A Hydrodynamic Mechanism for the Coalescence of Liquid Drops. Theory of Coalescence at a Planar Interface

Abstract

SIR: In a recent issue, Lang and Wilke (1971) have described some theoretical and experimental work on the coalescence of liquid drops at a planar interface. A cursory glance at their Figure 1 shows that the model of Figure 2 is oversimplified. The correct model is shown in Figure 1 of this correspondence.For the quasi-equilibrium they assume both horizontal and vertical resolutions of the forces may be made. The horizontal resolution of the Keumann triangle gives the equationwhich shows that u1 is less than the sum of the other two tensions. This means that their eq 1 is incorrect. Therefore, the entire theoretical development is questionable. However, it is questionable not just because of some minor variations in the constant 3 and the function Q(p) in their eq 1, but because the correct model raises some question about the very basis of surface chemistry. That question is-how thick is the interfacial region? Gibbs (1931) assumed that it was of molecular dimensions and that therefore the curvature contribution to the total surface free energy could be set equal to zero. If it can be shown that the thickness of the interfacial region is large, Gibbs' assumption that the curvature energy is zero is no longer tenable. However, because of the equation above, it is evident that the two sides of the duplex film of the liquid drop, or of a foam bubble, influence each other; i.e., the thickness of the surface region is of the same order of magnitude as the thickness of the duplex film.SIR: Scholberg has raised questions concerning three parts of our theoretical development (Lang and M7ilke, 1971a) : (a) the geometry of the drop quasi-equilibrium, (b) the resolution of forces about the drop, and (c) the curvature energy of the phase-2 film.Scholberg has proposed a drop geometry in which the phase-2 film has been omitted so that the drop resembles a floating oil drop a t a water-air interface (a three-phase system). This model cannot represent a liquid drop in a liquidliquid system. The geometry shown in our Figure 2 is based on the idealization that the drop behaves like a solid sphere. More realistic drop shapes have been proposed by Chappelear (1961), Princen (1963), and others. As we discussed, all of these models predict a rate of film thinning proportional to the cube of the film thickness, differing from our eq 3 by small numerical constants. However, only the spherical drop geometry we used permits simple analytic stability analyses.The implied relationship between Scholberg's horizontal force balance and our vertical one is not a t all clear. However, because the phase-2 film was omitted from Scholberg's figure, it is not possible to derive from it a vertical force balance which included the force due to gravity. It is this latter force which causes the phase-2 film to thin and increases the likelihood of drop coalescence.Scholberg has also recommended the inclusion of a "curvature energy" term @, which he calls the 'flexural rigidity." I n his equations (Scholberg, 1971), the term @/a2 + y appears, where y i...