skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Time-dependent Tonks-Langmuir model is unstable

Publication Date:
Sponsoring Org.:
OSTI Identifier:
Grant/Contract Number:
Resource Type:
Journal Article: Publisher's Accepted Manuscript
Journal Name:
Physical Review E
Additional Journal Information:
Journal Volume: 96; Journal Issue: 5; Related Information: CHORUS Timestamp: 2017-11-07 11:16:20; Journal ID: ISSN 2470-0045
American Physical Society
Country of Publication:
United States

Citation Formats

Sheridan, T. E., and Baalrud, S. D. Time-dependent Tonks-Langmuir model is unstable. United States: N. p., 2017. Web. doi:10.1103/PhysRevE.96.053201.
Sheridan, T. E., & Baalrud, S. D. Time-dependent Tonks-Langmuir model is unstable. United States. doi:10.1103/PhysRevE.96.053201.
Sheridan, T. E., and Baalrud, S. D. 2017. "Time-dependent Tonks-Langmuir model is unstable". United States. doi:10.1103/PhysRevE.96.053201.
title = {Time-dependent Tonks-Langmuir model is unstable},
author = {Sheridan, T. E. and Baalrud, S. D.},
abstractNote = {},
doi = {10.1103/PhysRevE.96.053201},
journal = {Physical Review E},
number = 5,
volume = 96,
place = {United States},
year = 2017,
month =

Journal Article:
Free Publicly Available Full Text
This content will become publicly available on November 7, 2018
Publisher's Accepted Manuscript

Save / Share:
  • The famous Tonks-Langmuir model of a plane symmetric discharge is supplemented by a kinetic theory of the plasma-sheath transition on the 'intermediate scale'. The mathematical structure of the transition is identical to that found for the collision dominated charge exchange model [K.-U. Riemann, J. Phys. D: Appl. Phys. 36, 2811 (2003)]. It reveals the significance of the asymptotic sheath edge and clarifies the role of the kinetic Bohm criterion. From the matching conditions an approximation for the plasma eigenvalue problem is obtained that is correct up to terms of the order {lambda}{sub D}/L. Matched asymptotic potential curves are constructed andmore » compared with numerical solutions. They show an excellent agreement, exceeding by far the expected range of validity in {lambda}{sub D}/L.« less
  • The paper presents a comprehensive kinetic theory of the famous Tonks–Langmuir model of a plane symmetric discharge, taking into account the thermal motion of ion source particles. The ion kinetics is governed by the ionization of neutrals at electron impacts. The plasma consisting of Boltzmann distributed electrons and singly charged ions is in contact with the absorbing negative wall. The derivations are performed in the frame of the “asymptotic two-scale” approximation, when the ionization mean-free path L{sub i} is much larger than the electron Debye length λ{sub D}. In the limit (λ{sub D}/L{sub i})→0, the plasma-wall transition (PWT) layer canmore » be split into two sublayers: a quasineutral presheath (PS) (with the scale-length L{sub i}) and the Debye sheath (DS) (with the scale λ{sub D}). Such a subdivision of the PWT layer allows to investigate these sublayers separately and simplify the analysis of the influence of the ion source thermal motion (this has been neglected in the major part of publications up to now). The uniform description of the PWT layer as a single unit is complicated by the singular presheath and sheath structure and by a coupling with the eigenvalue problem originating from the plasma balance in the bounded system. The issue is clarified both analytically and numerically by construction of a matched asymptotic expressions. The equation and the length-scale governing the transition between neighboring PS and DS sublayers are derived. The eigenvalue problem combining the wall potential, the wall location, and the ionization mean-free path is discussed.« less
  • An analytical solution of the Tonks-Langmuir (T-L) problem with a bi-Maxwellian electron energy distribution function (EEDF) is obtained for a plasma slab. The solution shows that the ambipolar potential, the plasma density distribution, and the ion flux to the wall are mainly governed by the cold electrons, while the ionization rate and voltage drop across the wall sheath are governed by the hot electrons. The ionization rate by direct electron impact is found to be spatially rather uniform, contrary to the T-L solution where it is proportional to the plasma density distribution. The temperature of hot electrons defined by themore » ionization balance is found to be close to that of the T-L solution for a mono-Maxwellian EEDF, and is in reasonable agreement with experiments carried out in a low pressure capacitance RF discharge. The energy balance for cold electrons in this discharge shows that their heating by hot electrons via Coulomb interaction is equalized by the cold electrons` escape to the RF electrodes during collapse of the RF sheath.« less
  • A plasma-sheath transition analysis requires a reliable mathematical expression for the plasma potential profile {Phi}(x) near the sheath edge x{sub s} in the limit {epsilon}{identical_to}{lambda}{sub D}/l=0 (where {lambda}{sub D} is the Debye length and l is a proper characteristic length of the discharge). Such expressions have been explicitly calculated for the fluid model and the singular (cold ion source) kinetic model, where exact analytic solutions for plasma equation ({epsilon}=0) are known, but not for the regular (warm ion source) kinetic model, where no analytic solution of the plasma equation has ever been obtained. For the latter case, Riemann [J. Phys.more » D: Appl. Phys. 24, 493 (1991)] only predicted a general formula assuming relatively high ion-source temperatures, i.e., much higher than the plasma-sheath potential drop. Riemann's formula, however, according to him, never was confirmed in explicit solutions of particular models (e.g., that of Bissell and Johnson [Phys. Fluids 30, 779 (1987)] and Scheuer and Emmert [Phys. Fluids 31, 3645 (1988)]) since ''the accuracy of the classical solutions is not sufficient to analyze the sheath vicinity''[Riemann, in Proceedings of the 62nd Annual Gaseous Electronic Conference, APS Meeting Abstracts, Vol. 54 (APS, 2009)]. Therefore, for many years, there has been a need for explicit calculation that might confirm the Riemann's general formula regarding the potential profile at the sheath edge in the cases of regular very warm ion sources. Fortunately, now we are able to achieve a very high accuracy of results [see, e.g., Kos et al., Phys. Plasmas 16, 093503 (2009)]. We perform this task by using both the analytic and the numerical method with explicit Maxwellian and ''water-bag'' ion source velocity distributions. We find the potential profile near the plasma-sheath edge in the whole range of ion source temperatures of general interest to plasma physics, from zero to ''practical infinity.'' While within limits of ''very low'' and ''relatively high'' ion source temperatures, the potential is proportional to the space coordinate powered by rational numbers {alpha}=1/2 and {alpha}=2/3, with medium ion source temperatures. We found {alpha} between these values being a non-rational number strongly dependent on the ion source temperature. The range of the non-rational power-law turns out to be a very narrow one, at the expense of the extension of {alpha}=2/3 region towards unexpectedly low ion source temperatures.« less
  • The article formulates the stability problem of the plasma sheath in the Tonks–Langmuir discharge. Using the kinetic description of the ion gas, i.e., the stability of the potential shape in the quasi-neutral pre-sheath regarding the high and low frequency, the perturbations are investigated. The electrons are assumed to be Maxwell–Boltzmann distributed. Regarding high-frequency perturbations, the pre-sheath is shown to be stable. The stability problem regarding low-frequency perturbations can be reduced to an analysis of the “diffusion like” equation, which results in the instability of the potential distribution in the pre-sheath. By means of the Particle in Cell simulations, also themore » nonlinear stage of low frequency oscillations is investigated. Comparing the figure obtained with the figure for linear stage, one can find obvious similarity in the spatial-temporal behavior of the potential.« less