Plasma Sheath Lower than Expected

[gregstath99]

Hi

I'm trying to run a simulation where I observe the formation and stabilisation of a Debye plasma sheath. I am using the electrostatic solver and I have set the x and y boundaries to be periodic but the z boundaries to be Dirichlet such that they model 2 perfectly conducting plates. This simulation is being done as a validation that my code works so there is a specific sheath potential I expect to observe. At first I observe this potential but then after some time the sheath potential drastically drops. For context this is a proton/electron plasma. What I have noticed is when looking at the ion and electron densities across the domain (zmin to zmax) what happens is after some time the charge densities perfectly balance out but for a sheath to form and be sustained there needs to be a charge imbalance at both of the z ends where the proton density is higher than the electron density and then the rest of the plasma is quasineutral. This would lead to the sheath being much lower than expected but I'm trying to figure out why the simulation behaves like this and how to change it. My first thought is that I need to use a non linear electrostatic solver but it could also be something related to the boundary conditions. Has this issue occured before and are there any potential suggestions?

Thank You

[gregstath99]

Hi

I think it is because of the Dirichlet boundaries being modelled as PEC boundaries which mirror and reflect charge thus suppressing particle densities near the boundary, is there any way to set up the boundaries in a way that they are absorbing with a fixed potential but don't mirror charge, perhaps using embedded boundaries might do this?

[roelof-groenewald]

Hi @gregstath99. Indeed we have seen similar effects when the cell-size is set such that the Debye length is not very well resolved. Have you tried running the simulation with a smaller cell size?

As you noted, using an embedded boundary rather than the domain boundary would avoid the charge reflection. You can also specify the potential you want on an embedded boundary. That would be a great test to confirm that the decrease in sheath height you observe is coming from the charge reflection. Let us know if you need any guidance setting up the case with embedded boundaries.

[gregstath99]

Hi

So I don't think it's the cell size since 1) I've made sure it's less than the Debye length, 2) I've ensured the timestep is smaller than the inverse plasma frequency and 3) I've ensured the CFL condition is met. I added an embedded boundary at each z end by using a quadratic function and I've got that simulation running for the night so I'll see the result tomorrow morning. Fingers crossed 🤞

[gregstath99]

So I realised I had got the positive and negative parts of the boundary the wrong way round so I added the following quadratic function: "z^2 - 0.008z + 0.00000079" with solutions (z-0.0001)(z-0.0079) (domain length is 8mm) so that the solution is negative between z=0.0001 and z=0.0079 but when I now ran it I got this error: "Domain is completely covered ! ! !", what am I doing wrong?

[roelof-groenewald]

The parser is sometimes a bit finicky with spaces. Could you try using z*z-0.008z+0.00000079 (without the spaces)?

Also, what particle shape are you using? I'm finding if I use particle shape 1, the sheath behaves as expected.

[gregstath99]

The particle shape I use is "linear" so I assume that's the same as 1, the same thing happened again but maybe it's because I still had PEC boundaries just behind the embedded boundaries so I'm running the same simulation again with Neumann boundaries at either end instead of Dirichlet.

[gregstath99]

The issue is persisting, I made sure I set the potential to zero inside the embedded boundary as well: eb = picmi.EmbeddedBoundary(implicit_function="zz-0.008z+0.00000079", potential="z-z"). Would you be able to send me an example code where the sheath behaves as expected, maybe with a sheath forming at either end of the domain so I can see where I'm going wrong in my code.