On Growth and Form

by D'Arcy Wentworth Thompson

The contraction of a liquid surface and other phenomena of surface tension involve the doing of work, and the power to do work is what we call energy. It is obvious, in such a simple case as we have just considered, that the whole energy of the system is diffused throughout its molecules; but of this whole stock of energy it is only that part which comes into play at or very near to the surface which normally manifests itself in work, and hence we may speak (though the term is open to some objections) of a specific surface energy. The consideration of surface energy, and of the manner in which its amount is increased and multiplied by the multiplication of surfaces due to the subdivision of the organism into cells, is of the highest importance to the physiologist; and even the morphologist cannot wholly pass it by, if he desires to study the form of the cell in its relation to the phenomena of surface tension or “capillarity.” The case has been set forth with the utmost possible lucidity by Tait and by Clerk Maxwell, on whose teaching the following paragraphs are based: they having based their teaching upon that of Gauss,—who rested on Laplace.

Let E be the whole potential energy of a mass M of liquid; let e₀ be the energy per unit mass of the interior liquid (we may call it the internal energy); and let e be the energy per unit mass for a layer of the skin, of surface S, of thickness t, and density ρ (e being what we call the surface energy). It is obvious that the total energy consists of the internal plus the surface energy, and that the former is distributed through the whole mass, minus its surface layers. That is to say, in math­e­mat­i­cal language,

E = (M − S · Σ t ρ) e₀ + S · Σ t ρ e .

But this is equivalent to writing:

= M e₀ + S · Σ t ρ(e − e₀) ;

and this is as much as to say that the total energy of the system may be taken to consist of two portions, one uniform throughout the whole mass, and another, which is proportional on the one hand to the amount of surface, and on the other hand is proportional to the difference between e and e₀ , that is to say to the difference between the unit values of the internal and the surface energy. {208}

It was Gauss who first shewed after this fashion how, from the mutual attractions between all the particles, we are led to an expression which is what we now call the potential energy of the system; and we know, as a fundamental theorem of dynamics, that the potential energy of the system tends to a minimum, and in that minimum finds, as a matter of course, its stable equi­lib­rium.

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We see in our last equation that the term M e₀ is irreducible, save by a reduction of the mass itself. But the other term may be diminished (1) by a reduction in the area of surface, S, or (2) by a tendency towards equality of e and e₀ , that is to say by a diminution of the specific surface energy, e.

These then are the two methods by which the energy of the system will manifest itself in work. The one, which is much the more important for our purposes, leads always to a diminution of surface, to the so-called “principle of minimal areas”; the other, which leads to the lowering (under certain circumstances) of surface tension, is the basis of the theory of Adsorption, to which we shall have some occasion to refer as the modus operandi in the development of a cell-wall, and in a variety of other histological phenomena. In the technical phraseology of the day, the “capacity factor” is involved in the one case, and the “intensity factor” in the other.