Free Energy of Surfaces and Interfaces

The interfacial tension of a stable, two-phase system is always positive, otherwise the two phases would spontaneously mix since they lower their free energy by making more and more interface. Therefore, near the critical point for phase separation, where the two coexisting phases become indistinguishable, one expects the surface tension between the two phases to vanish. The addition of a third interfacially active component to a two-component mixture with a tendency to phase separate can also result in an effectively negative tension (related to the chemical potential of the third component) which can cause the two components to spontaneously form a dispersion with an amount of internal interface related to the amount of the interfacially active component. Such systems are described in Chapter 8. 

Consider a curved interface with different pressures on each side e.g., a liquid in equilibrium with its vapor), as shown in Fig. 2.1. The free energy is 

In equilibrium, the free energy is stationary with respect to variations in the position of the interface we thus require dFjdh = 0. This gives us the general relation for Ap — p — p2  

For solids, the surface tension is anisotropic — it is different for different crystal faces, defined by their normal n y = y h). The equilibrium crystal shape is not a sphere, but is determined by the Wulff construction (for a proof see Ref. 2) 

Consider a system described by an exact Hamiltonian, H. From Chapter 1, we know that the exact probability distribution, P, of the system is given by the Boltzmann factor, P this arises from the minimization of the 

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Figure S.1S. Upper Equilibrium polyhedrons of metal particles on a support for different combinations of free energies of surfaces and interfaces. Lower Transmission electron microscopy of structures obtained by depositing 5-10 ML of Pd on a MgO surface. [Adapted from www.fysik.dtu.dk/CAMP, and C.R. Henry, Surf. Sci. Rep. 21 (1998) l.j...

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