COMSOL 6.4 - The Plasma, Time Periodic Interface

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

	Except where described in this section, this physics interface shares it physics nodes with The Drift Diffusion Interface, The Heavy Species Transport Interface, and The Electrostatics Interface.

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 ptp.

Select a — or . When using the model, the mixture averaged diffusion coefficients are automatically computed based on the data specified for each species.

Select the checkboxes for which transport mechanisms to — , , , , or . The selection changes the number of Model Inputs requiring values on the page. Note the following:

•

For select that the thermodynamic properties of each reaction and species are computed automatically based on the thermodynamic properties of each species.

•

For it computes a more accurate expression for the Maxwell–Stefan diffusivities. Often the additional correction terms are negligible in which case the expressions are much simpler and the time taken to assemble the Jacobian matrix is reduced.

•

For additional terms are included in the definition of the mass flux vector to ensure that the same solution is obtained regardless of the choice of the species which comes from the mass constraint. This option makes the problem more nonlinear and strongly coupled, and is only necessary when the molecular weights of the species differ substantially (such as a mixture of sulfur hexafluoride and hydrogen).

•

For the tensor form of the ion transport properties when a static magnetic field is present is computed. This option only needs to be activated when a strong DC magnetic field exists and the operating pressure is very low (on the order of millitorr). When this option is activated an expression must be provided for the magnetic flux density, which typically another physics interface computes. This is set in the node.

Select the or , or checkboxes as needed.

Select to compute the tensor form of the electron mobility, electron diffusivity, energy mobility and energy diffusivity. This should only be used in cases where a strong DC magnetic field exists. Two quantities must be supplied, both of which are in the Plasma Model node. The DC mobility which is the value of the electron mobility in the absence of a DC magnetic field and the magnetic flux density which would typically be computed by another physics interface.

Select to specify the electron mobility, diffusivity, energy mobility and energy diffusivity in reduced form. The electron transport properties are computed from the reduced transport properties using:

where Nn the neutral number density defined as

where pA is the absolute pressure and T is the neutral gas temperature that are set in the node.

The checkbox adds an additional term to the definition of the electron current due to gradients in the electron diffusivity. If the diffusivity is a constant then including this does not affect the solution. It is only necessary to include this term if the electron diffusivity is a function of the electron temperature, and there are significant gradients in the electron temperature.

	For more information on the electron energy distribution function (EEDF), see the Theory for the Boltzmann Equation, Two-Term Approximation Interface.

If cross-section data is used to define source coefficients in the model then an electron energy distribution function (EEDF) needs to be selected. Select one of these options.

where φ is the mean electron energy (SI unit: eV), ε is the electron energy (SI unit: eV) and Γ is the incomplete gamma function:

•

. Use this option for a generalized distribution function where the EEDF is somewhere between Maxwellian and Druyvesteyn. For this option, specify a power law. This number is typically between 1 and 2. Mathematically, the EEDF takes the form:

•

. If a two-dimensional interpolation function has been added to the model, it can be used for the EEDF. In this case, the x-data should be the electron energy (eV) and the y-data should be the mean electron energy (eV).

	The two-dimensional interpolation function can be computed using a parametric sweep in The Boltzmann Equation, Two-Term Approximation Interface. This allows for modeling of discharges where the EEDF is far from Maxwellian. For step-by-step instructions on how to do this, refer to this blog entry: www.comsol.com/blogs/the-boltzmann-equation-two-term-approximation-interface See Interpolation in the COMSOL Multiphysics Reference Manual.

In all these cases the rate constants in the model are automatically computed based on the selected EEDF using the formula:

The rate coefficients when computed using cross-section data are a highly nonlinear function of the mean electron energy. COMSOL Multiphysics automatically computes the integral in Equation 6-1 and makes the result available for evaluation of the rate coefficient. The variation of the rate coefficient for any particular model can be plotted using <name>.kf_<reaction number>. For example, for reaction number 3 in the Plasma interface, with name plas, the rate coefficient is plotted using plas.kf_3.

Enter the Pxd (SI unit: s) over which the extra dimension should be constructed. The default is 1/(13.56[MHz]), which corresponds to the time for 1 RF period at the classical operating frequency of 13.56 MHz. This corresponds to the time, in seconds, of the lower excitation frequency in the system. Some care must be taken when specifying this value if there are multiple frequencies included in the model, see Using Consistent Source Frequencies and Period Settings for more details.

Enter the over which to discretize the extra dimension. The default value of 40 means that 40 mesh elements will be used to discretize the RF period in the extra dimension. In most cases, 40 should be enough for a single frequency. As for the setting, using more than one frequency typically means the must be increased.

The remaining settings describe in which space the nonelectron should be computed. See Choosing the Correct Space on Which to Solve for the Heavy Species for more details. For the , there are two options available:

•

The option means that all the nonelectron species in the model will be computed in the product space of the base geometry and the extra dimension. This option can generate a significant number of degrees of freedom and is better suited to 1D models.

•

For 2D models, it is usually better to solve for the heavy species in the . This keeps the number of degrees of freedom to a minimum, while sacrificing little in terms of accuracy.

If the is set to , there is an additional option for the . This can be either , in which case the time modulating electric field is used to compute the ion migrative fluxes, or , in which case the ions experience the period averaged electric field. Using the option is very computationally expensive, and is usually only used for testing simple 1D models.

To display this section, click the button (

) and select .

By default the is set to . With this option the gain of electrons due to secondary emission is made through the normal component of the electron flux at the wall. See The Wall Boundary Condition for more details. If the is set to the gain of electrons from secondary emission is distributed uniformly across the domain through the electron rate expression. This source of electrons is computed from

where the number of ionization events per secondary electron emitted is obtained from

where the sum is over all ionization reactions by electron impact, g is the , nk is the target number density, σk is the reaction cross section, and εb is the . In Equation 6-13 the integrand is the period-averaged secondary emission flux from all possible surface reactions, the integral is over all surfaces, and V is the reactor volume. Only contributions from individual surface reaction features are considered to compute Equation 6-13. The model is an extremely simplistic model that tries to capture the nonlocal behavior of ionization events produced by secondary electrons that are accelerated to very high energies. This phenomenon is important at low pressures below the 10 s of millitorr range. See Validity of Fluid Models for more detail on the regime of validity.

To display this section, click the button (

) and select .

The solver can run into difficulties as the species mass fractions approach zero. The checkbox (off by default) adds an additional source term to the rate expression for each species. In the ι field, enter a tuning parameter for the source stabilization. By default, no stabilization is added. However, if working with a model with no secondary electron emission included, it may be necessary to activate this. A value of 1 or higher is recommended, although if the plasma is high pressure (atmospheric) then it can help to lower this number to somewhere in the range 0.25–0.5.

The solver can also run into difficulties as the electron density or electron energy density approach zero. The checkbox (off by default) adds an additional source term to the equation for the electron density and electron energy density. The same values and guidelines as above apply.

To enable this section, click the button (

) and select in the dialog. This section is only available if the in is set to .

This method is equivalent to adding a term to the diffusion coefficient in order to dampen the effect of oscillations by making the system somewhat less dominated by convection or migration. If possible, minimize the use of the inconsistent stabilization method because by using it you no longer solve the original problem. By default, the checkboxes are not selected because this type of stabilization adds artificial diffusion and affects the accuracy of the original problem. However, this option can be used to get a good initial guess for underresolved problems.

If required, select the checkboxes and enter a δid,i. The default value is 0.25.

Select — (the default), or .

The dependent variables (field variables) are the , and , . The name can be changed but the names of fields and dependent variables must be unique within a model. The dependents variables that by default end with _per are only used to map the time periodic solution into a time-dependent solution using the Time Periodic to Time Dependent study.