IS THE LIPPMANN EQUATION APPLICABLE TO SOLID INTERFACE?

In electrochemical surface science now attention is drawn to several unsolved problems crucial for its future development, especially for electrocapillarity of solid-liquid interface. Electrocapillarity and the Lippmann equation have long been of fundamental importance in surface electrochemistry, and continue to play an important role in modern developments of surface technologies. However, quite often there is a problematic opinion that the classic Lippmann equation is suitable only for liquid surfaces, but for solid surfaces other equation should be found. The main problem in electrocapillarity of solid surfaces is the determining the surface stress because Gibbs never used the concept of surface stress, introducing only ‘surface tension’ for a liquid electrode and ‘closely related quantity’ for a solid electrode. Despite of lack of clarity, the attempts to introduce a surface stress into the electrocapillarity of a solid interface are continuing, as well as suggestions of “modernized” forms of the Lippmann equation for solid electrodes.

Our critical analysis shows that such attempts to introduce a thermodynamic definition of surface stress [Shuttleworth, Herring] as well as efforts to find electrocapillary relations for a solid electrode and “to generalize” Lippmann equation [Eriksson, Couchman, Gokhstein, Weissmüller, etc.] contain mathematical defects. It is shown that confusing interpretations of some Gibbs’ concepts encountered in the literature have led to “modifications” of the Lippmann equation based on the critical error in the Gibbs–Duhem relation due to the occurrence of an extensive variable, which is inadmissible.

Now we propose the simplest derivation of the Lippmann equation as a complete Legendre transform of Gibbs fundamental equation applicable to liquid or solid interface.