Plasma boundary layer

The transition area between a plasma and a delimiting wall is referred to as the boundary layer or Debye layer . The electrons in a plasma usually have a similar or higher temperature than the ions and are many times lighter. This means that they are at least the factor faster and therefore get lost on the wall faster: ${\displaystyle {\sqrt {m_{\mathrm {i} }/m_{\mathrm {e} }}}}$

${\displaystyle {\begin{aligned}E_{\mathrm {kin} ,e}&\geq E_{\mathrm {kin} ,i}\\\Leftrightarrow {\frac {1}{2}}\,m_{e}\,{v_{e}}^{2}&\geq {\frac {1}{2}}\,m_{i}\,{v_{i}}^{2}\\\Leftrightarrow {\frac {{v_{e}}^{2}}{{v_{i}}^{2}}}&\geq {\frac {m_{i}}{m_{e}}}\\\Leftrightarrow {\frac {v_{e}}{v_{i}}}&\geq {\sqrt {\frac {m_{i}}{m_{e}}}}>1\end{aligned}}}$

In order not to violate the principle of quasi-neutrality within the plasma, a negative potential builds up in the edge layer , which reflects the electrons and accelerates the ions towards the wall. Only a small fraction of the electrons can penetrate the potential barrier of the surface layer. This is known as ambipolar diffusion . ${\displaystyle V}$

Edge layers are typically on the order of a few Debye lengths . You yourself are characterized by a clear violation of the principle of quasi-neutrality, i.e. that is, they have an excess of positive space charge . In (quasi-) stationary plasmas (e.g. capacitively coupled RF plasmas) the potential difference over the edge layer is set in such a way that on average the same number of electrons and ions pass through the edge layer.

history

Edge layers were described for the first time in 1923 by the American physicist Irving Langmuir :

"Electrons are repelled by negative electrodes , while positive ions are attracted by them. Therefore, an edge layer with a clearly defined thickness forms around each negative electrode , which only contains positive ions and neutral atoms. [..] Electrons are removed from the outer shell of the edge layer reflects, while all positive ions that reach the surface are attracted to the electrode. [..] It follows that there is no change in the positive ion current that reaches the electrode. The electrode is perfectly shielded from the discharge by the surface , and neither its potential nor the current to the electrode are affected by phenomena in the discharge. "

Mathematical treatment

Particle density  n and electrostatic potential  V
in a surface layer (sheath) and the fore- edge layer (presheath)

The one-dimensional equation

The physics of the surface layer is determined by four phenomena:

• Conservation of energy of the ions: If, for the sake of simplicity, we assume cold ions of the mass , which enter the surface layer at the same speed , the following applies due to the conservation of energy : ${\displaystyle m_{\mathrm {i} }}$ ${\displaystyle u_{\mathrm {s} }}$

${\displaystyle {\frac {1}{2}}m_{\mathrm {i} }\,u^{2}={\frac {1}{2}}m_{\mathrm {i} }\,u_{\mathrm {s} }^{2}-e\,V(x)}$

with the elementary charge . ${\displaystyle e}$

• Ion number conservation: In a stationary plasma no ions are formed or destroyed, which is why the flow is the same everywhere:

${\displaystyle n_{\mathrm {s} }\,u_{\mathrm {s} }=n_{\mathrm {i} }(x)\,u(x)}$

• Boltzmann equation for the electrons: Since most electrons are reflected, their particle density is given by:

${\displaystyle n_{\mathrm {e} }(x)=n_{\mathrm {s} }\,\exp(e\,V/k_{\mathrm {B} }\,T_{\mathrm {e} })}$

with the Boltzmann constant . ${\displaystyle k_{\mathrm {B} }}$

• Poisson's equation : The curvature of the electrostatic potential is related to the net charge density as follows:

${\displaystyle {\frac {d^{2}V}{dx^{2}}}={\frac {1}{\varepsilon _{0}}}\,q\,(n_{\mathrm {e} }-n_{\mathrm {i} })}$

with the electric field constant . ${\displaystyle \varepsilon _{0}}$

Combine these equations and replace potential , position and ion velocity with the dimensionless quantities ${\displaystyle V}$ ${\displaystyle x}$ ${\displaystyle u_{\mathrm {s} }}$

${\displaystyle \chi =-{\frac {e\,V}{k_{\mathrm {B} }\,T_{\mathrm {e} }}}\quad \Leftrightarrow \quad V=-{\frac {\chi \,k_{\mathrm {B} }\,T_{\mathrm {e} }}{e}}}$

${\displaystyle \xi ={\frac {x}{\lambda _{\mathrm {D} }}}\quad \Leftrightarrow \quad x=\xi \,\lambda _{\mathrm {D} }}$with the Debye length ${\displaystyle \lambda _{\mathrm {D} }}$

${\displaystyle {\mathfrak {M}}={\frac {u_{\mathrm {s} }}{\sqrt {k_{\mathrm {B} }\,T_{\mathrm {e} }/m_{\mathrm {i} }}}}\quad \Leftrightarrow \quad u_{\mathrm {s} }={\mathfrak {M}}\,{\sqrt {k_{\mathrm {B} }\,T_{\mathrm {e} }/m_{\mathrm {i} }}},}$

this gives the equation for the surface layer:

${\displaystyle \Rightarrow \chi ''=\left(1-{\frac {2\chi }{{\mathfrak {M}}^{2}}}\right)^{-1/2}-e^{-\chi }}$

The Bohm-sheath Criterion

The boundary layer equation can be integrated if it is multiplied by: ${\displaystyle \chi '}$

${\displaystyle \int _{0}^{\xi }\chi '\chi ''\,d\xi _{1}=\int _{0}^{\xi }\left(1-{\frac {2\chi }{{\mathfrak {M}}^{2}}}\right)^{-1/2}\chi '\,d\xi _{1}-\int _{0}^{\xi }e^{-\chi }\chi '\,d\xi _{1}}$

At the boundary between the boundary layer and the plasma ( ), the potential is set to zero ( ) and the electric field is also assumed to be zero ( ). With these boundary conditions the integration results: ${\displaystyle \xi =0}$ ${\displaystyle \chi =0}$ ${\displaystyle \chi '=0}$

${\displaystyle {\frac {1}{2}}\chi '^{2}={\mathfrak {M}}^{2}\left[\left(1+{\frac {2\chi }{{\mathfrak {M}}^{2}}}\right)^{1/2}-1\right]+e^{-\chi }-1}$

This integral can simply be written in closed form, although it can only be solved numerically . Nevertheless, important analytical conclusions can be drawn from this: since the left side is a quadratic expression, the right side must also assume a positive value for every value of , especially for small values. With a Taylor expansion um one finds that the first term that does not vanish is the quadratic one. So we can assume that ${\displaystyle \chi }$ ${\displaystyle \chi =0}$

${\displaystyle {\begin{aligned}{\frac {1}{2}}\chi ^{2}\left(-{\frac {1}{{\mathfrak {M}}^{2}}}+1\right)&\geq 0\\\Leftrightarrow {\mathfrak {M}}^{2}&\geq 1\\\Rightarrow u_{\mathrm {s} }&\geq {\sqrt {k_{\mathrm {B} }T_{\mathrm {e} }/m_{\mathrm {i} }}}\end{aligned}}}$.

This inequality is known as the Bohm-sheath criterion , named after its discoverer, David Bohm . If the ions penetrate the boundary layer too slowly, the boundary layer potential expands into the plasma in order to accelerate it. Ultimately, a so-called pre-sheath is formed with a voltage drop in the order of magnitude of and an expansion which is determined by the ion source (often as large as the plasma itself). ${\displaystyle (k_{\mathrm {B} }\,T_{\mathrm {e} }/2e)}$

Footnotes

1. ↑ Langmuir, Irving: Positive Ion Currents from the Positive Column of Mercury Arcs (1923) Science , Volume 58, Issue 1502, pp. 290–291 bibcode : 1923Sci .... 58..290L