Electrostatic Probe in a Reacting Gas

A spherical perfectly catalytic conductor with negative potential bias lies in a quiescent slightly ionized gas whose density is large enough to warrant the use of continuum theory. The Debye number $\epsilon$ , the ratio of the Debye length to the conductor radius, is small. Principal attention is given to the case, amenable to asymptotic analysis, of probe potential ${\psi }_p\text{\hspace{0.17em}>\hspace{0.17em}0(ln\hspace{0.17em}ε)}$ . The relation between the current collected by the probe and the electrode potential is sought; the current virtually saturates (increases very slowly relative to increases in applied potential difference) for ${\psi }_p\text{\hspace{0.17em}≫\hspace{0.17em}0(ln\hspace{0.17em}ε)}$ . A reconstruction of the chemically frozen case with three novel features is furnished: a transformation of variables that renders the transition‐zone equations almost uniformly valid; a very accurate, easily obtained closed‐form approximation to the solution of the resulting boundary‐value problem; and a current‐potential relationship that is derived on the basis of the gross behavior of this approximate closed‐form solution. Then the increase in current is determined for the case of finite‐rate gas‐phase ionization and recombination in which the negative‐charge carrier is an electron and in which an electron does not serve as the third body for recombination. It is found that ionization can play a role in a thick sheath in addition to the more conventional effect of both ionization and recombination in the quasineutral region. When an electron serves as the third body for recombination, the sheath is frozen.