On Growth and Form

by D'Arcy Wentworth Thompson

Fig. 99.

the drop is concerned, then, equi­lib­rium depends on a proper balance between the energy, per unit area, which is resident in its own two surfaces, and that which is external thereto: that is to say, if we call E_bc the energy at the surface between the two fluids, and so on with the other two pairs of surface energies, the condition of equi­lib­rium, or of maintenance of the drop, is that

E_bc < E_ab + E_ac.

If, on the other hand, the fluid A happens to be oil and the fluid B, water, then the energy per unit area of the water-air surface is greater than that of the oil-air surface and that of the oil-water surface together; i.e.

E_wa > E_oa + E_ow.

Here there is no equi­lib­rium, and in order to obtain it the water-air surface must always tend to decrease and the other two interfacial surfaces to increase; which is as much as to say that the water tends to become covered by a spreading film of oil, and the water-air surface to be abolished. {295}

The surface energy of which we have here spoken is manifested in that contractile force, or “tension,” of which we have already had so much to say[341]. In any part of the free water surface, for instance, one surface particle attracts another surface particle, and the resultant of these multitudinous attractions is an equi­lib­rium of tension throughout this particular surface. In the case of our three bodies in contact with one another, and within a small area very near to the point of contact, a water particle (for instance) will be pulled outwards by another water particle; but on the opposite side, so to speak, there will be no water surface, and no water particle, to furnish the counterbalancing pull; this counterpull,

Fig. 100.

Fig. 101.

which is necessary for equi­lib­rium, must therefore be provided by the tensions existing in the other two surfaces of contact. In short, if we could imagine a single particle placed at the very point of contact, it would be drawn upon by three different forces, whose directions would lie in the three surface planes, and whose magnitude would be proportional to the specific tensions char­ac­ter­is­tic of the two bodies which in each case combine to form the “interfacial” surface. Now for three forces acting at a point to be in equi­lib­rium, they must be capable of representation, in magnitude and direction, by the three sides of a triangle, taken in order, in accordance with the elementary theorem of the Triangle of Forces. So, if we know the form of our floating drop (Fig. [100]), then by drawing tangents from O (the point of mutual contact), {296} we determine the three angles of our triangle (Fig. [101]), and we therefore know the relative magnitudes of the three surface tensions, which magnitudes are proportional to its sides; and conversely, if we know the magnitudes, or relative magnitudes, of the three sides of the triangle, we also know its angles, and these determine the form of the section of the drop. It is scarcely necessary to mention that, since all points on the edge of the drop are under similar conditions, one with another, the form of the drop, as we look down upon it from above, must be circular, and the whole drop must be a solid of revolution.