Computing Versus Assuming the EEDF
With the fluid model that COMSOL employs, the electron energy distribution function (EEDF) can either be an assumed function (such as Maxwellian or Druyvesteyn), or it can be computed using a two-term approximation of the Boltzmann equation. When choosing to solve the Boltzmann equation in the two-term approximation (BETT) in a space dependent or global model the BETT is solved fully coupled with the remaining equations of the plasma model. This means that for each time step, the BETT is solved with the local mole fraction of the different species (and the local ionization degree and electron density if e-e collisions are included) obtained from the fluid model equations. The EEDF obtained from the solution of the BETT is used to compute macroscopic rate constants and transport coefficients intervening in the model equations thus closing the loop.
The coupling of the model equations with the BETT can be made in two ways:
•
If the Local field approximation is used (and the model is a global model), the reduced field for which the BETT is solved must be given by the user. If the model is space dependent the reduced field is obtained from the solution of Poisson’s equation.
•
If the Local energy approximation is used, the reduced field is computed such that the mean electron energy (obtained by averaging the EEDF) is equal to the mean electron energy obtained from Equation 6-40.
When the BETT is solved, an extra dimension is attached to the reactor geometry (also referred here as base geometry). The extra dimension represents the electron energy coordinate and the BETT is solved in this space. It is important to note that there are as many BETT being solved as elements in the base geometry (if there are 1,000 elements in the base geometry, there are 1,000 BETT being solved). This makes this option very computationally expensive.
Since the recommended number of elements in the extra dimension should be larger than 50 the number of degrees of freedom of the product space (defined by the base geometry and the extra dimension) and the memory requirements increases considerable when the BETT is solved.
In summary, when computing the EEDF:
•
Computational time increases considerably. A typical ICP model using an analytic EEDF that normally takes 1 minute will take 24 hours when solving for the EEDF.
•
The memory requirements increase considerably. For 2D models, 64GB+ of memory is recommended.
•
The problem formulation is highly nonlinear, which is quite a challenge for the solver. Convergence might be very difficult for a given setup, or not even possible, depending on the exact geometry and operating conditions.
The following approaches are considered good practice when solving the BETT coupled with a space dependent or global model:
•
The model should be first setup using an analytic EEDF.
•
It is advised to do preliminary studies using the Boltzmann Equation, Two-Term Approximation interface to understand how the computed and analytic EEDFs differ and how they affect the quantities of interest.
•
If the model is space dependent start with a 1D model or a global model if possible.
•
An EEDF Initialization study is needed to provide initial conditions for the fully coupled problem. With this study only the BETT is solved. Therefore, this study can also be used to do preliminary analyzes. However, keep in mind that there are as many BETT being solved as elements in the base geometry, which implies that even the initialization study can take 10’s of minutes in 2D models.
In conclusion, special care must be taken when computing the EEDF, as it makes the problem much more non-linear and convergence is much more difficult, and often impossible. Computing the EEDF should only be used as a final “verification” stage, when the parameter space and operating conditions have been extensively studied using an assumed function. Due to the overwhelming computational demand, only limited technical support can be provided in these advanced models.
•
DC Glow Discharge Coupled with the Two-Term Boltzmann Equation: Application Library path Plasma_Module/Space-Dependent_EEDF_Modeling/positive_column_1d_boltzmann
•
GEC ICP Reactor, Fluid Model Coupled with the Two-Term Boltzmann Equation: Application Library path Plasma_Module/Space-Dependent_EEDF_Modeling/argon_gec_icp_boltzmann