THERMAL MOTION IN SOLIDS AND LIQUIDS AND THE THEORY OF MELTING

Ya. I. Frenkel, Leningrad

INTRODUCTION

Until recently, the solid and liquid states of matter were usually contrasted with one another as, in a certain sense, opposites. Above all, solids are characterized by the regular arrangement of the particles composing them, which is manifested outwardly in their crystalline structure and the regular faceting of individual crystals; liquids, on the contrary, were ascribed a completely irregular, amorphous structure. Further, the “hardness” of solids, i.e. their ability to resist a change of shape, was contrasted with the “fluidity” of liquids, which tend only to preserve their volume but offer no resistance to a change of shape. This last opposition, however, proved to be highly relative.

First, a number of substances are known that pass from the liquid state into the solid state continuously, without an ordering of their structure, becoming amorphous solids. It follows from this that fluidity and hardness are not mutually exclusive properties, but can to some extent coexist.

Second, solids in the crystalline state also possess a kind of fluidity, which usually manifests itself only under sufficiently large shearing stresses, in the form of sliding of one part of a crystal over another along definite crystallographic planes and directions.

Despite the presence of the indicated features of similarity between the liquid and solid states, they were nevertheless treated as fundamentally opposite, this opposition being associated with the different character of the thermal motion of the particles; namely, it was assumed that in a solid each particle is invariably bound to a definite position of equilibrium, about which it can perform only small oscillations, whereas the thermal motion of particles in a liquid was usually likened to the motion of particles of a strongly compressed gas, or, figuratively speaking, to the “crawling of a heap of worms.”

Studies of recent years have shown that both these conceptions do not correspond to reality. As regards the charac

both in the character of thermal motion and with respect to their structure. Liquids exhibit a series of transitions between gaseous and solid bodies, approaching the former at high temperatures and the latter at low ones. This circumstance has recently led me to the idea of the fundamental continuity of the transition from the liquid amorphous state to the solid crystalline state.

§ 1. Thermal Motion in Crystals

If the thermal motion of atoms in a crystal were reduced only to the oscillatory motion of each atom about an invariable equilibrium position, determined by the mean position of the neighbors surrounding it, then in the solid state the phenomenon of diffusion, i.e. the mutual penetration of atoms of two similarly constructed crystals when these crystals are in close contact with one another, would be impossible. Meanwhile, such a phenomenon is observed experimentally to a very slight degree at low temperatures, but is already quite noticeable at temperatures close to the melting temperature.

This phenomenon of mutual diffusion of two crystalline bodies can, according to Hevesy, be conceived as the result of a gradual transfer or “exchange of places” (Platzwechsel) of two different atoms that have found themselves next to one another. By means of such a transfer with one of its neighbors, a foreign atom that has reached the surface of a given crystal can gradually penetrate into its interior and travel throughout its entire volume.

An analogous phenomenon of transfer of neighboring atoms must evidently also occur in the case where the latter are identical, i.e. in a perfectly homogeneous crystal. Hence it is clear that the thermal motion of atoms cannot be reduced to small oscillations about invariable equilibrium positions, but must, at least in part, consist in the pairwise transfers considered above.

Many crystals, in particular crystals of various salts, are constructed not of atoms but of oppositely charged ions. Thus, for example, a crystal of common salt consists of positive sodium ions and negative chlorine ions, with each ion of either kind surrounded by six ions of the opposite kind. Such, at least, is the normal structure possessed by a crystal of common salt at low temperatures.

Experiment shows that, under the influence of an external electric field, in rock salt, as well as in crystals of other salts, an electric current appears, associated with chemical decomposition, quite analogous to what occurs in an aqueous solution of the same salts, i.e. in liquid electrolytes. This phenomenon (electrolysis) is due to the mobility of individual ions. It would be impossible in a solid crystal if the ions composing it were immobile or were bound to invariable equilibrium positions. However, the phenomenon of transfer of two neighboring ions of the same sign, which we were compelled to admit

for explaining diffusion is insufficient for explaining the electrical conductivity of ionic crystals. Indeed, such hopping does not produce any transport of electricity, whereas an electric current consists precisely in such transport.

To explain the passage of an electric current through ionic crystals, A. F. Ioffe, to whom we owe a detailed experimental investigation of this question, advanced the idea that some of the ions can break away from their normal positions at the nodes of the crystal lattice, or, as it is said, dissociate, and that precisely these dissociated ions can move throughout the entire volume of the crystal, giving it the properties of an electrical conductor. With increasing temperature the degree of dissociation, i.e. the relative number of dissociated ions, must increase, which explains the sharp increase in electrical conductivity observed experimentally with rising temperature.

This idea of A. F. Ioffe was quantitatively developed and extended by me in 1926[^1].

First of all the question arises—where do the dissociated ions go? There can be only one answer to this: breaking away from the nodes of the crystal lattice, the dissociated ions must become stuck in the “internodes,” i.e. in the spaces between atoms or ions that remain in their normal positions. These spaces may be too large for negative ions, for example for chlorine ions, which are distinguished by their large dimensions, but relatively small positive ions, for example sodium ions, can squeeze into them, pushing their neighbors slightly apart. It should not be thought, however, that a sodium ion stuck in some internode of the rock-salt lattice remains forever bound to this new, improper equilibrium position. If, owing to random fluctuations of the energy of thermal motion, some ion has managed to jump out of its normal position at a lattice node and find itself in a neighboring internode, which represents a certain improper equilibrium position, then it is clear that, owing to the same random fluctuations of thermal motion, expressed in deviations of the ion’s energy now to one side, now to the other from the mean value, the dissociated ion must sooner or later tear itself out of the internode it occupies and either return back or pass into a neighboring internode. Thus dissociated ions, passing from one internode to a neighboring one, wander throughout the entire volume of the crystal, like completely free ions in an aqueous solution of the corresponding salt.

Under ordinary conditions this wandering has a completely irregular character, representing a disorderly roaming with more or less prolonged stops in each internode.

However, in the presence of an external electric field, which draws the ions in a definite direction, this wandering acquires

to some degree a directed character, since ions more often move in the direction of the force acting on them than in the opposite direction. This partial ordering, or directedness, of their displacements is what constitutes an electric current.

When a sodium ion is torn away from its normal position at a site of the rock-salt lattice, this site remains vacant. After the detached ion has moved away from it to a more or less considerable distance, the vacant site remains symmetrically surrounded on all sides by other ions seated at their own sites. The following then becomes possible: one of the sodium ions surrounding the empty place left by the departing (dissociated) sodium ion may jump into that place, filling it and freeing its own. Such a process may be described as the transfer of an empty place with one of the neighboring ions of the corresponding sign. By means of such transfers the empty place, or “hole,” can move throughout the whole volume of the crystal, just as happens when an extraneous atom penetrates into it (see above). We see, therefore, that alongside the dissociated ions wandering through the interstices of the crystal, their partners—positive “holes”—also move in the latter; it is as though they were not “phantoms,” but real material particles. This similarity increases still more if we consider the action of an electric field on such holes. Moving in the absence of a field in a completely disordered fashion, in the presence of a field they must move predominantly in the direction opposite to that in which positive ions tend to move. Indeed, if, in the absence of external forces, any of the sodium ions surrounding a sodium hole had equal chances of jumping into it, then in the presence of a field acting on the sodium ions, say to the right, a sodium ion located to the left of the hole has a greater chance of jumping into it than a sodium ion located to the right; in other words, the hole will have a greater chance of moving to the left than to the right. This means that the positive holes left by dissociated sodium ions will behave like mobile negatively charged ions.

Thus the electric current in a crystal of rock salt, caused by the action of an external electric field, must consist not only in the motion of dissociated sodium ions in the direction of the electric field, but also in the motion of sodium “holes” in the opposite direction, so that these negative holes play, in the full sense of the word, the role of mobile negative ions.

In my work of 1926 it was assumed that the number of holes of each kind is equal to the number of dissociated atoms of the corresponding kind. With dissociation of both positive and negative ions we would have to have two kinds of holes,

of the corresponding two kinds of ions, and moreover in amounts equal to the number of dissociated ions of the corresponding kind.

In a recently published work (1935), Schottky pointed out the circumstance that such equality need not obtain. For the neutrality of the crystal it is sufficient that the sum of the number of dissociated positive ions and the number of negative holes (i.e., holes left by negative ions) be equal to the sum of the number of dissociated negative ions and the number of positive holes. In particular, one may imagine that in the crystal there are no dissociated ions at all, but that there are, instead, holes of both kinds in equal numbers.

Schottky was led to this idea by the consideration that in certain cases the space in the interstices proves too cramped both for negative and for positive ions. He did not, however, attempt to elucidate the mechanism by which such holes might be formed without their corresponding partners, i.e., without dissociated ions.

In this respect Schottky’s considerations are easily supplemented by the following simple representation. The space surrounding the crystal is, as it were, an unlimited reservoir of holes of either kind. By means of the above-considered process of “settling” with the ions of the crystal, these holes can gradually penetrate into the depths of the crystal, just as in the diffusion of foreign atoms. Instead of the formation of dissociated atoms, we then obtain only a gradual increase in the volume of the body. As for dissociated atoms, they too can be formed independently of holes by the detachment of one of the surface atoms from its normal position in a lattice site and its transition into the nearest interstice, whence it can then pass into the interior of the crystal, wandering through the interstices.

These phenomena of the formation of holes and dissociated atoms may occur not only in crystals of salts, but also in monatomic crystals, for example in metallic bodies. In this case the quantity of both the former and the latter must increase sharply with rising temperature, especially as it approaches the melting temperature. Thus, in a crystal at high temperature, we have a picture far more complex than the former “classical” picture of the structure of crystalline bodies. Whereas the majority of atoms, in accordance with this old picture, oscillate about certain equilibrium positions at the lattice sites, a part of these sites proves vacant, and some atoms oscillate in the interstices; moreover, neither the former nor the latter remain in one place, but travel throughout the entire crystal. The latter therefore represents a kind of solid solution of holes and dissociated atoms, and the more concentrated the higher the temperature. An ideally regular structure can exist only at absolute zero temperature. With an increase of the latter, a gradually progressing amorphiza-

tion, expressed in the above-mentioned loosening (i.e. the formation of holes), on the one hand, and in an increase in the number of incorrectly located or dissociated atoms, on the other.

It should be noted that up to the melting temperature the degree of amorphization of crystals of the type of metals or binary salts (such as rock salt) remains insignificant, not exceeding, as an approximate calculation shows, tenths and even hundredths of a percent.

This calculation is based on Boltzmann’s well-known expression for the relative number of dissociated atoms and the number of corresponding holes (under the assumption of equality between these numbers):

\[ x = e^{-\frac{U}{2kT}}, \tag{1} \]

where \(U\) is the energy of dissociation (i.e. of formation of a pair: an incorrectly located ion \(+\) a hole), \(T\) is the absolute temperature, and \(k\) is Boltzmann’s constant. This expression determines, in its main features, also the dependence of the electrical conductivity of a crystal on temperature. Thus it becomes possible to determine experimentally the value of \(U\).

It should, however, be noted that the electrical conductivity is determined by the product of the quantity \(n_0 x\) (where \(n_0\) is the total number of ions in \(1\ \mathrm{cm}^3\)) by the mobility of the ions (or holes), i.e. by the mean velocity of their displacement in the direction of the electric field per unit strength of the latter. This mobility \(w\) is connected with the diffusion coefficient of the ions \(D\) by Einstein’s well-known formula

\[ \frac{DE}{w} = kT, \tag{2} \]

where \(E\) is the charge of the ion (or hole). As for the diffusion coefficient \(D\), it can be calculated from the formula

\[ D = \frac{a^2}{2\tau}, \tag{3} \]

where \(a\) is the mean distance between two neighboring interstices (or two neighboring lattice sites in the case of a hole), and \(\tau\) is the mean time during which a dissociated ion remains in one and the same interstice (or a hole in one and the same lattice site).

This time, as I have shown in connection with the theory of adsorption phenomena, can be calculated with the aid of the following approximate formula

\[ \tau = \tau_0 e^{-\frac{U'}{kT}}. \tag{4} \]

where \(U'\) is the “activation energy” for transition from one interstice to a neighboring one, and \(\tau_0\) is the period of the oscillations performed by an ion about its temporary equilibrium position.

The theory as yet provides no possibility of determining the energies \(U\) and \(U'\). However, the values of the distance \(a\) and the time \(\tau\) may be regarded as approximately known \((a \simeq 10^{-8}\ \mathrm{cm},\ \tau \simeq 10^{-13}\ \mathrm{sec})\). Thus, in the theoretical formulas for the diffusion coefficient \(D\) and the specific electrical conductivity \(\sigma\) as functions of temperature,

\[ D=\frac{a^2}{6\tau_0}e^{-\frac{U'}{kT}}=D_0e^{-\frac{U'}{kT}}, \tag{5} \]

\[ \sigma=\frac{E^2 n_i a^2 e}{6\tau_0 kT}=\,\,\frac{U+2U'}{2kT}=\sigma_0 e^{-\frac{U+2U'}{2kT}} \tag{6} \]

the values of the coefficients \(D_0\) and \(\sigma_0\) can be calculated. These calculated values turn out to be in satisfactory agreement with experimental data, which also confirm the correctness of the temperature dependence of the quantities \(D\) and \(\sigma\) determined by the preceding formulas. This agreement is a confirmation of the correctness of the ideas set forth above.

§ 2. Thermal motion in liquids and in amorphous solids

The heat capacity of monatomic solids, as is known, is twice as large as the heat capacity of the same bodies in the gaseous state. This circumstance is explained by the fact that in solids the particles (atoms or molecules) possess not only kinetic energy, but also potential energy corresponding to the forces of interaction of each particle with its neighbors. Assuming that these forces are proportional to the relative displacements of the particles, one may show that their potential energy, since the motion of the particles is restricted to small oscillations, is on the average equal to the kinetic energy. And since the kinetic energy of thermal motion remains the same in a solid and in a gas (at sufficiently high temperatures), it follows that in a solid the thermal energy, or, more precisely, its change when the temperature is raised by one degree, i.e. the heat capacity, is twice as large as that of the same substance in the gaseous state. In the melting of solids, especially such simple bodies as metals, their heat capacity not only does not decrease, but even increases somewhat (for mercury, for example, by \(4\%\), for lead by \(27\%\)); this means that the character of the thermal motion of atoms upon melting does not undergo any substantial change. Hence the inevitable conclusion follows that the thermal motion of atoms in a molten metal, just as in a solid metal, consists

in its main features in small oscillations about certain equilibrium positions.

Since liquids lack a regular crystalline structure, the differences between them and solids from the point of view of thermal motion must reduce only to the fact that, in the case of a liquid, these equilibrium positions do not form a regular crystalline lattice. Thus, in the case of a liquid, there is no need to distinguish between regular and irregular equilibrium positions: here all equilibrium positions may be regarded as more or less irregular, like the positions of dissociated atoms of a crystal in the interstices (cf., however, below).

Dissociated atoms remain in each interstice only for a limited time and, in the end, break out of it in order to pass into one of the neighboring interstices. In exactly the same way, in liquid bodies atoms do not remain forever bound to one and the same equilibrium position, but from time to time jump from one position to a neighboring one. This time $\tau$, i.e. the time of residence in some position, is the greater the lower the temperature of the liquid. With rising temperature it decreases extremely rapidly, tending in the limit to the period of oscillations $\tau_0$, i.e. approximately to $10^{-13}$ sec. At ordinary temperatures the time $\tau$ is a hundred times greater than the period of oscillations $\tau_0$, while at very low temperatures it may attain a very considerable value.

Thus liquids, considered at any given moment, in essence differ in no way from a solid body, except perhaps only by the irregular arrangement of the particles, or, more precisely, of those temporary equilibrium positions about which they oscillate. The question arises: from this point of view, how is one to explain the fluidity of a liquid, the absence of resistance to forces tending to change its shape? This question is very easy to answer. If a force tending to change the shape of a liquid, i.e. the arrangement of its particles, acts in one and the same direction for a sufficiently long time, i.e. for a time long in comparison with the time $(\tau)$ of residence of these particles, then during this time the latter will manage repeatedly to change their equilibrium positions, moving thereby predominantly in the direction of the forces acting upon them (just as happens with ions in a crystal in the presence of an external electric field). These discrete elementary displacements, following one another at extremely short intervals of time, merge for the observer into a practically continuous flow.

But if the liquid is subjected to the action of forces whose direction oscillates with time and, moreover, so rapidly that during the residence time of the particles it manages to change many times, then with respect to such forces the liquid will behave like a solid body, without at all exhibiting

of its fluidity and resisting a change of its shape. Hence it is clear that a liquid possesses not only fluidity, but also solidity, which under ordinary conditions is merely masked by fluidity and which can reveal itself in the case of rapidly oscillating forces with a period shorter than the time of settled existence of the particles of the liquid. The latter condition can be achieved not only by shortening the period of oscillations of the external forces (for example, by passing to ultrasonic oscillations of extremely high frequency), but equally by lowering the temperature of the liquid, a lowering which, as was already indicated above, is accompanied by a sharp lengthening of the time of settled existence of the particles.

Thus a body that behaves like a liquid at high temperatures with respect to rapidly oscillating forces will, at lower temperatures, with respect to these very same forces, behave like a solid, revealing its fluidity only with respect to forces oscillating considerably more slowly or, in particular, acting constantly in one and the same direction.

When liquids are cooled, in most cases they crystallize at a quite definite temperature. However, many liquids are capable of being supercooled below the crystallization temperature and of gradually solidifying in an amorphous state. The amorphous solid thus obtained is often treated as a “supercooled liquid.” This interpretation is quite correct, provided only that one does not associate with the concept of a liquid the notion of an absence of solidity, as is usually customary.

Solidity and fluidity are not properties that mutually exclude one another; they can coexist perfectly well, usually merely masking one another. One may say that, in a certain sense, the difference between the solid and liquid states—if one abstracts from the regularity or irregularity of structure and confines oneself to amorphous bodies—depends on the choice of the unit of time. If, as the unit of time, one chooses the time of settled existence of the particles in connection with one and the same position of equilibrium, then the body under consideration will appear liquid or solid depending on whether it is subjected to the action of a long-lasting or short-lived force, i.e. a force acting in one and the same direction for a time large or small in comparison with the time of settled existence of the particles. This time \((\tau)\) we shall call the relaxation time. It is easy to show that the coefficient of viscosity \(\eta\), which characterizes the magnitude inverse to the fluidity of a liquid, is connected with the modulus of its shear elasticity \(N\), which characterizes the resistance to change of shape, by the following simple relation

\[ \eta = N\tau . \tag{7} \]

If one assumes that the solidity of a liquid, masked by its fluidity, is approximately equal to the solidity of the corresponding sol—

of a solid body and, consequently, to substitute for \(N\) the value of the modulus of elasticity for solids and if, further, the relaxation time is expressed as a function of temperature by the formula

\[ \tau=\tau_0 e^{\frac{U'}{U'_t}}, \tag{8} \]

which can easily be derived from general theoretical considerations,* then for the coefficient of viscosity of a liquid one obtains a value of the correct order of magnitude and, moreover, one that varies with temperature in accordance with experimental data.

Thus the theory of thermal motion in liquid bodies presented above not only enables us, in a certain sense, to unify the conceptions of the solid and liquid states, but also makes it possible to calculate a quantity characterizing the fundamental property of liquids and amorphous solids—their fluidity.**

Let us note that an attempt to unify the concepts of hardness and fluidity was made as early as 60 years ago by Maxwell in a purely macroscopic form, without any ideas about the nature of thermal motion in solid and liquid bodies. The conception set forth above thus provides a molecular basis for this phenomenological theory of Maxwell.

It is usually assumed that a liquid offers resistance only to uniform compression. From the point of view of the theory set forth above, however, it must equally possess resistance to uniform extension, like solids. The uniform extension of a liquid by means of so-called negative pressure is a rather difficult experimental problem; nevertheless, to a certain extent this problem has been solved, and it has been found that liquids can withstand, without rupture, negative pressures exceeding \(100\) atm.

The capacity of a liquid to offer resistance to uniform extension is also assumed by the well-known theory of Van der Waals, which brings liquids closer to gases, and from the point of view of this theory this capacity, depending on the cohesion between the particles of the liquid, is a distinctive feature of the liquid state.

It should be noted that, from the point of view of the theory set forth in this paragraph, thermal motion in liquids upon an increase in

* Here \(\tau_0\) denotes the period of oscillation (of the order of \(10^{-13}\) sec.), and \(U'\) the activation energy for the transition of particles from one equilibrium position to a neighboring one [cf. formula (4)].

** In my work of 1926 I derived a formula for the coefficient of viscosity, somewhat different from (7), namely: by replacing the shear modulus \(N\) by an expression of the form \(nkt\), where \(n\) is the number of particles per unit volume; this expression is equal to the pressure that the particles of the liquid would exert in the absence of cohesion between them.

temperature gradually approaches, in its character, thermal motion in gaseous bodies. Indeed, as \(T\) increases, the time \(\tau\) according to formula (8) approaches the time of one oscillation \(\tau_0\). However, formula (8) thereby becomes insufficiently exact and expresses in a qualitative way only the fact that the particles lose any connection even with any temporary positions of equilibrium, whose role begins to be played by short-lived stops upon the meeting or “collision” of two particles with one another.

The notion of such “pair” collisions makes sense only in the case when the mean distance between the particles proves sufficiently large, as is the case in gases, even strongly compressed ones. As is known, above a certain “critical” temperature the difference between a liquid and a gas disappears. This circumstance is usually expressed in the incorrect assertion that above the critical temperature a substance can exist only in the gaseous state. In reality, under sufficiently strong compression, any substance, being at a temperature above the critical one, can be converted into the solid crystalline state. Bridgman first pointed out this circumstance; in fact, crystallization at a temperature above the critical one was achieved in 1929 by Simon, in the case of substances with the lowest critical temperature—neon, hydrogen, and helium.

Thus, in considering the thermal motion of particles in liquids, we discover a continuous series of transitions between the picture of oscillations about practically unchanged (or very rarely changing) positions of equilibrium, typical of solids, and the picture of free translational motion, disturbed by collisions, typical of gases. The transition of a liquid from a “solid-like” state to a “gas-like” one can take place continuously with a gradual increase in temperature, provided only that the body under study is not subjected to excessively strong compression. According to the van der Waals theory, an analogous influence must also be exerted by an increase in volume at not too low temperatures. We shall return to this question in § 4.

§ 3. Gradualness in the amorphization of a liquid with increasing temperature or with increasing volume

The typical, more precisely thermodynamically equilibrium, state of solids is the crystalline state, characterized by a regular arrangement of particles. This regularity, as we saw above, is violated only to an insignificant degree when approaching the melting point as a result of the formation of holes, or dissociation of particles. In the case of simple bodies or binary ionic compounds, to which the considerations of the two preceding paragraphs applied, the amorphous solid state has until now

could not be achieved. Upon gradual cooling of the corresponding liquids, the latter—sometimes after fairly considerable supercooling—necessarily crystallize.

The discontinuous character of the crystallization process has until recently been associated with the notion that liquids, even in the immediate vicinity of the crystallization temperature, are completely devoid of any regularity in their structure, i.e., in the sense of the arrangement of their particles (atoms, ions, molecules), are just as amorphous as gases.

This notion of the complete amorphousness of liquids was supported by the rapprochement between the liquid and gaseous states that was effected by the famous van der Waals theory and that has become firmly rooted in physics over the last decades.

But if it is true that, from the standpoint of thermal motion, liquids, when the temperature is lowered (or the volume decreased), gradually pass from a purely gaseous state into a state ever more and more “solid-like,” then the notion that, with respect to their structure, they nevertheless remain completely amorphous, i.e., absolutely “gas-like,” becomes more than doubtful.

Since, from a completely free, nomadic motion at high temperatures and large volumes, disturbed only by collisions with one another, the particles of a liquid pass to an ever more settled way of life, reducing to oscillations about equilibrium positions that change more and more rarely, these equilibrium positions must gradually acquire greater and greater regularity, approaching that regularity which characterizes the arrangement of particles—or, more precisely, of their equilibrium positions—in ideal crystalline bodies.

Thus we naturally arrive at the idea that the notion of the complete amorphousness of liquids is only an idealization, applicable to the limiting case of very high temperatures (close to the critical temperature) and large volumes, and that, as the temperature is lowered or as the volume is decreased, liquids gradually become solid-like not only with respect to the character of their thermal motion, but also with respect to the increasing regularity—or, if one wishes, “crystallinity”—of their structure.

In recent years this notion has received direct confirmation in the X-ray diffraction patterns produced by various liquids, both complex (Stuart) and simple (Keesom, Debye and Menke, Bernal and Fowler)*. Thus, for example, the study of X-ray diffraction patterns of liquid mercury shows that its atoms are arranged approximately in the same way as shot poured into a glass and pressed against one another by gravity and the pressure of the walls. In this case, as is known, a densely packed hexagonal structure is obtained, in which each shot

* See V. I. Danilov, Scattering of X-rays in Liquids, ONTI, 1935.

surrounded by 12 regularly arranged neighbors. If the relative arrangement of the atoms in liquid mercury exactly coincided with the arrangement of the shot in the glass, then in its structure liquid mercury would not differ in any way from an ideal crystalline body with the corresponding hexagonal lattice.

In reality, however, the X-ray diffraction pattern of mercury (as of all other liquids) consists not of separate spots corresponding to definitely oriented crystalline planes, but of several rings, resembling in this respect the X-ray diffraction patterns of crystalline powders or microcrystalline structures consisting of a large number of small crystallites in all possible orientations.

As is known, in the latter case the width of the diffraction rings in an X-ray photograph permits one to judge the sizes of the crystallites. In the case of X-ray photographs of liquids, in particular liquid mercury, these rings prove to be broader than those which would be obtained from aggregates of crystals of the smallest conceivable dimensions (for example, consisting of groups of 13 atoms—the central one and 12 neighbors). Further, the width of these rings increases with increasing scattering angle, as though the dimensions of the crystallites were thereby decreasing.

It is clear from this that a liquid—and in particular liquid mercury—is neither a single crystal nor an aggregate of small crystallites of definite size. Its structure is neither single-crystalline nor microcrystalline. But at the same time it is not purely amorphous either. In the latter case all positions of the atoms around any one of them would be equiprobable. Analysis of the X-ray diffraction pattern of liquid mercury shows that the first several layers of atoms surrounding any arbitrarily chosen atom are arranged around it predominantly at the same distances as in single crystals, but with a considerable scatter (i.e., some are somewhat closer, some somewhat farther than they ought to be).

As Kratky showed, this scatter increases rapidly with increasing distance, in accordance with the law of addition of random displacements, i.e., in such a way that the mean displacement of atoms from those positions which they should have occupied relative to the initial one in a regularly constructed crystal lattice increases proportionally to the square root of the distance; moreover, as V. E. Lashkarev showed, the distribution of displacements about this mean or most probable position is determined by the Gaussian law of random errors.

A structure of this kind we shall call “pseudocrystalline” or “crystal-like.”

Let us note that the degree of crystallinity, determined by the mean value of the “scatter” of the nearest neighbors of the initial atom, decreases with decreasing temperature, so that as the temperature approaches the crystallization temperature the liquid becomes more and more crystal-like.

Comparison of the crystal-like structure of liquids with the crystalline—more precisely, with the single-crystal—structure of solids can be carried out from two different points of view, depending on whether we direct our attention to the arrangement of atoms far from the initial one (“long-range correlation”) or close to it (“short-range correlation”). From the first point of view there is an impassable gulf between the structures of the two types. Indeed, in a single crystal the regularity of arrangement is preserved over arbitrarily large distances: in moving away from the initial atom along any straight line we obtain a regular alternation of atoms—or, more precisely, of their equilibrium positions—with one and the same periodicity over arbitrarily large distances. In a crystal-like structure, however, the regularity of arrangement with respect to the initial atom disappears more or less rapidly as one moves away from it, so that the system of crystallographic axes drawn from one atom to three of its neighbors, as the distance from the initial atom increases, is not only deformed in an irregular manner, but is also irregularly rotated (this is why the X-ray diagrams of liquids are, to some extent, similar to the X-ray diagrams of microcrystalline solids).

This point of view of “long-range correlation” is of importance for comparing the properties of a liquid with those properties of crystalline solids which depend essentially on the preservation or disappearance of a regular periodicity in the arrangement of particles at large distances. Such properties include electrical conductivity (in the case of metals and especially of electronic semiconductors) or mechanical properties, manifested as flow in liquids and as plastic slips in solids, especially in single crystals.

Let us note that, with respect to plastic properties, single crystals differ substantially from microcrystalline aggregates, which is in accord with the presence of long-range correlation in the former and its absence in the latter.

However, with respect to a whole series of other important physical properties—mechanical (for example, compressibility or tensile strength), thermal (for example, heat capacity or coefficient of expansion), electrical and magnetic—there are no substantial differences between a single crystal and a microcrystalline aggregate.*

When comparing a liquid with a solid with respect to properties of this kind, determined in their main features by the interaction of the nearest particles (as is known, interparticle forces decrease extremely rapidly with increasing distance), the point of view of long-range correlation loses all meaning. If, however, one adopts

* The classification of the properties of solids and liquids considered by us coincides with that which has long been applied to solids; in this case the properties of the first category are called “structure-sensitive,” and the properties of the second category “structure-insensitive.”

from the point of view of “near-range correlation,” the difference between the crystalline structure of a solid body and the “crystal-like” structure of a liquid—especially near the crystallization temperature—will prove to be not qualitative, but quantitative, gradually increasing with the rise of the temperature of the liquid or with the increase of its volume, until, finally, above the critical temperature and at a sufficiently large volume the liquid becomes completely amorphous.

The gradual increase in the crystallinity of a liquid upon lowering the temperature or decreasing the volume can be continued beyond the crystallization temperature as well—experimentally, by using supercooling of liquids; theoretically, by gradually reducing the statistical scatter in the positions of the particles around one of them and thereby introducing correlation, until, finally, we obtain a strictly crystalline structure.

It should be noted that the disorientation of the various elementary cells of a crystal-like structure (i.e., the unequal orientation of these cells, which causes the appearance of rings on the X-ray diffraction patterns of liquids) should not necessarily be regarded as the result of a gradual curvature of the crystal axes as they are continued from the initial cell to the next.

In a crystal-like structure such lines cannot be drawn in an unambiguous manner. The principal cause of the decoördination characterizing such a structure is, it seems to me, the formation of holes and dissociated atoms, considered by us in § 1: with a sufficiently large number of both and in the absence of any regularity in their distribution, no trace of long-range correlation will remain, and the lattice elements in different places will seem to be oriented in different ways. To make this circumstance clear, it is enough to imagine a square of soldiers, the orderly arrangement of which has been disturbed by, say, one quarter of the soldiers leaving their places by half the distance separating each of them from one of his four neighbors. With a random distribution of such “disturbers of order,” the orderly square will turn into a crowd in which we shall be able to discern some degree of coordination only in the immediate vicinity of each individual.

In the solid state—up to the melting temperature—the disorder of coordination, or the degree of amorphization, remains, as we saw above, extremely insignificant. In the liquid, above the crystallization temperature, this disorder of coordination at once proves to be very considerable, but nevertheless still very far from the limit corresponding to complete amorphization, which it approaches only gradually as the temperature rises or the volume of the liquid increases.

This gradual and, moreover (as X-ray diffraction patterns of liquids show), very rapid increase in the degree of amorphization of a liquid with rising temperature explains the circumstance

(as already noted in § 2) that the heat capacity of liquids at temperatures not too far from the crystallization temperature is always greater than the heat capacity of the corresponding solids (for mercury, for example, by several percent, for tin—by 13, for lead—by 25, and for water—by 100%). This additional heat capacity corresponds to the progressing “amorphization” of the crystal-like structure of the liquid, an amorphization that only begins upon melting (and is not completed by it, as is usually assumed). The increased heat capacity of liquids corresponds to the energy of amorphization, part—and moreover usually only a small part—of which is found in the form of the latent heat of fusion.

Another characteristic feature of liquids, connected with the relative looseness of their structure, is, as Bridgman showed, their anomalously high compressibility: at ordinary pressures it exceeds the compressibility of the corresponding solids by approximately 15 times. This increased compressibility disappears at pressures exceeding 2000 atm. It would be interesting to trace how the X-ray diffraction pattern of liquids changes in this case.

In conclusion to this paragraph we must note one more circumstance. The crystal-like structure of liquids does not always coincide in its type with the crystalline structure of the corresponding solids. Thus, for example, mercury in the solid state crystallizes in the form of a rhombohedral lattice, and not in a hexagonal close-packed lattice, which characterizes liquid mercury.

In exactly the same way, liquid water possesses a structure analogous to that of quartz, whereas ice has a tridymite-type lattice. Thus the melting of solids may be connected not only with the amorphization of the initial crystalline lattice, but in a number of cases—especially for simple substances—also with a change in lattice type, analogous to allotropic modifications in the solid phase.

§ 4. Theory of Melting and Crystallization

The mechanism of the melting of solids or the crystallization of liquids has until now remained entirely unclear. We have had only a formal thermodynamic theory, which made it possible to connect the melting (or crystallization) temperature with the external pressure. In this, however, the question remained entirely unexplained as to why melting—at a given external pressure—takes place in a discrete manner at a quite definite temperature, and not over some temperature interval.

So long as the liquid state was considered qualitatively different from the solid, so long as, both with respect to the character of thermal motion and with respect to structure, it was regarded as “gas-like” (differing from the latter only by a large ...).

density), and not “solid-like,” this discreteness of the transition from the solid state to the liquid state appeared quite natural.

We have seen, however, that in reality the difference between the solid and liquid states has rather a quantitative, and not a qualitative, character, like the difference between liquids and gases.

The latter circumstance does not, however, as we know, prevent the discreteness of the processes of evaporation (boiling) of a liquid and condensation of vapor. According to the Van der Waals theory, the liquid and gaseous states are limiting forms of a certain qualitatively unified, absolutely amorphous “liquid-gaseous” state, whose density or volume at a given temperature may assume all intermediate values between those corresponding to the two limiting forms.

At the same time, however, the change of volume is connected with a very peculiar change of the external pressure, which is shown by the solid curve in Fig. 1 and which is characterized by the presence of an intermediate region \(BD\), where the increase of volume occurs not with a decrease of pressure, but, on the contrary, with its increase. This region of states proves to be mechanically unstable. As a result, under an isothermal increase of the volume, the “liquid-gaseous” body at a certain point \(A\) splits into a liquid with specific volume \(V_1\) and a gas with specific volume \(V_2\), corresponding to the point \(E\), at which the Van der Waals curve intersects the straight line drawn from \(A\) parallel to the abscissa axis. The increase of volume is then accompanied by a gradual increase in the amount of the gaseous phase at the expense of the liquid phase, until finally, at \(V_1 = V_2\), all the liquid has turned into vapor. The position of the straight line \(AE\), characterizing the evaporation of the liquid, is determined in this case by the condition of equality of the areas \(ABC\) and \(CDE\).

Fig. 1.

If, in this way, the principal discontinuity between the gaseous and the (idealized) liquid state, in connection with the instability of the intermediate states of the “liquid-gaseous” body, in fact leads to a discontinuous process of evaporation of a liquid or condensation of vapor, then the question arises whether it is not possible in an analogous manner to explain the discontinuity of the transition from the liquid state to the solid one observed in experiment, taking into account the fact (established in the preceding paragraphs) that a liquid, on cooling or compression, gradually loses its similarity to a gas and becomes more and more “solid-like.”

In order to formulate this idea, it is necessary first of all to postulate—as is done in the van der Waals theory—the existence of a continuous series of states intermediate between liquid and solid, not only with respect to density (or volume), but also with respect to the character of thermal motion and the further ordering of the crystal-like structure that is observed in liquids near the crystallization temperature.

From the point of view of the ideas set forth above concerning the partial amorphization of solids near the melting temperature, this hypothesis appears quite natural.*

Next it is necessary to show that, with a gradual increase in the volume of such a “solid-liquid” body, the external pressure (at constant temperature) changes along a curve analogous to the van der Waals curve, passing first through a minimum and then through a maximum, as shown in Fig. 1.

If we gradually stretch a crystal uniformly in all directions, then, according to the usual notions, it must ultimately rupture. However, long before this rupture, which theoretically requires an increase in volume of about 15%, the crystal will begin to lose its regular structure, provided only that its temperature is above absolute zero. In fact, when the interatomic distances increase, the energy that must be expended for the dissociation of the crystal decreases, in particular for the formation in it of holes or dissociated atoms: as the interstices expand, more and more room appears in them for dissociated atoms. We see, therefore, that during an isothermal expansion of a crystal in all directions the degree of its amorphization must increase in the same way as with an increase in temperature; in other words, its structure must thereby gradually approach that of liquids.

If the pressure exerted by the crystal at a given volume decreased monotonically with increasing volume, we would have a continuous transition from a solid state with small volume and a high regularity of structure to a liquid state with larger volume and negligible regularity of structure.

It is easy, however, to show that the thermal oscillations of atoms about the equilibrium positions determined by their mean distances from one another lead to the fact that, at not too low temperatures, the pressure of a crystal, as its volume increases, first decreases to a certain minimum, then increases to a certain maximum, after which it begins to fall monotonically.

* The fact that in certain substances, for example mercury or water, melting is accompanied by a change in the type of crystal lattice does not contradict our hypothesis; one may assume that this change, entirely analogous to allotropic transformations in the solid phase, occurs at some intermediate state (volume) of the “solid-liquid” body.

This result is explained in the following way. The forces holding each atom near its mean position are asymmetric with respect to relative displacements in the direction of approach, namely: in the first case they increase more slowly than in the second. This explains the fact that upon heating, i.e. with an increase in the amplitude of the thermal vibrations of the atoms, solids expand. The same fact can be interpreted as the result of a pressure caused by thermal vibrations in connection with the negative pressure associated with the general expansion of the body, i.e. with an increase in the distance between the mean positions of the atoms. The thermal pressure is directly proportional to the absolute temperature and increases with increasing volume because this increase augments the above-mentioned asymmetry of the interatomic forces. The increase of the thermal pressure with increasing volume ceases only owing to the amorphization of the body (otherwise, i.e. if an ideally regular structure were preserved, it would increase without bound at the point of rupture). Superposed on the monotonic decrease of the static pressure (which depends on the mean distances between atoms), the thermal pressure also determines—at not too low temperatures—the shape of the isotherm (pressure—volume) for a “solid-liquid” body, similar to the shape of the Van der Waals isotherm for a “liquid-gaseous” body.

As a consequence of this, in the first case as well as in the second, the transition from the solid state to the liquid state is in fact accomplished discontinuously at a quite definite pressure, corresponding to the dashed horizontal line in Fig. 1. At the same time, a gradually increasing part of the body passes from an almost regular crystalline structure with a smaller specific volume to a “crystal-like” structure with a larger volume.

The difference between the theory of melting and crystallization set forth here and the Van der Waals theory of evaporation of a liquid and condensation of a vapor consists only in the following. According to the Van der Waals theory, above a certain temperature, called critical, the oscillation of the isotherms (pressure—volume) disappears. In this case the distinction between the gaseous and liquid states disappears; they merge into a single “liquid-gaseous” state, stable at all intermediate volumes. In our case, however, this oscillation of the isotherms, since it is directly caused by thermal motion, disappears not above a certain temperature, but, on the contrary, below a certain temperature. Thus, below this temperature, which may be called the critical temperature of melting or crystallization, the transition of a body from the crystalline state to the amorphous state (which, depending on temperature, may be both liquid and solid in the sense of § 2) must take place continuously. The two critical temperatures have, generally speaking, nothing in common with each other. Hence it is clear that any body can be obtained in a solid crystalline state at tempe-

…temperature higher than the critical temperature in the ordinary sense of the word (i.e., in the sense of the van der Waals theory), provided only that it is subjected to a sufficiently great pressure. This circumstance, as was already mentioned above, has in recent years been verified experimentally by Simon in the case of hydrogen and neon.

We thus see that the liquid state represents an intermediate link between the solid crystalline and gaseous states. In principle, the transition between all three states by a change of volume at constant temperature is continuous. In practice we observe in it one or two discontinuities because of the unstable character of the intermediate states associated with a decrease of pressure upon compression. There are substances for which an isotherm (the pressure–volume curve) within certain temperature limits gives not two, but a larger number of oscillations; such substances, along with the usual three states of aggregation, occur in states of intermediate character, for example in the form of “liquid crystals” (or “anisotropic liquid”).

It must be assumed that the liquid-crystalline phase may be connected with the solid crystalline and with the ordinary liquid phase by a continuous series of intermediate states of unstable type in the same way as the liquid phase connects the solid and gaseous phases with one another.

From our theory there follows yet another curious consequence, namely: the possibility of obtaining any substances in a solid amorphous state, which, as is known, in the case of simple substances, for example metals, has not so far been achieved. The reason for this failure should be sought in the same circumstance as the reason for the failure of earlier attempts to condense the so-called permanent gases. In the latter case the failure was explained by the fact that these gases were compressed at a temperature above their critical temperature.

The failure of attempts to obtain metals and other simple bodies in the solid amorphous state is explained, in all probability, by the fact that these substances were investigated either at temperatures above the critical temperature of crystallization, or only at ordinary positive pressure. Meanwhile, in order to obtain these substances in the amorphous state, it is necessary to subject them to negative pressure, after cooling them below the critical temperature of crystallization.

This prediction of the theory has not yet been subjected to experimental verification because of the difficulty of realizing a negative pressure of considerable magnitude.

§ 5. Additional Remarks

The considerations and results set forth above relate chiefly to simple bodies—metals and inert elements—and only in part to salt-like compounds and more complex substances.

THERMAL MOTION IN SOLID AND LIQUID BODIES

Without dwelling on a detailed consideration of the questions relating to this, I would like to note only one circumstance characteristic of substances consisting of simple molecules of small size (for example, HCl, HBr) or containing simple ionic radicals ($\mathrm{NH_3^+}$, $\mathrm{NO_2}$, etc.), namely: the structure of such substances is characterized not only by the positions of the centers of gravity of the molecules (or radicals), but to an equal degree also by their orientation with respect to one another.

In the crystalline state at low temperatures these orientations are distributed in just as regular a manner as are the positions of the centers of gravity. In saying this I have in mind the mean or equilibrium orientations, about which the molecules (or, more precisely, the radicals) can execute small oscillations, just as their centers of gravity oscillate about definite positions of equilibrium.

With an increase in temperature or an increase in the volume of the body (i.e., in the mean distance between molecules), the amplitude of these rotational oscillations must gradually increase, and in a catastrophic manner at that, since its increase is accompanied by a diminution of the mean orienting effect which each molecule experiences from the others.

The result of this may be a process which it is natural to call “orientational melting,” in view of its analogy with ordinary melting. This process amounts to the following: at a certain temperature, depending on the external pressure (or within a certain narrow temperature interval), the molecules pass from a regular, “crystalline” distribution of their equilibrium orientations to an irregular, or, more precisely, not quite regular, distribution, which may be interpreted as “crystalloid” and which must gradually lose the remnants of regularity with further increase of temperature (or increase of volume).

This phenomenon is in fact observed in a whole series of substances, usually manifesting itself in the form of a sharp anomaly of the heat capacity, which usually—within a very narrow temperature interval—first rapidly rises to a very large value and then drops precipitously, returning to values of the usual order of magnitude. The area of the anomaly is a measure of the energy of “orientational amorphization” and corresponds to the latent heat of melting. Thus the body under consideration remains crystalline with respect to the arrangement of the centers of gravity of its molecules, but becomes crystalloid or even amorphous with respect to their orientation.

The character of the thermal motion—or, more precisely, of that part of it which corresponds to the rotation of the molecules—may in this case undergo no substantial change, just as the character of the vibrational motion of the centers of gravity of the molecules does not undergo such a change in ordinary melting. Thus we must imagine that the “disoriented” molecules above the temperature of orientational melting

do not pass into free rotation, as occurs in the case of gases (and as was previously assumed by Pauling), but continue to execute small oscillations about certain mean orientations, changing from time to time.

This same character of the rotational motion of molecules must, generally speaking, be retained even after the ordinary melting of a crystal, i.e., in the liquid phase. In particular this applies, for example, to the molecules of such a liquid as water. The former notion of free “gas-like” rotation of molecules in liquid water must be replaced, as Debye has especially emphasized recently, by the notion of rotational librations of water molecules about certain more or less irregularly (or, conversely, more or less regularly) distributed equilibrium orientations, which from time to time change abruptly.

This notion leads to essentially new results in the question of the dielectric constant and dielectric losses in water and other polar liquids.