Plasma electron kinetics

Modelling plasma chemical systems is a complex task, because these system are far from thennodynamical equilibrium. A complete model includes the external electric circuit, the various physical volume and surface reactions, the space charges and the internal electric fields, the electron kinetics, the homogeneous chemical reactions in the plasma volume as well as the heterogeneous reactions at the walls or electrodes. These reactions are initiated primarily by the electrons. In most cases, plasma chemical reactors work with a flowing gas so that the flow conditions, laminar or turbulent, must be taken into account. As discussed before, the electron gas is not in thennodynamic equilibrium... 

Various plasma diagnostic techniques have been used to study the SiH discharges and results have helped in the understanding of the growth kinetics. These processes can be categorized as r-f discharge electron kinetics, plasma chemistry including transport, and surface deposition kinetics. 

Cordierite-supported perovskites, preparation impregnation, 36 251-253 plasma spraying, 36 253 Core electron kinetic energy of, 34 211 spectra of, 34 210... 

Figure 3. Electron mean free path as a function of electron kinetic energy through organic materials. Key V, PMMA , poly(p-xylene) O, plasma polymerized fluorocarbon , barium stearate 0, cadmium arachidate and A, carbon. (Reproduced, with permission from Ref. 35.)...

As noted above, solving the Boltzmann equation is problematic because of the multidimensionality of the problem. A promising approach to calculating the electron distribution function in low pressure plasmas is the so-called non-local approach to electron kinetics. This was proposed by Bernstein and Holstein [56] and popularized by Tsendin [57], who initially suggested this approach for the positive column of a DC discharge. Since then, the non-local approach has been applied to a variety of low pressure gas discharge systems [58]. 

III. Electron Kinetics in Time- and Space-Independent Plasmas. 32... 

Furthermore, the detailed procedure ultimately used to determine the velocity distribution sensitively depends on the type of plasma and is quite different when studying the electron kinetics in steady-state, time-dependent, or space-dependent plasmas. 

If the electric current and thus the density of electrons and excited atoms and molecules grows in the plasma, electron collisions with excited atoms and molecules and the Coulomb interaction between the electrons become increasingly important and have to be included in the kinetic study of the electron behavior. 

The kinetic equation is very complex and covers a tremendous number of special electron kinetic problems. Consequently, there does not seem to be any chance of finding some kind of general solution of this equation that can later be adapted to the specific plasma conditions of interest. 

Equations (12), simplified by the assumption of isotropic scattering in exciting and dissociating collisions, represent the basic equations for studying many quite different problems in electron kinetics. In particular, the additional simplification to steady-state, purely time-dependent, or purely space-dependent plasma conditions allows a detailed microphysical analysis of various electron kinetic problems related to each of these plasma conditions. 

To illustrate the behavior of electron kinetic quantities in steady-state conditions, weakly ionized plasmas in neon and molecular nitrogen are considered as typical representatives of atomic and molecular gas plasmas. 

The essential differences between these plasmas with respect to the electron kinetics are the energy regions where the electron collision processes in each gas... 

With respect to the application of electron kinetic quantities in an extended quantitative plasma description, the mean collision frequencies v (or the corresponding rate coefficients kf) related to the various inelastic electron collision processes are of particular importance. According to Eq. (29), these mean collision frequencies are determined by the isotropic distribution. The evolution of the various mean collision frequencies with growing field strength is presented in Fig. 6 for the neon (left) and nitrogen (right) plasmas. 

In particular, the addition of one or some molecular components to an atomic plasma presents the most sensitive case with respect to the change of the electron kinetics. For example, addition of only a few percent of molecular gases can cause about half the power input from the electric field to be dissipated by electron collisions into the molecular admixture components. 

If, in a plasma with a single gas component instead of a mixture plasma of different gases, collision processes with excited atoms or molecules of the same gas are additionally taken into account, each kind of excited particle has to be treated as a mixture component in the frame of the electron kinetics. Thus, the same equations, Eqs. (40) and (41), are the basis for the study of the electron kinetics influenced additionally by elastic and conservative inelastic electron collisions with excited particles of the same gas. 

According to the relevant power and momentum balance, Eqs. (38) and (39), the electron kinetics in steady-state plasmas is characterized by tbe conditions that at any instant the power and the momentum input from the electric field are dissipated by elastic and inelastic electron collisions into the translational and internal energy of the gas particles. This instantaneous complete compensation of the respective gain from the field and the loss in collisions usually does not occur in time-dependent plasmas, and often the collisional dissipation follows with a more or less large delay—for example, the temporally varying action of a time-dependent field. Thus, the temporal response of the electrons to certain disturbances in the initial value of their velocity distribution or to rapid changes of the electric field becomes more complicated, and the study of kinetic problems related to time-dependent plasmas naturally becomes more complex and sophisticated. Despite this extended interplay between the action of the binary electron collisions and the action of the electric field, the electron kinetics in time-... 

In particular, by this approach, the kinetics of the electrons in moving and standing striations, occurring at low discharge currents in dc column plasmas, can be investigated (Sigeneger e/ al., 1998). Under the action of the highly modulated, spatially periodic electric field of the striations, an undamped, spatially periodic evolution of all electron kinetic properties is established. 

A quite different situation with respect to the spatial behavior of the electron component arises if the inhomogeneity of the plasma is caused by a plasma confinement. The electron kinetics established in the radial direction of the positive column of dc glow discharges, which usually operates in a cylindrical discharge tube with an isolated wall, presents a representative example (Kortsha-gen, 1995 Uhrlandt and Winkler, 1996 Pfau et al, 1996 Alves et al, 1997) of such a condition. From the point of the electron kinetics, this space-dependent problem is somewhat more complex than those considered above. 

By the preceding representations, an attempt has been made to give, on the basis of the electron Boltzmann equation, an introduction to the kinetic treatment of the electron component in steady-state, time-dependent, and space-dependent plasmas and to illustrate by selected examples the large variety of electron kinetics in anisothermal weakly ionized plasmas. 

In this representation, particular emphasis has been placed on a uniform basis for the electron kinetics under different plasma conditions. The main points in this context concern the consistent treatment of the isotropic and anisotropic contributions to the velocity distribution, of the relations between these contributions and the various macroscopic properties of the electrons (such as transport properties, collisional energy- and momentum-transfer rates and rate coefficients), and of the macroscopic particle, power, and momentum balances. Fmthermore, speeial attention has been paid to presenting the basic equations for the kinetie treatment, briefly explaining their mathematical structure, giving some hints as to appropriate boundary and/or initial conditions, and indicating main aspects of a suitable solution approach. 

Another important point, closely connected with electron kinetics, concerns the self-consistent treatment of electron kinetics, of the particle and/or power balance equations for heavy particles (such as different ions and excited atoms or molecules), and of the Maxwell equations (or a reduced version such as the Poisson equation or appropriate electric circuit equations) to obtain a more complete description of all important plasma components as well as of the internal electric field. This self-consistent treatment is usually tricky and is based on an iterative approach to the solution of the various types of equations involved (Loffhagen and Winkler, 1994 Uhrlandt and Winkler, 1996 Yang and Wu, 1996). To integrate the electron kinetic equation in such an approach adequately, a very effective solution procedure for this equation is of particular importance, although remarkable progress with respect to the speed of computation has been achieved in recent years. 

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