Finite element simulations of the electrochemical double layer structure under microscopic probes of various geometries

Abstract:
A widespread technology for analysing solid surfaces is atomic force microscopy, whose probes can operate in aqueous environment. In electrolytes, applied potential differences lead to a potential-specific double layer structure. We investigate the impact of probe geometry on the ionic charge distributions within a confined electrolyte. We obtain the equilibrium distributions of cations and anions by solving the static Poisson-Nernst-Planck equations. To illuminate the relationship between probe geometry and ion distributions, we investigate three specific geometries: a paraboloid probe and an extruded parabolic probe above a planar electrode, and a reference system of two opposing planar electrodes without any protruding probe geometry. Within those three geometries, we solve Poisson‘s equation coupled to the Nernst-Planck equations capturing drift and diffusion of an arbitrary number of ion species by means of the classical Finite Element Method (FEM). Our calculations show that the systems of parabolic or paraboloid probe geometry assume a near-electroneutral plateau between substrate and probe apex for our choice of parameters. This agrees well with expectations based on the classical double layer model at a plate capacitor. Parabola and paraboloid show a potential offset between their respective plateaus. The higher the reference concentration the broader the electroneutral plateau. Reducing this concentration, we observe the point of geometry-induced divergence from one-dimensional behaviour to move towards the planar electrode. By finding geometry-induced differences in behaviour, we have reached a better understanding of how the shape of probe tips can manipulate the electrochemical double layer structure between substrate and probe in atomic force microscopy systems. Within the investigated parameter space, the geometry of the probe tip has a non-negligible influence. When comparing a rotational symmetry with an axial symmetry, the potential difference between an electroneutral point and the substrate is lower for the former case.