Plasma Reactors Theory
The Plasma interface couples the Drift Diffusion, Heavy Species and Electrostatic interfaces. A brief explanation of the underlying equations is given here.
For detailed information about electron transport see Theory for the Drift Diffusion Interface.
For detailed information about the transport of the nonelectron species see Theory for the Heavy Species Transport Interface.
For detailed information about electrostatics see The Electrostatics Interface.
The electron density and mean electron energy are computed by solving the following set of electron transport equations in the drift-diffusion approximation
If a strong DC magnetic field is present then the electron mobility can be a tensor:
where the inverse of the mobility has been used because the actual expression for the electron mobility cannot be written in a compact form. The quantity μdc is the electron mobility in the absence of a magnetic field. The electron diffusivity, energy mobility, and energy diffusivity are then calculated using:
The electron rate expression is defined as
where νe,j is the stoichiometric coefficient for electrons, and the reaction rate is defined as
where kjf is the forward rate constant and kjr is the reversed rate constant. Both the Electron Impact Reaction feature and Reaction feature can contribute to the electron rate expression. However, when using the Reaction feature it is important to note that the associated electron energy gain or loss is not included in the source term of the electron mean energy equation.
In the plasma community, rate constants are referred as rate coefficients to emphasize the fact that they are rarely constant when defining electron impact reactions, depending strongly on the electron energy.
When Townsend coefficients are used, the reaction rate is defined as
where αj/Nn is the reduced Townsend coefficient for reaction j (SI unit: m2) and Γe is the electron flux as defined above (SI unit: 1/(m2·s)). Townsend coefficients are defined for two-body reactions; thus, in practice, the product operator reduces to the concentration of the target species. Townsend coefficients can increase the stability of the numerical scheme when the electron flux is field driven as is the case with DC discharges.
The total electron energy loss or gained is calculated by summing the collisional energy changes from all reactions defined with the Electron Impact Reaction feature as
where Δεj is the energy loss from reaction j (SI unit: V) and F is the Faraday constant (SI unit: C/mol). For excitation and ionization collisions Δεj corresponds to the energy of the excited state being excited/deexcited or ionized, for attachment Δεj is set to zero, and for elastic collisions
The rate coefficients can be computed from cross-section data by the following integral:
For nonelectron species, the following equation is solved for the mass fraction of each species.
The electrostatic field is computed using the following equation:
The space charge density, ρ is automatically computed based on the plasma chemistry specified in the model using the formula:
.