Question

I want to analytically determine a simplified potential and electric field distribution around planar electrodes, and an 40nm square electrode in an 50nm insulating well.

I think that the Poisson-Boltzmann equation should be appropriate but I’m unsure about a few things:
how do I account for the presence of the stern layer?
how do I find the electric field distribution when the potential applied is high (e.g. 2V? Where the Debye Huckel approximation cannot be used).

Answer 1

The Poisson-Boltzmann equation is used for analyzing the potential and electric field distribution near electrodes but it becomes complex with the Stern layer and high potentials. The Stern layer can be included through boundary conditions and the non-linear charge distribution at high potentials necessitates numerical solutions.

Step-by-step explanation:

In the context of electrochemical systems and electrodes, when comparing a simplified potential and electric field distribution around planar electrodes, the Poisson-Boltzmann equation can indeed be an appropriate tool. However, it becomes complex when you try to account for the Stern layer, which is a layer of immobilized ions at the electrode's surface. Moreover, when applying a high potential such as 2V, the Debye-Huckel approximation is no longer valid. In such cases, one must use the full Poisson-Boltzmann equation without the linearization that leads to the Debye-Huckel approximation.

To include the effects of the Stern layer, one may need to implement boundary conditions that represent this specific region of non-linear charge accumulation. Advanced electrochemical models can include both the Stern layer and the diffuse layer that extends into the electrolyte. As for calculating the electric field distribution, you would need to solve the Poisson equation numerically, taking into account the non-linear distribution of charge that exists at high applied potentials.