Kinetic Theory of the Wall Sheath for Arbitrary Conditions in a Gas-Discharge Plasma

Abstract

The self-consistent problem of the structure of a perturbed wall sheath in a dc gas-discharge plasma near a flat surface under negative potential relative to the plasma has been solved for an arbitrary relation between the Debye radius and the ion mean free path. The solution has been obtained without artificial separation of this layer into the quasi-neutral “presheath” and the wall sheath in which quasi-neutrality is violated substantially. The actual ion distribution function in the unperturbed plasma, the dependence of the charge-exchange cross section on the ion energy, and the nonzero electric field in the unperturbed plasma have been considered. It is shown that when the average electron energy is conserved the structure of the perturbed wall sheath weakly depends on the form of the electron distribution function. It has been established that the mean energy of ions in the unperturbed plasma substantially affects the structure of the quasi-neutral presheath as well as the structure of a part of the wall sheath in which quasi-neutrality is not observed even under the assumption that the mean electron energy is much higher than the mean energy of ions. The calculations of ion flow parameters and the structure of the perturbed wall sheath are in conformity with experimental data obtained by other authors, which could not be adequately interpreted earlier.

APPENDIX

Let us consider relative ion energy ε = $\frac{{{{E}_{i}}}}{{k{{T}_{e}}}}$. We assume that the dependence of charge-exchange cross section on the ion energy is defined by relation (7). Then, system (6) can be written in form [39]

$$R{\kern 1pt} '(V) = \frac{{{{\kappa }_{2}}K(V)}}{{\sqrt {F(V)} }},$$

$$\begin{gathered} F{\kern 1pt} '(V) = \frac{{2\kappa _{2}^{2}}}{{\kappa _{1}^{2}}}\{ \exp [ - R(V) + {{\kappa }_{2}}V]{\text{erfc}}(\sqrt {{{\kappa }_{2}}V} ) \\ \, + Q(V) - \exp ( - V)\} , \\ Q(V)\, = \,\sqrt {\frac{{{{\kappa }_{2}}}}{\pi }} {\text{exp}}[ - R(V)] \\ \times \,\int\limits_0^V {{\text{exp}}[R({v}{\kern 1pt} ')]\frac{{{{F}^{{ - 0.5}}}({v}{\kern 1pt} ')B({v}{\kern 1pt} ')d{v}{\kern 1pt} '}}{{\sqrt {V - {v}{\kern 1pt} '} }}} . \\ \end{gathered} $$

((A.1))

Function K(V) is the solution to integral equation

$$\begin{gathered} K(V) = \exp [ - R(V)]\int\limits_0^\infty {B[2{{\kappa }_{2}}(\varepsilon + V)]{{f}_{{i0}}}(\varepsilon )d\varepsilon } \\ \, + {{\kappa }_{2}}\exp [ - R(V)]\int\limits_0^V {{{F}^{{ - 0.5}}}({v}{\kern 1pt} ')\exp [R(V{\kern 1pt} ')]} \\ \, \times B[2{{\kappa }_{2}}(V - V{\kern 1pt} ')]K(V{\kern 1pt} ')dV{\kern 1pt} ', \\ \end{gathered} $$

((A.2))

and has a simple physical meaning; it is function B(ε) averaged over the IDF for given potential V. Function f_i0(ε) on the right-hand side of expression (A.2) is the IDF in the unperturbed plasma. In the strong field approximation, when the energy of an ion acquired by it on its free path relative to the charge exchange is much higher than the thermal energy of atoms; according to [37], it is defined as

$${{f}_{{i0}}}(\varepsilon ) = C\exp ( - {{\kappa }_{2}}\varepsilon ){\text{erfc}}( - \sqrt {({{\varepsilon }_{0}} - 1){{\kappa }_{2}}\varepsilon } ),$$

((A.3))

where

$$C = \frac{{{{I}_{0}}}}{{{v}_{i}^{2}\left( {1 + \sqrt {\frac{{{{\varepsilon }_{0}} - 1}}{{{{\varepsilon }_{0}}}}} } \right)}} \approx \frac{{{{I}_{0}}}}{{2{v}_{i}^{2}}},$$

I₀ is the ion flux density to which IDF f_i0(v) is normalized;

$${\text{erfc}}( - x) = \frac{2}{{\sqrt \pi }}\int\limits_x^\infty {\exp ( - {{y}^{2}})dy;} $$

$${{\varepsilon }_{0}} = \frac{{eE{{\lambda }_{i}}}}{{2k{{T}_{a}}}},\quad {{{v}}_{i}} = \sqrt {\frac{{eE{{\lambda }_{i}}}}{M}} .$$

It was shown in [39] that provided that an ion in the PWS (i.e., in the quasi-neutral presheath and in the WS) experiences at least one collision (which is realized for almost any parameters in the plasma because, according to the results of calculations, the thickness of the presheath is always larger than the ion mean free path), the following solution holds for Eq. (A.2):

$$\begin{gathered} K(V) \equiv \bar {K}(V) + \left\{ {1 - {{\kappa }_{2}}\exp [ - R(V)]\int\limits_0^V {{{F}^{{ - 0.5}}}(V{\kern 1pt} ')} } \right. \\ \left. {_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}}}}}}}}}} \times \exp [R(V{\kern 1pt} ')]B[2{{\kappa }_{2}}(V - V{\kern 1pt} ')]\bar {K}(V{\kern 1pt} ')dV{\kern 1pt} '} \right\} \\ \times \left\{ {1 - {{\kappa }_{2}}\exp {{{[ - R(V)]}}_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}}}}}}}}}} \right. \\ {{\left. { \times \int\limits_0^V {{{F}^{{ - 0.5}}}(V{\kern 1pt} ')\exp [R(V{\kern 1pt} ')]B[2{{\kappa }_{2}}(V - V{\kern 1pt} ')]dV{\kern 1pt} '} } \right\}}^{{ - 1}}}. \\ \end{gathered} $$

((A.4))

As the zeroth approximation of the solution to system (6), we can take solution R⁽⁰⁾(V), F⁽⁰⁾(V), when K⁽⁰⁾(V) = B(V) is used for K(V).

For calculating the first and following approximations, we first calculate K⁽¹⁾(V) using on the right-hand side of Eq. (A.2) solutions R⁽⁰⁾(V), F⁽⁰⁾(V) obtained earlier. Then, solving Eq. (A.2) with determined function K⁽¹⁾(V), we obtain R⁽¹⁾(V), F⁽¹⁾(V), and so on. For almost any parameters κ₁ and κ₂, the procedure converges so rapidly that the first approximation is sufficient. To reduce the computation time, we can use the following approximate formulas obtained in [39]:

$$Q(V) \equiv {{Q}_{a}}(V) = {{F}^{{ - 0.25b(V)}}}{\text{erf}}[\sqrt {R(V)} ];$$

((A.5))

$$b(V) = 1\quad {\text{if}}\quad V < {{a}_{1}};$$

$$\begin{gathered} b(V) = 1 - 5\left[ {{{{\left( {\frac{V}{{{{a}_{1}}}}} \right)}}^{{0.075}}}{{{\left( {\frac{{{{a}_{1}}}}{V}} \right)}}^{{{{a}_{2}}}}}{{{\left( {\frac{{500}}{{{{\kappa }_{2}}}}} \right)}}^{{0.266}}} - 1} \right] \\ \times \,\,\,{{\left( {\frac{V}{{15}}} \right)}^{{{{a}_{3}}}}}{{\left( {\frac{{{{a}_{4}}}}{{{{\kappa }_{1}}}}} \right)}^{{0.7{{{\left( {\frac{{{{\kappa }_{1}}}}{{0.1}}} \right)}}^{{{{a}_{5}}}}}}}}; \\ \end{gathered} $$

$${{a}_{1}} = 8.899 \times {{10}^{{ - 4}}}{{\kappa }_{2}} + 1.611;$$

$${{a}_{2}} = - 1.11 \times {{10}^{{ - 4}}}{{\kappa }_{2}} + 1.211;$$

$${{a}_{3}} = 2.788 \times {{10}^{{ - 5}}}{{\kappa }_{2}} + 0.072;$$

$${{a}_{4}} = 0.01\quad {\text{at}}\quad {{\kappa }_{1}} \geqslant 0.005,$$

$${{a}_{4}} = - 2.5{{\kappa }_{1}} + 0.0225\quad {\text{at}}\quad {{\kappa }_{1}} < 0.005;$$

$${{a}_{5}} = 0.1\quad {\text{at}}\quad {{\kappa }_{1}} \geqslant 0.005,$$

$${{a}_{5}} = - 40{{\kappa }_{1}} + 0.32\quad {\text{at}}\quad {{\kappa }_{1}} < 0.005,$$

which are correct with an error not exceeding 5% in the following range of variation of parameters κ₁ and κ₂: κ₁ ≥ 3 × 10^–3,

$${{\kappa }_{2}} \geqslant {{\kappa }_{{2\min }}}({{\kappa }_{1}}) = \frac{{100}}{{1 + 0.8{{{\left( {\frac{{{{\kappa }_{1}}}}{{0.2}}} \right)}}^{5}}}}.$$

((A.6))