When the column is stationary, the tension in the surface can be considered as acting ver tically upward at all points in the circle of con tact of the surface with the wall of the tube. Representing by T the tension in the surface, or the force acting in the surface across a line of unit length, and by r the radius of the tube, we have the expression 2irrT for the upward force acting on the column. Representing by p the density of water, the acceleration of grav ity, and by h the height of the column above the general level of the water surface outside the tube, we have the expression for the weight of the column. Setting the two forces equal, we obtain and conclude that the height of the column is inversely as the radius of the tube. This law was made known by the experiments of Jurin (1718), and is gen erally known as Jurin's law.
By a slight extension of Young's conception of surface tension, we may deduce from it the constancy of the contact angle. We need only to suppose that a tension exists, in any surface separating two substances, which has a particu lar value for each pair of substances. Consider then three fluids in contact along a line. It is evident that the line of contact will be at rest when the angles made with each other, at that line, by the three surfaces in which the fluids meet in pairs, are such that the tensions in the three surfaces are in equilibrium. These angles are therefore obtained by constructing the tri angle of forces, with the three tensions as sides, and they are constant, for the three substances.
We may consider more particularly the spe cial case in which one of the three substances is a solid. Suppose, for convenience in statement, that the three substances are a liquid, air and a solid. Represent by Ti,, Ti, and T. the ten sions in the surfaces separating the liquid from air, the liquid from the solid, and air from the solid, respectively. Denote the angle of contact of the liquid-air surface with the solid by The line of contact will be at rest when the sum of all the tensions or components of tension in the plane of the solid is equal to zero, or when v.
The angle of contact is therefore given by cos = Tv and is constant. It is acute or obtuse, accord ing as T. is greater or less than Ti,. In the case of mercury and glass, Ti, is the greater, and is obtuse. In the case of most liquids and glass, the tension corresponding to is the greater, and 4) is acute. When T. equals or exceeds Ti, Ti,, the angle 'P becomes evanes cent.
Almost contemporaneously with Young, La place (1805) formally applied the hypothesis of molecular forces to the study of the forms of liquid surfaces. He considered the pressure at the end of a liquid filament, beginning in the surface and drawn normal to it, and terminating in the interior of the liquid. He proved that it may be expressed by the sum of two pressures. One of these, called the molecular pressure, is very great, and is constant at all points of the liquid that are not in the surface layer. This pressure is eliminated from all equations of equilibrium of liquids, and plays no part in determining the forms of liquid surfaces. The
other pressure depends upon the shape of the liquid surface, and is given by the formula (// 1 R 1 i , in which H is a constant, and R 2 R' and R' the two principal radii of curvature of the surface. This pressure, at any point under the surface layer, is in equilibrium with the hydro static pressure at that point. Under a flat sur face, and therefore under the level surface of a large expanse of liquid, the radii of curvature are infinite and this pressure vanishes. If h is the height of a point in the curved surface above the general level, we then have for H (i equilibrium the condition =-- pg 1,.
2 R R' This relation may be deduced from Young's hypothesis of surface tension, and it is found that Laplace's constant H is equal to 2T. As an example of the use of this equation, consider again the rise of water in a tube. The surface in the tube, if its bore is small enough, may be considered a hemisphere, and therefore r, the radius of the tube. Accordingly we have H pgrh, as we obtained before by Young's method.
Laplace's theory did not suffice to demon strate the constancy of the contact-angle, and Laplace was forced to assume it as a fact of observation.
A more profound and successful application of the hypothesis of molecular forces to the problem of capillarity was made by Gauss (1829). He showed, by means of the principle of virtual work, that a system of substances in contact possesses a certain amount of potential energy.arising from the molecular forces. For each pair of substances the energy is propor tional to the extent of surface separating them, and the factor of proportion is a characteristic constant for the two substances. This constant is called the surface energy. The existence of such a surface energy may readily be deduced from the hypothesis of molecular forces. Con sider, for example, a mass of liquid surrounded by another liquid of the same specific gravity. If its surface is enlarged, it can only be by the movement of some of its parts out of its in terior into the surface layer, and it is evident that, as they move out through the layer, work is done against the molecular forces, which will be proportional, generally, to the area by which the surface is increased. The liquid acquires potential energy equal to the work done in in creasing its surface. As the potential energy of a system in equilibrium is always a minimum, the condition of equilibrium of such a mass of liquid is, therefore, that the area of its surface shall be a minimum. If the liquid is entirely free, its surface will be spherical. If it is sub ject to conditions, so that portions of the sur face are limited by certain fixed boundary lines, it may be proved that the forms of the various portions of the surface, which will make the surface energy a minimum, are such that the sum of the reciprocals of the principal radii of curvature is the same for all parts of all the surfaces. We are thus led to the same rule for the form of a liquid surface as that reached by Laplace.