ANALYSIS AND SIMULATION BASED KINETIC STUDIES ON MAGNETIZED PLASMA WALL INTERACTION
Prabhat Kumar Dubey Christian Eminent College, Indore
Abstract - The physical setups where a material plasma, or the fourth state of matter, is confined or terminated by material boundaries, a special region of plasma appears adjacent to the boundary locations, commonly known as a plasma sheath. The conventional sheath and presheath theories are widely discussed in many arenas and it always at the centre to many investigations. The main function of plasma sheath is to ensures the quasineutrality of the bulk plasma by localizing the maximum potential drop close to the absorbing boundary. The formation of sheath structure achieves this by ensuring an equal flux of ions and electrons though the ions are much massive than the highly mobile electrons. The basic features of the plasma-sheath transition have been revealed in the early works of Langmuir in the kinetic analysis of the low pressure column.
1 INTRODUCTION
Anything which occupies space and has some volume is a matter. There are three states of matter, i.e. solid, liquid and gas. We are aware that solid is the state of matter in the atoms is closely pack hence they cannot move from their position but they can oscillate to their mean position. In liquid the molecules are loosely bounded but they cannot leave their surface. In gas the inter molecular forces are weak, hence they moves freely. We also know that when heat is given to solid they melt at their melting point and converted to liquid.
When heat is given to liquid it converted to gas. When the gas is heated to a higher temperature about 10,000 K, the gas get ionised producing ions and free electrons. This state of matter is termed as plasma and it is called fourth state of matter. Plasma is an ionized gas that consists of ions, electrons, and neutral atoms or molecules, they maintain charge neutrality. Electrons and ions are so close enough that each of them can influence many nearby charged particles within a radius called Debye screening length. Due to these charged particles in plasma responds collectively to the external fields. It have high density of free moving ions and electrons, hence plasma is highly electrically conductive. Except at boundary regions between plasma and wall, plasma contains the same amount of positive and negative charges. There is no space charge within the bulk of the plasma. The electron temperature is usually equal to or higher than that of ions. Since electrons are much lighter than ions, they can move from plasma at a higher speed than ions if there is no confining potential barrier. Once electrons are mostly depleted from the boundary interface between plasma and wall, a region which only has positive ions and neutrals will be formed. This region is called plasma sheath. Positive charges in plasma sheath can push more ions to diffuse out of plasma. It also produces a potential barrier to prevent electrons from diffusing out of plasma. The loss rate of electrons and ions will reach an equilibrium state. Plasma sheath also creates positive plasma potential with respect to the grounded walls. The potential drop across the plasma sheath can accelerate ions and create an ion sputtering effect in many applications.
F.F. Chen define that Plasma is a quasineutral gas which is the collection of charged as well as neutral particles. The sheath formation at the material boundary have great importance and it is used in many application where plasma is formed in the definite volume. In plasma region the mobility of electron is very high with respect to wall potential therefore the wall potential adjust itself at negative potential with respect to surrounding in this way a positively charged region is formed near to wall .This region is called Sheath. An electric field is formed in the sheath region directed towards the wall .This electric field accelerated the positive ions while the electron suffers the repulsive force hence they are retarded. The width of the sheath region determined by the electron Debye length.λ𝐷.The Debye length is very small in comparison to the other length like mean free path.
The early work on plasma sheath transition was done by Langmuir in 1929 then followed by Tonks and Langmuir. The clear explanation on the sheath region was given by Bohm in 1949. In the beginning the plasma sheath was related with collision-free plasma. In 1951 Boyd introduced the Bohm criterion in diffusion controlled theory of the collision dominated plasma. In 1962 Persson first recognise the role of mass of ion in the boundary region. This region is called Presheath. In 1991 Riemann gave the kinetic analysis on Collisional Presheath.
2 DEBYE SHIELDING
The potential variance near the material boundary can be determined with the help of Poisson’s equation or the Guass’s law.
Consider a plasma region having positive ions as well as electrons.
Consider a test charge 𝑞𝑇 in the plasma region. let the potential around the charge particle is ∅. If we assume the field is electrostatic then, the electric field due to test charge particle is
𝐸 = −∇∅
From the Guass’s law
∇. 𝐸 =𝜌
𝜀0 or −∇2𝜙 =𝜌 𝜀0
The charge density is due to the test charge as wel as the due to polarisation charges The charge density due to stationary test charge is 𝑞𝑇𝛿(𝑋 − 𝑋𝑇) and due to polarisation charges is 𝛿𝜌𝑃
From the Gibb’s distribution of particles which is 𝑓 = 𝑒−𝐻𝐾𝑇 but for density distribution function it looks like
𝑛 = 𝑛0𝑒−𝑞𝜙𝐾𝑇 When it is expanded we get,
𝑛 = 𝑛0 1 −𝑞𝜙 𝐾𝑇 Hence the polarising charge density
𝛿𝜌𝑃= 𝑛𝑗𝑞𝑗 𝑗
or
𝛿𝜌𝑃 = 𝑞𝑗𝑛0𝑗 1 −𝑞𝑗𝜙 𝐾𝑇𝑗
𝑗
But for the quasineutrality 𝑞𝑗 𝑗𝑛0𝑗 = 0 Therefore,
𝛿𝜌𝑃 = − 𝑛0𝑗𝑞𝑗2𝜙 𝐾𝑇𝑗 𝑗
From the Poisson’s equation
−∇2𝜙 =𝜌 𝜀0
−∇2𝜙 = 1
𝜀0 𝑞𝑇𝛿(𝑋 − 𝑋𝑇) − 𝑛0𝑗𝑞𝑗2∅ 𝐾𝑇𝑗
𝑗
∇2− 𝑛0𝑗𝑞𝑗2 𝐾𝑇𝑗 𝑗
𝜙 = −1 𝜀0
𝑞𝑇𝛿(𝑋 − 𝑋𝑇)
1
λ𝐷2 = 𝑛0𝑗𝑞𝑗2 𝐾𝑇𝑗
𝑗
The term 𝜆𝐷 is called Debye length ∇2− 1
λ𝐷2 𝜙 = −1 𝜀0
𝑞𝑇𝛿(𝑋 − 𝑋𝑇) The potential will be
𝜙 𝑥 = 𝑞𝑇
4𝜋𝜀0 𝑋−𝑋0 𝑒− 𝑋−𝑋0 𝜆𝐷
From the above equation, we see that the potential is not a linear function, It decreases exponentially.
3 SHEATH EQUATION
With the help of Poisson’s equation we have derive the expression for the Debye length𝜆𝐷. We have seen that the potential 𝜙 is not a linear and it is a nonlinear potential. As given by Chen, at the plane 𝑥 = 0 near the wall let the ions enters in the sheath region from the bulk plasma with some velocity 𝑢0.Let we assume that temperature of all the positive ions is zero, hence all the ions are moving with velocity 𝑢0 at 𝑥 = 0. The sheath region is collisonfree region, as the value of 𝑥 increases the potential decreases. Inside the sheath region,from the law of conservation of energy.
1
2𝑚𝑢02= 1
2𝑚𝑢2 𝑥 + 𝑒𝜙(𝑥) 1
2𝑚𝑢2 𝑥 =1
2𝑚𝑢02− 𝑒𝜙(𝑥) or
𝑢2 𝑥 = 𝑢02−2𝑒 𝑚𝜙(𝑥) 𝑢 𝑥 = 𝑢02−2𝑒
𝑚𝜙(𝑥) 𝑢 𝑥 = 𝑢02−2𝑒
𝑚𝜙(𝑥)
1 2
The ion density in sheath region and in the bulk plasma must be equal hence, 𝑛0𝑢0= 𝑛𝑖 𝑥 𝑢(𝑥)
When we put the value of 𝑢 𝑥 the above equation looks like 𝑛0𝑢0= 𝑛𝑖 𝑥 𝑢02−2𝑒
𝑚𝜙(𝑥)
1 2
𝑛0𝑢0= 𝑛𝑖 𝑥 𝑢0 1 − 2𝑒 𝑢02𝑚𝜙(𝑥)
1 2
𝑛𝑖 𝑥 = 𝑛0 1 − 2𝑒 𝑢02𝑚𝜙(𝑥)
−1 2
From Boltzman relation for the electrons
𝑛𝑒 𝑥 = 𝑛0𝑒𝑒𝜙 𝑘 𝑇𝑒 From the Poisson’s equation
−∇2𝜙 =𝜌 𝜀0
−𝜀0
𝑑2∅ 𝑑𝑥2= 𝜌 But 𝜌 = −𝑒 𝑛𝑒− 𝑛𝑖
On putting the values of 𝑛𝑒and 𝑛𝑖 the above equation will be 𝜀0𝑑2∅
𝑑𝑥2= 𝑛0𝑒 𝑒𝑒𝜙 𝑘𝑇𝑒 − 1 − 2𝑒 𝑢02𝑚𝜙(𝑥)
−1 2
The above equation looks like, If we consider the following
𝜒 ≡ − 𝑒𝜙 𝑘𝑇𝑒, 𝜉 ≡ 𝑥
𝜆𝐷 or 𝜉 ≡ 𝑥 𝑛0𝑒2
𝜀0𝐾𝑇𝑒
1 2and 𝜇2= 𝑢02
𝐾𝑇𝑒 𝑚 or 𝜇2=𝑚𝑢02
𝐾𝑇𝑒
On differentiating the 𝜒 with respect to 𝑥we get, 𝑑𝜒 𝑑𝑥= −𝑒
𝐾𝑇𝑒
𝑑∅
𝑑𝑥
𝑑𝜒 𝑑𝑥 =𝑑𝜒
𝑑𝜉 𝑑𝜉
𝑑𝑥 but 𝑑𝜉
𝑑𝑥= 𝑛0𝑒2
𝜀0𝐾𝑇𝑒
1 2hence
−𝑒 𝐾𝑇𝑒
𝑑∅
𝑑𝑥=𝑑𝜒 𝑑𝜉
𝑛0𝑒2 𝜀0𝐾𝑇𝑒
1 2
Again on differentiating the above equation with respect to 𝑥
−𝑒 𝐾𝑇𝑒
𝑑2∅ 𝑑𝑥2= 𝑑
𝑑𝑥 𝑑𝜒 𝑑𝜉
𝑛0𝑒2 𝜀0𝐾𝑇𝑒
1 2 or
−𝑒 𝐾𝑇𝑒
𝑑2∅ 𝑑𝑥2 =𝑑2𝜒
𝑑𝜉2 𝑛0𝑒2 𝜀0𝐾𝑇𝑒
𝑑2∅
𝑑𝑥2=−𝑛0𝑒 𝜀0
𝑑2𝜒 𝑑𝜉2
−𝑛0𝑒𝑑2𝜒
𝑑𝜉2= 𝑛0𝑒 𝑒𝑒𝜙 𝑘𝑇𝑒− 1 − 2𝑒 𝑢02𝑚𝜙(𝑥)
−1 2
or
𝑑2𝜒
𝑑𝜉2 = − 𝑒−𝜒− 1 +2𝜒 𝜇2
−1 2
𝑑2𝜒
𝑑𝜉2 = 1 +2𝜒 𝜇2
−1 2
− 𝑒−𝜒
The above equation is a non linear equation called Sheath Equation.
4 SIMULATION BASED KINETIC STUDIES ON MAGNETIZED PWI
A kinetic simulation of an oblique magnetized presheath , results of which are also used in the analysis of the present paper, was presented by D. Sharma . This study presented resolution of kinetic finite ion Larmour radius effect in the presheath and provided the normal flows generated by the additional drifts in an obliquely incident magnetic field setup. Sharma presented the kinetic solutions of a magnetized oblique presheath driven by stationery Maxwellian sources. Kinetic effects of open and closed orbits could be exactly accounted for by following the evolution of ion distribution function fi along the Lagrangian phase-space characteristics of the Vlasov equation in an associated self-consistent presheath potential distribution. The self-consistently determined three-dimensional ion velocity distribution function shows development of parallel and cross-field flows and strong deviation from the equilibrium distribution depending upon the inclination of magnetic field with respect to the solid surface. Moments of the ion distribution function, determined in a
uniformly sampled four-dimensional phase space, could be computed to obtain the profiles of flow velocity components and those of parallel and perpendicular temperatures in the oblique magnetized presheath. In the magnetized presheath, scaling over several gyroradii, dominant kinetic effects emerge which cannot be described suitably in a fluid approach.
The case of perfectly normal incidence where the cross-field flows vanish identically in the entire presheath region, the solutions approach the unmagnetized fluid results. However, for oblique incidences, the flow along the 𝐸 × 𝐵 direction shows growth in the entire presheath region and its boundary value becomes comparable to that of parallel flow for sufficiently smaller θ. The flow u⊥, which contributes to the normal flow ux, shows strong growth close to the solid surface at the smaller θ. The parallel and perpendicular temperatures could be seen to drop preferentially in two different limits of normal and grazing incidence, respectively. The behavior of presheath plasma in simulation in [28], as expected, indicates a finite contribution of the drop in perpendicular temperature in the determining the flow velocity profiles and values at the electrostatic sheath entrance in the limits of grazing incidence. A similar simulation based approach was adopted by Devaux and Manfridi who studied plasma-wall interactions in the case of a magnetized and weakly collisional plasma. The physical regimes chosen for the simulations by Devaux and Manfridi are relevant to low-pressure laboratory plasmas and tokamak edge plasmas. Kinetic Vlasov simulations were performed by them using an accurate Eulerian code. The use of a nonuniform grid allowed them to simulate the entire transition from the equilibrium plasma to the wall with a moderate number of grid points. The resulting code enabled them to obtain smooth phase-space distributions along the entire plasma-wall transition region.
Their results showed that, even for relatively strong magnetic fields, the angle of incidence of the impinging ions is never as grazing as the angle between the magnetic field and the wall. In the regimes considered by Devaux and Manfridi, the electric field always manages to partially redirect the ions normally to the surface. They also observed that two different ion populations can be present at the wall, giving rise to a two-peak velocity distribution.
These two populations correspond to ions that have been drifting without collisions along the magnetic field lines and to ions that have experienced several collisions before hitting the wall.
REFERENCES
1. R. Chodura, Phys. Plasmas 12, 013502 (2005).
2. D. Bohm, in The Characteristics of Electrical Discharges in Magnetic Field, edited by A. Guthrie and R. K.
Wakerling (McGraw-Hill, New York, 1949), chap. 3.
3. K. U. Riemann, IEEE Trans. Plasma Sci. 23, 709 (1995).
4. N. Hershkowitz, C. S. Yip, and G. D. Severn, Phys. Plasmas 18, 057102 (2011).
5. C.-S. Yip, N. Hershkowitz, and G. Severn, Phys. Rev. Lett. 104, 225003 (2010).
6. Y. B. Zeldovich and Y. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena (Academic, New York, 1967, 1967).
7. D. Sharma and P. K. Kaw, Phys. Plasmas 19, 113507 (2013).
8. D. Lee, L. Oksuz, and N. Hershkowitz, Phys. Rev. Lett. 99, 155004 (2007).
9. M. J. Druyvesteyn and F. M. Penning, Rev. Mod. Phys. 12, 87 (1940).
10. S. D. Baalrud, C. C. Hegna, and J. D. Callen, Phy. D. Sharma. Phys. Plasmas, 12:
0103506, 2005.
