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1
MOLECULAR THEORY OF CAPILLARITY
1.1 Introduction
If a glass tube with a bore as small as the width of a hair (Latin: capillus) is dipped into water then the liquid rises in the tube to a height greater than that at which it stands outside. The effect is not small; the rise is about 3 cm in a tube with a bore of 1 mm. This apparent defiance of the laws of hydrostatics (which were an achievement of the seventeenth century) led to an increasing interest in capillary phenomena as the eighteenth century advanced. The interest was two-fold. The first was to see if one could characterize the surfaces of liquids and solids by some simple mechanical property, such as a state of tension, that could explain the observed phenomena. The things to be explained were, for example, why does water rise in a tube while mercury falls, why is the rise of water between parallel plates only a half of that in a tube with a diameter equal to the separation of the plates, and why is the rise inversely proportional to this diameter? The second cause of interest was the realization that here were effects which must arise from cohesive forces between the intimate particles of matter, and that the study of these effects should therefore tell something of those forces, and possibly of the particles themselves. In this book we follow the first question only sufficiently far to show that a satisfactory set of answers has been found; our interest lies, as did that of many of the best nineteenth-century physicists, in the second and more difficult question, or, more precisely, in its inverse—how are capillary phenomena to be explained in terms of intermolecular forces.
We could attempt an answer by summarizing the experimental results and then bringing to bear on them at once the whole armoury of modern thermodynamics and statistical mechanics. To do this, however, would be to throw away much of the insight that has been gained slowly over the last two centuries. Indeed the way we now look at capillary phenomena, and more generally at the properties of liquids, is conditioned by the history of the subject. In the opening chapters we follow the way the subject has developed, not with the aim of writing a strict history, but in order to trace the many strands of thought that have led to our present understanding.
In this first chapter we describe the early attempts to explain capillarity which were based on an inevitably inadequate understanding of the molecular structure and physics of fluids. Most of the equations of this chapter are therefore only crude approximations which are superseded by exact or, at least, more accurate equations in the later chapters.
1.2 Molecular mechanics
That matter was not indefinitely divisible but had an atomic or molecular structure was a working hypothesis for most scientists from the eighteenth century onwards. There was a minor reaction towards the end of the nineteenth century when a group of physicists who professed a positivist philosophy pointed out how indirect was the evidence for the existence of atoms, and their objections were not finally overcome until the early years of this century. If in retrospect, their doubts seem to us to be unreasonable we should, perhaps, remember that almost all those who then believed in atoms believed equally strongly in the material existence of an electromagnetic ether and, in the first half of the nineteenth century, often of a caloric fluid also. Nevertheless those who contributed most to the theories of gases and liquids did so with an assumption, usually explicit, of a discrete structure of matter. The units might be named atoms or molecules (e.g. Laplace) or merely particles (Young), but we will follow modern convention and use the word molecule for the constituent element of a gas, liquid, or solid.
The forces that might exist between molecules were as obscure as the particles themselves at the opening of the nineteenth century. The only force about which there was no doubt was Newtonian gravity. This acted between celestial bodies; it obviously acted between one such body (the Earth) and another of laboratory mass (e.g. an apple); Cavendish1 had recently shown that it acted equally between two of laboratory mass, and so it was presumed to act aplso between molecules, in early work on liquids we find the masses of molecules and mass densities entering into equations where we should now write numbers of molecules and number densities. In a pure liquid all molecules have the same mass so the difference is unimportant. It was, however, clear before 1800 that gravitational forces were inadequate to explain capillary phenomena and other properties of liquids. The rise of a liquid in a glass tube is independent of the thickness of the glass;2 thus only the forces from the molecules in the surface layer of the glass act on those in the liquid. Gravitational forces, however, fall off only as the inverse square of the distance and were known to act freely through intervening matter.
The nature of the intermolecular forces other than gravity was quite obscure, but speculation was not lacking. The Jesuit priest Roger Boscovich4 believed that molecules repel at very short distances, attract at slightly larger separations and then show alternate repulsions and attractions of ever decreasing magnitude as the separation becomes ever larger. His ideas influenced both Faraday5 and Kelvin6 in the next century but were too elaborate to be directly useful to those who were to study the theory of capillarity. They wisely contented themselves with simpler hypotheses.
The cohesion of liquids and solids, the condensation of vapours to liquids, the wetting of solids by liquids and many other simple properties of matter all pointed to the presence of forces of attraction many times stronger than gravity but acting only at very short separations of the molecules. Laplace said that the only condition imposed on these forces by the phenomena were that they were insensible at sensible distances’. Little more could in fact be said until 1929.
The repulsive forces gave more trouble. Their presence could not be denied; they must balance the attractive forces and prevent the total collapse of matter, but their nature was quite obscure. Tw...
