OF LIQUID FILMS.

403

clined at 120°, as with the liquid films, and also in the honeycomb.

The corners of the rhombic dodecahedron are of two kinds, where (i) three, (ii) four edges and faces meet; and these corners can be constructed by planes through the edges of a tetrahedron or cube, meeting in the centre.

This can be realised practically with liquid films proceeding from the edges of a wire tetrahedron or cube, which has been dipped in soapy water; but whereas the arrangement is stable in the tetrahedron, a small cubelet, with curved edges and faces, is generally formed at the centre of the cube; and on examination the interior arrangement of the liquid film in froth will be found to be composed of these elementary arrangements.

If a right prism whose ends are equilateral triangles is dipped into the liquid, two corners of the first kind can be formed if the height is greater than 6 times a side of the triangle.

298. If the substance B is made solid, with a plane face, the liquid C will form a drop on it in the shape of a spherical segment or meniscus, meeting the plane at an angle a, called the angle of contact, given by

TAC COS a+TBC=TAB, or cos a = (TAB-TBC)/TAC the condition of equilibrium of the edge of contact; the normal component Tac sin a being balanced by the reaction of the plane; thus with mercury on glass it is found that a is about 140°.

AC

The air bubble in a spirit level is another illustration of a drop in contact with a solid.

АС

But if TAB exceeds the sum of TAC and TBC, the liquid C cannot stand as a drop, but will be drawn out into an attenuated film over the surface, like oil on water.

404

SURFACE TENSION.

In a V-shaped groove in B, straight or circular, the liquid C will gather in a cylindrical or ring shape, having the same angle of contact.

299. Suppose a volume V of the liquid C is between two parallel planes B and B', a small distance d apart; it will meet these planes at the angle of contact a and form a film, of area A suppose.

Then, neglecting the curvature of the outline of the film, the pressure in the liquid C will be less than the pressure outside by 2T cos a/d, T denoting the surface tension of C; and the planes will be pressed together in consequence by a thrust

2AT cos a/d=2VT cos a/d2,

which becomes considerable when d is small.

A parallel plane, dividing the liquid into two parts of volumes V1 and V2, will have a position of equilibrium, at distances x and y from the fixed planes, where

1

but this position of equilibrium will be unstable, and the plane will stick to one or other of the two fixed planes.

The Regelation of Ice may be explained in this manner; and also the sticking together of two accurate plane surfaces, in consequence of the capillarity of the air.

300. When a large number n of rain drops of radius r coalesce into a single drop of radius R, then the volume of water being unchanged,

R=n3r.

πR3 =π3, or R-nr.

The diminution of surface is thus

4π(nr2 — R2) = 4π(n— n3)r2 = 4π(n3 — 1)R2;

so that surface energy has been liberated amounting to 4π(n−n3),2T=4π(n3 — 1)R2T ;

RISE IN CAPILLARY TUBE.

405

and it is supposed that this is the source of the electric energy in a thunderstorm.

Thus if a thousand million rain drops coalesce to form a single drop 0·1 inch in diameter, n = 10o and 4R2 = 10−2; and, with T-3.23 grains/inch, this energy amounts to

(103-1)10-2x 3.23 inch-grains, or 000123 ft-lb.

A cubic foot of water will make about 330 millions of such large drops, so that the corresponding energy would be about 400 thousand ft-lb.

301. When we return to the surface of the earth, and restore gravity, the shape of the liquids will be considerably altered, as shown for instance in fig. 89, p. 400; but the conditions of equilibrium of the edges will remain the same as before.

To determine the height h (ins) which a liquid of density w (grains/in3), and surface tension T (grains/inch), will ascend in a capillary tube of internal bore d (ins), when a is the angle of contact of the liquid with the solid of the tube; take h to denote the mean height of the column above the level of the liquid outside, so that dhw is the weight of the column in grains.

on putting T/w=c2; so that h is inversely as d.

302. If the tube is slightly conical, ẞ denoting the semi-vertical angle, and the vertex is at a height a above the outside surface, then

27(a-h)tan B. T cos(a-ẞ) = wπh(a− h)2tan2ß,

ah-h2 = 2c2cos(a — ẞ)cot ß,

406

CAPILLARY ATTRACTION.

a quadratic for h, of which the smaller root gives the stable position of equilibrium; the larger root will give a position of equilibrium which will tend to fill the cone. The potential energy of the liquid raised is

wrastan2ß. a-wπ(a-h)3tan2ß{h+1(a− h)}

reducing to a constant, 2wc cosa, for a cylindrical tube.

The height h to which the liquid ascends is independent of the shape of the vessel, except at the part near the upper surface.

In this way the rise of sap in trees may be explained.

303. For the rise h between two parallel, vertical, plane plates, a distance d apart, we have

2T cos a=wah, or h=2T cos a/wd, half the rise in a circular tube, as is easily observed.

If the plates are vertical, but not quite parallel, then d varies as x the distance from the line of intersection of

THE CAPILLARY CURVE.

407

the plates; so that the elevation y is inversely as x, showing that the upper surface of the liquid between the plates will be a curve in the form of a hyperbola, with vertical and horizontal asymptotes, one the line of intersection of the planes, and the other the line of intersection with the free liquid; this is easily verified experimentally. 304. The Capillary Curve.

The vertical cross section of the cylindrical surface formed by the free surface of a liquid in contact with a plane boundary, or of a broad drop on a horizontal plane, is called the capillary curve.

Supposing the angle of contact is obtuse, as with mercury, the free surface is depressed below the asymptotic horizontal plane with which it coalesces at a distance from the edge; and the depression x and the slope of the capillary curve are connected by the simple relation,

x=2c sin 4,

where c=(T/w), T denoting the surface tension, and w the density of the liquid.

For considering the equilibrium of the liquid in unit length of the capillary cylindrical surface cut off by a horizontal plane through a point P on the capillary curve, the horizontal hydrostatic thrust wy2 must be balanced by the tension T of the asymptotic horizontal surface, and by the horizontal component, Tcos p, of the tension at P; so that

}wy2 = T(1 — cos 4) = 2T sin214.

305. Thus if, in a large drop of mercury on a horizontal plate, the depth k below the flat top of the point K where the tangent plane is vertical is measured (fig. 90), then

T=wk2, or k=c/2.