All Particle Simulation of a Cathodic Arc Plasma

I.J. Cooper

D. R. McKenzie

Tim Ruppin and Andrew Rigby

Traces left by an arc on tungsten cathode

3

Vacuum Arc• High Current, Low

Voltage discharge in vacuum ambient

• Current conducted in metal vapor plasma produced by discharge itself from evaporated electrode material

Usually plasma production concentrated at cathode spots

Ion flow rapid heating of micro-protrusion shock wave traveling to base explosion of micro-protrusion

Liquid drops, energetic electrons, ions and atoms ejected from cathode leaving a micro-crater

Atoms ionized by electron impact or if density sufficient, self ionization

Time Evolution of Cathode Spot Cell

Expanding hot dense plasma cell in non-thermal equilibrium layer

New ion flow to cathode

Ion flow to anode

Micro-protrusions on cathode surface

Cathodic arc plasma Subspots (fragments) Cells

Initial confinement of plasmaL = 1×10-8 mV = 1×10-24 m3

Number of ions 10 to 100Densitymax ~ 1026 ions.m3

Hot e- Te =3x104 K

Cold ions

3

( ) ( ) ( , )( )

4 ( , )x

oj i

q i q j x i jF i

r i j

All particle N body simulation

Coulomb forces between electrons and ions

( ) ( ) 1

4 ( , )oj i

q i q jU

r i j

212

( ) ( ) ( )i i

K m i v i K i

E U K Up > 0 Ue > 0 Upe < 0

2

3

( , )

2 ( , ) ( , )

( ) ( , ) ( , )( )

4 ( ) ( , , )o j i

x i t t

x i t x i t t

q j x i t x j tq i t

m i r i j t

r(i, j, t) small problems

r(i, j, t) r(i, j, t) +

Problem: Lots of particles – lots of calculations

Can modify equations to include external electric and magnetic fields

xj(t+1): qj Ex t2

FB = q v x B Bx =0, By = 0, Bz vz = 0

x(t+1):

G2[2x(t) + (G12-1)x(t-1) + 2G1y(t) – 2G1y(t-1)]

G1 = t Bz /2m G2 = 1 / (1+G12)

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04Motion of a proton: B = 0.8 T and E = 0 V/m

x (m)

y (

m)

Software MATLAB slow need to remove loops by using array operations

qq = meshgrid(q,q)

xx = meshgrid(x_1,x_1);

yy = meshgrid(y_1,y_1);

zz = meshgrid(z_1,z_1);

xd = xx - xx';

yd = yy - yy';

zd = zz - zz';

rd = sqrt(xd.^2 + yd.^2 + zd.^2);

rd = rd + rdMin;

rd3 = rd.^3;

Sx = (qq.*xd) ./rd3;

Sy = (qq.*yd) ./rd3;

Sz = (qq.*zd) ./rd3;

SSx = -A2 .* sum(Sx');

SSy = -A2 .* sum(Sy');

SSz = -A2 .* sum(Sz');

xfp = 2.*x_1 - x_2 + SSx;

yfp = 2.*y_1 - y_2 + SSy;

zfp = 2.*z_1 - z_2 + SSz;

For each time step t ~ 1x10-18 s Nsteps ~ 107:

SIMULATIONS single, multiple and mixed charged states H C Ti

10 ps 50 Ti+ 50 e-

10 ps

100 Ti+

100 e-

10 ps

100 Ti+

100 e-

0.10 ps

50 Ti+

50 e-

10 ps 50 ions 50 e-

10 ps

100 Ti+

100 e-

10 ps 100 Ti+ 100 e-

10 ps

100 Ti+

100 e-

10 ps1026 ion.m-3 Kavg ~ 3.8 eV

Kavg(real) ~ 60 eV 1028 ion.m-3

Initial Volume

(m3)

Initial Ion

density (ion.m-3)

No. of

e-

No. of Ti ions

Average ion KE (eV)

1.0×10-24 100×1024 100 100 Ti+ 3.8 0.5

1.0×10-24 30×1024 30 30 Ti+ 1.7 0.6

1.0×10-24 30×1024 60 30 Ti2+ 9.6 1.2

1.0×10-24 30×1024 6010 Ti+

10 Ti2+

10 Ti3+

1.8 0.68.2 1.211.8 1.6

10 ps

R = Ti2+ / Ti+