Defined by: \[\mu _{i}^{\sigma }=(\frac{\partial A^{\sigma }}{\partial n_{i}^{\sigma }})_{T,A_{\rm{S}},n_{j}^{\sigma }}=(\frac{\partial G^{\sigma }}{\partial n_{i}^{\sigma }})_{T,p,\gamma ,n_{j}^{\sigma }}\] \[\mu _{i}^{\rm{S}}=(\frac{\partial A^{\rm{S}}}{\partial n_{i}^{\rm{S}}})_{T,V^{\rm{S}},A_{\rm{S}},n_{j}^{\rm{S}}}=(\frac{\partial G^{\rm{S}}}{\partial n_{i}^{\rm{S}}})_{T,p,\gamma ,n_{j}^{\rm{S}}}\] where \(A^{\sigma }\) is the surface excess Helmholtz energy, \(G^{\sigma }\) is the surface excess Gibbs energy, \(A^{\rm{S}}\) is the interfacial Helmholtz energy, \(G^{\rm{S}}\) is the interfacial Gibbs energy, and \(A_{\rm{S}}\) is the surface area. The quantities thus defined can be shown to be identical, and the conditions of equilibrium of component \(i\) in the system to be \[\mu _{i}^{\alpha }=\mu _{i}^{\sigma }=\mu _{i}^{\rm{S}}=\mu _{i}^{\beta }\] where \(\mu _{i}^{\alpha }\) and \(\mu _{i}^{\beta }\) are the chemical potentials of \(i\) in the bulk phases $\ce{\upalpha\!}$ and $\ce{\upbeta\!}$. (\(\mu _{i}^{\alpha }\) or \(\mu _{i}^{\beta }\) have to be omitted from this equlibrium condition if component \(i\) is not present in the respective bulk phase.) The surface chemical potentials are related to the Gibbs energy functions by the equations \[G^{\sigma }=\sum _{\begin{array}{c} i \end{array}}n_{i}^{\sigma }\ \mu _{i}^{\sigma }\] \[G^{\rm{S}}=\sum _{\begin{array}{c} i \end{array}}n_{i}^{\rm{S}}\ \mu _{i}^{\rm{S}}\]