Flat interface small deformations

The first approach to the understanding of the capillary forces in equilibrium states involves the approximation of small interfacial deformations about a reference flat state. It will be understood under this approximation that both the departures from the flat interface and the spatial derivatives of the departure are as small as necessary to retain only the lowest-order terms in an expansion in those small quantities. This approximation simplifies considerably the analytical calculations and it also seems to describe well many of the experiments performed so far Because of the typically large values of y, considerable large forces already come into play upon tiny interfacial deformations additionally, the geometrical configurations usually considered are simple enough that no large derivatives appear. 

The height of the interface over the reference flat interface is described by a 2D scalar field M(rn) (see Figure 2.4 for the definitions). The force Fj , defined in Equation 2.3 is easily evaluated up to second order in the small-deformation approximation [34] ... 

If the flnid interface is already curved in the absence of particles, the approach just discussed has to be modified to consider small deformations about an unperturbed curved interface. The application of the tools of differential geometry, as discussed in, for example, reference [68], is a promising approach that still remains to be developed. So far, several particular cases have been considered like the interfacial deformations of a spherical droplet by the electric field of a particle in the limit of a large droplet radins [34,37,69-71], or of an unperturbed interface that is itself a slight deformation of a flat interface [40]. In these two cases, the key assumption is a good separation of length scales the typical radius of curvature of the unperturbed, smooth interface is much larger than the typical... 

---