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Molecular Theory of Capillarity
By J. S. Rowlinson, B. Widom
Dover Publications, Inc.Copyright © 1982 J. S. Rowlinson and B. Widom
All rights reserved.
MOLECULAR THEORY OF CAPILLARITY
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); Cavendish 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 Boscovich 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 Faraday and Kelvin 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. Two misunderstandings complicated the issue. First, heat was often held to be the agent of repulsion for, so the argument ran, if a liquid is heated it first expands and then boils, thus separating the molecules to much greater distances than in the solid. The second arose from a belief which went back to Newton that the observed pressure of a gas arose from static repulsions between the molecules, and not, as Daniel Bernoulli had argued in vain, from their collisions with the walls of the vessel.
With this background it was natural that the first attempts to explain capillarity, or more generally the cohesion of liquids, were based on a static view of matter. Mechanics was the theoretical branch of science that was well understood; thermodynamics and kinetic theory lay still in the future. The key assumption in this mechanical treatment was that of strong but short-ranged attractive forces. Liquids at rest, whether in a capillary tube or not, are clearly at equilibrium, so these attractive forces must be balanced by repulsions. Since even less could be guessed about these than about the attractive forces, they were often passed over in silence, and, in Rayleigh's phrase, 'the attractive forces were left to perform the impossible feat of balancing themselves'. Laplace was the first to deal with the problem satisfactorily by supposing that the repulsive forces (of heat, as he supposed) could be replaced by an internal or intrinsic pressure that acted throughout an incompressible liquid. (This supposition leads to occasional uncertainty in nineteenth-century work as to exactly what is meant by the pressure in a liquid.) Our first task is to follow Laplace's calculation of the internal pressure. It must balance the cohesive force in the liquid and he identified this with the force per unit area that resists the pulling asunder of an infinite body of liquid into two semi-infinite bodies bounded by plane surfaces. Our derivation is closer to that of Maxwell and Rayleigh than to Laplace's original form, but there is no essential difference in the argument.
Consider two semi-infinite bodies of liquid with sharp plane faces separated by vapour of negligible density of thickness l (Fig. 1.1), and let us take an element of volume in each. The first is in the upper body and is at height r above the plane surface of the lower body; its volume is dx dy dr. The second is in the lower body and its volume is s2 sin θ, ds dθ, dφ, where the origin of the polar coordinates is the position of the first elementary volume. Let f(s) be the force between two molecules at separation s, and let its range be d. Since the force is always attractive we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.1)
If the number density of molecules is p in both bodies then the vertical component of the force between the two elements of volume is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.2)
The total force of attraction per unit area (a positive quantity) is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.3)
Let u(s) be the potential of the intermolecular force
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.5)
Integrate by parts again,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.6)
Laplace's internal pressure K is the attractive force per unit area between two planar surfaces in contact, that is, F(0).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)
where dr is a volume element which may be written here 4πr2 dr. Since u(r) is everywhere negative or zero, by supposition, then K is positive. Laplace believed it to be large in relation to the atmospheric pressure, but it was left to Young (§ 1.6) to make the first realistic numerical estimate. The derivation above rests on the implicit assumption that the molecules are distributed uniformly with a density ρ, that is, that the liquid has no discernible structure on a scale of length measured by the range of the forces, d. Without this assumption we could not write (1.2) and (1.3) in these simple forms, but would have to ask how the presence of a molecule in the first volume element affected the probability of there being a molecule in the second. We return to this point in § 1.6.
By 1800 the concept of surface tension was commonplace; indeed it is almost irresistible to anyone who has tried to float a pin on water or perform similar experiments in childhood. What was lacking was a quantitative relation between this tension and the supposed intermolecular forces. A tension per unit length along an arbitrary line on the surface of a liquid must, in a coherent set of units, be equal to the work done in creating a unit area of free surface. This follows from the experiment of drawing out a film of liquid (Fig. 1.2). A measure of this work can be obtained at once from the expression above for F(l), (1.6). If we take the two semi-infinite bodies to be in contact, and then draw them apart until their separation exceeds the range of the intermolecular forces, the work done per unit area is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.8)
The separation has produced two free surfaces and so the work done can be equated to twice the surface energy per unit area, which is equal to the surface tension σ; that is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.9)
Thus K is the integral of the intermolecular potential, or its zeroth moment, and H is its first moment. Whereas K is not directly accessible to experiment, H can be found if we can measure the surface tension. Before we turn to this point let us consider some further implications of the results so far obtained.
Let φ be the cohesive energy density at a point in the fluid, that is, the ratio (δU/δV), where δU is the internal energy of a small sample of fluid δV which contains the point. For the molecular model we are using it is given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.10)
where r is the distance from the point in question. Then, following Rayleigh, we can identify Laplace's K as the difference of this potential 2φ between a point on the plane surface of the liquid, 2φs, and a point in the interior 2φ1. At the surface the integration in (1.10) is restricted to a hemisphere of radius d, whilst in the interior it is taken over a complete sphere. Hence φs is half φ1, or
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.11)
Consider now a drop of radius R. The calculation of φ1 is unchanged, but the integration to obtain φs is now over an even more restricted volume because of the curvature of the surface. That is, if θ is the angle between the vector r and a fixed radius R,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.12)
The internal pressure in the interior of the drop is therefore
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.13)
where H is given by (1.9). Had we taken not a spherical drop but a portion of liquid with a convex surface defined by its two principal radii of curvature R1 and R2, then we should have obtained an internal pressure of
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.14)
By a theorem of Euler, the sum (R1-1 + R2- 1) is equal to the sum of the reciprocals of the radii of curvature of the surface along any two orthogonal tangents.
Since K and H are positive, and R is positive for a convex surface, it follows from (1.13) that the internal pressure in a drop is higher than that in a liquid with a plane surface. Conversely the internal pressure within a liquid bounded by a concave spherical surface is lower than that in the liquid with a plane surface since R is now negative. These results are the foundation of Laplace's theory of capillarity. The equation for the difference of pressure between p1 that of the liquid inside a spherical drop of radius R, and pg, that of the gas outside, is now called Laplace's equation:
p1 - pg = 2σ/R (1.15)
It is quite general and not restricted to this molecular model. It is re-derived by purely thermodynamic arguments in Chapter 2 and used repeatedly in later chapters.
1.3 Capillary phenomena
It is interesting and useful to extend these results to a three-phase system of a solid in contact with a liquid and a vapour (again, of negligible density). The solid is of different chemical constitution from the liquid, and quite insoluble in it. We assume moreover, that it is a perfectly rigid molecularly uniform array of density Ï 2. The intermolecular potential between two molecules of the species in the liquid is denoted u11, between two of those in the solid u22, and that between a molecule of each u12. By the argument of the last section, the force per unit area between a slab of liquid and one of solid at separation l is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.16)
and the work to separate the liquid and solid is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.17)
This work is equal to the sum of the surface energies of the two new surfaces formed, liquid-gas and solid-gas, less that of the surface destroyed, liquid-solid;
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.18)
Thus the surface tension of the liquid-solid interface is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.19)
with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where, as in (1.9),
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.20)
Excerpted from Molecular Theory of Capillarity by J. S. Rowlinson, B. Widom. Copyright © 1982 J. S. Rowlinson and B. Widom. Excerpted by permission of Dover Publications, Inc..
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