Remembering that θ(0) is a finite quantity, and that dθ/dz = − ψ(z), we find
T = 4πρ² ∫c0 zψ(z) dz. (27)
When c is greater than ε this is equivalent to 2H in the equation of Laplace. Hence the tension is the same for all films thicker than ε, the range of the molecular forces. For thinner films
| dT | = 4πρ²cψ(c). |
| dc |
Hence if ψ(c) is positive, the tension and the thickness will increase together. Now 2πmρψ(c) represents the attraction between a particle m and the plane surface of an infinite mass of the liquid, when the distance of the particle outside the surface is c. Now, the force between the particle and the liquid is certainly, on the whole, attractive; but if between any two small values of c it should be repulsive, then for films whose thickness lies between these values the tension will increase as the thickness diminishes, but for all other cases the tension will diminish as the thickness diminishes.
We have given several examples in which the density is assumed to be uniform, because Poisson has asserted that capillary phenomena would not take place unless the density varied rapidly near the surface. In this assertion we think he was mathematically wrong, though in his own hypothesis that the density does actually vary, he was probably right. In fact, the quantity 4πρ2K, which we may call with van der Waals the molecular pressure, is so great for most liquids (5000 atmospheres for water), that in the parts near the surface, where the molecular pressure varies rapidly, we may expect considerable variation of density, even when we take into account the smallness of the compressibility of liquids.
The pressure at any point of the liquid arises from two causes, the external pressure P to which the liquid is subjected, and the pressure arising from the mutual attraction of its molecules. If we suppose that the number of molecules within the range of the attraction of a given molecule is very large, the part of the pressure arising from attraction will be proportional to the square of the number of molecules in unit of volume, that is, to the square of the density. Hence we may write
p = P + Aρ²,
where A is a constant [equal to Laplace’s intrinsic pressure K. But this equation is applicable only at points in the interior, where ρ is not varying.]
[The intrinsic pressure and the surface-tension of a uniform mass are perhaps more easily found by the following process. The former can be found at once by calculating the mutual attraction of the parts of a large mass which lie on opposite sides of an imaginary plane interface. If the density be σ, the attraction between the whole of one side and a layer upon the other distant z from the plane and of thickness dz is 2 π σ² ψ(z) dz, reckoned per unit of area. The expression for the intrinsic pressure is thus simply