Extremely confused about EQS boundary conditions

[Sancor]

I'm having a tremendously hard time understanding the connection between macro and micro scale electrostatics and how (if?) they're described EQS boundary conditions. I understand that in a medium with mobile ions, an applied current or field will lead to the establishment of an electric double layer with thickness on the order of a debye length, and this explains the micro scale side of EQS and why we see that the boundary conditions are violated since potential is not continuous across a DL.

But does this mean that the boundary condition given by the conservation of charge is not very meaningful when you are trying to consider the micro scale? If we consider the interface between an insulator and a conducting electrolyte, where the system is experiencing a DC potential drop, you necessarily have no electric field in the conducting medium, and your conductivity in the insulator is zero. But this clearly has to be describing the macro scale electric field, correct? Because there is still an field due to the DL at the interface. But it doesn't seem like there's any equations that allow Laplace's equation to describe the field due to the DL.

I'm also really confused about what AC current does when introduced to these systems. It seems like if the characteristic time of the current is much greater than the characteristic time of charge relaxation you can treat it as a DC current, but does this mean the exact same DL will form in your conducting media as when you were using DC? Or does this mean that your DL will form but will switch interfaces over time?

If anyone at all is knowledgeable I would really appreciate any clarification you could offer, I've been scouring the internet for hours on end every day for the past week trying to understand this material and haven't gotten anywhere.

 

[Valerie Park]

The connection between macro and micro scale electrostatics can be described by the electric double layer (EDL). The EDL is a thin layer of charge that forms at the interface between an electrolyte solution and a conducting surface. This layer is comprised of both positive and negative ions and is responsible for the conductivity of the solution. The thickness of this layer is on the order of the Debye length, which is a measure of the strength of the electrostatic interactions in the medium. At the macro scale, the presence of the EDL is accounted for by applying the appropriate boundary conditions to Laplace's equation. In particular, a voltage drop across the EDL is assumed, which leads to a discontinuity in the electrical potential across the interface. This means that the electric field will not be continuous across the EDL, and hence the conservation of charge boundary condition will be violated. At the micro scale, the EDL can be further used to understand how an applied current or field affects the medium. When a DC current is applied, it will cause the EDL to form, leading to an increased conductivity in the electrolyte solution. However, when an AC current is applied, the characteristics of the current must be taken into account. If the characteristic time of the current is much larger than the characteristic time of charge relaxation, then the EDL will form and remain relatively constant over time. On the other hand, if the characteristic time of the current is much smaller than the characteristic time of charge relaxation, then the EDL will switch interfaces over time. In summary, the connection between macro and micro scale electrostatics can be described by the EDL. At the macro scale, the presence of the EDL is accounted for by applying the appropriate boundary conditions to Laplace's equation, while at the micro scale, the EDL can be used to understand how an applied current or field affects the medium. Furthermore, when an AC current is applied, the characteristics of the current must be taken into account in order to understand how the EDL will form and change over time.