Grand-Canonical Approach to Density Functional Theory of Electrocatalytic Systems: Thermodynamics of Solid-Liquid Interfaces at Constant Ion and Electrode Potentials

Properties of solid-liquid interfaces are of immense importance for electrocatalytic and electrochemical systems but modelling such interfaces at the atomic level presents a serious challenge and approaches beyond standard methodologies are needed. An atomistic computational scheme needs treat at least part of the system quantum mechanically to include adsorption and reactions while the entire system is in thermal equilibrium. The experimentally relevant macroscopic control variables are temperature, electrode potential, choice of the solvent and ions and these need to be explicitly included in the computational model as well; this calls for an thermodynamic ensemble with fixed ion and electrode potentials. In this work a general framework within density functional theory with fixed electron and ion chemical potentials in the grand canonical ensemble is established for modelling electrocatalytic and electrochemical interfaces. Starting from a fully quantum mechanical description of nuclei and electrons, a systematic coarse-graining is employed to establish various computational schemes including i) the combination of classical and electronic density functional theories within the grand canonical ensemble and ii) on the simplest level a chemically and physically sound way to obtain the (modified) Poisson-Boltzmann (mPB) implicit solvent model. The detailed and rigorous derivation clearly establishes which approximations are needed for coarse-graining as well as highlights which details and interactions are omitted in vein of computational feasibility. The transparent approximations also allow removing some the constraints and coarse-graining if needed. We implement various mPB models in the GPAW code and test their capabilities to model capacitance of electrochemical interfaces as well as study different approaches for modelling partly periodic charged systems. Our rigorous and well-defined DFT coarse-graining scheme to continuum electrolytes highlights the inadequacy of current linear dielectric models for treating properties of the electrochemical interface.