mathrm{abs}}\), i.e., at the melting temperature of glass. There is nothing surprising in the fact that, under these conditions, an electric field can tear out ions fixed in the lattice.

In the very instructive table given on p. 151, certain data relating to the fields of \(1.5 \times 10^8\ \mathrm{V/cm}\) now attainable are compared with the corresponding data for limiting fields under ordinary conditions, i.e., for \(3 \times 10^5\ \mathrm{V/cm}\).

The possibility of obtaining within a dielectric electric forces comparable in magnitude with cohesive forces and with chemical forces makes it possible to solve a number of both theoretical and purely practical problems.

At present we are studying: the dielectric constant, elasticity, residual rays, the Stark effect, surface forces, etc., in dielectrics at limiting values of electric fields. The phenomenon described has been made the basis for the manufacture of a new type of insulators, capacitors, etc.