GENERAL THEORY OF CAPILLARY ACTION It is found by experiment that the forces between molecules to which cohesion is due only act pre ceptibly across very short distances. If we regard the forces between two particles as acting according to a law depending only upon their positions it must vary inversely as the distance according to a higher power than that of the inverse square which is obeyed by gravitation. The experiments of Quincke and others show that the extreme range through which sensible effect is produced by them is certainly much less than a thousandth of a centimetre and it is probably less than one millionth.
In order to illustrate the important bearing of this limitation in range consider a molecule, A, well within a substance. A sphere of a thousandth of a centimetre radius may be drawn round it and the molecules outside this sphere will have no sensible in fluence on A. Those within the sphere attract A but in the case of a homogeneous isotropic body their pulls will be uniformly distributed and the resultant force on A will be zero. There will, however, be a pressure throughout the sphere due to the attrac tion: each spherical layer, being attracted, will compress the particles lying inside it. But if we consider a particle B very near the surface—nearer in fact than the range of action—the state of things will be found to be different. There will be more particles pulling B downwards than upwards so that all such particles as B will experience a resultant-force downwards. There will also be a pressure at B but its value is less the nearer B is to the surface. There is, therefore, a thin, indefinitely bounded, layer near the surface which is in a different condition from the main body of the substances. Many of the properties will be different in this surface layer from elsewhere. For example, corresponding to the different pressure, the density will be different. Since work is done in producing compression the potential energy will differ in the two regions. In the main body it can be written where V = volume, p density and x the potential energy per unit mass ; in the film, owing to its minute thickness, e, which we di vide up into still thinner layers, de, and surface S we write while the total potential energy (in terms of x its value per unit mass) is or Multiplying the total mass by Xo and subtracting from the last expression The right hand side is therefore an expression for that part of the energy which depends upon the existence of a surface S. The
integral which multiplies it is the constant (or rather, factor) known as the surface tension. This integral is a measure of the work done in increasing the area of the surface. It must be carefully distinguished from the corresponding increase in the total energy because heat is drawn in simultaneously if the change takes place at constant temperature. It is in strictness a measure of the change in the "free-energy." (See THERMODYNAMICS : Applied to Chemistry.) By the principles of energy the potential energy (at constant temperature) tends to a minimum; in other words, any change that takes place spontaneously involves a diminution of this energy. Hence, whenever the surface tension is positive there will be a tendency for the surface to decrease, and the diminution will in fact take place unless it is resisted by other forces. This diminution takes place not by a contraction of the liquid but by a passage of the surface molecules into the body of the liquid. The properties of the body are not changed thereby. In this respect the phenomenon is quite different from the case of a stretched india-rubber film. In the case of the liquid the surface tension remains constant during the contraction of the surface area ; in the case of the stretched india-rubber it diminishes along with the contraction.
We may express it otherwise by saying that if an imaginary straight line be drawn anywhere in the surface, when equilibrium exists there must be a force acting across the line in such a direction as to prevent further contraction of area. It is easy to show that this force per unit length of line is numerically the same as the surface-tension as defined above.