CAPILLARY ACTION 61 distance z from a stratum whose surface-density is a; and whose principal radii of curvature are Rj and K. 2 . To find the work done when m is brought to the point P in the neighbourhood of a solid body, the density of which is a function of the depth v below the surface, we have only to write instead of a pdz, and to integrate where, in general, we must suppose p a function of z. This expres sion, when integrated, gives (1) the work done on a particle m while it is brought from an infinite distance to the point P, or (2) the attraction on a long slender column normal to the surface and terminating at P, the mass of unit of length of the column being m. In the form of the theory given by Laplace, the density of the liquid was supposed to be uniform. Hence if we write the pressure of a column of the fluid itself terminating at the sur face will be and the work done by the attractive forces when a particle m is brought to the surface "of the fluid from an infinite distance will be If we write then ZirmpO(z) will express the work done by the attractive forces, while a particle m is brought from an infinite" distance to a distance z from the plane surface of a mass of the substance of density p and infinitely thick. The function 6(z) is insensible for all sensible values of z. For insensible values it may become sensible, but it must remain finite even when 2 = 0, in which case 6(0) = K. If x is the potential energy of unit of mass of the substance in vapour, then at a distance z from the plane surface of the liquid x = x -2>rpe(~~). At the surface At a distance z within the surface If the liquid forms a stratum of thickness c, then X = x - 4wp0(0) + 2irp0(z) + 2irp0(z - c) . The surface-density of this stratum is <T = C P . unit of area is The energy per = cp( x - 4*p0(0) ) + 2irp s y 6(z)ch + 1*f 6(c - z)dz . Since the two sides of the stratum are similar the last two terms are equal, and e = cp( x - 4irp0(0) ) + 4V/7 e ( z ) dz Differentiating with respect to c, we find Hence the surface-tension Integrating the first term within brackets by parts, it becomes d0 Remembering that 9(0) is a finite quantity, and that ^ = - we find When c is greater than this is equivalent to 2H in the equation of Laplace. Hence the tension is the same for all films thicker than r, the range of the molecular forces. For thinner filma flence if ty(c] is positive, the tension and the thickness will increase
- ogether. Now 2ir??ip^(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 jetween 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 pheno mena 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 47rp 2 K, 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 rf 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 attrac tion of a given molecule is very large, the part of the pressure aris- from attraction will be proportional to the square of the number ofmolecules in unit of volume, that is, to the square of the density. Hence we may write 7> = P + A P 2 ...... (1), where A is a constant. But by the equations of equilibrium of the liquid dp^-pdx ....... (2). ITence - P (7 x = 2Ap^p ...... (3), and x - X --2Ap-2n ..... (4). where B is another constant. Near the plane surface of a liquid we may assume p a function of ?. We have then for the value of x at the point where z-c, where e is the range beyond which the attraction of a mass of liquid bounded by a plane surface becomes insensible. The value of x depends, therefore, on those values only of p which correspond to strata for which z is nearly equal to c. We may, therefore, expand p in terms of z-c, or writing a; for z- c, where the suffix (c) denotes that in the quantity to which it is applied after differentiation, z is to be made equal to c. We may now write The function $(x) has equal values for + x and - x. Hence /*"*" e x*$(x)dx vanishes if n is odd. But if we write This is the expression for x on the hypo thesis that the value of p can be expanded in a series of powers of z - c within the limits z-e and 2+ e. It is only when the point P is within the distance of the surface of the liquid that this ceases to be possible. If we now substitute for x its value from equation 4, we obtain a linear differential equation in p, the solution of which is where n lf 7T 2 , n 3 , n 4 are the roots of the equation
- Mn 4 + L>t 2 + K-A = 0.
