Thermodynamics of electrolyte solutions near charged surfaces: Constant surface charge vs constant surface potential

Electric double layers are ubiquitous in science and engineering and are of current interest, owing to their applications in the stabilization of colloidal suspensions and as supercapacitors. While the structure and properties of electric double layers in electrolyte solutions near a charged surface are well characterized, there are subtleties in calculating thermodynamic properties from the free energy of a system with charged surfaces. These subtleties arise from the difference in the free energy between systems with constant surface charge and constant surface potential. In this work, we present a systematic, pedagogical framework to properly account for the different specifications on charged bodies in electrolyte solutions. Our approach is fully variational-that is, all free energies, boundary conditions, relevant electrostatic equations, and thermodynamic quantities are systematically derived using variational principles of thermodynamics. We illustrate our approach by considering a simple electrolyte solution between two charged surfaces using the Poisson-Boltzmann theory. Our results highlight the importance of using the proper thermodynamic potential and provide a general framework for calculating thermodynamic properties of electrolyte solutions near charged surfaces. Specifically, we present the calculation of the pressure and the surface tension between two charged surfaces for different boundary conditions, including mixed boundary conditions.