Statistical Mechanics Volume 1

A Review of the Recent Literature Published Up to July 1972

By K. Singer

The Royal Society of Chemistry

Copyright © 1973 The Chemical Society
All rights reserved.
ISBN: 978-0-85186-750-2

Contents

Chapter 1 Integral Equation Approximations in the Theory of Fluids By R. O. Walts,
Chapter 2 Perturbation Theory in Classical Statistical Mechanics of Fluids By W. R. Smith,
Chapter 3 Equilibrium Theory of Liquid Mixtures By I. R. McDonald,
Chapter 4 Thermal Transport Coefficients for Dense Fluids By K. E. Gubbins,
Author Index, 254,

CHAPTER 1

Integral Equation Approximations in the Theory of Fluids

BY R. O. WATTS

1 Introduction

When the Senior Reporter approached the author of this chapter he asked that the Report be used to give an account of advances in the use of integral equation approximations in the theory of fluids rather than a summary of recent papers. Consequently an attempt has been made at producing an historical account of the use of such methods, with some emphasis on advances in recent years. During the preparation of this article it became apparent that the usefulness of integral equation approximations is on the decline. This is primarily due to two reasons, firstly the difficulty of calculating second-order correction terms and secondly the rapid increase in the power of digital computers, resulting in strong competition from the more powerful simulation methods. As a result it is unlikely that many more significant advances will be produced in this area and consequently this article is in part a summing-up of the field.

The structure of the Report is traditional, and the following section gives a short account of the basic theory required in the article. After this the first integral equation theories are introduced, those due to Born and Green and Kirkwood, and an account is given of recent work on the hierarchy of equations, first introduced by Yvon, which were the basis of these equations The Percus–Yevick (PY) approximation introduces the functional Taylor series derivation and it is indicated how more complete approximations can be obtained by including higher-order terms in the expansion. A section on the hypernetted chain (HNC) approximation is used to introduce the cluster expansion derivation of integral equation approximations, and it is shown how the introduction of further diagrams may be used to improve this type of approximation. Following this a number of other integral equation approximations are introduced, not all of which have been examined for realistic systems. Having given an account of the major routes by which integral equation approaches are obtained, a section is devoted to the various numerical methods used to solve them. An attempt is made to give a critical evaluation of the several alternatives. At this stage it is possible to begin evaluating the numerous approximations, and initially this is done by considering the first few virial coefficients. The ability of several of the approximations to predict the structure of dense fluids is considered in a section on the radial distribution function, and then the predicted thermodynamic properties are discussed. A separate section is given to the use of the PY approximation with the energy equation, for this has proven to be the most successful integral equation route to thermodynamic properties. In recent years a number of extremely successful perturbation theories (reviewed in Chapter 2 of this volume) have been based on the properties of hard-spheres and so we devote a section to this model system. Possibly, this area will represent the most important use of integral equation theories in future work. The review is practically completed by two sections, one on the use of PY and HNC approximations in ionic solution theory and the other on a number of topics including the theory of liquid metals and the ground state of liquid helium-4. In the last section a summary of the most likely future developments in the area of integral equation approximations is given. Throughout the Report emphasis is given to results obtained for physically reasonable potential functions, and no attempt is made to discuss one- and two-dimensional systems or lattice statistics. Two further points: firstly, the Reporter finds an extensive literature survey rather tiresome and so it is certain that a number of papers on integral equations will have been missed; secondly, this does not pretend to be a review of liquid state theory, and for such an article the reader is referred to a recent publication of Barker and Henderson.

2 Basic Theory

The fundamental problem in statistical mechanics is to use a knowledge of the interparticle interactions in a system to predict the thermodynamic properties of that system. In recent years the ability to predict accurately thermodynamic properties of liquids for a given model potential has enabled us to obtain accurate potential functions, at least for spherically symmetric particles, but we are not concerned with that problem in this Report. Intermolecular potential theory is based upon two apparently well-founded assumptions, first that intermolecular potentials are a function of internuclear separations alone (Born–Oppenhiemer approximation) and secondly that the total potential energy of a system of N atoms whose nuclei are at positions r1, …, rN may be written (in the absence of external fields) as a sum of two-body, three-body, four-body, … potential functions:

[MATHEMATICAL EXPRESSION OMITTED] (1)

Here the various functions are known as two-body potentials [φ(2)], three-body potentials[φ(3)], etc. and in this Report are assumed to be given. In general, we shall be concerned with two potential functions, the simple hardsphere model,

[MATHEMATICAL EXPRESSION OMITTED] (2)

and the more realistic Lennard-Jones (12, 6) potential,

[MATHEMATICAL EXPRESSION OMITTED] (3)

In both potentials σ is a measure of the diameter of the particle and for the Lennard-Jones potential ε gives the value of the potential at its minimum. The hard-sphere model is not used to represent real systems; the Lennard-Jones potential is, and the values of ε and σ used for many different particles are well documented. Throughout this Report temperatures will be given in terms of the dimensionless parameter T* – kT/ε and densities in terms of p* = pσ3. In one later section we shall also be concerned with more accurate potential functions, but they will be introduced at that point. During most of the Report we shall be making the assumption that three-body and higher-order interactions are zero – that is, we shall truncate the expansion given in equation (l) after the first term. On those occasions where three-body interactions are needed, they will be specifically dealt with.

In principle, once we know the potential energy of a system as a function of the positions of the constituent particles we may obtain the equilibrium properties of a thermodynamically closed system from the equation

A – – kT log ZN (4)

where A is the Helmholtz free energy of the system, k is Boltzmann’s constant and T the temperature. The canonical partition function, ZN, is given by

[MATHEMATICAL EXPRESSION OMITTED] (5)

where λ = (h2/2πmkT)[??]. It is assumed that the particles are obeying the laws of classical mechanics. Similar expressions may be written for other thermodynamic systems (isolated, open, etc.) and for systems containing several species. Unfortunately, except for one or two very special cases it is not possible to evaluate equation (5) analytically, and so we resort to various approximate schemes. The integral equation methods belong to a particular class of such approximations.

Other than for a crystalline solid at very low temperatures it is not meaningful to speak of the structure of a system, if we mean by this some rigid arrangement of atoms. The thermal motions in the system are such that we can only refer to the probable distribution of particles. If we know the potential energy of the system as a function of positions, then statistical mechanics tells us that the distribution of particles at a particular temperature is proportional to the function exp[ -Φ(r1, …, rN)/kT]. In general we are not concerned with this N-particle distribution function, and for the study of dense fluids we are mainly concerned with the one-particle distribution function, the number density ρ, and with a particular form of the two-body distribution function, the radial distribution function, g(r). The radial distribution function is determined theoretically from the equation for the two-body distribution function,

[MATHEMATICAL EXPRESSION OMITTED] (6)

where we have used the uniformity of the fluid to write g(r) as a function of position only. If we are dealing with a system of spherically symmetric particles, then the radial distribution function is a function of interparticle distances alone. The radial distribution function is a particularly interesting measure of the structure of a fluid as it is open to experimental determination. Suppose S(k) is the structure factor determined from an X-ray diffraction study of a fluid. Then this function is related to the Fourier transform of the radial distribution function,

[MATHEMATICAL EXPRESSION OMITTED] (7)

The radial distribution function can also be obtained from other radiation-scattering experiments. When dealing with many-body effects, for example with a three-body potential function, it is often necessary to deal with three-body and higher-order distribution functions. However, we will not deal with these functions in detail in this Report.

As well as giving information about the distribution of particles in a fluid, the radial distribution function may be used to obtain several thermodynamic properties. Suppose that we are dealing with a systen of particles interacting through two-body central forces alone (e.g. Lennard-Jones potential). Then we may use the equation

p = – (δA/δV)τ (8)

where p is the pressure and V the volume, together with equations (4) and (5) to obtain an expression for the equation of state of the system,

[MATHEMATICAL EXPRESSION OMITTED] (9)

This equation will be frequently referred to as the Pressure Equation in later sections. A second equation that uses the radial distribution function to obtain thermodynamic properties is the Energy Equation. This equation is derived from equations (4) and (5) by making use of the thermodynamic relation

U = δ(A/T)/δ(1/T) (10)

where U is the internal energy of the system, to give

[MATHEMATICAL EXPRESSION OMITTED] (11)

Finally, if we consider fluctuations in the Grand Canonical Ensemble we may derive an expression for the isothermal compressibility of the system, the Compressibility Equation,

[MATHEMATICAL EXPRESSION OMITTED] (12)

If we consider equation (7) with [absolute value of k] = 0, it follows that we have the relation

S(O) = kT (δp/δp)τ (13)

giving the structure factor at zero wavenumber in terms of the isothermal compressibility. In later sections we shall use all three equations relating the radial distribution function to the thermodynamic properties of a fluid to test various approximations. We shall see that in general the results obtained from one equation will not be the same as those obtained from another and consequently that the size of these inconsistencies may be used to judge a particular approximation.

A relatively simple method of examining an approximation in statistical mechanics is to consider successive terms in the virial expansion of the equation of state of a dilute gas. The experimental approach of analysing the equation of state in terms of a power series was first introduced by Kammerlingh Onnes, who used the representation

[MATHEMATICAL EXPRESSION OMITTED] (14)

where B, C, … are the second, third, … virial coefficients. For an ideal gas all the virial coefficients (other than the first) are zero. A corresponding expansion may be obtained for the statistical mechanical equation of state, when it is found that the virial coefficients are related to certain integrals. These are usually depicted graphically as follows. Define the Mayer [Florin]-bond for a two-body potential by the line

[MATHEMATICAL EXPRESSION OMITTED] (15)

Then if the variable r12 is integrated out, we drop the open circles. Using this convention we may write the virial coefficients in terms of these graphs; for example

[MATHEMATICAL EXPRESSION OMITTED] (16)

[MATHEMATICAL EXPRESSION OMITTED] (17)

[MATHEMATICAL EXPRESSION OMITTED] (18)

A particular approximation will ignore certain diagrams in the expansion and consequently we may examine its effectiveness by comparing the predicted virial coefficients with those calculated using the exact expression, as will be seen later.

We may also examine the effectiveness of a particular approximation by comparing the results for dense fluids with known accurate results. These accurate results may be of two types. Firstly, provided it is known that the potential being considered is very accurate for a particular real substance (e.g. Barker, Fisher, and Watts potential for argon interactions) we may compare the approximate results directly with experiment. However, in most cases to date the approximate calculations have been carried out for a model system. Under these circumstances it is difficult to separate errors due to the approximation used from those due to the poor potential function. Consequently some other standard must be employed, most often a comparison with computer simulation studies of the model. Computer simulation studies are of two types, the Monte Carlo method and the molecular dynamics method. In the first of these methods a large number of configurations of a system of N particles in a periodic box are generated so that the probability of a particular configuration being examined is proportional to its Boltzmann factor

[MATHEMATICAL EXPRESSION OMITTED] (19)

The thermodynamic properties are then calculated by averaging the appropriate quantity over the con.figurations; for example

[MATHEMATICAL EXPRESSION OMITTED] (20)

[MATHEMATICAL EXPRESSION OMITTED] (21)

Equivalent expressions may be obtained for the specific heat and radial distribution function. The molecular dynamics method is basically similar to the Monte Carlo method except that configurations are generated by solving Newton’s equations of motion and the various thermodynamic quantities are obtained by time-averaging rather than by ensemble-averaging. Both methods have the useful property that they can be as accurate as one wishes, the accuracy being basically dependent on the expenditure of computer time. Consequently the accuracy of the various approximations may be determined by comparing their results with those of computer simulation calculations.

3 Derivation of Integral Equations

In this section we will discuss the more important methods of obtaining integral equation approximations for the radial distribution function, particular emphasis being given to methods that indicate possible routes to improvement. Thus, despite the importance of the Percus–Ycvick theory, the original derivation will not be described, but it will be introduced in terms of the functional Taylor series method.

Born–Green and Kirkwood Approximations. — The Born–Green and Kirkwood approximations are based upon the truncation of two equivalent hierarchies of equations. We saw in equation (6) that the radial distribution function is given exactly by an integral over the function exp[-Φ(r1, … rN)/kT]. Suppose we make the assumption that the total potential energy of the system can be written as a sum over pair potentials,

[MATHEMATICAL EXPRESSION OMITTED]

introduce this into equation (6) and then differentiate both sides with respect to the position r1. We obtain the equation

[MATHEMATICAL EXPRESSION OMITTED] (22)

where [Florin](3) (r1, r2, r3), the three-body distribution function, is defined by an equation analogous to equation (6). This equation is exact, and relates the two-body and three-body distribution functions. We can continue the hierarchy of equations by relating [Florin](3) to [Florin](4), the four-body distribution function, and so on. This equation was derived originally by Yvon and independently at a later date by Born and Green.

Kirkwood obtained a similar hierarchy of equations by introducing a parameter [xi] which couples molecule I to the rest of the system,

[MATHEMATICAL EXPRESSION OMITTED] (23)

where [xi] takes values between 0 and 1. If this expression is introduced into equation (6) and both sides are differentiated with respect to [xi], the following equation relating [Florin](2) and [Florin](3) is obtained,

[MATHEMATICAL EXPRESSION OMITTED] (24)

This is the first member in the Kirkwood hierarchy of equations, and again higher terms in the series may be obtained by considering derivatives of [Florin](3) and so on. At this point both equation (22) and equation (24) are exact, but as we do not know [Florin](3)we are no further forward, unless we can introduce some suitable closure relation.

(Continues…)Excerpted from Statistical Mechanics Volume 1 by K. Singer. Copyright © 1973 The Chemical Society. Excerpted by permission of The Royal Society of Chemistry.
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