Interfaces reducing area/curvature

The interfacial tension gives rise to a force in the interface, directed to reduce the interfacial area. In a spherical droplet of one immiscible fluid in another, the interface is curved. The interfacial tension exerts a force perpendicular to the interface, directed to the concave side of the interface. The size of the force is proportional to the interfacial area, and hence it can be described as a pressure. This pressure is called the Laplace pressure. This pressure is proportional to the curvature of the interface. If the interface is flat (zero curvature), there is no force perpendicular to the interface. A cylinder is curved in one direction (with curvature radius R ) and straight in another, and the resulting pressure is equal to... 

Now, consider again a spherical liquid drop. Because of the curvature of the interface, there is a pressure difference between the inside and outside of the drop. This difference exists because of the interfacial tension, which tends to reduce the area of the liquid system, so that equilibrium is maintained with a higher pressure inside the drop than the atmospheric pressure outside. If the radius of the drop is r, its surface area is 4nr. The incremental work dvrs done in increasing the radius by dr is... 

To further elaborate the underlying idea of a configurational force, we appeal to the examples indicated schematically in fig. 2.8. Fig. 2.8(a) shows an interface within a solid and illustrates that by virtue of interfacial motion the area of the interface can be reduced. If we adopt a model of the interfacial energy in which it is assumed that this energy is isotropic (i.e. y does not depend upon the local interface normal n), the driving force is related simply to the local curvature of that interface. Within the theory of dislocations, we will encounter the notion of image dislocations as a way of guaranteeing that the elastic fields for dislocations in finite... 

Let us now consider fluid interfaces composed of chemical components which are soluble in (and equilibrated with) the adjacent bulk phases. If a new area is created at constant T, x p, H, and D, this does not correspond to any change of the physical state of the interface. In such a case, dtoVda = 0, and from Eq. (101), one realizes that y = to. Then, Eq. (101) reduces to a generalized form of the Gibbs adsorption equation cf. Eq. (1). It is now evident that the bending and torsion moments, B and , are connected with the curvature dependence of the interfacial tension. 

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