Title: Modelling of ProtoSphera experiments
A modelling activity has been undertaken in order to reproduce experimental results on the central plasma column of the ProtoSphera device. The activity started from scratch. At first we tried to adapt the model of V.A. Nemchinsky and Y. Raises [1] to the case of ProtoSphera but the program was diverging. Probably this paper fits only small arcs in nanomaterials.
After that, in order to find the elements for an appropriate model, we reviewed the fundamentals of the theory of gas discharges for the cases of glow and arc discharges. The aim was to find in the literature the most appropriate formulas for the cathode fall, the electric field at the cathode and the plasma density as a function of the cathode potential and also to use explicit formulas for the ionization coefficient, the ratio E/p (E the electric field, p the neutral gas pressure) used for estimating all the other quantities. The formulas change according to different authors because many different discharge regimes exist, so we collected some formulas from the book of Sanborn C. Brown [2], the book of Yuri Raizer [3], and the paper of Druyvestein and Penning [4] for a comparison. In particular we were looking for the influence of the magnetic field on the cathode fall and the electric field at the cathode, and on transport in the positive plasma column. Selected results are summarized in [5].
We examined various models of gas discharge which are appropriate to our low-pressure conditions [6, 7] and chose one which was expanded in the inverse of the magnetic field [7], according to the high-magnetization experimental conditions. This model fits well the ProtoSphera central pinch because it is based on the independence of the motion along the z axis and the r axis, in fact it is possible to solve it through variable separation. But the solution is available only if one makes an 1/B expansion and retains the terms of higher order [8]. Solutions in a cylinder of finite length (a first approximation to ProtoSphera geometry) were calculated. In the first graph there appears the azimuthal velocity, the radial velocity and the density along r. Velocities are normalized to the Bohm speed. In the second picture the axial velocity (top frame) and the density are shown as functions of the axial coordinate z.
References
[1] V.A Nemchinsky and Y. Raises, Anode sheath transition in an anodic arc for synthesis of nano materials, Plasma Sources Sci. Technol, 25 ,2016, 035003)
[2] Sanborn C. Brown, Introduction to electrical discharges in gases, Wiley series in Plasma Physics, 1966)
[3] Yuri Raizer, Gas discharges physics, John Allen Editors 1991
[4] Druyvestein and Penning Review of Modern Physics V.12,2, 88-173 1940
[5] B. Tirozzi , P. Buratti, Glow equations, preprint, 2016
[6] N. Sternberg, V. Godyak, D. Hoffman, Magnetic effects on gas discharge plasmas, Phys. Plasmas 13, 063511 (2006)
[7] A. Fruchtman, G. Makrinich and J. Ashkenazy, Plasma Sources Sci. Technol, 14, 152-167, (2005)
[8] B. Tirozzi, P. Buratti, F. Alladio, P. Micozzi, 1/B expansion of models of gas discharges preprint 2017
Figure 1.
Figure 2. Axial profiles of velocity normalized to Bohm speed (top) and normalized density versus distance ….