DIELECTRIC PROPERTIES OF PURE LIQUIDS*

P. Debye

1. Understanding the dielectric properties of gases (at sufficiently low pressures) presents no substantial difficulties if one takes into account that molecules are not only capable of being polarized, but may also be oriented owing to the presence of a permanent electric dipole. An analogous situation also exists for dilute solutions of polar molecules in nonpolar solvents, if one neglects small corrections due to the action of the solvent. Matters are quite different, however, when one attempts, on the basis of the theory developed for gases, to understand the dielectric properties of ordinary pure liquids, in which the molecules are situated in immediate proximity to one another. This complicating effect appears especially sharply in the case of liquids with relatively small, strongly polar molecules. It must be emphasized that the difficulties are encountered not only in this limiting case. They appear—though, to be sure, not in so pronounced a form—in all liquids, even nonpolar ones, and their cause is the mutual influence of neighboring molecules, which is not necessarily due to the existence of permanent dipoles.

2. It is quite obvious that in the study of pure liquids the principal problem consists in explaining the interaction between molecules. A vivid example of the general propositions stated here is water. The molecule H₂O is polar and has a dipole moment equal to \(1.84 \cdot 10^{-18}\) electrostatic units, according to measurements on water vapor. This makes it possible to calculate the orientational polarization of the molecules with the aid of the well-known equation

\[ P_0=\frac{4\pi}{3}N\frac{\mu^2}{3kT}, \tag{1} \]

where \(N\) is Avogadro’s number, \(\mu\) is the moment, \(k\) is Boltzmann’s constant, and \(T\) is the absolute temperature. The thus obtained theoretical—

* Chem. Rev., December 1936, translated by S. Levina.

tical value \(P_0\) is \(71\ \mathrm{cm}^3\). However, experiments with liquid water give the dielectric constant \(\varepsilon=81\),

\[ P=\frac{\varepsilon-1}{\varepsilon+2}\frac{M}{d}, \tag{2} \]

and hence from the equation for liquid water the total molecular polarization is obtained as \(P=17\ \mathrm{cm}^3\) (in this equation \(M\) denotes the molecular weight, and \(d\) the density). Part of the obtained quantity should be attributed to the polarizability of the molecules, and not to their orientation. As is known from measurements of refractive indices, this part amounts to about \(4\ \mathrm{cm}^3\). Thus from experiments with liquid water one obtains the orientational polarization \(P_0=13\ \mathrm{cm}^3\); this is in contradiction with the calculated value \(P_0=71\ \mathrm{cm}^3\). It is difficult to believe that the dielectric properties of the water molecule would be so strongly influenced by the direct proximity of other molecules. It is more natural to suppose that the influence of neighbors on the orientation of a molecule in a liquid manifests itself during measurements in an electric field. Consequently, equation (1), which assumes perfectly free rotation, cannot be applied to liquids without substantial corrections. Eversheim’s measurements (1902) of the dielectric constant of carbon disulfide are an excellent example of how great a practical significance these corrections acquire when the molecules are situated close to one another. At a temperature of \(150^\circ\) and a density \(\delta=0.76\ \mathrm{g/cm}^3\), for the orientational polarization the value \(P_0=33\ \mathrm{cm}^3\) was found, while from the values of the dipole there is obtained by calculation \(P_0=37\ \mathrm{cm}^3\). In this case both these quantities are fairly close. However, at a temperature of \(14.5^\circ\) and a density \(\delta=1.39\ \mathrm{g/cm}^3\), one obtains \(P_0=27\ \mathrm{cm}^3\) whereas the theoretical value is \(P_0=55\ \mathrm{cm}^3\).

The study of electric double refraction (the Kerr effect) shows clearly that also in the case of nonpolar liquids free rotation is strongly hindered, as was already noted above. As an example of such molecules one may cite carbon disulfide. Measurements of electric double refraction in vapors can be used to calculate the three kinds of polarization of the molecules: \(\alpha_1=15.1\cdot10^{-24}\) in the direction of the molecular axis \(S—C—S\), and \(\alpha_2=\alpha_3=5.54\cdot10^{-24}\) in directions perpendicular to the molecular axis. From these data one can calculate the Kerr constant for liquids according to the equation

\[ \frac{n_p-n_s}{n_0}=KE^2, \tag{3} \]

where \(K=19.7\cdot10^{-12}\). In this equation \(n_p\) is the refractive index parallel to the field, \(n_s\) is the refractive index perpendicular to the field, and \(E\) is the field strength in electrostatic units. Direct measurements in the liquid give a value \(40\%\) lower, namely: \(K=11.8\cdot10^{-12}\). Nevertheless, the mean

the polarizability of the molecule

\[ \alpha = \frac{1}{3}(\alpha_1+\alpha_2+\alpha_3) \]

does not change upon transition to the liquid state, as may be concluded by comparing the refractive index of the gas and of the liquid, using the Lorentz–Lorenz equation. Thus an alternative is created: either to assume that the differences in the polarizability of the molecule in different directions decrease upon transition to the liquid state, while their sum remains unchanged, or else that the molecule in a liquid cannot orient itself freely in an external field, as it does in a gas. Obviously, the choice must be made in favor of the second assumption.

1. All the preceding considerations focus attention on the bonds between the molecules of a liquid. These bonds are known from the study of the solid state, where they determine the elastic properties of crystals. From the standpoint of continuity between gases and liquids, in the van der Waals sense, the understanding of these forces becomes more difficult. However, even from the standpoint of classical theory one can raise the question of the extent to which, in a liquid, an approximation to the solid state is achieved. In the course of the last several years, experimental material has accumulated indicating that it is legitimate to speak of a quasicrystalline structure of a liquid. After the first joint experiments of the author and Sherrer (1916), which showed the appearance of interference maxima in the scattering of X-rays in liquids, the appearance of the principal interference maxima was explained in a number of investigations by other authors, among whom it is necessary to mention Keesom’s work with liquefied gases and Stuart’s excellent investigations. In a liquid there must be small regions within which the relative orientation of the molecules approaches the uniform distribution in the solid state; the only difference is that in liquids this orientation depends on time. An especially clear picture can be obtained in the study of X-ray scattering by monatomic liquids, for example mercury. In those cases where the individual atoms determine the intensity of the scattered rays, which decreases monotonically with increasing angle between the primary and secondary rays, these same atoms, present in the liquid, cause noticeable interference maxima and minima. In this case it is possible even to determine the mutual orientation and the distance between atoms on the basis of the probability curve obtained from scattering measurements. Thus it becomes clear that, in the sense of the orientation of molecules, liquids behave like quasicrystals.

As experiments on the scattering of light of ordinary frequencies show, the similarity between liquids and crystals exists not only in orientation, but also in the types of molecular motion. If one studies the scattering of monochromatic light in a liquid by means of pri-

of an instrument with high resolving power, the original monochromatic line proves to be split, with the formation of a triplet. The magnitude of the separation of the components of the triplet depends on the direction in which the observation is made, and is greatest at a scattering angle of \(180^\circ\). The central line of the triplet has the same frequency as the original light. The separation of the two outer symmetric components of the triplet is the greater, the higher the velocity of sound in the liquid. These facts, established in the experiments of Gross, Meyer, Raman, and others, would be entirely incomprehensible if the molecules could move freely in the liquid independently of one another. In that case molecular motion would produce not a splitting of the original line into a triplet, but only its broadening. It is known that in an ideally solid body the motion of an atom must be regarded as a superposition of motions that give rise to a large number of thermal sound waves propagating in the body in all directions. An experimental confirmation of this is the applicability of the \(T^3\) law for the specific heat of solids at low temperatures. Brillouin carried out a theoretical investigation of the scattering of light in an ideally solid body; he found that the scattering should be regarded as Bragg reflection by sound waves and that, as a consequence of the combined action of the Doppler effect, the primary spectral line is split into a doublet.

Liquids give not a broadening but a splitting of the lines. Hence, on the basis of Brillouin’s calculations, one may conclude that the motions of neighboring molecules in a liquid are connected with one another in the same way as in solids. The fact that, in addition to the doublet, in agreement with Brillouin’s theory, a third, unshifted line is observed in liquids should be explained by the fact that a liquid is not an ideally solid body. As is well known, even for crystals the ideally solid state is only a limiting case, which can never be realized in practice. Using analogous arguments, Placzek and Landau, in a recent letter to the author, express the view that there is a connection between the intensity of the middle component of the scattered light and the difference in the heat capacity of the liquid at constant pressure and at constant volume. Recent experiments in Leipzig have confirmed this interpretation of the distribution of intensity among the three lines of the scattered light. It may be concluded with complete definiteness that the orientation and bonds of molecules in a liquid are very similar to the bonding of molecules in solid crystals.

1. In order to obtain an idea of the character of the motion of an atom in a monatomic liquid from the point of view of the ideas developed here, it is necessary to emphasize the following propositions. The atom oscillates as in a solid crystal and therefore has, on the average, approximately the same amount of kinetic and potential energy. However, it is typical of a liquid that the center of oscillation is not constant, but moves slowly in the liquid, as if obeying Brownian motion. In a beautiful

in agreement with this is the observation that the heat capacity of liquid mercury has a value that accords well with the Dulong and Petit law. Of considerably greater interest, as compared with monatomic liquids, is the behavior of liquids consisting of more complex molecules. In this case what is very interesting is not the translational motion of the center of gravity of the molecule, but its rotation. The latter must be regarded not as free rotation, but as a kind of torsional vibration, the axis of which undergoes slow rotation in space instead of being rigidly fixed, as in torsional vibration in crystals.

An attempt to apply these ideas to a mathematical treatment of the orientation of dipoles in liquids leads to the conclusion that molecules in liquids are subject to the action of two categories of forces. First, the molecules experience the action of forces due to the external electric field; second, there exists a potential energy that tends to hold the molecule in the position which at the given instant is fixed by the medium surrounding it. This potential energy may be expressed, in the form of the simplest approximation, as follows:

\[ u=-E\cos\theta, \tag{4} \]

where \(\theta\) is the angle between the axis of the permanent electric moment and the instantaneous axis fixed by the surrounding medium. It is quite obvious that an equation of the type (4) is not capable of representing the detailed situation in all special cases. Nevertheless, this equation may be regarded as the first term of a series of spherical harmonics, and one may hope that even this first term is a good approximation to reality. The total energy of a dipole in the field \(F\) is obtained by adding to the energy given by equation (4) also the field energy \(\mu F\cos\theta'\), where \(\theta'\) is the angle between the axis of the dipole and the external field. One can easily calculate the mean component of the dipole in the direction of the field \(F\). However, this depends to a large extent on the angle between the field and the axis that is conditioned by the medium surrounding the molecule. The mean observed moment in the direction \(F\) can be obtained by taking the average over all orientations conditioned by the medium surrounding the molecule. The quantitative expression for the moment in the direction \(F\) is given by

\[ m=\frac{\mu^2}{3}\frac{E}{kT}\left[1-L^2(\beta)\right], \tag{5} \]

where \(\beta\) is the reduced expression: \(\beta=\frac{E}{kT}\), and \(L\) is the well-known Langevin function, so important in the theory of magnetism, namely:

\[ L(\beta)=\operatorname{ctg}\beta-\frac{1}{\beta}. \tag{5'} \]

Equation (5) shows, in accordance with the assumptions stated, that the interaction between the molecules of a liquid

encounters the orientation of dipoles in an external field. The classical value \(m=\dfrac{\mu^2 F}{3kT}\), obtained for completely freely rotating molecules, must be multiplied by the coefficient \([1-L^2(\beta)]\), which is less than unity. For the case when the binding forces are small \((E \ll kT)\), this coefficient is very close to unity, while for large forces \((E \gg kT)\) it tends to the value

\[ \frac{2}{\beta}=\frac{2kT}{E}, \]

which it reaches at very large values of the binding energy, and thus the effect caused by permanent dipoles is wholly eliminated. This coefficient may be taken equal to unity, apart from the ratio between the value of the mean moment in the direction of the instantaneous axis in the absence of an external field and the absolute value of the moment. It is very significant that this result remains valid independently of the special assumptions made in deriving equation (4), and is valid for any conceivable law of forces.

We shall begin the consideration of the applicability of equation (5) with water. The orientational polarization of the molecule determined experimentally is \(13\ \text{cm}^3\), whereas the value calculated theoretically for the case of freely rotating molecules is \(71\ \text{cm}^3\). Thus the correction coefficient is \(13/71=2/11\); according to the theory set forth above, this coefficient, for the case of strong binding occurring here, is equal to \(2/\beta\). If one assigns to the binding energy in liquid water, in accordance with the definition of \(\beta\), the value \(E=11\,kT\), then the result corresponding to experiment is obtained. This is a very large quantity; it causes the same hindrance to orientation as if a field of 72 million V/cm were acting. From the molecular point of view this value of the energy is quite understandable, since a field of strength 72 million V/cm may be produced by the dipole of water at a distance of \(2.5\cdot 10^{-8}\ \text{cm}\). Approximately the same value of \(E\) is observed, for example, also for nitrobenzene, whereas for liquid diethyl ether the correction coefficient is approximately equal to unity, so that \(E\) is close to zero, and the molecules must possess almost free rotation. Between these limiting cases lie the alcohols. Thus, for ethyl alcohol the correction coefficient is equal to 0.56; this means, in accordance with equations (5) and (5′), that the binding energy is \(E=2.9\,kT\). With increasing length of the carbon chain in alcohols this binding energy decreases.

1. The data presented indicate that the magnitude of the binding energy is an accurate measure of that type of association in liquids which acts upon orientation. The expression of the binding energy by means of equation (4), and its use in further calculations, should be regarded as a first, extremely rough approximation to reality. In this connection it is interesting to trace how far this expression is capable of explaining other properties of a liquid. The decrease of the dielectric constant is very

characteristic of dipole orientation. Especially in the case of saturation, for associated liquids one should expect an enormous difference between theory and experiment. The first exact measurements of the saturation effect were made by Herweg in ethyl ether. These measurements, with the introduction of corrections for double electric refraction and electrostriction, were analyzed by means of a formula obtained from the theory of dipoles and applied to molecules with free rotation, namely:

\[ \varepsilon=\varepsilon_0-\frac{4\pi}{15}\,n\,\frac{\mu^4}{k^3T^3}\left(\frac{\varepsilon_0+2}{3}\right)^4 E^2. \tag{6} \]

In this formula \(\varepsilon\) is the dielectric constant measured in the field \(E\), and \(n\) is the number of molecules in \(1\ \mathrm{cm}^3\) of liquid. For ethyl ether, which served as the sole object of Herweg’s experiments, a remarkable agreement between theory and experiment was observed. Subsequently, Malyshev suggested that the performance of the experiments would be easier in liquids with a high dielectric constant, since in this case the strength of the internal field is greater than that of the external field by

\[ \frac{\varepsilon_0+2}{3}. \]

The difficulty was the circumstance that all liquids with high dielectric constants are conductors. However, these difficulties were overcome by means of a special method. The experimental results turned out to be in complete contradiction with the theory. Thus, for example, from equation (6) for water one obtains

\[ \frac{\varepsilon-\varepsilon_0}{\varepsilon_0} = -3.9\left(\frac{E_v}{100\,000}\right)^2, \]

if the strength of the external field \(E_v\) is measured in volts per centimeter. Contrary to this, the experimental results gave

\[ \frac{\varepsilon-\varepsilon_0}{\varepsilon_0} = -1.1\cdot 10^{-3}\left(\frac{E}{100\,000}\right)^2. \]

It is thus seen that for water the effect obtained experimentally is 3500 times smaller than the calculated one. In other analogous cases discrepancies are also observed, though not so large. Thus, for example, for ethyl alcohol the observed effect is 14 times smaller than the calculated one.

It is a question whether these differences are due to the quasi-crystalline structure of the liquid and whether they can be quantitatively explained with the aid of the considerations given above. If for the binding energy one adopts the same formula, i.e. \(u=-E\cos\theta\), and calculates the orientational polarization with a higher degree of approximation, then in equation (6) the second term must be multiplied by a coefficient \(R\). This coefficient can be represented by means of

formula

\[ R(\beta)=3\left[(1-4L^2+3L^4)+4\,\frac{L}{\beta}(2L^2-1)+6\,\frac{L^2}{\beta^2}\right], \tag{7} \]

in which \(L\) is the Langevin function. For large values of \(\beta=\dfrac{E}{kT}\), \(R\) is given by the expression

\[ R(\beta)=\frac{3}{\beta^4}. \tag{7'} \]

If for water one adopts Malyshev’s results, namely: \(\dfrac{3}{\beta^4}=\dfrac{1}{3500}\) and \(\beta=10\), then \(E=10kT\). This means that the same binding energy which must be assumed in order to interpret the value of the orientational polarization, and which is 5.5 times smaller than the calculated value, at the same time explains why the saturation effect found experimentally is 3500 times smaller than the calculated one. In fact, there is a small difference between \(E=11kT\), obtained from the orientational polarization, and the \(E=10kT\) derived in the present case; this discrepancy does not go beyond the limits of error of Malyshev’s measurements. The results obtained for ethyl alcohol and ethyl ether are also satisfactory. In the first case, where the saturation effect is 14 times smaller than the calculated one, it is found that \(\beta=2.9\). This agrees with the value obtained for \(\beta\) from the orientational polarization. In the second case we know that the orientational polarization is almost equal to the value calculated for free molecules, and it becomes clear how Herweg was able to express his experimental results by means of equation (6), without introducing any correction coefficients.

1. For substances with polar molecules, in the region of long electrical waves, anomalous dispersion and absorption discovered by Drude are characteristic. The theory which has hitherto been applied to the case of free dipoles gives an explanation of this effect, stating that at high frequencies there exists a phase difference between the electric field and the dipole orientation. This may be expressed by an equation giving the mean moment \(m\) in a field of strength \(Fe^{i\omega t}\), as follows:

\[ m=\frac{\mu^2}{3kT}\,\frac{Fe^{i\omega t}}{1+\dfrac{i\rho\omega}{2kT}}. \tag{8} \]

The constant \(\rho\) in this formula is a measure of the friction during rotation; if the molecule is represented in the form of a sphere of radius \(a\), and if \(\eta\) is the coefficient of viscosity of the liquid, then

\[ \rho=8\pi\eta a^3. \tag{8'} \]

Thus we express not only the fact that the dispersion and absorption effect is limited to the case of polar liquids, but also that the frequency at which these effects occur is closely related to the viscosity of the liquid. It has been shown experimentally by many investigators that the relation between viscosity and frequency is qualitatively correct, but in some cases it was necessary to introduce deliberately reduced values for the molecular radius. A characteristic example of such a case is glycerin. It may be said that for particles of molecular dimensions there is no information concerning their friction constants, and it is possible that in this case Stokes’ formula with the usual viscosity constant is not applicable. On the other hand, one must not forget that the temperature variation of the dispersion and absorption effect is closely connected with the temperature variation of the viscosity. From this point of view it is of interest to calculate how the fact that the molecules are not free and that one has to deal with a quasicrystalline structure affects the theoretical result.

The calculations are considerably more complicated than the preceding ones; here only the main principles of the calculation will be indicated. It is necessary to find the equation for the distribution function \(f\) of the dipole axis; the latter is at the same time under the influence of the molecular binding energy \(u_0\) and of the alternating field \(F e^{i\omega t}\), which determines the orientation energy

\[ u_1=-\mu F e^{i\omega t}\cos\theta'. \]

As in the case of free dipoles, one may apply the equation of rotation for Brownian motion.

If this effect is taken into account, and if it is borne in mind that the external field can exert only a slight orienting influence, then the problem can be formulated in the following way.

If the total energy of a dipole under the influence of the surrounding medium and the electric field is expressed as

\[ u=u_0+u_1 e^{i\omega t} \]

and it is assumed that

\[ f=f_0+f_1 e^{i\omega t}, \]

then one obtains

\[ f_0=C e^{-\frac{u_0}{kT}}. \]

Here the function \(f_1\) is the solution of the equation

\[ \frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\sin\theta \left[ \frac{\partial f_1}{\partial\theta}+\frac{f_1}{kT}\frac{\partial u_0}{\partial\theta} \right] + \frac{1}{\sin^2\theta}\frac{\partial}{\partial\Phi} \left[ \frac{\partial f_1}{\partial\Phi}+\frac{f_1}{kT}\frac{\partial u_0}{\partial\Phi} \right] - \frac{i\omega\rho}{kT}f_1 = -\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\sin\theta \left[ \frac{f_0}{kT}\frac{\partial u_1}{\partial\theta} \right] -\frac{1}{\sin^2\theta}\frac{\partial}{\partial\Phi} \left[ \frac{f_0}{kT}\frac{\partial u_1}{\partial\Phi} \right], \tag{9} \]

in which \(\theta\) and \(\Phi\) are the usual spherical coordinates.

The solution may be represented in the following way. Starting from the equation

\[ \frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\sin\theta \left[\frac{\partial X_n}{\partial\theta}+\frac{X_n}{kT}\frac{\partial u_0}{\partial\theta}\right] + \frac{1}{\sin^2\theta}\frac{\partial}{\partial\Phi} \left[\frac{\partial X_n}{\partial\Phi}+\frac{X_n}{kT}\frac{\partial u_0}{\partial\Phi}\right] + \lambda_n X_n=0, \tag{10} \]

one must first find all such values of \(\lambda_n\) that \(X_n\) can be regarded as eigenfunctions on the sphere. In this way one obtains a generalization of the ordinary spherical harmonics, which express \(X_n\) for \(u_0=0\) when \(\lambda_n=n(n+1)\). If these functions are known, then the right-hand side of equation (9) can be represented by expansion in a series of the form

\[ \sum c_n X_n, \]

and in this case one obtains

\[ f_1=\sum \frac{c_n X_n}{\lambda_n+\dfrac{i\omega\rho}{kT}}. \]

This path was followed after adopting equation (4) for the binding energy. However, it proved possible only to represent the function by an expansion in powers of

\[ \beta=\frac{E}{kT}. \]

For small values of \(\beta\), and also small \(\dfrac{\omega\rho}{kT}\), the mean moment \(m\) is given by the expression

\[ m=\frac{\mu^2 F e^{i\omega t}}{3kT} \left[ \left(1+\frac{\beta^2}{9}\right) - \frac{i\omega\rho}{2kT} \left(1-\frac{11}{54}\beta^2\right) \right]. \tag{11} \]

If the values of \(\beta\) are large, then the calculations are simplified, because in this case the motion of the dipole may be approximated as a simple oscillation about an axis determined by the interaction of the surrounding molecules.

In this limiting case one obtains

\[ m=\frac{\mu^2 F e^{i\omega t}}{3kT}\cdot \frac{1}{\dfrac{\beta}{2}+\dfrac{i\omega\rho}{2kT}}. \tag{12} \]

It is very easy to estimate the results obtained from these formulas for the limiting cases \(\beta=0\) and \(\beta\gg 1\). If they are represented in the form

\[ m= \frac{\dfrac{\mu^2 F e^{i\omega t}}{3kT}} {1+\dfrac{i\omega\rho}{2kT}} \quad(\beta=0); \qquad m= \frac{\dfrac{\mu^2 F e^{i\omega t}}{3\,\dfrac{E}{2}}} {1+\dfrac{i\omega\rho}{2\,\dfrac{E}{2}}}, \tag{12'} \]

it is clear that the transition from free dipoles to strongly bound dipoles can be effected by replacing the thermal energy \(kT\) by the considerably larger energy \(\dfrac{E}{2}\). From the formulas for dielectric losses and the dispersion effect it follows that, at the same time, there is a reduction of the dipole action both in the dielectric constant itself and in the losses. The binding energy acts (at large values of \(\beta\)) in such a way as though a decrease of the constant were taking place

\[ \rho = 8\pi\eta a^3. \]

This can be explained either by the viscosity being less than normal, or by the radius of the molecules being smaller than assumed. Further experiments are necessary; however, the fact that the greatest discrepancy between theory and experiment is observed in highly associated liquids makes it probable that the difficulties arising in this case can likewise be overcome if the quasi-crystalline structure of these liquids is taken into account.