Equilibrium Condition for Curved Interfaces

The equilibrium conditions for systems with curved interfaces [3] are in part identical to those defined earlier for heterogeneous phase equilibria where surface effects where negligible  

Note that the chemical potential of a given component at the interface is equal to that in the two adjacent phases. This is important since this implies that adsorption can be treated as a chemical equilibrium, as we will discuss in Section 6.3. 

To establish the equilibrium conditions for pressure we will consider a movement of the dividing surface between the two phases a and [i. The dividing surface moves a distance d/ along its normal while the entropy, the total volume and the number of moles n, are kept constant. An infinitesimal change in the internal energy is now given by 

At equilibrium (dt/) v n. = 0, which leads to the equilibrium condition for pressure expressed in terms of the two principal curvatures or alternatively in terms of the two principal radii of curvature  

Equation (6.27) is the Laplace equation, or Young-Laplace equation, which defines the equilibrium condition for the pressure difference over a curved surface. In Section 6.2 we will examine the consequences of surface or interface curvature for some important heterogeneous phase equilibria. 

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The purpose of this chapter is to introduce the effect of surfaces and interfaces on the thermodynamics of materials. While interface is a general term used for solid-solid, solid-liquid, liquid-liquid, solid-gas and liquid-gas boundaries, surface is the term normally used for the two latter types of phase boundary. The thermodynamic theory of interfaces between isotropic phases were first formulated by Gibbs [1], The treatment of such systems is based on the definition of an isotropic surface tension, cr, which is an excess surface stress per unit surface area. The Gibbs surface model for fluid surfaces is presented in Section 6.1 along with the derivation of the equilibrium conditions for curved interfaces, the Laplace equation. 

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