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Ion Momentum and Drift Diffusion Approximation in VizGlow

February 7th, 2018

When setting up a new non-equilibrium plasma simulation in VizGlow, the user must select an ion momentum model. Two available options in VizGlow are “Solve ion momentum equation” and “Use drift-diffusion approximation.” This discussion is intended to help a new user understand the differences between these options and to determine which selection is appropriate for their problem.

The drift-diffusion relationships are derived from the species momentum equations and characterizes the spatial transport of particles. Drift refers to a velocity induced by the presence of an electric field. Diffusion refers to velocity induced by a density or concentration gradient.  Both drift and diffusion are collisional in nature, e.g. both are governed by an average motion due to a driving electric field or density gradient force along with numerous collisions with a dominant background species.

The first option, “Solve ion momentum equation,” solves a separate species momentum equation for each ion, while using the drift-diffusion approximation for neutral species and electrons. The second option “Use drift-diffusion approximation” uses the simplified drift-diffusion approximation for all ions, neutral species, and electrons.

We first consider the species momentum equation:

The left side describes the time dependent and convective accelerations. The right side describes the forces acting on a particle that includes electric field, pressure gradient, and particle collisions, respectively.

The drift-diffusion approximation is a simplification of the momentum equation which assumes that collisional processes (the terms on the right-hand side) are dominant.  It can be derived from the momentum equation as follows.  First, ion inertial effects are neglected.  Second, assume that time variations are slow such that   term can be neglected.  Pressure can be expressed using the ideal gas equation of state.  If time variations are slow, then temperatures will have time to equilibrate with their surroundings and T can be assumed to be constant (isothermal).

With these simplifications, the momentum equation becomes

This results in the following relationship such that drift-diffusion species flux  is function of electric field strength and density gradient.

Where

Note that the background collision frequency appears in both the drift and diffusion variables which confirms that they are collisional dependent processes.

Because this equation is an algebraic function rather than a PDE, it is far less computationally expensive to solve (and also tends to be slightly more stable in practice from a numerical standpoint).

The need for solving the full ion momentum equation for ions arises in low pressure discharge (P < 100’s mTorr) or in small-scale micro-discharges despite the high pressures. In these situations, the number of collisions is reduced and ion inertial effects between collisions should be considered.

The drift-diffusion approximation is applied to both electrons and neutral species for either of the above options. The neutrals are not influenced by electric fields therefore the drift-diffusion approximation remains valid. The approximation is valid for electrons due to their very small mass and large thermal energies, even in strong electric field regions of a discharge.

Selections for a few typical non-equilibrium plasma models:

  • Low pressure CCP (P < 100 mTorr)  –  Solve ion momentum
  • High pressure CCP (P > 100 mTorr)  –  Drift-diffusion approximation
  • ICP coupled plasmas  –  Drift-diffusion approximation
  • Atmospheric plasma jet  –  Drift-diffusion approximation
  • High pressure plasma streamer  –  Drift-diffusion approximation
  • Low pressure magnetron reactor  –  Solve ion momentum
  • Dielectric barrier discharge  –  Drift-diffusion approximate
  • Micro discharge  –  Solve ion momentum

The important factor to consider is the influence of ion inertial effects and collisions to the particle momentum. If you have any further questions, please contact us at info@esgeetech.com.