Recent Developments in the Theory of the Liquid State1

Ya. I. Frenkel, Leningrad

The liquid state is intermediate in its properties between the solid and gaseous states. At a low temperature, close to the crystallization temperature, it approaches the solid state, and at high temperatures, close to the critical temperature, it approaches the gaseous state.

In the gaseous state the molecules move independently of one another; their thermal motion has a purely individual character. In contrast to this, in solids the molecules are closely bound to one another and can move only collectively. In the liquid state we observe an intermediate degree of collectivism in the character of the thermal motion and in the properties connected with it. The concept of the “degree of collectivism” in the motion of particles must be made the basis for the classification both of the various states and of the processes occurring in each of them.

The degree of collectivism may be different depending on the character of the motion under consideration. For example, in a gaseous body, where the molecules move almost independently of one another, the propagation of sound vibrations is possible; these constitute a typical example of a collective process—a process in which all particles participate in a similar way. True, we know that the vibrational motion of particles in gases during the propagation of sound waves constitutes only an insignificant part of their motion in comparison with the main part, which has a completely disordered character.

In the solid state the motion of particles has an entirely collective character, since not a single particle can move without carrying other particles along with it; thus the thermal motion itself in solids should be described as a superposition of waves in which all particles participate to an equal degree. The disorder in the thermal motion of the particles of a solid body is connected with the disorder in the distribution of the phases of waves of different directions and wavelengths.

In liquids we observe various degrees of collectivity in the motion of particles, and this collectivity may manifest itself to different degrees in different properties and processes. At temperatures close to the crystallization temperature, the structure of a liquid, i.e. the relative arrangement of particles—at least in the case of simple monatomic liquids (for example, molten metals)—is known to approach the arrangement of particles in the corresponding crystalline bodies. This similarity consists above all in the equality of the coordination number, i.e. the number of nearest neighbors possessed by each atom. Apparently, this coordination number does not change when a crystal melts. The difference lies in the fact that the arrangement of neighboring particles around any given particle of a liquid is not quite regular. Strictly speaking, even in a crystal there is no ideal regularity in the arrangement of particles, since they take part in thermal motion and oscillate about positions of equilibrium. However, in a crystal these positions themselves are distributed regularly, whereas in a liquid they are distributed irregularly. But this irregularity is not so great that features of similarity to the structure of a solid cannot be detected in the structure of a liquid.

At the present time there are two different points of view on the question of the structure of liquids and, in particular, of liquid metals. The first of them might be called the “quasicrystalline theory”; it amounts to the assertion that the relative arrangement of particles in a liquid approaches that existing in a crystal, with deviations from regularity increasing systematically with distance from the initial atom. Therefore, at large distances we find no traces of the regularity characteristic of a crystal. In other words, in the structure of liquids there is a complete absence of “long-range order,” i.e. order over large distances, which characterizes the structure of crystals. However, at small distances in liquids there is found “short-range order” of the same type as in crystalline bodies. This is the essence of the “quasicrystalline” point of view.

The other—the “microcrystalline”—point of view, which was first put forward by Stewart, then abandoned and is now again being advanced by V. I. Danilov1, amounts to the assertion that a liquid, like a microcrystalline body, consists of separate crystallites, but ones still much smaller than in the case of solids, i.e. consisting of only a few hundred or even a few tens of atoms.

In his original conception Stewart assumed that these microcrystalline or, in his terminology, “cybotactic” regions are separated from one another by layers of sufficiently great thickness, so that the liquid represents, as it were, a two-phase system consisting of small crystallites floating in an amorphous medium. Such a point of view is undoubtedly incorrect and cannot be reconciled with the data of X-ray analysis of the structure of liquids.

Another version of this microcrystalline theory, supported by V. I. Danilov, consists in the assertion that the layers between...

microcrystallites are extremely thin, so that these crystallites are in direct contact with one another.

The data of X-ray analysis do not contradict this microcrystalline interpretation, for the interference of rays scattered by atoms belonging to different (neighboring) microcrystallites leads to a blurring of the intensity maxima on the X-ray diagram of the same kind as that obtained theoretically from a quasicrystalline structure.

Thus, the data of X-ray analysis cannot at present decide the question definitively in favor of one point of view or the other. Additional investigations are required for this.

It must be said that the difference between these two points of view is not as great as it may seem at first glance. Everything depends on whether we consider the arrangement of the atoms of a liquid at some definite instant of time, or investigate their average distribution over a large interval of time. In the latter case the microcrystalline interpretation gives approximately the same picture as the quasicrystalline interpretation. In the course of time the existing microcrystals disintegrate, the boundaries between them shift, and atoms that at first belonged to one and the same crystallite may soon find themselves in different ones. If the distance between any two atoms is averaged over time, we shall obtain a picture close to that to which the quasicrystalline theory leads.

On instantaneous photographs the two theories give different results; on photographs with a sufficiently long exposure they give identical ones. In fact, an X-ray diagram is taken with so long an exposure that during the exposure the atoms have time to pass many times from one microcrystalline aggregate into another. Thus, whatever microcrystallinity may perhaps exist in the structure of a liquid at each instant of time cannot manifest itself noticeably on the X-ray diagram.

However, there exist indirect data which make it possible to choose between the two conceptions indicated above. One of the arguments in favor of a microcrystalline structure of molten metals follows from the X-ray analysis of the structure of metallic alloys in the liquid state. It turns out that, in the case of many binary alloys, the X-ray diagrams represent a superposition of the X-ray diagrams belonging to both components. This means that the two components do not truly mix “atomically,” but form a eutectic structure, the grain size being extremely small, yet in any case sufficient for the structure characteristic of each of the two metals to appear.

However, if in a mixture of two metals the atoms of each kind combine into separate groups, then the preservation of some separateness of the atomic groups in each of the metals individually becomes very probable as well—separateness corresponding to separate microcrystallites of extremely small size.

A very interesting question is that of the linear dimensions of microcrystallites; these dimensions may serve as a measure of the degree of short-range order in the structure of liquids from the microcrystalline point of view. If one adopts the standpoint of a quasicrystalline structure, then the linear dimensions of the microcrystallites must be replaced by the distance over which short-range order is still perceptible in the arrangement of the atoms. This quantity is of fundamental importance for understanding various properties of liquids, as well as the process of their crystallization. Unfortunately, however, we as yet have no theoretical calculation of this quantity. Prins and Petersen² attempted to compute it by considering thermal fluctuations in the arrangement of atoms in a one-dimensional model of a crystal (i.e. a chain of quasielastically bound particles with equally spaced equilibrium positions). However, this theory of Prins has no relation to the question of disorder in a liquid that interests us, although it does lead to the correct conclusion that the mean relative displacement of two atoms increases in proportion to the square root of the mean value of the distance between them.

The earlier theory of Prins and Tsernike³ deserves more attention; in it a liquid, in contrast to a crystal, is characterized by the presence of a certain free volume, which can be defined as the average magnitude of the “gap” between neighboring atoms. At any instant the gaps between neighboring atoms may be unequal. As a result there is a smearing of the relative equilibrium positions of the atoms, the square of which (as also in the theory considered above) grows in proportion to the mean distance between them, and, consequently, the smearing itself grows in proportion to the square root of this distance.

In the case of the simplest linear model of a liquid, consisting of a row of solid spheres, with an average gap \(\lambda\) between them \((\lambda = a - p,\) where \(a\) is the average distance between the centers of neighboring spheres, and \(d\) is their diameter), the probability that the displacement of the \((n+1)\)-st sphere relative to its mean position \(\bar{x}_n = na\) (the first sphere is assumed fixed) lies between \(x = \bar{x}_n + \xi\) and \(x + dx = \bar{x}_n + \xi + d\xi\), is expressed by the formula

\[ e^{-\frac{\xi}{\lambda}}\frac{1}{n!}\left(\frac{\xi}{\lambda}\right)^n \frac{d\xi}{\lambda}. \]

This formula also expresses the probability that, in \(n\) successive free paths, a gas particle will travel a path lying between \(\xi\) and \(\xi + d\xi\), if the mean value of the free-path length is equal to \(\lambda\). Both formulas can be derived in exactly the same way. For \(n \to \infty\), the preceding formula may be represented approximately by the Gaussian curve

\[ \frac{1}{\sqrt{\pi}} e^{-\frac{\xi^2}{2n\lambda^2}}\frac{d\xi}{\lambda}. \]

In this case, for the mean value of the square of \(\xi\) one obtains the expression

\[ \overline{\xi^2}=n\lambda^2. \]

The quantity \(\sqrt{\overline{\xi^2}}=\lambda\sqrt{n}\) is a measure of the blurring. The distance \(an\), for which this blurring is of order \(a\), i.e. \(\dfrac{a^3}{\lambda^2}\), can serve as a measure of the degree of order in the sense indicated above.

Unfortunately, these results have not yet been generalized to the case of a three-dimensional model of a liquid, where the gap \(\lambda\) must be replaced by the free volume \(\Delta V\) falling to one particle of the liquid or to its total mass. This free volume is represented by many authors (especially by Eyring and his collaborators\(^{4}\)) in the form of separate holes, i.e. places not filled by atoms. The concept of a hole in a crystal lattice was first introduced by me in 1926 in connection with the study of the electrical conductivity of ionic crystals. I think, however, that the notion of holes in a liquid, where the atoms are not located at the nodes of a regular spatial lattice, is essentially illegitimate. The free volume, which is specific to the liquid state, is realized not in the form of separate holes, but continuously, in the form of a system of cavities—slits, cracks, which permeate the entire structure of the liquid.

Shot pellets compressed under pressure into a compact mass and arranged in the form of a cubic face-centered or hexagonal lattice have no mobility whatever. If one pellet is pulled out of this compact mass, a hole will be obtained, the presence of which, however, will still not ensure the fluidity that characterizes the liquid state. In order to realize something like the liquid state, it is necessary to pull out a sufficiently large number of pellets from this compact mass; it must be sufficiently loosened. But when we loosen it, the “holes” will cease to exist as definite localized voids and will dissolve in the form of cavities of various types—cracks, which permeate the entire structure of the body.

Proceeding from these considerations, one might even approach the definition of the melting temperature as that temperature at which the volume of the body is sufficient for the free space to be able to distribute itself in it continuously; until the free space is still small, it can be realized only in the form of separate holes.

At the end of 1939 there appeared a theoretical work by Kirkwood\(^{5}\), who attempted to establish the arrangement of atoms in simple liquids without proceeding from any model notions of a quasicrystalline or microcrystalline structure, but directly from the general principles of statistical mechanics, treating the atoms as hard spheres located in a volume \(V\) greater than the minimum volume \(V_0\) necessary for their placement. It turned out that the curve of particle density,

depending on the distance from the center of one of them, has the same form (of rapidly damped oscillations) as is obtained from the theory of quasicrystalline or microcrystalline structure, and also follows from the analysis of X-ray patterns of liquids. Such a density distribution is obtained, however, only so long as the free volume \(V - V_0\) is sufficiently large. When this volume becomes smaller than a certain value, the equation determining the sought density distribution has no solution. Kirkwood interprets this result as an expression of the circumstance that, for a sufficiently small free volume, only the crystalline state (with separate holes) can exist; the liquid state becomes possible only when the free volume is sufficiently large.

It must be noted that atoms cannot, strictly speaking, be treated as incompressible solid spheres. If this were so, then at sufficiently small volumes all bodies would have to be in the crystalline state at arbitrarily high temperatures. In reality, however, Bridgman observed the melting of bodies with increasing temperature at volumes considerably (by \(10\%\)) smaller than those which correspond to ordinary conditions (i.e. atmospheric pressure).

This circumstance is explained by the fact that atoms in reality possess a certain compressibility, which is made use of when the temperature is raised, when they, flying at one another with great velocity, flatten out, increasing the free volume of the liquid by their own compression.

Such is the present state of the question of the structure of the simplest atomic liquids.

As for more complex liquids, consisting of molecules rather than separate atoms, almost nothing has yet been done in this direction, apart from certain experiments with models of elongated molecules\(^6\). We can only say that in the case of liquids consisting of molecules not of spherical shape, there must exist a certain short-range order not only in the arrangement of the centers of gravity of the molecules, but also in their orientation. In the case of water, such a point of view was developed in detail by Fowler and Bernal, who showed that one can explain the properties of water in the liquid state if one assumes that in this state its molecules have a tetrahedral arrangement corresponding to the structure of quartz, and that they are oriented in such a way that the negative pole of one molecule is adjacent to the positive pole of a neighboring molecule. The work of Bernal and Fowler is perhaps the only work done in this direction in all the past years. I shall have to return below to the question of the structure of more complex liquids; here we must note only the fact that in this case it is necessary to distinguish not only short-range order in the arrangement of the centers of gravity of the molecules, but also order in their orientation.

I have repeatedly compared the liquid state with the solid; however, of interest is not only such a comparison, but also

clarification of the causes and mechanism of transition from one state to another, i.e., the mechanism of the processes of melting and crystallization.

On this question, many different works have appeared recently, which may be divided into two groups. The works of the first group proceed from a consideration of the crystalline state and do not take into account distortions in the regular structure of the crystal when the temperature is raised, but consider only the change in volume by which this rise in temperature is accompanied in the case of constant pressure, or else consider the change in pressure when the volume is increased in the case of constant temperature. In 1934 Herzfeld and Goeppert-Mayer \(^{7}\) showed that, at constant temperature and with an increase in the volume of the crystal, the pressure exerted by it changes in such a way that it passes through a minimum. This minimum is explained by the fact that the pressure of the crystal is composed of two parts: first, of the elastic stress (equivalent to a negative pressure, if the crystal is subjected to an all-sided stretching), and of the thermal pressure due to thermal motion. The latter may be represented by the formula:

\[ p=-3RT\frac{\partial \lg \nu}{\partial V}, \]

where \(\nu\) is the mean frequency of atomic vibrations, decreasing with an increase in the volume of the crystal. In accordance with this, the thermal pressure proves to be a substantially positive quantity and, moreover, one that increases rapidly with increasing volume (at constant temperature), since the decrease of \(\nu\) then proceeds faster and faster. The superposition of the elastic pressure on the thermal pressure leads to the result that the curve of the resultant pressure passes through a minimum and then begins to increase with increasing volume. In view of the instability of states of this kind, Herzfeld and Goeppert-Mayer proposed to consider that the pressure minimum corresponds to the destruction of the crystal lattice, so that the corresponding volume characterizes the melting point at the given temperature. In this, volume is treated as an independent variable, and temperature as a constant parameter. Having made the corresponding recalculation, one can, conversely, compute the melting temperature at a given volume or a given pressure. The melting point determined in this way cannot be regarded as correct, since it is incompatible with the condition of equality of the thermodynamic potentials of the two phases. The latter is evident from the circumstance that the Herzfeld—Goeppert-Mayer theory proceeds from consideration of only one crystalline phase, and not from the conditions of its equilibrium with the liquid. Thus this theory indicates only the inevitability of the melting process, but cannot determine the moment of its onset. The theory of Herzfeld and Goeppert-Mayer was further developed by me in 1935 \(^{8}\), where I noted that the increase of pressure which is obtained according to the theory after it passes through the minimum cannot continue without limit. The pressure must reach a certain maximum and then begin to decrease again. Thus, the isotherms of the crystal must have the same form as the isotherms

of a “gas–liquid” body in the van der Waals theory. It is clear, moreover, that the intermediate states, which are characterized by an increase of pressure with increasing volume at constant temperature, are unstable, just as in the van der Waals theory, in the transition from the liquid state to the gaseous one. In reality this transition is effected along a certain horizontal straight line, which corresponds to a combination of two different states (liquid and gaseous, or solid and liquid) in a variable proportion.

From this point of view there must exist a certain “critical melting temperature,” below which the isotherm loses its wavy character and becomes monotonic. Indeed, the waviness of the isotherm, i.e. the appearance on it of a pressure maximum, is connected with an increase of the thermal pressure when the volume increases. And since the thermal pressure is proportional to the absolute temperature, at sufficiently low temperatures it becomes insignificant and the “hump” caused by it disappears.

Recently there has appeared a theory of Born’s,9 devoted to the same question of the melting of crystals. Like Herzfeld, Born assumes that the crystal lattice preserves its regularity when the temperature is raised and the volume is increased without limit. Born then investigates how the moduli of the lattice change, in particular the shear modulus, with increasing temperature at a given external pressure.

Taking into account the change in the mean frequency of oscillation of the particles of the crystal when its volume changes, Born shows that, with increasing temperature (and constant pressure), the shear modulus of the crystal decreases and at a certain temperature becomes zero. This temperature is then defined by him as the melting temperature. Can the liquid state, however, be regarded as a state for which the shear modulus is zero? This is a debatable question; in any case, with respect to high-frequency oscillations liquids possess a modulus of elasticity different from zero. This is especially clearly revealed in the case of very viscous liquids, for example molten glasses, where the relaxation time is large. It is necessary, however, to distinguish the static shear modulus, which decreases with increasing temperature, tending to zero, from the dynamic one, which may remain different from zero even in liquids. According to Born, the melting point corresponds to the vanishing of the static modulus. This interpretation has the same defect as that of Herzfeld and Eppert-Mayer: it takes into account only the initial state, whereas the melting temperature corresponds to equilibrium between both states and is determined by the equality of their thermodynamic potentials. Without regard to the liquid state, it is obviously impossible to fix the melting temperature. In Born’s theory, however, the inevitability of the melting process is shown as the result of the progressive softening of the lattice.

The theories of Herzfeld, mine, and Born’s form the first group of theories which approach the question of melting from the standpoint of the softening of a regular crystal lattice with increasing volume or

with increasing temperature and constant pressure. This softening is expressed in a decrease of the mean frequency of oscillations of the lattice. Another group of theories approaches the same question from an entirely different point of view, considering not the softening of the crystal lattice, in the assumption that it preserves its regularity, but the diminution of the regularity of the crystal lattice which occurs when the temperature is raised and the volume increases, accompanying this rise in the case of constant pressure.

All these theories, without exception, make use of the concept of “holes,” but only in different senses. For example, Lennard-Jones’s theory^10 reduces to the following simple idea: a crystal may be regarded, if desired, as a binary alloy of atoms (at the lattice sites) and holes in the interstices, into which atoms that have jumped from their places at the lattice sites may pass (which is equivalent to the passage of holes from the interstices to the sites). Thus, for example, aluminum, which crystallizes in a face-centered cubic lattice, may be treated as an analogue of a rock-salt crystal, with the role of the sodium ions played by the aluminum atoms, and the role of the chlorine ions by the “holes.” At absolute zero of temperature the atoms are arranged in complete order, alternating with holes throughout the entire volume of the crystal.

Let us denote the metal atoms by \(A\), and the holes by \(O\). In the state of ideal regularity we have a regular alternation of \(A\) and \(O\). As the temperature is raised, the order in this alternation must be disturbed more and more often. This disturbance is also promoted by the increase of volume at a given (nonzero) temperature.

It is known that binary alloys of the type \(AB\), when the temperature is raised, pass from an ordered state into a disordered one. When the short-range order in the alternation of atoms \(A\) and \(B\) becomes insufficiently complete, the long-range order is destroyed. The long-range order, which is determined by the ratio of the number of atoms occupying their own places to the total number of atoms, decreases with increasing temperature, at first slowly and then faster and faster, becoming zero at a certain critical temperature, just as the spontaneous magnetization of ferromagnetic bodies disappears at the Curie temperature.

This conception, first developed by Bragg, Williams, and others, was applied by Lennard-Jones to a monatomic metallic lattice, treated as a solid solution of atoms and holes^1).

Melting, from Lennard-Jones’s point of view, is the liquidation of the long-range order in this alternation of atoms and holes. Consequently, the melting temperature may be interpreted as a kind of Curie temperature. In reality we do not have complete similarity here: as is known, melting is characterized by the presence of a latent heat of transition, whereas the Curie point is only an anomaly of the heat capacity, which in its vicinity is represented by a curve resembling—

^1) Quite independently of Lennard-Jones, this conception was put forward by one of my students, now a postgraduate student, Stilbans.

bearing the Greek letter “lambda” ($\Lambda$), i.e., characterized by the presence of a more or less sharp maximum. The area bounded by this $\Lambda$-shaped part of the heat-capacity curve plays the role of the latent heat of transition; the latter, consequently, occurs not at a quite definite temperature, as in the case of ordinary melting, but over a certain interval of temperature.

It is clear that the Lennard-Jones model is extremely crude; it is, however, surprising that this crude model leads to rather good results in the sense of agreement with experimental data.

Another theory, which was recently published by Bresler,^11 proceeds from the idea of the loosening of the crystal lattice by the formation of holes in it, with the point being not fictitious holes forming interstices, but real holes arising at the nodes of the crystal lattice and due to the fact that some atoms leave their places at the nodes, leaving the latter vacant. Previously it had been assumed that the formation of holes in a crystal is possible only when the corresponding atoms pass into the interstices. Several years ago Schottky^12 showed that such holes can also arise without atoms passing into interstices, by a process that might be called a probing of the void by the crystal. Instead of dissociated atoms we then obtain excess atoms on the surface of the crystal.

Bresler starts from the consideration of such holes and shows that when their percentage becomes sufficiently large, for example if, in the neighborhood of each atom, instead of twelve, there are on the average only ten or nine neighbors, the lattice becomes unstable. The energy required for the formation of holes decreases as their number increases. If this decrease occurs according to a linear law, then it can be shown that the number of holes at first grows very slowly, and then, as the temperature is raised, faster and faster.

Qualitatively this theory gives the same results as the Lennard-Jones theory. Its fundamental shortcoming consists in the incorrect application of the notion of holes to a liquid.

It is clear, however, that the essence of the melting process is also described more or less correctly from this point of view.

What should a true theory of melting look like? It must, evidently, take into account both elements—the softening of the crystal lattice with increasing temperature or increasing volume, on the one hand, and the decrease of order in the lattice, in connection with the formation of holes in it or in connection with the partial transition of atoms into interstices, on the other.

Although the premises of such a theory are at present clear, nevertheless in quantitative form it has so far not yet been developed by anyone.

The considerations set forth apply to simple monatomic substances, chiefly to metals. What is the situation in the case of a substance with complex molecules? In complex substances in the crystalline state we are dealing not only with a regular arrangement of molecules, but also with their regular orientation. Usually both kinds

of regularity, or “long-range order”—in the arrangement of the centers of gravity and in the orientation of the molecules—disappear simultaneously upon melting. But this rule has quite numerous and, moreover, very interesting exceptions of a twofold kind. Namely, cases are observed in which long-range order in the orientation of the molecules disappears, while long-range order in the arrangement of the centers of gravity is still preserved (or conversely). This phenomenon is usually revealed in a heat-capacity anomaly of the Curie-point type or of the “lambda-point” type. It is observed, for example, in the hydrogen halides. Pauling^13 in 1930, and Simon (in an unpublished work), supposed that this heat-capacity anomaly corresponds to the transition of hydrogen-halide molecules (for example, HCl) from rotational oscillations to free rotation. If this were correct, then the heat capacity of the body above the Curie point would be less than the heat capacity below it, when the rotational motion of the molecules is associated not only with kinetic but also with potential energy corresponding to the rotational oscillations. In the transition from rotational oscillations to free rotation the heat capacity should decrease (by approximately \(2\ \mathrm{cal}/\mathrm{mole}\)). In reality, however, at this transition it increases somewhat.

This circumstance led me in 1936 to express the idea^14 that in the transition through the lambda point (or Curie point) the rotational motion of the molecules retains the character of rotational oscillations, and only the long-range order in the distribution of the equilibrium orientations, about which these rotational oscillations take place, is lost.

Thus we are dealing here with a process analogous to ordinary melting; I proposed calling this process “orientational melting.” Despite the disappearance of long-range order in the equilibrium orientations of the molecules, the long-range order in the arrangement of their centers of gravity is preserved, i.e. ordinary melting does not occur.

In the case of a dipolar substance, the temperature of orientational melting can be readily calculated. For this it must be taken into account that each dipole is oriented by the electric field \(E\) created by the surrounding dipoles. The mean value of the cosine of the angle \(\theta\) by which the dipole deviates from the equilibrium direction can be calculated with the aid of the usual Langevin or Debye theory, i.e. from the formula

\[ \overline{\cos\theta}=L\left(\frac{pE}{kT}\right), \]

where \(p\) is the dipole moment, and \(L(x)=\operatorname{cth}x-\frac{1}{x}\) is the Langevin function.

On the other hand, it is clear that the orienting field \(E\) must be proportional to \(\overline{\cos\theta}\) (since it is determined only by the longitudinal components of the electric moments of the corresponding dipoles). If in the preceding formula we put \(E=E_0\overline{\cos\theta}\), then for the mean value \(\cos\theta\) as a function of the temperature \(T\) a curve is obtained that coincides with the one which characterizes the change in spontaneous

magnetization of ferromagnetic bodies. At the temperature \(T_c = \dfrac{pE_0}{3k \cos \vartheta}\), corresponding to the ferromagnetic Curie point, the mean value \(\overline{\cos \vartheta}\) becomes zero. This, however, by no means signifies (contrary to Pauling’s opinion) that the dipoles begin to rotate freely. It means only that the regular distribution of orientations, about which they perform their rotational oscillations, disappears. In other words, the long-range order in the distribution of these equilibrium orientations disappears, and only short-range order remains. The latter may in this case persist even further, upon passing through the temperature of ordinary melting, when long-range order disappears in the arrangement of the centers of gravity of the molecules.

Recently V. N. Tsvetkov applied analogous considerations to the question of the structure of liquid crystals[^15]. It is necessary to note that, from the point of view set forth here, a liquid crystal, or anisotropic liquid, should be regarded as a substance in which, owing to the strongly elongated form of the molecules, which determines the tendency toward a parallel arrangement of their axes, the temperature of orientational melting is higher than the temperature of ordinary melting. Therefore, in such a substance the long-range order in the distribution of the orientations of the molecules is preserved even after the disappearance of long-range order in the arrangement of their centers of gravity.

The transition from the liquid-crystalline state to the ordinary amorphous phase may in this case be interpreted as orientational melting in the sense considered above.

In the case of non-dipolar molecules of an anisotropic liquid, the degree of long-range order may be characterized by the mean value of the spherical function of the second order

\[ P_2(\cos \vartheta) = \frac{1}{2}\left[\cos^2 \vartheta - \frac{1}{3}\right], \]

where \(\vartheta\) is the angle formed by the axis of the molecule with the equilibrium direction (i.e. the mean direction of the molecular axes at the corresponding point).

Proceeding from Ornstein’s hypothesis that to a deviation \(\vartheta\) there corresponds a potential energy proportional to \(\vartheta^2\), V. N. Tsvetkov obtained the dependence of \(\overline{P_2(\cos \vartheta)}\) on temperature, in good agreement with experimental data.

Everything that has been said relates only to the statics and thermodynamics of melting and other phase transformations. However, the question of the kinetics of these transformations is also of great interest, i.e. of the rate at which they proceed in time. Developing Folmer’s ideas[^16], I approached this question by means of a statistical (or thermodynamic) method based on the conception of “heterophase fluctuations” and “pre-transitional states”[^17]. Similar ideas were expressed in a more special form, as applied to the phenomena of vapor condensation or crystallization of liquids, still earlier, for example, by Kaischew[^18].

The essence of these ideas is as follows: nuclei of the liquid phase appear in the crystal even before the thermodynamic melting temperature is reached, in the form of small droplets. They may be bounded by crystalline faces, i.e., they may be not droplets but, as it were, “negative crystals.” As the melting temperature is approached, the number of these liquid nuclei in the crystal and their average sizes increase. The formation of liquid droplets in a crystalline body below the melting temperature is an event of low probability, but, as is known, in the presence of thermal equilibrium, less probable states also occur (alongside the most probable ones), though much more rarely. The phenomenon under consideration becomes significant within 5–10° near the melting temperature, manifesting itself in a catastrophic increase in the heat capacity of the body and in its coefficient of expansion, which, according to Strelkov’s measurements[^19], increases by 20 times or more in comparison with its normal value. This increase is connected with the “premelting” of the body.

An analogous process occurs during crystallization of a liquid. Even before the conditional crystallization temperature is reached, nuclei of the crystalline phase arise in the liquid, the number of which decreases (in a geometric progression) as their sizes increase.

This idea should not be confused with the idea of the microcrystalline structure of a liquid. Here we are speaking of the formation not of microcrystallites consisting of several tens or hundreds of atoms and practically in direct contact with one another, but of microcrystalline inclusions in an amorphous liquid medium. These crystalline nuclei, which exist in the liquid even before the crystallization temperature is reached, begin to grow systematically upon passing through this temperature. In doing so they must pass through certain “critical dimensions,” above which their growth becomes irreversible and thermodynamically inevitable. Before this “critical growth,” owing to the increase in the surface of the crystallites with increasing linear dimensions and to the associated increase in surface energy, the growth of the nuclei is impeded.

As Volmer[^16] showed already in 1927, the critical dimensions of a nucleus depend on temperature. At the thermodynamic crystallization temperature these critical dimensions are equal to infinity; therefore the rate of crystallization at the indicated temperature is equal to zero. When the temperature is lowered, the critical dimensions of the nuclei decrease, and the rate of their formation and growth increases. This increase, upon further lowering of the temperature, is limited by the following circumstance. In order that particles of the liquid may attach themselves to the crystal nucleus, they must first separate from one another; this requires a certain activation energy \(U\), which characterizes the viscosity of the liquid (according to the formula

\[ \eta = \operatorname{const}\, e^{\frac{U}{kT}}). \]

This circumstance retards crystallization;

slowing it down in proportion to \(e^{-\frac{U}{kT}}\). As a result, the rate of nucleus formation and crystallization, as the degree of supercooling increases, at first rises (owing to the decrease in the critical sizes of the nuclei), and then, at some optimal temperature—which may lie considerably below the crystallization temperature (by several tens and even hundreds of degrees)—passes through a maximum and begins to decrease; the lower the optimal temperature lies relative to the crystallization point, the smaller the maximum rate of the latter.

The greater the activation energy, the greater the viscosity, the greater the optimal degree of supercooling of the liquid and the smaller the maximum rate of its crystallization. This explains, in particular, the existence of “glass-like” bodies, which are supercooled liquids that have not had time to crystallize. Their optimal rate of crystallization is so small that, in the best case, months and years are required for crystallization to be completed; if the temperature is lowered below the optimal one, crystallization practically does not occur at all.

In this connection I should like briefly to dwell on certain curious phenomena which have long been known, but are only now acquiring a clear meaning. It is known, for example, that the degree of supercooling which a liquid can withstand without crystallizing depends on its preliminary heat treatment. If the liquid is thoroughly heated above the melting temperature, it proves capable of very strong supercooling. On the contrary, a liquid obtained by melting a crystal and kept (not too long) near the melting temperature is supercooled only very weakly.

It is also known that, when a crystal is melted and the liquid subsequently crystallizes, a crystal of the original orientation is often obtained (the phenomenon of “memory”).

The dependence of the maximum degree of supercooling of a liquid on its initial heating is explained by the fact that crystalline nuclei of relatively large size remain in the liquid for some time after melting; at a not too high temperature this time may be quite considerable. If, therefore, the heating of the liquid is insufficiently intense or prolonged, the crystallites do not have time to melt and, upon secondary cooling of the liquid, serve as centers of crystallization, increasing the rate of the latter and eliminating the possibility of strong supercooling. If, however, the liquid is kept for a long time at a sufficiently high temperature, then the crystallites initially contained in it have time either to be completely destroyed or to melt down to negligibly small sizes corresponding to equilibrium at that temperature.

From this point of view it is not difficult to explain also the above-mentioned phenomenon of “memory” (which is observed, incidentally, only when the melt is heated insignificantly).

The rate of transition from the liquid state to the solid depends essentially on the surface tension at the boundary—

between the solid and liquid phases. This quantity remains, however, as yet undetermined not only theoretically, but also experimentally.

The question of the surface tension of a liquid at its boundary with vapor or air is of independent interest. Until recently, as it seems to me, there was no correct theory on this question. There existed only empirical formulas, such as Eötvös’ formula, which states that the surface tension \(\sigma\) is a linear function of temperature, vanishing at the critical temperature \(T_0\):

\[ \sigma=\gamma(T_0-T). \]

Here \(\gamma\) denotes a coefficient proportional to the \(2/3\) power of the molecular volume. Recently I succeeded in obtaining a very simple derivation of this formula, which in a few words may be set out as follows\({}^{20}\).

The surface tension of a liquid is nothing other than the free energy per unit surface. The free energy is expressed as the difference between the total—or, practically, the potential—energy per unit surface \(u\) and the product of the temperature \(T\) by the entropy \(s\), likewise referred to unit surface.

In order to calculate the quantity \(s\), it is necessary to establish the character of the thermal motion associated with the surface of the liquid.

As was first shown by Mandelstam in his theory of the diffuse reflection of light from the surface of liquids\({}^{21}\), the thermal motion of the latter can be described as the result of the superposition of capillary waves of different lengths, furrowing the surface in all directions, quite similarly to what is done in Debye’s theory of the heat capacity of solids, where the role of capillary waves is played by acoustic waves. From this point of view, for the entropy \(s\) one obtains the following expression (analogous to that given by Debye’s theory for the volume entropy of a solid):

\[ s=nk\lg\frac{kT}{h\nu}, \]

where \(n\) is the number of molecules per unit surface, \(h\) is Planck’s constant, and \(\nu\) is the mean frequency of the capillary waves. It is assumed here that \(h\nu \ll kT\) (at ordinary temperatures this condition is always fulfilled). Hence, for the dependence of the surface tension of a liquid \(\sigma\) on temperature, the following formula is obtained:

\[ \sigma=u-nkT\lg\frac{kT}{h\nu}. \]

This formula is very close to Eötvös’ formula. The coefficient \(\gamma\) proves to be equal to \(nk\lg\frac{kT}{h\nu}\); its dependence on temperature is expressed so weakly that it may be regarded as practically constant. Its numerical value is in good agreement with experimental data.

In conclusion I would like to dwell briefly on the question of the viscosity of liquids, which, as was shown above, is of substantial importance for the phenomena of their crystallization. I do not intend here to expound the theory which I published many years ago\(^{22}\) and which, I think, is known to most of those present. I wish to point out only certain new points of view and new facts that are now emerging in this field.

Between the viscous flow of a liquid and the plastic deformation of crystals there is a very substantial difference, consisting in the following. In the case of a liquid, the motion of atoms occurs in a more or less individual manner, with a “drift” due to external forces superposed on the individual thermal motion of the separate particles. On the other hand, in the case of plastic deformation of crystals we are dealing with the organized collective displacement of entire atomic layers. Between these two extremes there apparently exists a series of intermediate gradations, which are characterized by various degrees of collectivity in the motion of the particles of a liquid. In particular, A. P. Aleksandrov has recently shown\(^{1}\) that, in the case of complex liquids, the coefficient of viscosity drops sharply with an increase in the applied force (or in the rate of flow). This circumstance should be interpreted as an indication that, with increasing force, the displacement of the particles of the liquid acquires an increasingly organized character, becomes less and less individual and more and more collective. Only in the case of very small forces does this motion retain that individual character, directly connected with diffusion, which is assumed in my old theory.

It is curious that, in the case of homologous substances, for example of the paraffin series, the coefficient of viscosity increases exponentially with an increase in the number of links in the molecular chain. This result was theoretically predicted by me about a year and a half ago, on the basis that the activation energy for the displacement of polymer molecules must be proportional to the number of links of which they consist.

In the case of liquids with complex molecules, the question of viscosity is complicated by the circumstance that, besides the translational motion of the molecules, it proves necessary to consider their rotational motion as well, the two kinds of motion being closely connected with one another.

In the classical theory of elasticity and in Navier–Stokes hydrodynamics, molecules are treated as material points, so that the concepts of orientation and rotational motion turn out to be inapplicable to them. Connected with this circumstance is one of the fundamental conclusions of the theory concerning the symmetry of the tensor of elastic and viscous stresses:

\[ T_{xy}=T_{yx}\ \text{and so on.} \]

In fact, the difference \(T_{xy}-T_{yx}\), multiplied by the volume element of the body \(dV\), is equal, as is known, to the moment of the rotational forces which on

\(^{1}\) Not published.

act on it. If the particles composing it are material points capable of having only translational motion, then the equation of rotation about the \(Z\) axis must have the form:

\[ J \frac{d\omega}{dt} = (T_{xy} - T_{yx})\, dV, \]

where \(\omega\) is the angular velocity, and \(J\) is the moment of inertia. The latter is proportional to the mass and to the square of the linear dimensions; since, further, the mass is proportional to \(dV\), the moment of inertia is proportional to \((dV)^{5/3}\). Thus, as \(dV \to 0\), \(\frac{d\omega_z}{dt} \to \infty\), if \(T_{xy} - T_{yx} \ne 0\), which is obviously impossible.

It follows from this that the tensor \(T\) must be symmetric. If, however, the molecules of the body, in addition to translational motion, can also have rotational motion about their own axis (“spin”), then the angular momentum of the molecules in the volume element \(dV\), corresponding to this “spin” of the molecules, turns out to be proportional to \(dV\). Under such conditions the relation \(T_{xy} = T_{yx}\) need not be satisfied.

When considering a liquid consisting of complex molecules, it is further necessary to take into account the connection between rotational and translational motion. Thus, for example, in the case of a liquid crystal with parallel-oriented molecules, rotation of the latter about their own axes would lead to the formation of gaps (voids) or to the climbing of some molecules onto others, if it were not accompanied by a compensating translation, i.e. by a definite displacement of the centers of gravity of the molecules. In modern hydrodynamics and in the theory of elasticity this circumstance is not taken into account, because the internal degrees of freedom of the particles are not considered. If they are taken into account, a considerable complication of the theory results. Alongside the coordinates of the centers of gravity of the particles and the components of their translational velocity, it becomes necessary to introduce into consideration their orientation and rotation about their own axes, kinematically linking rotation with translation. This program still remains unfulfilled[^23]; its fulfillment is, in my opinion, one of the most urgent tasks of molecular physics and, in particular, of the physics of the liquid state.

REFERENCES

1. V. I. Danilov, dissertation, Dnepropetrovsk, 1940; Sow. Phys., 12, 757, 1937.
2. Petersen u. Prins, Physica, 3, 147, 1936.
3. Prins u. Zernicke, Z. Physik, 41, 184, 1927.
4. Eyring and Hirshfelder, J. Chem. Phys., 41, 249, 1937.
5. Kirkwood, J. Chem. Phys., 7, 919, 1939.
6. Kast u. Stuart, Physik. Z., 40, 714, 1939.
7. Hertzfeld and Göppert-Mayer, Phys. Rev., 46, 995, 1934.
8. J. Frenkel, Acta Physicochimica USSR, 3, 635, 913, 1935.
9. M. Born, J. Chem. Phys., 7, 591, 1939.
10. Lennard-Jones, Proc. Roy. Soc., 169, 317, 1939.

1. S. E. Bresler, ZhETF, 6, 711, 1939.
2. W. Schottky, Z. phys. Chem., 29, 335, 1935; see also J. Frenkel, Acta Physicochimica USSR, 4, 567, 1936.
3. L. Pauling, Phys. Rev., 36, 430, 1930.
4. Ya. I. Frenkel, ZhETF, 6, 902, 1936.
5. V. N. Tsvetkov, dissertation, Leningrad, 1940.
6. O. Volmer, Z. Elektrochem., 35, 555, 1929.
7. Ya. I. Frenkel, ZhETF, 9, 952, 1939.
8. Kaischew, Ann. Physik, 30, 184, 1937.
9. P. T. Strelkov, Sow. Phys., 12, 23, 73, 1937.
10. Ya. I. Frenkel, ZhETF (in press).
11. I. Mandelstam, Ann. Physik, 41, 609, 1913.
12. J. Frenkel, Z. Physik, 35, 662, 1926.
13. See, however, Oseen, Trans. Farad. Soc., 29, 883, 1933.

1. Report delivered in Leningrad at the Conference of the Academy of Sciences of the USSR on the liquid state, 9/VI 1940. ↩↩