Introduction

We saw in the previous chapter how in the limit of low densities we were able to obtain a convergent expansion for the configurational partition function, and thereby determine all the thermodynamic functions of the system including the virial equation of state. However, what we did not have was a detailed knowledge, or indeed any knowledge, of the structure of the system.

Clearly, at liquid densities the series expansions will fail to converge(1), and anyway the computational labour in evaluating the multi-dimensional high-order cluster integrals which would develop in the highly connected fluid would be overwhelming, even in the economical Ree-Hoover formalism.

So, we are forced to adopt a new mathematical approach-the formalism of molecular distribution functions, or correlation functions. Instead of trying to evaluate the N-body configurational integral directly, the theory describes the probability of configurational groupings of two, three, and more particles. Further, it may be shown that we can still obtain the same amount of information concerning the system as is obtained from the study of the statistical integral itself. Moreover, in this way we obtain direct information on the molecular structure of the system being studied. Of course the method of correlation functions applies equally well to gases and solids, but we would not generally adopt that approach by choice since other characteristics of these phases suggest a more direct route to the partition function. The formalism is adopted, however, in the case of amorphous solids.

We shall find that the one- and two-body distribution functions, generally denoted by g(1) and g(2) respectively, will be of central importance in the equilibrium theory of liquids.

Introduction

In the previous chapters approximate integral equations were developed for the pair distribution. From this function we were able to establish virtually all the thermodynamic functions of interest, although it was clear that certain approximations had, of necessity, to be invoked either on the grounds of mathematical expediency as in the KBGY class of equations, or on more physical grounds in the case of the PY-HNC cluster expansions. The development of approximate integral equations was, of course, enforced by the impossibility of obtaining a direct evaluation of the N-body partition function for dense fluids, and as yet there appears to be no general method of calculating this quantity. It will be recalled that the difficulty was mathematical rather than conceptual, originating in the collective nature of the total potential ΦN even in the pairwise additive approximation. Powerful numerical techniques have been brought to bear upon the various integral equations, and these are generally evaluated by iterative techniques. The question arises as to whether these complex calculational techniques should not be used directly in rigorous variants of the theory. An ideal case would, of course, be the possibility of calculating directly the configuration integral of a system of a large number of particles.

It appears that with the advent of large electronic computers we have at our disposal a means of calculating an ensemble average in terms of the accessibility of states of the system. The ‘states’ of the system may be purely configurational and determinate as in the molecular dynamics approach, where the representative point evolves in accordance with a classical Liouville equation, subject to constraints of microscopic reversibility and ergodicity.

Introduction

Numerical evaluation and comparison of the various integrodifferential equations developed in chapter 2 may be conveniently divided into three sections. First the low density solutions will be discussed. In this case comparison is not made in terms of the form of the pair distribution function but rather by evaluating the equation of state, based on the pressure and compressibility relations, and comparing the virial coefficients so determined with the exact results of chapter I. Of course, if the theory were self-consistent the equation of state and virial coefficients would be independent of the method of approach-the pressure and compressibility equations would yield the same results. Some effort has gone into forcing self-consistency by choosing a form for c(r) which ensures identical results regardless of the approach (the self-consistent approximation: SCA). This device of enforcing thermodynamic consistency cannot be regarded as an advance of physical understanding; nevertheless, the results are excellent at least to the sixth virial coefficient for hard spheres.

At liquid densities direct comparison of the radial distribution function may be made. It will be seen that all the theories discussed in the previous chapter agree in the qualitative form of the pair distribution, but the quantitative discrepancy is large. Again, appeal to the equation of state is made. The extreme sensitivity of the pressure equation to the precise form of g(2)(r) (and, indeed, to the assumed form of the pair potential) provides a severe test of the theory. It is computationally convenient to work in terms of an idealized potential such as the hard sphere or square-well interactions- these models have the important advantage that the resulting equation of state may be compared directly with machine simulations.

This book, which is about chemical physics as well as about ice, has been written with two general classes of readers in mind. The principal group, as indicated by the inclusion of the book in the present series, consists of graduate or advanced undergraduate students in physics or physical chemistry who have already taken the usual courses in quantum mechanics and solid–state physics and are looking for something on which to try their teeth. The detailed study of a reasonably simple material like ice is admirable for this purpose; not only does it employ a whole range of the techniques learnt in more formal courses but also it shows these techniques and concepts in relation to one another. For these people I have tried at all stages to show the connexions between the various properties discussed and, in particular, the way in which they all derive from the structure of the water molecule itself.

The second group is made up of those who are interested in ice for its own sake—glaciologists, cloud physicists and the like—and for these I have taken the view that what is really required is an exposition of what is now generally regarded as understood and accepted about the chemical physics of ice.

In many parts of the world, ice is an important engineering material. Either it must serve as a substrate to bear the weight of buildings, vehicles and aircraft or else it is an obstacle which must be dug, drilled or blasted to reach something lying below. Sometimes the ice involved is merely compacted snow and sometimes it is very impure, as when formed from sea water, but often, as in glaciers, one deals with masses of almost pure ice of large grain size and with very little included air. In this latter case the properties of the whole mass can be understood quite well from fundamental studies on pure single crystals, while the former situations involve considerations peculiar to the specific environment.

The present chapter is divided into four parts. After a review of elasticity theory, we shall discuss simple elastic behaviour of single crystals of ice, then examine the relaxation processes which take place for periodically varying stresses and finally take up a brief treatment of plastic deformation, creep and related topics.

Elasticity theory

Since treatments of the elastic properties of crystals are often very brief, or at best limited to a discussion of cubic crystals, let us begin by giving a short review of this subject as it applies to hexagonal crystals like ice. We shall follow the development given by Nye (1957), to whose book the reader is referred for a fuller treatment.

Ice crystals can grow in two simple distinct ways: either by the freezing of liquid water or by direct sublimation from the vapour phase. In each case the mechanisms which determine the rate and habit of growth are the transport of water molecules to the point of growth and their accommodation into the growing interface, together with the transport of latent heat away from this interface. Many different physical situations can occur, of course, but they are all controlled by these basic mechanisms.

If the system contains another component in addition to water, the situation can be much more complicated because crystal growth depends upon the transport of this component as well and there may also be competing processes occurring at the interface with the growing ice crystal. Some additional components, such as air in growth from the vapour, have a relatively small and simply understandable effect, but more complicated phenomena occur, for example, in the freezing of brine or sugar solutions. In this chapter we shall have very little to say about such cases but concentrate upon understanding the simpler systems.

Basic theory

The thermodynamic force driving any crystallization process is an excess of the chemical potential of molecules in the environment relative to its value in a bulk crystal. This excess may be specified as a supercooling, in the case of freezing, or as a supersaturation in the case of growth from the vapour or from solution.

Our understanding of the structure and properties of liquids is in a much less well developed state than is the theory of solids and, for that reason if no other, it is difficult to give a concise view of the subject. Simple liquids like argon or sodium, which behave to a first approximation like an assembly of hard spherical atoms, can be treated reasonably satisfactorily by current methods but the non–spherical nature of the water molecule leads to molecular association in the liquid state which complicates the problem immensely.

In this book, which is primarily about ice, we shall be concerned with only a few aspects of the structure and behaviour of liquid water; a comprehensive discussion of water and aqueous solutions would occupy several volumes. In particular we shall discuss current views on the structure of water at temperatures not too far removed from the normal freezing point and then go on to consider in some detail the phase transition involved in freezing. The actual kinetics of crystal growth will be reserved for discussion in chapter 5. Among reviews of the liquid state which provide useful background are those of Green (i960), Furukawa (1962), Barker (1963), Kavanau (1964) and Pryde (1966).

Experimental information on water structure

In the case of a crystalline solid it is possible to determine, by diffraction methods, the equilibrium positions and vibrational amplitudes of all the atoms involved and this information specifies the structure of the crystal.

The electrical properties of ice have been studied, in recent years, more extensively than any other of its attributes. One reason is that a large variety of phenomena can be conveniently classified under this heading, while another is that, because many electrical properties are very sensitive to the purity of the crystal, the number of possible results is greatly multiplied and a detailed interpretation, though necessarily complicated, gives a considerable amount of insight into the molecular processes involved. Before we examine any topic in detail, therefore, let us make a brief survey of the field to be discussed to see what the phenomena are and, in general terms, how they can be interpreted.

Introductory survey

First consider the dielectric constant of pure ice as a function of frequency, as shown schematically in fig. 9.1. The static dielectric constant εs is very large, close to 100, and the mechanism responsible for it may reasonably be identified as the orientation of molecular dipoles preferentially in the direction of the field. The value of εs does not vary greatly with a small amount of impurity in the specimen, in agreement with this interpretation. As the frequency is increased through the kilohertz range, ε falls dramatically to a high–frequency value ε∞ = 3.2 which is maintained through the microwave region (Lamb & Turney, 1949).

So far in this book we have been concerned almost entirely with the properties of perfect crystals of ice. The properties discussed were, in fact, not structure–sensitive and both the theory and necessarily the measurements apply equally to real crystals. In the remaining chapters, however, we shall examine attributes of ice which do depend sensitively upon the perfection of the particular crystal involved. There are many imperfections which occur in real crystals—surfaces, impurities, dislocations, vacancies and so on—with which we shall necessarily be concerned but in the present chapter we focus attention on the various possible kinds of point defects.

The structure of a perfect ice crystal is, as we have seen, of a statistical kind in which many different configurations are allowed provided they satisfy the three rules: (i) each lattice position is occupied by a water molecule tetrahedrally bonded to its four nearest neighbours; (ii) water molecules are intact so that there are just two protons near each oxygen; (iii) there is just one proton on each bond. Violation of the first rule leads to a vacancy, an interstitial or an impurity atom, while violation of the second or third gives rather more subtle defects peculiar to the ice structure. The very existence of these rules implies an energy penalty for their violation but, in any real crystal at a finite temperature, there will be a Boltzmann probability for finding such exceptions.

Following on our survey of the properties of the water molecule, we now come to consider the ways in which these molecules link together to form a liquid or a solid phase. This is a large subject and we shall treat it in several stages. In the present chapter we look at ordinary ice—Ice Ih—and see what is known about its crystal structure and cohesive energy. With this discussion as a background, we shall then go on to consider the other forms of ice which can exist at low temperatures or at high pressures and to take an over–all view of the phase diagram of the material called water.

Crystal structure of Ice Ih

The equilibrium structure to which a material crystallizes under given conditions of temperature and pressure is completely determined by the interaction forces between its molecules and these, in turn, are determined by the electronic structure of those molecules, as we have seen. Unfortunately it is almost impossible to proceed uniquely in this way from molecules to crystals in any but the very simplest solids, because the interaction forces are not known with sufficient accuracy. We must therefore usually content ourselves with observing that the crystal structure which actually occurs is consistent with what is known of the molecular interactions, and with comparing the value which we can calculate for its cohesive energy with that found from experiment.

In this chapter we shall discuss those properties of ice crystals which derive essentially from the thermal motions of water molecules within the crystal structure. In broad outline the theory describing these phenomena is simple and well known and leads to simple generalizations like the Debye theory of specific heats. However, because of the structure of the water molecule and, deriving from it, the structure of the ice crystal, such theories in their simple form represent only a first approximation to the observed behaviour. The coefficient of thermal expansion, for example, is negative at low temperatures and the specific heat is only poorly described by a Debye curve. It will be in tracing the reasons for some of these deviations from simple behaviour that most of our interest will lie.

Thermal expansion

The coefficient of thermal expansion can be determined in two different ways. The first, and most direct, is simply to take a single crystal of ice and measure its dimensions as a function of temperature. From the symmetry of the crystal the results can be expressed in terms of two linear expansion coefficients: α = l1(dl/dT) in directions parallel to the c–axis and to an a–axis respectively. Alternatively X–ray diffraction methods can be used to measure the c and a dimensions of the unit cell as a function of temperature.

To gain a proper understanding of the behaviour of a complex system we must first appreciate the structure and properties of the elementary units of which it is composed. In the study of ice this means that we must begin with a study of the water molecule, for it is from the individuality of the structure of that molecule that most of the unusual properties of ice and water arise. Without such a relation back to the fundamentals of molecular structure, the study of a particular material becomes simply a catalogue of its properties—useful, no doubt, but not very illuminating. In this book we shall try, at all stages, to show this relation so that a coherent picture emerges. Similar pictures can be built up for all solids; the outlines, it is true, have many variations but they all follow in the same sort of way from the basic elements of which they are built.

Water is among the simplest of molecules and for that reason if no other it has been the subject of a large amount of theoretical work, computation and experiment. Molecular structure is not a simple subject, however, and our understanding is still far from complete, but it is enough to give a picture of the water molecule which is reasonably accurate and sufficient for our present purposes.