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.

164. The difficulties in the way of an exact mathematical treatment of diffusion are similar to those which occurred in the problems of viscosity and heat conduction. Following the procedure we adopted in discussing these earlier problems, we shall begin by giving a simple, but mathematically inexact, treatment of the question.

We imagine two gases diffusing through one another in a direction parallel to the axis of z, the motion being the same at all points in a plane perpendicular to the axis of z. The gases are accordingly arranged in layers perpendicular to this axis.

The simplest case arises when the molecules of the two gases are similar in mass and size–like the red and white billiard balls we discussed in § 6. In other cases differences in the mass and size of the molecules tend, as the motion of the molecules proceeds, to set up differences of pressure in the gas. The gas adjusts itself against these by a slow mass-motion, which will of course be along the axis of z at every point.

Let us denote the mass-velocity in the direction of z increasing by w0, and let the molecular densities of the two gases be ν1, ν2. Then ν1, ν2 and w0 are functions of z only.

132. At a collision between two molecules, energy, momentum and mass are all conserved. Energy, for instance, is neither created nor destroyed; a certain amount is transferred from one of the colliding molecules to the other. Thus the moving molecules may be regarded as transporters of energy, which they may hand on to other molecules when they collide with them. As the result of a long chain of collisions, energy may be transported from a region where the molecules have much energy to one where they have but little energy: studying such a chain of collisions we have in effect been studying the conduction of heat in a gas. If we examine the transport of momentum we shall find that we have been studying the viscosity of a gas–the subject of the present chapter. For viscosity represents a tendency for two contiguous layers of fluid to assume the same velocity, and this is effected by a transport of momentum from one layer to the other. Finally if we examine the transfer of the molecules themselves we study diffusion.

For the moment, we must study the transport of momentum. We think of the traversing of a free path of length λ as the transport of a certain amount of momentum through a distance λ. If the gas were in a steady state, every such transport would be exactly balanced by an equal and opposite transport in the reverse direction, so that the net transport would always be nil.

The Origins of the Theory

1. As soon as man began to think of abstract problems at all, it was only natural that speculations as to the nature and ultimate structure of the material world should figure largely in his writings and philosophies.

Among the earliest speculations which have survived are those of Thales of Miletus (about 640-547 B.C.), many of whose ideas may well have been derived from still earlier legends of Egyptian origin. He conjectured that the whole material universe consisted only of water and of substances derived from water by physical transformation. Earth was produced by the condensation of water, and air by its rarefaction, while air when heated became fire. About 500B.C. Heraclitus advanced the alternative view that earth, air, fire and water were not transformable one into the other, but constituted four distinct unalterable “elements”, and that all material substances were composed of these four elements mixed in varying proportions–a sort of dim anticipation of modern chemical theory. At a somewhat later date, Leucippus and Democritus maintained that matter consisted of minute hard particles moving as separate units in empty space, and that there were as many kinds of particles as there are different substances.

Unhappily nothing now remains of the writings of either Democritus or Leucippus; their opinions are known to us only through second-hand accounts. From these we learn that they imagined their particles to be eternal and invisible, and so small that their size could not be diminished; hence the name ἄτιμις–indivisible.