A Hydrodynamic Mechanism for the Coalescence of liquid Drops. Theory of Coalescence at a Planar Interface SIR: In a recent issue, Lang and Wilke (1971) have described some theoretical and experimental work on the coalescence of liquid drops a t a planar interface. A cursory glance a t 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 equation g1 COS
el
= g2 COS
e2 + u3 COS e3
which 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 is the surface tension and a is the radius of curvature of the interface. In the water-air system, @ varied
+
430
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
Figure 1
Because the thickness of the duplex film is about two to three orders of magnitude greater than molecular dimensions, it is necessary to take the curvature energy into account. This has been done (Scholberg, 1971) with the definition of a new energy term, 4, which has the dimensions ergs per radian, and which, by analogy with the strength of materials, has been called the flexural rigidity. This is in accord with the terminology “surface tension” for the area dependent energy. Literature Cited
Gibbs, J. W., “Collected Works,” Vol. 1, p 227, Longmans, Green and Co., New York, K. Y., 1931. Lane. S. B.. Wilke. C. R.,IND.ENG. CHEM..FUNDAM. 10. 32-9 (1971 j.
Scholberg, H. M., Surface Sci. 28, 229 (1971).
H . M . Scholberg Comstock & Wescott, Inc. Cambridge, X a s s . 02138
from 0.811 X to 4.11 X 10-3 erg. Thus for any measurable contribution of “curvature energy” to surface energy, the radius of curvature would have to be less than about 0.05 cm, a value much smaller than the radii of the drops studied in our experimental work (Lang and Wilke, 1971b). Further, he states that the inclusion of the “curvature energy” term implies that the two sides of the phase-2 film influence one another. We believe that such interactions on a molecular scale should be described in molecular terms as well as by phenomenonological models. Our analysis of possible molecular interactions did not suggest any “curvature energy” effects. In conclusion, we find no grounds for questioning the physical basis of our coalescence model. Literature Cited
Chappelear, D. C., J.Colloid Sci. 16, 186 (1961). Lane, S. B., Wilke, C. R., IND.ENG.CHEM.,FUNDAM. 10, 329 (137la).
’
Lang, S. B., Wilke, C. R., IND.ENG.CHEM.,FUNDAM. 10, 341 (1971b).
Princen, H. M., J . Colloid Sci. 18, 178 (1963). Scholberg, H. M., Surface Sci. 28, 229 (1971). Sidney B . Lang Department of Chemical Engineering McGill University Montreal 110, Quebec, Canada Department of Chemical Engineering University of California Berkeley, Calif. 94720
C. R. Wilke*
