Dependence of surface stress, surface energy and surface tension on potential and charge.

A toy model and simple model functions are used to exemplify the relation between surface tension, surface energy and surface stress given by Shuttleworth's equation. Variations of the surface tension of charged interfaces must obey Lippmann's equation. Variations Deltaf of the surface stress of electrodes would be either equal to or proportional to and smaller than those variations Deltagamma of the surface tension, if the potential of zero charge (pzc) did not depend on the surface strain epsilon. However, since the pzc E(0) is a function of strain epsilon, the basic dependence of the surface stress on the charge, f(q), is described by a sum of three terms: the first one is the surface stress of the uncharged surface. The second one varies linearly with the surface tension, gamma(q), as long as the amount of specific adsorption remains constant, and is quadratic in E and q for a potential-independent double layer capacitance. The third summand that contributes to f(q) is linear in q and is a direct consequence of the potential dependence E(0)(epsilon) of the pzc. This result should help to resolve the seeming discrepancy between previous work on the surface stress changes of electrodes: Most experimental and theoretical results supported either the view that the variations of surface stress are identical or similar to that of surface tension, i.e. have essentially a quadratic dependence on the electrode potential, or that the basic dependence is a linear function of the charge. The more comprehensive model description presented here allows an explanation of the different results and their seeming discrepancies without assuming experimental errors or fundamental thermodynamic problems. Furthermore, an estimate of the potential dependence of the surface modulus can be obtained.