Colloquium: Physically based fluid modeling of collisionally dominated low-temperature plasmas

This colloquium examines the theoretical modeling of nonequilibrium low-temperature (tens of thousands of degrees) plasmas, which involves a juxtaposition of three distinct fields: atomic and molecular physics, for the input of scattering cross sections; statistical mechanics, for the kinetic modeling; and electromagnetic theory, for the simultaneous solution of Maxwell’s equations. Cross sections come either from single-scattering beam experiments or, at very low energies $(<0.5\mathrm{eV})$, from multiple-scattering experiments on “swarms” in gases—the free diffusion or large Debye length limit of a plasma, where they are embedded in transport coefficient data. The same Boltzmann kinetic theory that has been developed to a high level of sophistication over the past $50\text{years}$, specifically for the purpose of unfolding these transport data, can be employed for low-temperature plasmas with appropriate modification to allow for self-consistent rather than externally prescribed fields. A full kinetic treatment of low-temperature plasmas is, however, a daunting task and remains at the developmental level. Fortunately, since the accuracy requirements for modeling plasmas are generally much less stringent than for swarms, such a sophisticated phase-space treatment is not always necessary or desirable, and a computationally more efficient but correspondingly less accurate macroscopic theoretical model in configuration space at the fluid level is often considered sufficient. There has been a proliferation of such fluid modeling in recent times and this approach is now routinely used in the design and development of a large variety of plasma technologies, ranging from plasma display panels to plasma etching reactors for microelectronic device fabrication. However, many of these models have been developed empirically with specific applications in mind, and rigor and sophistication vary accordingly. In this colloquium, starting from the governing Boltzmann kinetic equation, a unified, general formulation of fluid equations is given for both ions and electrons in gaseous media with transparent and internally consistent approximations, all benchmarked against established results. Thereby a fluid model is obtained that is appropriate for practical application but at the same time is based on a firmer physical foundation.

Figure 1

(Color online) The steps involved in constructing fluid equations for both swarms and low-temperature plasmas, starting from the common Boltzmann equation for each of the ion and electron species, and proceeding through a common set of approximations for collision terms and closure. The symbol $S$ denotes the swarm limit (low charge density, external fields, linear equations). Swarm fluid equations follow either directly as a limiting case of the plasma fluid equations or indirectly as moment equations of the swarm limit of Boltzmann’s equation. In this picture, the plasma fluid equations are thus consistent with the numerous established results of swarm analysis (see Sec. 5D). In addition, it is legitimate in this scheme to employ measured or theoretically calculated swarm transport coefficients to replace approximate terms in either the swarm or plasma fluid equations, thereby considerably enhancing computational accuracy (see Sec. 5C).

Figure 2

The typical empirical fluid model of a low-temperature plasma suffers from the following defects: (i) any logical connection with exact moment equations derived from the Boltzmann equation is unclear; (ii) approximations for collision terms, and the closure hypothesis involving the heat flux, are not properly justified; and (iii) any commonality with swarm analysis and results is thus severed. Such an empirical model can therefore, in general, reproduce neither the swarm fluid equations nor the many well-known benchmark results and formulas established in the literature over many years (Sec. 5D). Furthermore, after having broken the nexus, the implementation of swarm transport coefficients in the empirical fluid equation model loses its physical foundation and can lead to unforeseen errors (see Sec. 5C).

Figure 3

An idealized model of the steady-state Townsend experiment, in which electrons emitted at a constant rate from an infinite plane source drift and diffuse in a uniform gas under the influence of uniform external fields.