When an external electric field is applied to a poly crystal, a current is induced, but charge will also pile up on grain boundaries. Here, we show how GOPF can be used to calcualte the charge densities and the resulting current profile as well as the effective macroscopic conductivity. In this example, we consider a material with conductivity tensor

The induced potential Φ arising from induced charges is given by the Poisson equation

where ρ is the charge density and ε₀ is the vacuum permittivity. The fourier transformed potential is therefore given by

By using the relation E = -∇ Φ, the total electric field is given by

where E_ext is the external electric field. From Ohm's law j_i = σ_ijE_j, and charge conservation, the final equation that needs to be solve is

We solved the equation above in a system consisting of several grains that are rotated with respect to each other. The initial charge density was zero everywhere.

The effective macroscopic conductivity can be found by averaging the current acros the simulation cell. j_avg = σ_effE_ext. By carrying the averaging we find that σ_eff = 2.6.