Bias Response of Electrode Surfaces and Surface Stress―First-Principles Analysis with the Effective Screening Medium Method―

The bias-voltage dependence of surface stress on metal electrodes is a long-standing observable that bridges electrochemistry and surface mechanics, yet its first-principles modelling has long been hampered by the difficulty of defining a constant–electron-chemical-potential ensemble under three-dimensional periodic boundary conditions. In this article we revisit our recent work [Hagiwara, Ishibashi, and Otani, Phys. Rev. B 110, 155409 (2024)] from a thermodynamic and conceptual perspective. We first clarify why the macroscopic stress tensor is a derivative of the Helmholtz free energy and why Nielsen–Martin's strain Hellmann–Feynman construction survives when an effective-screening-medium (ESM) boundary condition is applied. We then show how the grand-potential view naturally reproduces the Lippmann and Gokhshtein relations from first principles, and discuss the layer-resolved local stress that reveals a Friedel-oscillation-correlated response several layers deep. We close with prospects toward solid–liquid interfaces, where electric double layers, specific adsorption, and dynamic effects remain open challenges.