The thermodynamical stability of the heterogeneous system with a spherical interface

A one component system, a liquid drop in equilibrium with its vapor, is studied in the discussion of the stability of a heterogeneous system with a spherical interface. Due to the complexity of the second order functional differentiation of the thermodynamical potential function, a simpler treatment with the use of surface thermodynamics is provided instead. The Gibbs free energy and the Grand potential are found to be at a saddle point if respective natural thermodynamical variables are held constant. The Helmholtz free energy for this system at constant T, V, and N is either at a saddle point or a minimum depending on the relative ratio of the two phases. This is in agreement with a previous result. The method developed here can be used in a homogeneous nucleation to study the initial fluctuation which results in the formation of the critical nucleus. It can also be generalized to study the stability of the multicomponent system such as an emulsion. The use of the surface thermodynamics to study a kinetic process is discussed briefly.