# Page:EB1911 - Volume 05.djvu/275

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262
CAPILLLARY ACTION

density, they deduced its variations from the observed dilatation of the liquid by heat. This assumption, however, does not appear to be verified by the experiments of Brunner and Wolff on the rise of water in tubes at different temperatures.

3. The tension of the surface separating two liquids which do not mix cannot be deduced by any known method from the tensions of the surfaces of the liquids when separately in contact with air.

When the surface is curved, the effect of the surface-tension is to make the pressure on the concave side exceed the pressure on the convex side by ${\displaystyle {\mbox{T}}(1/{\mbox{R}}_{1}+1/{\mbox{R}}_{2})}$, where ${\displaystyle {\mbox{T}}}$ is the intensity of the surface-tension and ${\displaystyle {\mbox{R}}_{1}}$, ${\displaystyle {\mbox{R}}_{2}}$ are the radii of curvature of any two sections normal to the surface and to each other.

Fig. 3.

If three fluids which do not mix are in contact with each other, the three surfaces of separation meet in a line, straight or curved. Let ${\displaystyle {\mbox{O}}}$ (fig. 3) be a point in this line, and let the plane of the paper be supposed to be normal to the line at the point ${\displaystyle {\mbox{O}}}$. The three angles between the tangent planes to the three surfaces of separation at the point ${\displaystyle {\mbox{O}}}$ are completely determined by the tensions of the three surfaces. For if in the triangle ${\displaystyle abc}$ the side ${\displaystyle ab}$ is taken so as to represent on a given scale the tension of the surface of contact of the fluids ${\displaystyle a}$ and ${\displaystyle b}$, and if the other sides ${\displaystyle bc}$ and ${\displaystyle ca}$ are taken so as to represent on the same scale the tensions of the surfaces between ${\displaystyle b}$ and ${\displaystyle c}$ and between ${\displaystyle c}$ and ${\displaystyle a}$ respectively, then the condition of equilibrium at ${\displaystyle {\mbox{O}}}$ for the corresponding tensions ${\displaystyle {\mbox{R}}}$, ${\displaystyle {\mbox{P}}}$ and ${\displaystyle {\mbox{Q}}}$ is that the angle ${\displaystyle {\mbox{ROP}}}$ shall be the supplement of ${\displaystyle abc}$, ${\displaystyle {\mbox{POQ}}}$ of ${\displaystyle bca}$, and, therefore, ${\displaystyle {\mbox{QOR}}}$ of ${\displaystyle cab}$. Thus the angles at which the surfaces of separation meet are the same at all parts of the line of concourse of the three fluids. When three films of the same liquid meet, their tensions are equal, and, therefore, they make angles of 120° with each other. The froth of soap-suds or beaten-up eggs consists of a multitude of small films which meet each other at angles of 120°.

If four fluids, ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$, ${\displaystyle d}$, meet in a point ${\displaystyle {\mbox{O}}}$, and if a tetrahedron ${\displaystyle {\mbox{ABCD}}}$ is formed so that its edge ${\displaystyle {\mbox{AB}}}$ represents the tension of the surface of contact of the liquids a and ${\displaystyle b}$, ${\displaystyle {\mbox{BC}}}$ that of ${\displaystyle b}$ and ${\displaystyle c}$, and so on; then if we place this tetrahedron so that the face ${\displaystyle {\mbox{ABC}}}$ is normal to the tangent at ${\displaystyle {\mbox{O}}}$ to the line of concourse of the fluids ${\displaystyle abc}$, and turn it so that the edge ${\displaystyle {\mbox{AB}}}$ is normal to the tangent plane at ${\displaystyle {\mbox{O}}}$ to the surface of contact of the fluids ${\displaystyle a}$ and ${\displaystyle b}$, then the other three faces of the tetrahedron will be normal to the tangents at ${\displaystyle {\mbox{O}}}$ to the other three lines of concourse of the liquids, and the other five edges of the tetrahedron will be normal to the tangent planes at ${\displaystyle {\mbox{O}}}$ to the other five surfaces of contact.

If six films of the same liquid meet in a point the corresponding tetrahedron is a regular tetrahedron, and each film, where it meets the others, has an angle whose cosine is ${\displaystyle -{\tfrac {1}{3}}}$. Hence if we take two nets of wire with hexagonal meshes, and place one on the other so that the point of concourse of three hexagons of one net coincides with the middle of a hexagon of the other, and if we then, after dipping them in Plateau’s liquid, place them horizontally, and gently raise the upper one, we shall develop a system of plane laminae arranged as the walls and floors of the cells are arranged in a honeycomb. We must not, however, raise the upper net too much, or the system of films will become unstable.

When a drop of one liquid, ${\displaystyle {\mbox{B}}}$, is placed on the surface of another, ${\displaystyle {\mbox{A}}}$, the phenomena which take place depend on the relative magnitude of the three surface-tensions corresponding to the surface between ${\displaystyle {\mbox{A}}}$ and air, between ${\displaystyle {\mbox{B}}}$ and air, and between ${\displaystyle {\mbox{A}}}$ and ${\displaystyle {\mbox{B}}}$. If no one of these tensions is greater than the sum of the other two, the drop will assume the form of a lens, the angles which the upper and lower surfaces of the lens make with the free surface of ${\displaystyle {\mbox{A}}}$ and with each other being equal to the external angles of the triangle of forces. Such lenses are often seen formed by drops of fat floating on the surface of hot water, soup or gravy. But when the surface-tension of ${\displaystyle {\mbox{A}}}$ exceeds the sum of the tensions of the surfaces of contact of ${\displaystyle {\mbox{B}}}$ with air and with ${\displaystyle {\mbox{A}}}$, it is impossible to construct the triangle of forces; so that equilibrium becomes impossible. The edge of the drop is drawn out by the surface-tension of ${\displaystyle {\mbox{A}}}$ with a force greater than the sum of the tensions of the two surfaces of the drop. The drop, therefore, spreads itself out, with great velocity, over the surface of ${\displaystyle {\mbox{A}}}$ till it covers an enormous area, and is reduced to such extreme tenuity that it is not probable that it retains the same properties of surface-tension which it has in a large mass. Thus a drop of train oil will spread itself over the surface of the sea till it shows the colours of thin plates. These rapidly descend in Newton’s scale and at last disappear, showing that the thickness of the film is less than the tenth part of the length of a wave of light. But even when thus attenuated, the film may be proved to be present, since the surface-tension of the liquid is considerably less than that of pure water. This may be shown by placing another drop of oil on the surface. This drop will not spread out like the first drop, but will take the form of a flat lens with a distinct circular edge, showing that the surface-tension of what is still apparently pure water is now less than the sum of the tensions of the surfaces separating oil from air and water.

The spreading of drops on the surface of a liquid has formed the subject of a very extensive series of experiments by Charles Tomlinson; van der Mensbrugghe has also written a very complete memoir on this subject (Sur la tension superficielle des liquides, Bruxelles, 1873).

Fig. 4.

When a solid body is in contact with two fluids, the surface of the solid cannot alter its form, but the angle at which the surface of contact of the two fluids meets the surface of the solid depends on the values of the three surface-tensions. If ${\displaystyle a}$ and ${\displaystyle b}$ are the two fluids and ${\displaystyle c}$ the solid then the equilibrium of the tensions at the point ${\displaystyle {\mbox{O}}}$ depends only on that of thin components parallel to the surface, because the surface-tensions normal to the surface are balanced by the resistance of the solid. Hence if the angle ${\displaystyle {\mbox{ROQ}}}$ (fig. 4) at which the surface of contact ${\displaystyle {\mbox{OP}}}$ meets the solid is denoted by ${\displaystyle \alpha }$,

${\displaystyle {\mbox{T}}_{bc}-{\mbox{T}}_{ca}-{\mbox{T}}_{ab}\cos \alpha =0,}$

Whence

${\displaystyle \cos \alpha =({\mbox{T}}_{bc}-{\mbox{T}}_{ca})/{\mbox{T}}_{ab}.}$

As an experiment on the angle of contact only gives us the difference of the surface-tensions at the solid surface, we cannot determine their actual value. It is theoretically probable that they are often negative, and may be called surface-pressures.

The constancy of the angle of contact between the surface of a fluid and a solid was first pointed out by Dr Young, who states that the angle of contact between mercury and glass is about 140°. Quincke makes it 128° 52′.

If the tension of the surface between the solid and one of the fluids exceeds the sum of the other two tensions, the point of contact will not be in equilibrium, but will be dragged towards the side on which the tension is greatest. If the quantity of the first fluid is small it will stand in a drop on the surface of the solid without wetting it. If the quantity of the second fluid is small it will spread itself over the surface and wet the solid. The angle of contact of the first fluid is 180° and that of the second is zero.

If a drop of alcohol be made to touch one side of a drop of oil on a glass plate, the alcohol will appear to chase the oil over the plate, and if a drop of water and a drop of bisulphide of carbon be placed in contact in a horizontal capillary tube, the bisulphide of carbon will chase the water along the tube. In both cases the liquids move in the direction in which the surface-pressure at the solid is least.

[In order to express the dependence of the tension at the interface of two bodies in terms of the forces exercised by the bodies upon themselves and upon one another, we cannot do better than follow the method of Dupré. If ${\displaystyle {\mbox{T}}_{12}}$ denote the interfacial tension, the energy corresponding to unit of area of the interface