Steady-state voltage distribution in three-dimensional cusp-shaped funnels modeled by PNP.

We study here the bulk electro-diffusion properties of micro- and nanodomains containing a cusp-shaped structure in three-dimensions when the cation concentration dominates over the anions. To determine the consequences on the voltage distribution, we use the steady-state Poisson-Nernst-Planck equation with an integral constraint for the number of charges. A non-homogeneous Neumann boundary condition is imposed on the boundary. We construct an asymptotic approximation for certain surface charge distribution that agree with numerical simulations. Finally, we analyze the consequences of several piecewise constant non-homogeneous surface charge densities, motivated by designing new nanopipettes. To conclude, when electro-neutrality is broken at the scale of hundreds of nanometers, the geometry of cusp-shaped domains influences the voltage profile, specifically inside the cusp structure. The main results are summarized in the form of new three-dimensional electrostatic laws for non-electroneutral electrolytes. These formula provide a refined characterization of voltage distribution at steady-state in neuronal microdomains such as dendritic spines, but can also be used to design nanometric patch-pipettes.