Modified Poisson-Nernst-Planck theory for ion transport in polymeric electrolytes

We present a numerical method to describe the transport of ions through polymeric electrolytes under the influence of an applied electrostatic field. The transport of ions results in an ion concentration profile with a large gradient near charged interfaces, the so-called diffuse layer. We show that the Poisson-Nernst-Planck theory, which generally describes the transport of ions, can be simplified significantly if the dimensions of the bulk material are much larger than the thickness of the diffuse layer. In this case, charge transport through the bulk of the material is driven by the electric field only (Ohm's law). At all interfaces an analytical mathematical relation between the surface charge and the voltage difference over the diffuse layer can be used. Several of these relations are used, including relations that account for ion volume effects. We use experimental data from literature [J.A. van der Pol, R.T.H. Rongen, H.J. Bruggers, Microelectron.Reliab.40 (2000) 1267] of charge accumulation at the surface of an integrated circuit (IC) resulting from the transport of ions through the polymeric material encapsulating the device. The data are very well described by a two-dimensional finite element model based on the simplified Poisson-Nernst-Planck theory. In addition, the theory predicts that the accumulated charge at the circuit interface will slowly drain away after having reached the experimentally observed maximum.