The Boltzmann Equation, Two-Term Approximation Interface

The interface (

), found under the branch (

) when adding a physics interface, studies the electron energy distribution function (EEDF) and calculates it from a set of collision cross sections for some mean discharge conditions. Electron source coefficients and transport properties are also computed. The physics interface is unique because the geometry and mesh are hidden during setup of the model. The Boltzmann Equation, Two-Term Approximation interface can be used as a preprocessing step before solving space-dependent models. This physics interface is available for 0D components.

When this physics interface is added, these default nodes are also added to the : and . Then, from the toolbar, add other nodes that implement. You can also right-click to select physics features from the context menu.

	The Reduced Electric Fields and Mean Energies studies are available for this physics interface and described in the COMSOL Multiphysics Reference Manual.

The is the default physics interface name.

The is used primarily as a scope prefix for variables defined by the physics interface. Refer to such physics interface variables in expressions using the pattern <name>.<variable_name>. In order to distinguish between variables belonging to different physics interfaces, the name string must be unique. Only letters, numbers, and underscores (_) are permitted in the field. The first character must be a letter.

The default (for the first physics interface in the model) is be.

Select an (EEDF) — (the default), , , , or .

and

is the mean electron energy (eV), ε is the electron energy (eV) and Γ is the upper incomplete gamma function.

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Select for a generalized distribution function where the EEDF is somewhere between Maxwellian and Druyvesteyn. For enter a value for the g (dimensionless). The default is 1, and this number should be between 1 and 2. Mathematically, the EEDF takes the form:

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Select , or to solve a two-term approximation to the Boltzmann equation. When selected, the EEDF is computed from a partial differential equation instead of by taking an assumed function. The two-term Boltzmann equation is a complicated, nonlocal integral equation.

These following options are available when , or is selected as the . These settings also enable available settings on the window for feature.

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Select (off by default) if the ionization degree of the discharge is high. Electron-electron collisions tend to make the distribution function more Maxwellian and they also increase the complexity of the problem.

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Select (on by default) to describe how the energy is split between two electrons when an ionization collision occurs. If selected, then both electrons take an equal energy after the collision. If not selected, the secondary electron created in an ionization collision has zero energy and the ionizing electron carries all the excess energy.

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Select if the reduced angular frequency of the discharge is high, which is typically only true for microwave discharges. The reduced angular frequency is the ratio of the angular frequency and the neutral number density, ω/N.

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If the property is active, enter a ω/N (SI unit: m3/s). The default is 10−13 m3/s. If the reduced angular frequency is high, the proportion of electrons with high energies is substantially increased for the same mean electron energy. This is because in DC fields, collisional momentum transfer impedes electrons acquiring higher energies but high frequency fields have the opposite effect.

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Enter the N (SI unit: dimensionless) to specify the number of mesh elements to use to discretize the underlying energy space. The default is 100, but models with complex gas mixtures may require more.

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Enter the R (SI unit: dimensionless) to specify the rate at which the mesh coarsens away from the zero energy coordinate. The default is 10, and this doesn’t usually need to be changed. Higher values mean that the mesh will be finer closer to the zero energy, and coarser at higher energies.

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Select to have the software automatically compute the maximum energy coordinate based on certain assumptions about the EEDF. This is a powerful feature, in that the maximum energy coordinate does not need to be specified, but it does make the problem more nonlinear and thus difficult to solve.

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Enter the

(SI unit: V) to specify the maximum coordinate in energy space on which we are computing the EEDF. When computing the EEDF at high mean energies, this value may need increasing from its default value of 100 V. This option is only available if is not selected.

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Enter the

(SI unit: dimensionless) to specify how much the maximum energy coordinate should be scaled. This option is only available if is selected and is only needed for the study type.

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Enter the

(SI unit: dimensionless) to specify the minimum value that the EEDF should take at the maximum energy coordinate. In order to avoid divide by zero problems, the value should be small and positive. The default value of 1E-15 rarely needs to be changed.

The dependent variable (field variable) is the . The name can be changed but the names of fields and dependent variables must be unique within a model.