Surfaces, Interfaces, and Changing Shapes in Multilayered Films
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Drops Spherical Drop Consideration of the equilibrium of a spherical drop of radius R with surface free energy y shows that pressure inside the droplet is higher than outside. The difference is given by the well-known Laplace equation: AP = ^ . r MRS BULLETIN/FEBRUARY 1999
(1)
This result can be obtained by equating work done against internal and external pressure during an infinitesimal change of radius with the work of creating a new surface. By considering the mechanical equilibrium (i.e., no net force) of a portion of the same droplet obtained by a planar cut, the surface free energy is seen to be equivalent to a force-per-unitlength pulling in a plane tangent to the surface and perpendicular to the edge (see Figure la). In this way it cancels the force associated with the pressure differ ence between inside and outside. The surface free energy is therefore also referred to as the surface tension. By considering volume-conserving perturbations of the surface, one can show that in equilibrium the sum fCT of the two principal curvatures must be uniform over the entire surface. For a sphere, the principal curvatures are 1/r, and their sum 2/r is constant over the entire surface. Comparison to Equation 1 shows that AP = KTy, a result that is gen erally valid.* *This discussion applies only to fluids. For an isotropic solid sphere, Equation 1 becomes AP = 2//r, where/is the surface stress. The surface tension is the work to create a unit area at constant strain; the surface stress is re lated to the work to strain the surface elastically. For fluids, 1 ' 2 /= y. For the equilibrium shape problems in this article, the relevant quantity is still the interface tension, even for solids if there is sufficient plastic deformation (usually diffusional) to permit creation and elimination of interfaces at constant strain. If the solid particles are larger than about 10 nm, the bulk values of the interfacial tensions can be used. For smaller particles, the strains induced by the interface stress may shift the value of the interfacial tension slightly.
Consider a junction (Figure lb) of three interfaces with interface free energies that are independent of orientation. The equilibrium configuration, found by requiring that a Virtual displacement of the triple junction causes no change in the total interface free energy, corresponds to that for which the net force on the junction from the tensions of the in terfaces is zero.
Liquid Drop on a Solid Surface In the absence of gravity, a liquid drop on a solid surface has the shape of a spherical cap (constant curvature) and contacts the surface at an angle deter mined by the free energies of the three interfaces. At a fixed volume of liquid, the radius of the spherical cap and the height of the center of the corresponding sphere relative to the surface are uniquely determined. If the solid surface is rigid and planar (Figure lc), the contact angle is d e t e r m i n e d by b a l a n c i n g just the forces parallel to the surface. (Note: This
Figure 1. (a) Surface tension
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