  Modeling microstructure evolution in electrochemical systems is vital for
understanding the mechanism of various electrochemical processes. In this work,
we propose a general phase field framework that is fully variational and thus
guarantees that the energy decreases upon evolution in an isothermal system.
The bulk and interface free energies are decoupled using a grand potential
formulation to enhance numerical efficiency. The variational definition of the
overpotential is used, and the reaction kinetics is incorporated into the
evolution equation for the phase field to correctly capture capillary effects
and eliminate additional model parameter calibrations. A higher-order kinetic
correction is derived to accurately reproduce general reaction models such as
the Butler-Volmer, Marcus, and Marcus-Hush-Chidsey models. Electrostatic
potentials in the electrode and the electrolyte are considered separately as
independent variables, providing additional freedom to capture the interfacial
potential jump. To handle realistic materials and processing parameters for
practical applications, a driving force extension method is used to enhance the
grid size by three orders of magnitude. Finally, we comprehensively verify our
phase field model using classical electrochemical theory.
