Planar sheath and presheath

Transcription

1 5/11/1 Flui-Poisson System Planar sheath an presheath 1 Planar sheath an presheath A plasma between plane parallel walls evelops a positive potential which equalizes the rate of loss of electrons an ions. The potential profile in the plasma can be foun by integrating Poisson's equation from the plasma miplane to the wall. Moels are neee for the electron an ion ensities. It is customary to moel the electron ensity using a Mawellian istribution for which the ensity is given by a Boltzmann factor. The ions are moele using flui equations with a source term. The ion current to the wall is J an the charge ensity of ions is foun by iviing J by the flui velocity u. 1. Equations i Potential profile ( ) J ( ) e( ) o noq ep u( ) Te nu R Poisson's equation Continuity equation with source R It is assume that the ions are create uniformly throughout the plasma volume with a rate R per unit volume per unit time. For our one imensional plasma in a steay state, the current at a istance from the miplane must be sufficient to take away the ions that are create between the miplane an, which implies J = R. This result is obtaine by integrating once the continuity equation. The momentum equation is: t nu u nu nu u q m E where E is the electric fiel. Using the continuity equation, we can simplify the momentum equation: u nu q ne Ru m Note that the new ions are a sink for momentum because they are create with zero velocity an must be given momentum Ru per unit time in orer to move at the flui velocity. These ifferential equations can be solve simultaneously using Runge-Kutta or some other metho. It is useful to change to imensionless variables: / D R RD / nc s q / Te E ee / D T e J J / n o qc u i u / c s s J R ni J / u e n e where D T e / nq

2 5/11/1 Flui-Poisson System Planar sheath an presheath The equations for the problem, in imensionless form are then: ( ( ep ( E ) J i ) ( ) ( ) ) ( ) u E i Through substitutions, the momentum equation can be rewritten: u E R u n i u E u u In the Mathca version of the equation, the tiles are roppe for convenience.. Bounary conitions We must begin the integration slightly away from = to avoi the ivision by zero that woul occur in the momentum equation. u increases approimately linearly with, thus the term u/ has a finite value near the origin. There must be starting values for, E, an u a short istance from the origin. Symmetry requires that shoul be a function of even powers of. A first approimation for is then ( ) Thus E ( ) where is a constant to be etermine. 1 n e is assume at the miplane. From Poisson's equation we fin that (1 ) n i at the miplane. There is a slight ecess of ions at the center because the ions are less mobile. Using that J R n i u we fin for points near the origin R u R n This last relation provies us with a starting value for u at a small istance from the origin. The following three relations give us the starting values for E,, an u at the starting istance : u R E i 3. A relation between R an A further look at the momentum equation, using values vali near the origin, shows that R an cannot be chosen inepenently. The momentum equation can be solve for E to obtain R E ( ) n i Thus it follows that, but previously we showe R (1 ) E ( ) When R is sufficiently small: R

3 5/11/1 Flui-Poisson System Planar sheath an presheath 3 4. A relation between the plasma size an R A result of the sheath moel is that the current at the wall is approimately.5 in imensionless units. This means R =.5 at the wall. If the wall is a istance L from the miplane, RL =.5 an R =.5 / L, approimately. Thus the value of R to use can be foun from the size of the plasma L. 5. Defining the problem First we choose a size for the plasma of 1 Debye lengths: L 1 Then.5 R R α R α L We must start the integration a short istance from the origin: We use a istance 1% of the epecte istance L. 1 We put the starting values of, E an u into a vector y. Recall that the starting values are E u R The starting values will go in a vector y with the components, E an u: ystart α α R is y E is y1 u is y ystart is y E, is y1 u is y We will en the integration a little further than the estimate istance L: en ceil( L 16) 6. The erivatives for the Runge-Kutta integrator The meaning of this is: DY( y ) y 1 R e y y y y 1 y /= -E E/ = n i - n e = J/u - ep() u/ = E/u - u/ npoints en 1 the number of gri points. Note that the spacing is one Debye length. i 1 npoints

4 5/11/1 Flui-Poisson System Planar sheath an presheath 4 7. Solution by aaptive Runge-Kutta: M Rkaapt ystart ( ennpointsdy) For convenience, we assign the values in the answer matri M to the variable names use above: i Φ i u i M i M i1 M i3 n ei e Φ i n ii R i u i M () E() u() Below is a plot of the potential profile in imensionless units. The vertical scale is q/t an the horizontal scale is istance in Debye lengths. Potential profile Φ i Φ.618 Our estimate istance L is approimately where = -.5. L Ion velocity u as a function of istance. u i The ions become supersonic, u > 1, at approimately the bounary between the presheath an the sheath. u.916 L The bounary between the presheath an the sheath is somewhat arbitrary.

5 5/11/1 Flui-Poisson System Planar sheath an presheath 5 Electron an ion ensities 1.8 n ii n ei Note that the electron ensity falls more rapily as the wall is approache. The plasma is approimately quasineutral, up to the istance for which = -.5. Ion current to the wall R i L This trivial looking curve remains the same as L is change. In imensionless units, J is approimately.5 at the wall, always. The constancy of the curve shows that the ion current to the wall is nearly an invariant. The value of J =.5 n q c s is often calle the ion saturation current ensity. Try it: Change L from 1 to 1. It may be necessary to alter the efintion of en by a few Debye lengths so that the graph of escens to about -4, a typical value. Notes: 1. It is ifficult to fin plots of this type in the literature.. There are problems with numerical stability if L is mae larger than about 3. Stability is improve by using the Bulirsch-Stoer integrator instea of the Runge-Kutta integrator. Simply substitute: M Bulstoer( ystartennpointsdy)

6 5/11/1 Flui-Poisson System Planar sheath an presheath 6 3. There are iscussions in the literature of solutions mae by patching together solutions in the quasineutral region (mae assuming n i = n e eactly, sometimes calle the "plasma approimation") an solutions mae using Poisson's equation at the bounary. Moern computers can solve the full set of equations an there is no longer any motivation for patche solutions. 4. Eperimentalists usually assign zero potential to the walls, thus the center of the plasma is at a positive potential. Theoretical work is often one with the center of the plasma taken as the zero of the potential scale. The zero point of the potential scale is arbitrary. 5. At what place shoul the integration be ene? The potential in the center of a plasma is typically 3 to 5 T e /q greater than the wall potential. In moels, a value for is chosen so that the rate of loss of electrons an ions is equal. The simplest approimation is to fin where the ion current J = Rs equal to the ranom current of electrons, reuce by the factor ep[q/t e ]. This simple assumption, however, is not likely to be vali because the tail of the electron istribution is not necessarily Mawellian. 6. Can we have the integration en at = -4, for eample? There is no convenient way to have the integrator stop when a variable reaches a particular value. The loop below eecutes Runge-Kutta one step at a time an tests after each step whether or not has reache the specifie value. After each step, the new value for the variables is ae to the en of the answer matri N using the stack comman an the values for an ystart are upate so they can be use in the net iteration. 1 N M Rkaapt( ystart 1 DY) N M ystart submatri( M ) T M 1 while M 4 11 M Rkaapt( ystart 1 DY) ystart submatri( M ) T M 1 N stack( Nsubmatri( M1 1 3 )) N y shoul be a vector of one column, thus we use the matri transpose operation to convert the last row of the answer matri M to a column vector. Reference: Z. Sternovsky, Plasma Sources Sci. Technol. 14, 3-35 (5).