Theories of the Liquid State

I. Z. Fisher

§ 1. Introduction

The aim of the present brief review is to supplement the review article by J. de Boer1 printed above with the results of certain works that were not included in that article or were published after it had been written. Such a supplement seems to us all the more necessary because, of the two main directions in the contemporary development of the theory of the liquid state, de Boer’s article gives a detailed account only of the direction of “free-volume theories,” while the works based on the study of the radial distribution function receive quite inadequate treatment. Meanwhile, by the present time results have been obtained in this latter direction that are of undoubted interest for the theory of the liquid state, even though they represent only the first steps of a new theory.

The development of the theory of the liquid state is one of the most important problems of modern statistical physics. At present this problem is still very far from any complete solution, but nevertheless, over roughly the last 10–15 years, the efforts of many investigators have indicated paths toward its solution.

Among theories based on taking intermolecular forces into account, until recently noticeable successes had been achieved only by the free-volume theory, and therein lies its chief value. But the significance and possibilities of this theory, developed mainly by English and American investigators, are undoubtedly overestimated by its authors. At the basis of the free-volume theory lies an idea that is in itself correct, according to which, in calculating the configurational integral, because of the mathematical difficulties that arise, one may approximately replace integration over the entire phase space of the system by integration over comparatively small regions with the greatest statistical weight, neglecting regions of phase space that are large in size but insignificant in their statistical weight. For example, Debye–Born’s theory of the crystalline state is based on this. However, the correct choice of these

selecting regions in phase space is a very difficult matter, and the implementation of this choice in the theory of free volume raises objections. Here this choice is made on the basis of a “lattice” model of the liquid state and with the aid of various kinds of additional restrictions, often arbitrary ones. The insufficient rigor in the choice of these regions of phase space is reflected in a lowered value of the statistical weight of the liquid state (and, consequently, of the entropy of the liquid) in all variants of the theory of free volume. A rigorous analysis from consistently statistical positions of the theory of free volume is given in work², where the crude approximate character of this theory is shown.

Moreover (and in physical meaning—above all), the most important task of the physical theory of the liquid state should be the explanation of the actually existing molecular structure of a liquid and of its origin. In the theory of free volume the structure of a liquid is not explained, but is postulated in an arbitrary manner. The circumstance that, on the basis of the lattice model of a liquid, in some variants of the theory of free volume one succeeds in obtaining the “correct” value of the integral of states not only near the melting point, but also far from it—at small densities, up to ideal-gas densities—should evoke not so much a feeling of satisfaction as surprise at the “fortunate” choice of more or less arbitrary assumptions.

Among the shortcomings of the same origin should be included the fact that in the theory of free volume the phase transitions liquid—gas and crystal—liquid are represented by S-shaped bends of the \(p—V\) isotherms, whereas in a correct statistical theory the integral of states cannot contain unstable states of the system. A model in this respect may be furnished by the modern theory of real gases, where the transition region between gas and liquid at temperatures below the critical one corresponds to a horizontal section of the \(p—V\) isotherm. (See, for example,³.)

A less substantial shortcoming of the theory of free volume is that it has been developed as applied to a very special form of intermolecular potential, taken in the form \(\Phi(r)=4\varepsilon[(a/r)^{12}-(a/r)^6]\), which at present cannot be considered satisfactory either at large or at small distances between particles. This question will be discussed in more detail below.

With all these, as well as other, shortcomings, the theory of free volume is of known interest as a theory showing how one can, by comparatively very simple means, arrive in a first approximation at the solution of the problem of the liquid state. This theory also has the great merit consisting in the visual clarity of its representations, in the simplicity of the physical models underlying it. But at the same time it must be remembered that

The most valuable model of a phenomenon is that which follows from the theory, and does not more or less successfully precede it.

Without denying the value—in the indicated sense—of the free-volume theory, we must at the same time look more attentively than was done in de Boer’s survey at other existing statistical theories of the liquid state. It seems very probable that precisely the theories based on the study of the radial distribution function will in the near future be able to advance us further in understanding the nature of the liquid state and to provide more rigorous statistical methods for studying the liquid state. The development of this direction in the theory of liquids is closely connected with the investigations of N. N. Bogolyubov\(^{4,5}\), who provided rigorous mathematical methods whose application to the theory of the liquid state is very promising. So far only the first steps have been taken in this direction; despite many difficulties, they should be regarded as encouraging. The present article is devoted to the exposition and discussion of the existing works in this direction.

The emergence of modern statistical theories of the liquid state, with a more or less well-developed mathematical apparatus that makes it possible to obtain not only qualitative but also quantitative results, necessarily had to be preceded by the development of general physical ideas about the nature of liquids. In this connection it is necessary to note the exceptional merits of the late Ya. I. Frenkel in creating the now generally accepted ideas about the physical nature of the liquid state and the processes in that state. Ya. I. Frenkel’s many years of investigations on the theory of liquids are collected in his widely known monograph\(^{6}\), devoted to the kinetic theory of liquids.

In contrast to the formerly prevailing view of the closeness of the liquid and gaseous states, which became established mainly thanks to the successes of the van der Waals theory, Ya. I. Frenkel as early as 1925 for the first time put forward the fruitful idea of the closeness of the liquid and solid states. The approximate equality of the densities of a solid body and of its melt leads to the conclusion that in both cases the interatomic distances and interatomic interactions are approximately the same. Hence follows the closeness of many thermodynamic properties of solids and liquids (for example, heat capacities), and—what is much more important—the idea of a quasicrystalline structure of liquids (in the sense of the existence of short-range order in the arrangement and orientation of the particles of a liquid) and of the somewhat analogous character of the thermal motion of atoms or molecules in both states. In essence, Ya. I. Frenkel was the first to pose the problem of the molecular structure of liquids. As is known, experiments on the scattering of X-rays in liquids did indeed confirm the existence of short-range order in liquids. These experiments in a short time led to the creation of rather well-developed, more...

more detailed than had originally been predicted by Ya. I. Frenkel’s notions concerning the structure of liquids. No small role in the development of these “structural” theories also belonged to Ya. I. Frenkel.

Ya. I. Frenkel’s ideas on the closeness of the liquid and solid states made possible the emergence of “lattice” theories of the liquid state, in particular the theory of free volume. To Ya. I. Frenkel also belongs the idea of considering the melting process as a process of disordering. The melting theory of Lennard-Jones and Devonshire (see above, § 7 of de Boer’s survey) is, in essence, a repetition of the theory, developed earlier by Ya. I. Frenkel, of the dissociation of atoms (ions) and the formation of holes in real crystals.

Exceptionally fruitful consequences of the views developed by Ya. I. Frenkel on molecular structure and on the character of thermal motion in liquids were obtained by him in the field of the theory of transport processes in liquids and of the kinetics of crystallization. These works were subsequently developed by many physicists and have firmly entered the practice of modern science. However, these questions go far beyond the scope of the present article, and we cannot dwell on them here.

Turning to the exposition and discussion of works on the theory of the liquid state based on the study of the radial distribution function, we shall make the following remark. As in de Boer’s survey, below it is assumed throughout that the discussion concerns a liquid whose particles interact according to the law of central forces and whose total potential energy is, with sufficient accuracy, represented by the sum of the interaction energies of individual pairs of particles.

§ 2. RADIAL DISTRIBUTION FUNCTION

The radial distribution function was introduced into physics in connection with the problem of the angular distribution of the intensity of X-ray scattering by liquids\(^7\). This intensity distribution is determined by the mutual distribution of the particles of the liquid in space. If we denote by \(dw(r)\) the probability of finding an arbitrarily chosen pair of particles in a liquid at a distance from \(r\) to \(r+dr\) between their centers, then one may write \(dw(r)=g(r)\cdot 4\pi r^2 dr/V\), where \(g(r)\) is the so-called radial distribution function, and \(V\) is the total volume of the liquid. The function \(g(r)\) depends (as on parameters) on the temperature \(T\) and the density \(1/v\) of the liquid: \(g=g(r;T,v)\). The relative intensity of X-ray scattering is uniquely determined by the function \(g(r)\). It turned out that the converse is also true: from the known data for the intensity of X-ray scattering one can uniquely determine the form of the function \(g(r)\). (Here one has in mind a liquid of the type stipulated above.) In this way the functions \(g(r)\) were obtained for many liquids, which played a very important role in the development of our knowledge of the stru—

…of liquids (see, for example, \(^8\) and the review of experimental data \(^9\)). At the present time, scattering of neutron beams by liquids is also used for this purpose.\(^{10}\)

The special significance of the radial distribution function consists in the fact that it not only gives direct information about the molecular structure of the system under study, but knowledge of it also makes it possible to calculate the energy equation and the equation of state of a liquid or gas according to the equations (for the derivation see \(^4\)):

\[ U=N\left\{\frac{3}{2}kT+\frac{2\pi}{v}\int\limits_0^\infty \Phi(r)g(r)r^2\,dr\right\}, \tag{1} \]

\[ p=\frac{kT}{v}\left\{1-\frac{2\pi}{3vkT}\int\limits_0^\infty \Phi'(r)g(r)r^3\,dr\right\}, \tag{2} \]

where \(N\) is the total number of particles in the system, \(v=V/N\) is the volume per particle, and \(\Phi(r)\) is the intermolecular potential. Knowledge of \(U(V,T)\) and \(p(V,T)\) makes it possible, by known methods, to calculate all thermodynamic functions of a liquid or gas. With the aid of \(g(r)\) one can also calculate fluctuations of various quantities. For example, for fluctuations of the number of particles of a gas or liquid in an arbitrary macroscopic region \(G\), small in comparison with the total volume of the system \(V\), one obtains

\[ \overline{(\Delta N_G)^2}=\overline{N_G}\left\{1+\frac{4\pi}{v}\int\limits_0^\infty (g(r)-1)r^2\,dr\right\}, \tag{3} \]

where \(\overline{N_G}\) is the mean number of particles in \(G\).

The function \(g(r)\) is one of the set of “partial distribution functions” \(F_s(q_1,\ldots,q_s)\) \((s=1,2,3,\ldots,N)\), which determine the probability of a specified configuration of an arbitrarily chosen set of \(s\) particles of the system \(dw(q_1,\ldots,q_s)\) according to the relation

\[ dw(q_1,\ldots,q_s)=F_s(q_1,\ldots,q_s)\frac{dq_1,\ldots,dq_s}{V^s}, \tag{4} \]

where \(q_k=(q_k^1,q_k^2,q_k^3)\) is the set of Cartesian coordinates of the \(k\)-th particle and \(dq_k=dq_k^1dq_k^2dq_k^3\). For a gas or liquid, neglecting wall effects, \(F_1(q)=1\) and \(F_2(q_1,q_2)=g(|q_2-q_1|)\). In addition, for \(s=N\) the function \(F_N\) is proportional to the configurational part of the Gibbs distribution function:

\[ F_N(q_1,\ldots,q_N)=V^N D_N(q_1,\ldots,q_N), \tag{5} \]

\[ D_N(q_1,\ldots,q_N)=Q_N^{-1}\exp\left\{-\frac{1}{kT}\sum_{i>k}\sum \Phi(|q_i-q_k|)\right\}, \tag{6} \]

where \(Q_N\) is the configurational part of the state integral (configurational integral).

N. N. Bogolyubov was one of the first to introduce into statistical physics the apparatus of partial distribution functions and thereby gave the most complete and rigorous investigation of them.

Referring for details to the monograph of N. N. Bogolyubov\(^{4}\), we shall here confine ourselves to reporting some results of this investigation that are needed for what follows.

Differentiating the obvious relation

\[ F_s(q_1,\ldots,q_s)=V^s\int\cdots\int D_N(q_1,\ldots,q_N)\,dq_{s+1}\ldots dq_N \tag{7} \]

with respect to the coordinates of one of the particles, for example the first, one easily arrives at equations relating the functions \(F_s\) and \(F_{s+1}\). In the limit \(V,\,N\to\infty\) (but \(v=V/N=\mathrm{const}\)) these equations have the form

\[ kT\frac{\partial F_s}{\partial q_1^\alpha} +\frac{\partial U_s}{\partial q_1^\alpha}F_s +\frac{1}{v}\int \frac{\partial \Phi_{1,s+1}}{\partial q_1^\alpha} F_{s+1}\,dq_{s+1}=0, \tag{8} \]

\[ \alpha=1,2,3;\qquad s=1,2,3,\ldots, \]

where \(\Phi_{1,s+1}=\Phi(|q_{s+1}-q_1|)\) and

\[ U_s=\sum_{1\le i<k\le s}\Phi(|q_i-q_k|) \]

(in the absence of external force fields, which is assumed throughout).

These are the fundamental equations of N. N. Bogolyubov for determining the partial distribution functions. They constitute an infinite system of linear integro-differential equations.

The functions \(F_s(q_1,\ldots,q_s)\) determined by equations (8) must, moreover, satisfy the following conditions, following from their definition:

1. They must be symmetric functions of their arguments.

2. They must satisfy normalization conditions of the form

\[ \lim_{V\to\infty}\frac{1}{V}\int F_s(q_1,\ldots,q_s)\,dq_s = F_{s-1}(q_1,\ldots,q_{s-1}), \tag{9} \]

where the approach to the limit must be sufficiently rapid (see below).

1. They must satisfy the condition of weakening of correlations in the positions of particles as their mutual separation increases:

\[ \left. \begin{aligned} \lim F_s(q_1,\ldots,q_s)&=\prod_{1\le i\le s} F_1(q_i),\\ \text{when all } |q_i-q_k|&\to\infty\quad (i,k\le s). \end{aligned} \right\} \tag{10} \]

In work\(^{11}\) it was shown that the fundamental equations (8) can be interpreted as conditions for the absence of a flux of particles

in three-dimensional space. Namely, for a given \(s\), each of equations (8) is the condition for the absence of a flux of particles in the system, if \((s-1)\) particles in it are regarded as fixed.

In the case of a gas system, equations (8) can be solved by expanding the functions \(F_s\) in series in powers of the density \(1/v\), which is a small parameter. In this case, as was shown by N. N. Bogoliubov himself, results are obtained that are identical with those of the Ursell–Mayer theory for real gases\(^3\). In the case of a liquid such a method is no longer applicable. But since the principal quantity of interest to us is \(F_2(q_1,q_2)=g(|q_2-q_1|)\), satisfying the equation

\[ kT\frac{\partial F_2}{\partial q_1^\alpha} +\frac{\partial \Phi_{12}}{\partial q_1^\alpha}F_2 +\frac{1}{v}\int \frac{\partial \Phi_{13}}{\partial q_1^\alpha}F_3\,dq_3=0, \tag{11} \]

the problem of determining \(F_2\) would in principle be solved if \(F_3\) were expressed through \(F_2\), at least approximately. The simplest, though not the only possible, approximation of this kind is the so-called “superposition approximation,” already mentioned above in the article by de Boer:

\[ F_1(q_1)F_1(q_2)F_1(q_3)F_3(q_1,q_2,q_3)= \]
\[ =F_2(q_1,q_2)F_2(q_2,q_3)F_2(q_1,q_3), \tag{12} \]

or, in the case of a liquid or gas,

\[ F_3(q_1,q_2,q_3)=g(|q_2-q_1|)\,g(|q_3-q_1|)\,g(|q_2-q_3|). \tag{13} \]

If this is used, then, after the corresponding simplifications, equation (11) is reduced to the form (see [4])

\[ -kT\ln g(r)=\Phi(r)+\frac{2\pi}{v}\cdot \frac{1}{r}\int_0^\infty (g(\rho)-1)\times \]
\[ \times\left\{\int_{|r-\rho|}^{r+\rho}\mathcal{G}(t)t\,dt\right\}\rho\,d\rho, \tag{14} \]

where

\[ \mathcal{G}(t)=\int_\infty^t \Phi'(t)g(t)\,dt. \tag{15} \]

This is precisely N. N. Bogoliubov’s approximate equation for determining the radial distribution function \(g(r,T,v)\).

Condition (10) (\(g(r)\to 1\) as \(r\to\infty\)) is satisfied automatically by virtue of (14). The normalization condition (9) is now written as

\[ \lim_{R\to\infty}\frac{3}{R^3}\int_0^R g(r)r^2\,dr=1. \tag{16} \]

But it has already been mentioned above that the passage to the limit in (16) must

be sufficiently strong, which is connected with the existence of the fluctuation formula (3). By virtue of a very general law of statistical physics for fluctuations in the number of particles in any region \(G\) that is small in comparison with the total volume of the system \(V\) (see, for example, \(^{12}\)):

\[ \overline{(\Delta N_G)^2} = kT\left(\frac{\partial \overline{N_G}}{\partial \mu}\right)_{T,V}, \tag{17} \]

where \(\mu\) is the chemical potential, for ordinary molecular systems one must necessarily have \(\overline{(\Delta N_G)^2}\sim \overline{N_G}\). This, generally speaking, proves to be incorrect for systems with long-range Coulomb interaction. See, for example, \(^{13}\). But according to (3), for this it is necessary that the integral entering into (3) be finite (where, more precisely, instead of the upper limit \(\infty\) there should have stood \(R\), the radius of the region \(G\)). Thus the necessary, more stringent normalization condition for a solution of equation (14) is

\[ \int_{0}^{\infty} (g(r)-1) r^2\,dr < +\infty . \tag{18} \]

We must now dwell in somewhat greater detail on the “superposition approximation.”

First of all, recalling the probabilistic meaning of the functions \(F_s\), it is easy to see that assumption (13) is equivalent to the assumption that the density of the conditional probability \(W_{q_1,q_2}(q_3)\) that particle \(q_3\) will have a definite position when the positions of particles \(q_1\) and \(q_2\) are already known and fixed can be represented in the form

\[ W_{q_1,q_2}(q_3) \doteq W_{q_1}(q_3) W_{q_2}(q_3), \tag{19} \]

where \(W_{q_1}(q_3)\) and \(W_{q_2}(q_3)\) are the densities of the conditional probabilities that particle \(q_3\) has the specified position when the position of only one of the two other particles is known and fixed. Although expression (19) is not quite exact, it may nevertheless be assumed to be suitable as a first approximation. Accordingly, the theory of the liquid state constructed on the “superposition approximation” must be regarded and evaluated precisely as a first-approximation theory.

Next we note that assumption (13) obviously satisfies the symmetry requirement for the function \(F_3\) and, as is not difficult to verify, also satisfies the normalization conditions (9) and the weakening of correlation (10) for \(F_3\), provided only that these conditions are fulfilled for \(g(r)\). It follows from this that assumption (13) is valid at large distances between particles (greater than several molecular diameters). But, on the other hand, assumption (13) proves to be valid also at sufficiently small distances between particles. Indeed, if the form of the potential \(\Phi(r)\) is such that it ensures the “impenetrability” of the particles, then the correct functions \(g(r)\)

must vanish for \(r\) smaller than the effective diameter of the particle. But then this “impenetrability” is fully reflected also in \(F_3\) according to (13), since \(F_3 = 0\) if at least one of the functions \(g(|q_i - q_k|)\) \((i, k = 1, 2, 3)\) vanishes, i.e., the probability of such an arrangement of three particles in the system, when at least two of them are superposed on one another (“penetrate” one another), is equal to zero.

Since, in this way, assumption (13) proves to be valid both at large and at small distances between particles and, moreover, satisfies the normalization condition for the function \(F_3\), one may suppose that it is also approximately valid at intermediate distances between particles.

In works \(^{14,15}\) it was shown that, for the model of hard spheres, the fourth virial coefficient computed from (11), i.e., with the aid of assumption (13), differs somewhat from its exact value, which gave the authors of these works grounds to reject the superposition approximation. This was also reflected in § 8 of de Boer’s article. Meanwhile, the absence of absolute exactness in assumption (13) was clear in advance. The question can only concern the degree of applicability of this approximation. An answer to this question can be given only after a sufficient number of consequences concerning the properties of the liquid state have been derived from assumption (13) and compared with experiment. A priori it is entirely unclear whether these consequences will correspond to reality worse or better than the consequences of any other approximate theory of the liquid state, for example, the free-volume theory.

In conclusion of this paragraph we note that equations analogous to Bogoliubov’s were also obtained by other authors, but by less rigorous methods. The equations of Born and Green, \(^{16}\) obtained somewhat later, coincide with Bogoliubov’s, but were obtained from the continuity equation for the flux of phase points in the phase space of a subsystem of \(s = 1, 2, 3, \ldots\) particles. Special attention must be given to Kirkwood’s equations, \(^{17,18}\) which have already been mentioned in de Boer’s article. Although in general analogous to Bogoliubov’s equations, they do not coincide with them exactly. As Kirkwood’s own calculations \(^{20,21}\) showed, this also leads to a certain difference in the results of the theory. Since the “superposition approximation” is used to the same extent in both theories, and apart from this the Bogoliubov method strictly follows from the general laws of Gibbsian statistical mechanics, Bogoliubov’s equations should be recognized as the more correct ones. Apparently Kirkwood himself is inclined to this view, since he carries out the main body of his calculations precisely by Bogoliubov’s equations, and not by his own.

Finally, we note that the works of N. N. Bogoliubov are passed over in silence by foreign authors, and his equations are incorrectly called the Born–Green equations. Such an attitude toward the works of Soviet scientists has lately become customary abroad.

§ 3. THE PROBLEM OF HARD SPHERES

The simplest system for which one may try to solve the basic equation (14) is a system consisting of a large number of hard, noninteracting spheres of a given diameter \(a\). In this case equation (14) proves to be the simplest. It should be kept in mind from the outset that, in its thermodynamic properties, this system is very far from an ordinary liquid. Nevertheless, this problem is of interest from the point of view of clarifying the possibility of investigating the molecular structure (short-range order) of liquids by means of the integral equation (14).

If we pass to dimensionless distances \(r \to r' = r/a\) and define the potential \(\Phi(r')\) so that

\[ \begin{aligned} e^{-\Phi/kT} &= 1,\quad \text{if } r' > 1,\\ e^{-\Phi/kT} &= 0,\quad \text{if } r' < 1, \end{aligned} \tag{20} \]

then equation (14) is brought, by a simple route, to the form (see, for example, 4; we omit the primes on \(r\))

\[ g(r)=0 \quad \text{for } r<1, \tag{21} \]

\[ \ln g(r)=\frac{\lambda}{4r}\int_{r-1}^{r+1}\{(r-\rho)^2-1\}(g(\rho)-1)\rho\,d\rho \tag{22} \]

for \(r>1\). Here

\[ \lambda=\frac{24v_0}{v}g(1+0), \tag{23} \]

where \(v_0=(\pi/6)a^3\) is the proper volume of a particle—a sphere. In view of (20), the temperature has dropped out of the equations. It is required to determine a solution of (22)—(23) for \(1<r<\infty\) in the form \(g(r)=g(r;v)\), satisfying the normalization condition (18).

Let us first consider the solution at large distances, when \(r\to\infty\). Since then \(g(r)\to 1\), we may put

\[ g(r)=1+\frac{\varphi(r)}{r} \tag{24} \]

and regard here the second term as a small correction. Then, with sufficient accuracy, one may linearize the left-hand side of (22), and we obtain the homogeneous linear integral equation

\[ \varphi(r)=\frac{\lambda}{4}\int_{r-1}^{r+1}\{(r-\rho)^2-1\}\varphi(\rho)\,d\rho. \tag{25} \]

We shall seek its solutions, other than the trivial one \(\varphi(r)=0\), in the form

\[ \varphi(r)=\mathrm{const}\cdot e^{-kr}. \tag{26} \]

Substitution into (25) then leads to the equation for determining \(k = k(\lambda)\):

\[ \lambda\{k \operatorname{ch} k-\operatorname{sh} k\}+k^3=0. \tag{27} \]

This equation has, for any \(\lambda\), an infinite number of roots \(k_n\) \((n=1,2,\ldots)\), and it is not difficult to see that all these roots for \(\lambda>0\) are complex numbers, paired as conjugates. These roots can be computed graphically or by another method. Table I gives the values of the first four roots: \(k_1=\alpha_1+i\beta_1\), \(k_2=\alpha_1-i\beta_1\), \(k_3=\alpha_3+i\beta_3\), and \(k_4=\alpha_3-i\beta_3\), for certain values of \(\lambda\), according to the works \(^{18,19}\).

Table I

Roots of equation (27)

For the value \(\lambda=34.8\), \(\alpha_1\) disappears and \(k_1\) and \(k_2\) prove to be purely imaginary. Then, for \(\lambda=158.6\), \(\alpha_3\) disappears and \(k_3\) and \(k_4\) prove to be purely imaginary, and so on.

Thus, at large distances the radial distribution function has the form

\[ g(r)=1+\frac{1}{r}\sum_n A_n e^{-\alpha_n r}\cos(\beta_n r+\delta_n), \tag{28} \]

where \(A_n\) and \(\delta_n\) are numbers that cannot be determined from the homogeneous equation (25).

Let now \(R\) be some sufficiently large number. Then the normalization requirement (18) leads to the condition

\[ \sum_n A_n\int_R^\infty e^{-\alpha_n r}\cos(\beta_n r+\delta_n)r\,dr<+\infty, \tag{29} \]

that, by virtue of the property of the roots \(k_n(\lambda)\) noted above, is possible only for \(\lambda<\lambda_0=34.8\), since otherwise the integral (29) diverges.

Thus, normalized solutions of equation (22) exist only for \(\lambda<\lambda_0=34.8\), i.e., only at densities smaller than a certain limiting density, as is clear from (23). This density corresponds to the limit of stability of the liquid phase (for a liquid consisting of noninteracting hard-sphere particles). Still higher densities correspond to the crystalline state. For its investigation, however, equation (22) is no longer suitable. (For somewhat more detail on this, see below, § 6.)

A solution in the form (28) is valid not only at large distances (for all admissible densities), but also at small distances, if the density \(1/v\), or simply \(\lambda\), is sufficiently small, since then \(g(r)\) everywhere differs little from unity (for \(r>1\)). However, the determination of the coefficients \(A_n\) and \(\delta_n\) as functions of \(\lambda\) causes great difficulties. Their approximate determination in work \(^{18}\) proved to be erroneous.

In the work of J. Kirkwood, E. Maun, and B. Alder \(^{20}\), results are given for the numerical integration of equation (22) for several values of \(\lambda\) \((=5;\ 10;\ 20;\ 27.4;\ 33)\), carried out with the aid of an electronic computing machine. For this purpose a rather

Fig. 1.

Visible labels in the figure: \([g(r)-1]r\), \(g(r)\), \(r\), \(\lambda=33\), \(\lambda=27.4\), \(\lambda=20\), \(\lambda=10\), \(\lambda=5\).

complex method of successive approximations was developed, and the approximation actually realized, according to the authors’ estimate, gives an accuracy in \(g(r)\) of about \(1\%\). Without dwelling on the details of the calculations, for which we refer to the original paper, we shall present here the final results. In doing so we shall touch only on the results of integrating equation (22) in Bogolyubov’s form, and on the results of the inte-

THEORY OF THE LIQUID STATE

... we shall not dwell on the integration of the analogous Kirkwood equation. As has already been noted, these results of the two theories are not quite identical.

In Fig. 1 the behavior of the function \(g(r,\lambda)\) is shown for the values \(\lambda=5;\ 20;\ 33\), as well as the behavior of the functions \(r[g(r)-1]\) for the values \(\lambda=5;\ 10;\ 20;\ 27.4;\ 33\), according to \({}^{20}\). Qualitatively these curves agree with the well-known behavior of experimentally determined radial distribution functions, except for the shape of the first maximum, which turned out to be too sharp because of the “hardness” of the particles. The depth of the oscillations of \(g(r)\) and the distance over which these oscillations are still noticeable (which together characterize the degree of short-range order) increase with increasing density. The period of the oscillations decreases somewhat with increasing density. These properties of the radial distribution function qualitatively correctly reflect the properties of liquids.

The values \(g(1+0)\) for various \(\lambda\) are given in Table II. The values \(v/v_0\), calculated from (23), corresponding to the indicated values of \(\lambda\), are also given there.

Table II

Volume \(v\) as a function of \(\lambda\)

The data in the last row of this table are especially interesting. Since \(\lambda=34.8\) corresponds to the stability limit of the homogeneous distribution in a system of hard spherical particles, we see that the minimum volume of the “liquid” in this model is equal to

\[ v_{\min}=2v_0=\left(\frac{\pi}{3}\right)a^3 . \]

Since the volume per particle in a closely packed system of spheres of diameter \(a\) is \(a^3/\sqrt{2}\), \(v_{\min}\) is \((\pi\sqrt{2}/3)=1.48\) times greater than the volume per particle in a closely packed (cubic face-centered or hexagonal) system of spheres. Since \((\pi/3)\simeq1\), \(v_{\min}\simeq a^3\), and thus the stability limit of the liquid phase according to the theory presented approximately corresponds to the density that occurs in a simple cubic structure.

As for the behavior of the thermodynamic functions of the system under consideration, then, by virtue of the properties of the chosen model itself, they are of little interest for the theory of the liquid state. From (1) and (2) together with (20) it is not difficult to obtain:

\[ p=\frac{kT}{v}\left(1+\frac{1}{6}\lambda\right), \tag{30} \]

\[ U=\frac{3}{2}NkT, \tag{31} \]

so that

\[ \left(\frac{\partial U}{\partial v}\right)_T=0;\qquad \left(\frac{\partial^2 p}{\partial T^2}\right)_v=0. \tag{32} \]

It is quite possible that the model under consideration may serve as a certain approximation for describing the thermodynamic properties of highly compressed gases at temperatures far exceeding the critical temperature. At high pressures and increasing temperatures, an approach of the properties of gases to (32) is indeed observed. However, these questions have no direct bearing on the theory of the liquid state, and therefore we shall not dwell any further on the properties of the model under discussion that follow from (30) and (31).

§ 4. RESULTS OF NUMERICAL INTEGRATION OF THE EQUATION FOR RADIAL DISTRIBUTION FUNCTIONS

In view of the complexity of the basic equation (14), its integration for small values of \(v\) encounters great difficulties. In 1952, results were published for a numerical integration of this equation by J. Kirkwood, V. Lewinson, and B. Alder\({}^{21}\), carried out with the aid of an electronic computing machine. Below we present the principal results of this work; moreover, once again we shall dwell only on the results of solving the Bogoliubov equation and shall omit the solutions of the Kirkwood equation.

The intermolecular potential in this work was chosen in the form

\[ \begin{aligned} \Phi(r)&=+\infty, && \text{if } r<a,\\ \Phi(r)&=4\varepsilon\left[\left(\frac{a}{r}\right)^{12}-\left(\frac{a}{r}\right)^6\right], && \text{if } r>a, \end{aligned} \tag{33} \]

so that the particles are hard spheres of diameter \(a\), interacting by means of the Lennard-Jones potential if they are not in contact. This at once entails, according to (14), that \(g(r)=0\) for \(r<a\). For \(r>a\) the calculations were carried out, in general, according to the following scheme. Alongside the function \(g(r)\), the function \(\psi(r)\) is introduced in accordance with the relation \(\ln g(r)=\psi(r)\), and then the functions \(\psi(r)\), \(g(r)\), and \(\mathcal{G}(r)\) are expanded in series in powers of \((\varepsilon/kT)\):

\[ \psi(r)=\psi_0(r)+\frac{\varepsilon}{kT}\psi_1(r)+\left(\frac{\varepsilon}{kT}\right)^2\psi_2(r)+\cdots, \tag{34} \]

and correspondingly for \(g(r)\) and \(\mathcal{G}(r)\). After substitution into (14) and collection of terms with identical powers of \((\varepsilon/kT)\), this leads to a system of equations for the successive determination of \(\psi_k(r)\) \((k=0, 1, 2,\ldots)\). For \(\psi_0(r)\) (or \(g_0(r)\)) this leads to the same equation as in the problem of hard noninteracting spheres considered above. For the functions \(\psi_k(r)\) with \(k=1, 2, 3,\ldots\), after the transition

to dimensionless units of length \(r \to r' = r/a\). This leads to inhomogeneous linear integral equations of the form (the primes on \(r\) are omitted):

\[ \psi_k(r)=f_k(r)+\frac{\lambda}{4}\int_{r-1}^{r+1}\{(r-\rho)^2-1\}\,g_0(\rho)\psi_k(\rho)d\rho, \tag{35} \]

where

\[ \lambda=\frac{24v_0}{\sigma}\,g_0'(1+0) \tag{36} \]

and where the functions \(f_k(r)\) can be expressed, although very cumbersomely, in terms of integrals of \(\psi_{k-1}, \psi_{k-2}, \ldots\).

As the initial approximation, \(\psi_0(r)\) (or \(g_0(r)\)) was taken from the work \(^{20}\) described above in § 3, and then the computations were carried out up to and including the term \(\psi_2(r)\), so that the approximate expression for \(g(r)\) was obtained in the form

\[ \left. \begin{aligned} g(r)&=0, &&\text{if } r<1,\\ g(r)&=\exp\left\{\frac{1}{r}\left[\psi_0(r)+\frac{\varepsilon}{kT}\psi_1(r)+\left(\frac{\varepsilon}{kT}\right)^2\psi_2(r)\right]\right\}, &&\\ &&&\text{if } r>1. \end{aligned} \right\} \tag{37} \]

According to the estimate of the authors of work \(^{21}\), this gives sufficient accuracy in \(g(r)\) in the temperature interval from \(T=\infty\) to \(T\sim(2/3)T_{\mathrm{cr}}\). Such solutions were obtained for the values \(\lambda=5; 10; 20; 27.4; 33\), which covers the density interval \(v_0/v\) approximately from \((1/3)(v_0/v_{\mathrm{cr}})\) to \(3(v_0/v_{\mathrm{cr}})\) (\(T_{\mathrm{cr}}\) and \(v_{\mathrm{cr}}\) are the temperature and volume per particle at the critical point). Referring for all further details of the calculations to the original, we now turn to the presentation of the principal results of this work.

Fig. 2.

The result of the computations, above all, was a family of functions \(g(r)\) for various \(T\) and \(v\), tabulated in detailed tables. In contrast to the problem of hard noninteracting spheres, the functions \(g(r)\) now have, for \(T\ne\infty\), a rounded first maximum at a distance slightly exceeding \(r=1\). How well these functions correspond to reality is seen from

Fig. 2, where the theoretical radial distribution function (solid curve) for \(\lambda=27.4\) and \((\varepsilon/kT)=1.2\) is compared with the experimental values of this same function for liquid argon according to work \(^{22}\) (white circles in the figure), obtained at \(T=91.8^\circ\mathrm{K}\) and \(p=1.8\) atm, which corresponds approximately to the indicated \(\lambda\) and \((\varepsilon/kT)\). In this case the unit of length \(a\) for the theoretical curve was chosen so that the abscissas of the first maxima of the computed and experimentally obtained curves would coincide. The difference between the theoretical curve and the experimental points does not exceed the limits of experimental accuracy.

Of particular interest are changes in the function \(g(r;T,v)\) when the thermodynamic parameters \(T\) and \(v\) are varied. Fig. 3 shows three theoretical functions \(g(r)\), corresponding to

Fig. 3.

one and the same temperature \((\varepsilon/kT)=0.8\), but different densities, namely: curve \(A\): \(\lambda=27.4\), curve \(B\): \(\lambda=20\), curve \(C\): \(\lambda=5^*\). It is seen from the figure that an increase in density (according to (36) a larger \(\lambda\) corresponds to a greater density) leads to an increase in the magnitude of the oscillations of \(g(r)\) and to growth of the correlation radius in the positions of two particles. Fig. 4 shows that approximately the same result is produced by a decrease in temperature. This figure shows three theoretical functions \(g(r)\), corresponding to one and the same value \(\lambda=20\), but at different temperatures, namely—curve \(A\): \((\varepsilon/kT)=0\), curve \(B\): \((\varepsilon/kT)=0.6\), curve \(C\): \((\varepsilon/kT)=1.2\). The behavior of the radial distribution function shown in these figures is qualitatively in full agreement with what could be expected on the basis of the physical meaning of this function and is in agreement with experimental facts.

*) In the original, there is a misprint in the text for this figure: the values of \(\lambda\) for curves \(A\), \(B\), and \(C\) are given in the reverse order, which distorts the entire meaning of the figure.

Let us now turn to the thermodynamic properties of the model under consideration. From the known \(g(r; T, v)\) and \(\Phi(r)\), the equation of state and the energy equation are computed directly, if beforehand, according to (36), a recalculation is made from the quantities \(\lambda\) to \(v\).

Fig. 4.
Axes: \(g(r)\); curves \(A\), \(B\), \(C\).

The degree of suitability of the method under discussion and of the model under discussion is illustrated by Fig. 5, which shows the isotherms of gaseous argon at \(0^\circ C\). The dimensionless quantity \(pV/RT\) is plotted as a function of density in Amagat units three times: curve \(A\)—experimental, according to \(^{23}\); curve \(B\)—(dashed) according to the theory presented here; and curve \(C\)—according to the free-volume theory, according to \(^{24}\). It is seen from the figure that, at not too high densities, the theory presented here agrees with experiment better than the free-volume theory. Nevertheless, the theoretical curve still runs too gently, and at high densities this leads to a considerable discrepancy with experiment. At lower temperatures the \(p\)–\(V\) isotherms, for some intermediate densities, proceed in such a way that, on the side of higher densities, the points on the isotherm lie lower than on the side of lower densities. This corresponds to the gas–liquid phase transition. From the conditions \(p_{\text{liq}}=p_{\text{gas}}\) and \(\mu_{\text{liq}}=\mu_{\text{gas}}\),

Fig. 5.
Ordinate: \(pV/NkT\). Abscissa: density. Curves \(A\), \(B\), \(C\).

where \(\mu\) is the chemical potential, one can find, for each temperature, the boundaries of the transition region. Fig. 6 shows a family of isotherms of the liquid and gas, including the transition region. The pressure, volume, and temperature are given here in dimensionless units

Fig. 6.

\[ p^{*}=p\frac{a^{3}}{\varepsilon};\qquad v^{*}=\frac{v}{a^{3}};\qquad T^{*}=\frac{kT}{\varepsilon}. \tag{38} \]

(on the horizontal axis, instead of \(v^{*}\), \(v^{*}/4\pi=1/\lambda_{0}\) is plotted). And this figure shows that, at least qualitatively, the theory gives a good description of the real properties of liquids. For the critical point, the following values are obtained:

\[ p^{*}_{\mathrm{cr}}=0.199;\qquad v^{*}_{\mathrm{cr}}=2.585;\qquad T^{*}_{\mathrm{cr}}=1.433 \tag{39} \]

instead of the experimental values \(0.12;\ 3.16;\ 1.26\), respectively. For the dimensionless ratio \(pV/RT\) at the critical point, the theory presented gives the value \(0.358\). This quantity is compared with experimental data and with the data of other theories in Table III.

Table III

Value of \((pV/RT)_{\mathrm{cr}}\)

The situation is approximately the same with all the other thermodynamic quantities of the liquid in the problem under consideration: pressure, energy, entropy, etc., at densities approximately equal to the critical density, differ from the experimental values by 30–40%, and at higher densities the discrepancy with experiment increases still more. For this reason we shall refrain from presenting calculated data for the energy, entropy, etc., given in work \(^{21}\), and in the following paragraph we shall try to discuss the situation that has arisen.

§ 5. DISCUSSION OF THE RESULTS OF NUMERICAL INTEGRATION

The results of the preceding two paragraphs show that the theory based on the integral equation (14) qualitatively correctly reflects the structure (the presence of short-range order) of the liquid and its main thermodynamic properties. At the same time, however, the quantitative conclusions of this theory lead, in the problem analyzed above, to unacceptable values of the thermodynamic quantities of the liquid state. The numerical values of the pressure and of the volume-dependent part of the internal energy in the liquid state proper are approximately 40% or more too low in comparison with the experimental values of these quantities for liquid argon.

It is unquestionable that one of the reasons for this is the “superposition approximation” (13) lying at the basis of the theory expounded. In work \(^{15}\) it was shown that, in the problem of hard spheres, the “superposition approximation” necessarily leads to an underestimated value of the pressure. However, it appears difficult to estimate precisely to what extent the assumption (13) is responsible for the indicated discrepancy between theory and experiment in the problem analyzed, with interacting particles. It is therefore extremely important to determine what reasons, besides assumption (13), may be responsible for this discrepancy, and whether by eliminating them one cannot obtain better results.

In order to estimate the degree of validity of the “superposition approximation” in the problem analyzed, it would have been necessary to compare the results of this problem not with experimental data for some particular liquid, but with the results of an exact calculation of the configuration integral for a system with the intermolecular potential (33). Since such a comparison is not yet possible, the discrepancy under discussion between the data of theory and experiment may equally well be attributed either to assumption (13) or to the unsuitability of the potential (33).

The potential in the Lennard-Jones form

\[ \Phi(r)=4\varepsilon\left[\left(\frac{a}{r}\right)^{12}-\left(\frac{a}{r}\right)^6\right]. \tag{40} \]

represents, as is known, merely an interpolation formula which qualitatively correctly reflects the general course of the actual intermolecular potential. If, at large distances, the dependence of \(\Phi(r)\) on the distance \(\sim -1/r^6\) agrees with theoretical ideas about dispersion forces, then the dependence \(\Phi \sim 1/r^{12}\) at small distances is entirely arbitrary and even contradicts the data of quantum mechanics, according to which it would be natural to expect an exponential, in the main, dependence of \(\Phi\) on \(r\) at small distances. As is known, the power 12 in (40) appears as the simplest, but not the only possible, one from the data for the second virial coefficient (as also do the data for \(\varepsilon\) and \(a\). See, for example, \(^{26}\)). However, from the equation for the second virial coefficient

\[ B=-2\pi \int_0^\infty \left(1-e^{-\frac{\Phi(r)}{kT}}\right)r^2\,dr \tag{41} \]

it is quite clear that it depends only on the area under the curve \(f(r)=1-e^{-\Phi/kT}\), or, roughly speaking, on the area under the curve \(\Phi(r)\) in the region of its minimum. In this case the profile of the curve itself is quite immaterial. Consequently, one may propose a large number of interpolation formulas for \(\Phi(r)\) of the most varied form, differing from one another by the shape of the potential well in \(\Phi(r)\), but equally suitable for the “correct” calculation of the second virial coefficient.

Let us note, incidentally, that the dependence \(\Phi\sim -1/r^6\) at large distances at the present time also cannot be regarded as theoretically substantiated, and according to works \(^{27,28}\) the dependence \(\Phi\sim -1/r^7\) is more correct. The increase of the power here is obtained at the expense of the finite velocity of propagation of electrical interactions. If this is taken into account, the interpolation formula (40) turns out to be even more inconsistent with reality.

Meanwhile, from the equation for the pressure (2) it is quite obvious that, at high densities, the pressure is very sensitive to the behavior of the functions \(\Phi(r)\) and \(g(r)\) in the neighborhood of the point \(r=a\), which gives the main contribution to the integral appearing in (2). A small change in the profile of the curve \(\Phi(r)\) for \(r \gtrsim a\) (especially on the interval between \(r=a\) and \(r=r_{\min}\), where \(\Phi'(r_{\min})=0\)) can substantially change the numerical values of the pressure of the system at high densities.

To illustrate what has been said, we shall give two examples from work \(^{21}\). For the density corresponding in the problem of § 4 to \(\lambda=5\), \((\varepsilon/kT)=0.6\), the function \(g(r)\) was calculated for a potential taken not in the form (33), where \(\Phi=+\infty\) for \(r<a\), but in the form of the complete potential (40). In Fig. 7,a both potentials are shown; in Fig. 7,b—the course of the corres-

corresponding functions \(g(r)\). (Both figures are in dimensionless units of length.) For \(r>1\) both functions practically coincide, and to draw the difference between them on a figure of such scale is impossible. The calculation of the pressure for both cases leads to the result that, with the “shortened” potential, the pressure is approximately \(10\%\)

Fig. 7.

greater than with the full Lennard-Jones potential. It is quite possible that at higher density the difference in the pressures would increase still more.

Next it turns out that if one replaces the integral in the expression for the pressure by the following:

\[ \int_0^\infty \Phi'(r)\, g(r+b)\, r^3 dr, \]

i.e., shifts the function \(g(r)\) by some amount \(b\), then already for \(b=0.02\) this leads, at high densities, to a change in the value of the pressure by hundreds of atmospheres.

Both examples show that the numerical value of the pressure is indeed very sensitive to the detailed behavior of the potential \(\Phi(r)\) and of the function \(g(r)\), which itself is determined by the potential \(\Phi(r)\) (and by the density). Since, on the other hand, the potential used in [21] is surely far from the actual intermolecular potential in real liquids (for example, in liquid argon), one may suppose that a noticeable part of the discrepancy between the data of theory and experiment is due precisely to an unfortunate choice of the potential.

In conclusion of this paragraph, let us note that from the theoretical point of view the question of the manner of representing the gas–liquid phase transition in a theory based on the radial distribution function is very interesting. As we have seen, the free-volume theory leads to S-shaped bends of the \(\rho - V\) isotherms, which is an undoubted defect of a theory claiming any sort of correct calculation of the partition integral, since unstable states are not contained in this integral (as macroscopically observable states). The transition region between liquid and gas must correspond to a horizontal segment of the \(p - V\)

isotherms. In work \(^{29}\) the suggestion was made that, if at fixed \(T\) one regards \(1,v\) (or \(v\)) as a parameter of the integral equation (14), then for \(T<T_{\mathrm{cr}}\) a break occurs in the spectrum of eigenvalues of this parameter, so that the density spectrum proves to consist of two continuous, non-overlapping pieces, corresponding to gas and liquid (possibly including the metastable states of these phases). For intermediate values of the densities the function \(g(r)\) does not exist, which should be the representation of the liquid—gas phase transition. The results of the numerical integration discussed above do not contradict this view. At the same time, however, the data of work \(^{21}\), unfortunately, do not allow this view to be considered confirmed, just as in general they do not allow one to judge definitely the manner in which the liquid—gas phase transition is described in the theory under discussion. The reason for this is the scarcity of points taken along the density axis and the absence of a study of the convergence of the successive approximations used. The isotherms were obtained by joining comparatively sparse calculated points and, moreover, such that for \(T<T_{\mathrm{cr}}\) their “liquid” ends on the side of large volumes pass lower (even in the region of negative pressures) than the “gas” ends on the side of smaller volumes. For the reasons indicated above, it is impossible to judge definitely whether these ends of the isotherms then join in the form of S-shaped arcs (as in the van der Waals theory) or whether they break off and remain unconnected.

§ 6. THEORY OF MELTING

At the present time there is no more or less complete theory of the liquid—crystal phase transition based on the study of partial distribution functions. Such a theory would require knowledge of the distribution functions not only in the liquid, but also in the crystal. Such investigations have not yet been carried out. However, one can point to several interesting attempts at an approach to this problem.

Born and Green \(^{30}\) attempted to connect the crystallization of a liquid with the requirement of periodicity of the function \(g(r)\) at high densities, while preserving the condition \(F_1(q)=1\). No notable successes were achieved along this path, and the very formulation of the problem must be regarded as incorrect, since in a crystal \(F_1(q)\ne 1\) and \(F_2(q_1,q_2)\ne g(|q_2-q_1|)\).

A large number of works on the theory of the crystalline state and crystallization were carried out by A. A. Vlasov and his students (see, chiefly, the monograph \(^{31}\)). However, the method and the basic ideas of these works diverge from the method and basic ideas of Gibbsian statistical mechanics, to which we adhere here. Therefore we shall not discuss these works in detail in the present article.

It suffices to note the following. In these works the behavior is discussed of the function \(F_1(q)\), which, according to A. A. Vlasov’s theory, must satisfy the equation

\[ kT \ln \{\lambda F_1(q)\}+\int_{-\infty}^{+\infty} K(|q-q'|)\,F_1(q')\,dq'=0, \tag{42} \]

where \(K(r)\) is a not quite precisely defined “potential function of particle interaction” and \(\lambda\) is a normalizing constant. Equation (42) has the trivial solution \(F_1(q)=1\), corresponding to a gas or a liquid. However, it turns out that for values of \(T\) smaller than some \(T_0\), a spatially periodic solution branches off from the trivial solution; in the theory being presented, this solution is identified with the distribution function of particles in the crystalline state, and the temperature \(T_0\) with the crystallization temperature. Considering the appearance of a solution with an infinitely large period as the beginning of crystallization, A. A. Vlasov arrives at the condition

\[ kT_0=-4\pi\int_0^\infty K(r)\,r^2dr. \tag{43} \]

This is the criterion for the onset of crystallization in A. A. Vlasov’s theory. The crystalline state corresponds to the temperature region \(T<T_0\). It is obvious that condition (43) leads to \(T_0>0\) only in the case when the integral appearing here is negative, i.e., when the attractive forces between particles predominate over the repulsive forces.

The method of branching of solutions of nonlinear integral equations was also used by S. V. Tyablikov\({}^{32}\) in an attempt to construct a theory of crystallization on the basis of N. N. Bogolyubov’s theory. It is assumed that, with sufficient accuracy, one may put

\[ F_2(q_1,q_2)=F_1(q_1)\cdot F_1(q_2)\,g(|q_2-q_1|). \tag{44} \]

In the case of a gas or liquid \(F_1(q)=1\), and \(g(|q_2-q_1|)\) is the radial distribution function. In the case of a crystal \(F_1(q)\ne 1\), and \(g(|q_2-q_1|)\) is the correlation function, whose existence is assumed precisely in this form (i.e., it is considered that it must depend only on the distance between particles). Substitution of (44) into equation (8) for \(s=1\), after simple transformations, leads to a nonlinear integral equation of the same type as (42):

\[ kT \ln \{\lambda F_1(q)\}+\frac{1}{v}\int \mathscr{E}(|q'-q|)\,F_1(q')\,dq'=0, \tag{45} \]

where the function \(\mathscr{G}(r)\) is the same as in (15), and \(\lambda\) is a normalizing constant. And this equation admits, branching off from the trivial solution \(F_1(q)\equiv 1\), periodic solutions which in the linear approximation have the form

\[ T_1(q)=1+\text{const.}\sqrt{T_0-T}\,e^{i(\mathbf a q)}. \tag{46} \]

For a given \(|\mathbf a|\), the temperature \(T_0\), below which periodic solutions are possible, is determined from the condition

\[ kT_0=-\frac{4\pi}{v}\int_0^\infty \mathscr{G}(r)\, \frac{\sin(|\mathbf a|r)}{|\mathbf a|}\,r\,dr. \tag{47} \]

With the aid of (15) it is not difficult to verify that in Tjaplikov’s theory as well, for the existence of a crystal it is necessary that the attractive forces between particles predominate over the repulsive forces. Substituting into (47) the experimental values of \(v\) and \(|\mathbf a|\), and taking for the calculation of \(\mathscr{G}(r)\) from (15), as \(g(r)\), the experimentally determined radial distribution function of the liquid near the melting point, Tjaplikov found for argon \(T_0=96^\circ\mathrm K\), and for mercury \(T_0=260^\circ\mathrm K\), whereas the experimental values of the melting temperatures are respectively \(T_{\mathrm{m}}=83.4^\circ\mathrm K\) and \(T_{\mathrm{m}}=232^\circ\mathrm K\) (all numerical data refer to a pressure of \(1\) atm). The agreement between theory and experiment, as we see, is quite good.

However, the application of the branching method to the problem of crystallization raises a whole series of objections. The most important of them is that the crystallization criterion, both according to Vlasov and according to Tjaplikov, permits the presence of a spatially periodic structure in a system of particles only in the case where attractive forces between particles predominate over repulsive forces. Meanwhile it is quite beyond doubt that if the particles possess, to a sufficient degree, the property of “hardness” or “impenetrability,” then a periodic structure can arise under the action of external compressive forces alone, even if the particles attract one another only weakly or even not at all. Such, for example, is the case of hard spheres already discussed above, where by external compression one can produce their densest packing into a hexagonal or cubic face-centered structure. But for this—the simplest—problem both crystallization criteria, (43) and (47), prove inapplicable, since by virtue of (20) the temperature drops out altogether from the equation for \(F_1(q)\). Meanwhile it has been shown that in the problem of hard spheres a periodic structure at high densities can be obtained by direct calculation of the configuration integral (calculations have been carried out for the one-dimensional case).

Unsatisfactory in the branching method is also the fact that, as is seen from (46), as \(T \to T_0\) the liquid continuously transforms into a crystal. To explain in this way the discontinuous change of density upon melting appears, at the very least, extremely difficult.

For these, as well as for other reasons, it seems to us that the branching method cannot serve as the basis of a correct statistical theory of crystallization and of the crystalline state. In Tablikov’s formulation, the criterion of crystallization by the branching method should be regarded only as an approximate criterion for the limit of stability of the liquid state, whose validity is restricted to the region of low temperatures, since only in this case can the role of external pressure in the process of crystallization be neglected.

More promising appear to be the possibilities for further development of the results already noted in § 3. As we saw in the problem of hard spheres, the theory leads to the existence of a minimal volume per particle at which a homogeneous phase is still possible (i.e. with \(F_1(q)=1\)). It seems very plausible that in the case of systems of particles with interaction, if only the particles possess to a sufficient degree the property of “hardness,” there will likewise appear an analogous minimal volume, generally speaking depending on temperature, and according to (2) this will make it possible to calculate the “melting curve.” It is not clear in advance whether this will actually be the melting curve or the boundary of the region of stability of the supercooled liquid. However, this is not essential for us now, all the more since for the liquids of interest to us—liquid noble gases—the region of supercooling is very small.

Such calculations have not yet been carried out by anyone. However, the following qualitative considerations make the idea expressed here very plausible. Since in the problem of § 3 one had \(v_{\min}=2v_0\), where \(v_0\) is the proper volume of a particle, and \(\lambda_0=34.8\), it follows from (30) that the “melting curve” for the model of hard spheres is

\[ p(T)=3.4\,\frac{kT}{v_0}. \tag{48} \]

Let us now consider a real liquid at very high temperatures, say, far above the critical temperature. As is known, such a liquid too can be crystallized by applying a sufficiently high pressure. At such high temperatures the attractive forces between particles play an insignificant role, and they may approximately be neglected. But in doing so it is absolutely necessary to take into account the proper compressibility of the particles. We can do this approximately by putting

\[ v_0=v_0(T) \]

and by defining the effective diameter of the particle \(a_{\mathrm{eff}}\sim [v_0(T)]^{1/3}\) from the condition

\[ \Phi(a_{\mathrm{eff}})=\frac{1}{2}kT, \tag{49} \]

so that \(a_{\mathrm{eff}}\) is the smallest distance to which two particles can approach, the relative kinetic energy of approach of which is \(\frac{1}{2}kT\). If, for simplicity, we assume at small distances \(\Phi\sim 1/r^{12}\) (as in the Lennard-Jones potential), then this gives us

\[ a_{\mathrm{eff}}\sim T^{-1/12},\qquad v_0(T)\sim T^{-1/4}, \tag{50} \]

and substitution into (48) leads to the result

\[ p(T)\sim T^{5/4} \tag{51} \]

in good agreement with experiment\(^{33}\).

In conclusion, let us note that in the problem of hard spheres the question remains not entirely clear as to what type of phase transition corresponds to the “melting curve” (48). Kirkwood\(^{20}\) expresses the opinion that this transition from a homogeneous phase to a crystalline one for the hard-sphere model should be a transition of the second kind, owing to the absence of interaction between the particles. However, such an argument is not convincing, since the appearance of a heat of transition may turn out to be a purely entropic effect and need not necessarily be connected with a change in internal energy.

§ 7. BRIEF CONCLUSION

Summing up the review of the available results of theories of the liquid state based on the study of the radial distribution function, one must state that the situation cannot be assessed as pessimistically as was done in de Boer’s article\(^{1}\). Although this theory is only in its infancy, it has nevertheless already achieved notable successes. Among these, first of all, should be included the fact that, proceeding from the general laws of statistical mechanics, the theory leads to the existence of short-range order in liquids, qualitatively correctly reflected by the radial distribution functions calculated by the theory. Let us recall that in all other existing theories of the liquid state, including the free-volume theory, the structure of the liquid is postulated.

The quantitative results of the theory are as yet not very satisfactory. But, as we have seen, to some extent this may be attributed to ignorance of the exact form of the intermolecular potential. A considerable part of the difficulties is undoubtedly associated with the “superposition approximation” (13). But if it should subsequently turn out that

theory nevertheless is able to describe correctly all the principal properties of liquids, including the liquid–gas and liquid–crystal phase transitions, then, despite the quantitative discrepancies when compared with experiment, it is hard to imagine what more can still be demanded of a theory of the first approximation. It seems to us that the aim of a physical theory should be, first of all, a correct reflection of the basic physical regularities of the phenomenon under consideration, and not an attempt at any cost to arrive at quantitatively exact estimates of quantities. Such a pursuit of the “computational” side of a theory, often to the detriment of its physical side, is quite characteristic of many foreign works in the field of statistical physics.

The “superposition approximation” is not, in itself, a necessity that necessarily follows from the theory. In principle it can be replaced by a multitude of other approximate expressions relating \(F_3\) to \(F_2\), and it is quite possible that an expression more successful at high densities than (13) will be found.

CITED LITERATURE

1. I. de Boer, UFN, this issue, p. 41.
2. J. G. Kirkwood, J. Chem. Phys. 18, 380 (1950).
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1. J. de Boer. ↩