Capillary Forces Calculated with the Circular Approximation

Capillary Forces Calculated with the Circular Approximation 

Two particles in contact necessarily create a narrow slit around the contact area. If the surfaces are lyophilic with respect to a surrounding vapor, some vapor will condense and form a meniscus [510]. The meniscus causes an attractive force. This is for two reasons [480, 481, 511, 512]. First, the direct action of the surface tension of the liquids around the periphery of the meniscus pulls the particles together. Second, the curved surface of the liquid causes a Laplace pressure, which is negative with respect to the outer pressure. This negative pressure acts over the cross-sectional area of the meniscus and attracts the particles toward each other. 

We start this chapter by discussing the capillary force between a perfectly smooth sphere and a plane. Spheres are the first approximation for real particles. The calculation for a sphere and a plane is instructive and the results are simple. Then, we generalize the treatment to two spheres of arbitrary radius and contact angle. Though spheres are a model for the interaction between particles, some fundamental properties are unique for spheres. For example, the capillary force between perfectly spherical surfaces does not (or only weakly) depend on the vapor pressure. Therefore, we also discus other geometries and in particular the influence of roughness. 

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Capillary Forces Calculated with the Circular Approximation 1137... 

Capillary forces for axial symmetric menisci, which are much smaller than the capillary constant, can be well calculated with the circular approximation. In the circular approximation, the two radii of curvature are approximated by circles. For two spherical particles, the capillary force is given by F = 2jtYl cos . Two spheres are a unique case. Unlike other contact geometries, the capillary force does not (or only weakly) depend on the vapor pressure. In general, the capillary force depends sensitively on the vapor pressure. 

In order to facilitate the calculation of capillary forces, several approximations on the meniscus shape have been proposed. They are mainly applied for experimental conditions where the radius of curvature of the meniscus interface is much smaller than the radius of curvature of the solid surfaces. This is relevant for the surface force apparatus where the surface has centimetric radius, while the meniscus is typically tens of hundreds of nanometers. The most used approximation is the toroidal approximation assuming the liquid interface has a circular profile. Obviously, such a meniscus does not exhibit a constant curvature. Nevertheless, this approximation gave good results, in particular for small contact angles, and is therefore widespread (see Ref. 15 for its application in various geometries and section 9.3.1.1 for an example of its application in atomic force microscopy [AFM]). In the case of capillary condensation between a plane and a sphere with a large radius of curvature R, in contact, the tension term of the capillary force is negligible and the Laplace term leads to the simple formula F = AnRy cos 9 A parabolic profile is also sometimes used to eliminate some numerical difficulties inherent in circle approximation. 

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