T1 = T2 = T′12;
and T12 = 0, as it should do.
Laplace does not treat systematically the question of interfacial tension, but he gives incidentally in terms of his quantity H a relation analogous to (47).
If 2T′12 > T1 + T2, T12 would be negative, so that the interface would of itself tend to increase. In this case the fluids must mix. Conversely, if two fluids mix, it would seem that T′12 must exceed the mean of T1 and T2; otherwise work would have to be expended to effect a close alternate stratification of the two bodies, such as we may suppose to constitute a first step in the process of mixture (Dupré, Théorie mécanique de la chaleur, p. 372; Kelvin, Popular Lectures, p. 53).
The value of T′12 has already been calculated (32). We may write
T′12 = πσ1σ2 ∫∞0 θ(z) dz = 1⁄8 πσ1σ2 ∫∞0 z4φ(z) dz; (48)
| Fig. 5. |
and in general the functions θ, or φ, must be regarded as capable of assuming different forms. Under these circumstances there is no limitation upon the values of the interfacial tensions for three fluids, which we may denote by T12, T23, T31. If the three fluids can remain in contact with one another, the sum of any two of the quantities must exceed the third, and by Neumann’s rule the directions of the interfaces at the common edge must be parallel to the sides of a triangle, taken proportional to T12, T23, T31. If the above-mentioned condition be not satisfied, the triangle is imaginary, and the three fluids cannot rest in contact, the two weaker tensions, even if acting in full concert, being incapable of balancing the strongest. For instance, if T31 > T12 + T23, the second fluid spreads itself indefinitely upon the interface of the first and third fluids.
The experimenters who have dealt with this question, C.G.M. Marangoni, van der Mensbrugghe, Quincke, have all arrived at results inconsistent with the reality of Neumann’s triangle. Thus Marangoni says (Pogg. Annalen, cxliii. p. 348, 1871):—“Die gemeinschaftliche Oberfläche zweier Flüssigkeiten hat eine geringere Oberflächenspannung als die Differenz der Oberflächenspannung der Flüssigkeiten selbst (mit Ausnahme des Quecksilbers).” Three pure bodies (of which one may be air) cannot accordingly remain in contact. If a drop of oil stands in lenticular form upon a surface of water, it is because the water-surface is already contaminated with a greasy film.
On the theoretical side the question is open until we introduce some limitation upon the generality of the functions. By far the simplest supposition open to us is that the functions are the same in all cases, the attractions differing merely by coefficients analogous to densities in the theory of gravitation. This hypothesis was suggested by Laplace, and may conveniently be named after him. It was also tacitly adopted by Young, in connexion with the still more special hypothesis which Young probably had in view, namely that the force in each case was constant within a limited range, the same in all cases, and vanished outside that range.