Download Diffusion and Transport Transport equations •For given volume

Diffusion and Transport
Transport equations
•For given volume define flux  through surface (number of
particles per area and time)
• this gives the particle balance:
 
N
   dA   S dV
t
Source term S :particles that are born in plasma volume,
e.g. by ionisation.
•Particle balance as differential equation:
 
N
   dA   S dV
t

n
 dV       dV   S dV
t

n

     S
t
• similar equation for energy balance
• as usual in hydrodynamics we use the ansatz


   Dn  nv
i.e. diffusion and convection separated
Diffusion
• diffusion coefficients from „random walk ansatz“:
red: Binomial distribution

x2 

green: exp 
2
 2 N x 
•Step size : x , time for step: t
Average time, to reach point x :
For given t, x: confinement time (radius)2
t  x2
t
x 2
Diffusion
• random walk:no net flux
•With density gradient:net flux towards smaller densities
1 nAx
N
 
 2
 t A
 t A
1 (n  n)Ax
N
 
 2
 t A
 t A
 n x 2
1 x
       n  
  Dn
2 t
x 2t
x 2
D
2t
Particle diffusion
„random walk“ ansatz for diffusion coefficient:
x: average mean free path
t: time in between two collisions (inverse collision time)
2
2


v
m
D  2     th2 

 s 
kT

m 
Particle diffusion and mobility
Equation of motion for a particle in a plasma:
For stationary plasma we obtain the particle flux:
n  n  x  
p  q 

n E
m   m  
 n 
 kT 

  kT
  n  
m
m








Limit low temperature plasma
Particle diffusion and mobility
n   Dn  n   n  n  E
Diffusion coefficient
Mobility constant
D
kT


q
Diffusion coefficient agrees with random walk result!
Ambipolar Diffusion:
assume a low temperature plasma: ions are pulled by the
electrons via an electric field against the friction of neutrals
Total flux of positive und negative particles from plasma must be equal
(quasi neutrality!)
e    e  ne E  Dene  i  i  ni E  Di ni
This is estabilshed by E-field:
Ea 
Di  De n

i  e n
(ne=ni)
ambipolar particle flux:
 i De   e Di

 n   Da  n
i  e
 i De   e Di
Da 
i  e
ambipolar diffusion
since:
and using
e >> i
D
kT


q
i
Da  Di 
 De
e
 Te 
Da  Di  1  
 Ti 
In low temperature plasmas, we often have: Te>> Ti
i     i  n  Ea
e      e  n  Ea  Den
electrons pull ions
ions try to ‚hold‘ electrons
Ambipolare diffusion
The outflux of electrons is substantially reduced
e      e  n  Ea  Den  0
Ambipolar diffusion
e      e  n  Ea  Den  0
 e  n  Ea   Den
n / n   Ea   e / De   eEa / kT
Electron density is Boltzmann-distribution:
ne  ne 0  e 
 e Ea ( x )dx / kT
Heat transport
heat transport: similar ansatz as
for particle transport:
Heat transport coefficient
(conductivity) per particle
qn   n  Tn


n
 1 Ws 
m s  m 


 m2 
 
 s 
Electron heat conductivity usually dominates:
e 
v the
 ee
2
 Ce
5/ 2

kTe 

ne
Ci / Ce  me / mi  Z 2
Electric resistance of plasmas
Ohm‘s law :
j  E
bzw.
j
E

j   e  ne  v   e     e  ne    E
 e 2 ne 
j     E
m  
 e

resistivity:
me
 || 
 stoß
2
ne  e
Electric resistance of plasmas
For ionised plasma: consider only Coulomb collisions:
 me1/ 2  e1/ 2 
Z
 ||  K  

ln



3/ 2
2


T
0
e


Te in eV
 0,52 10 5  ln  
Z
Te
3/ 2
  m 
• is independent of particle density
• decreases stronlgy with increasing electron temperature (~Te3/2)
For low ionisation fraction, friction with neutrals has to be considered
 Neutra lg as
me

 ( ei   en )
2
ne  e
Summary:
diffusion and transport
n   Dn  n   n  n  E
ambipolarity constraint gives:
 i De   e Di
Da 
i  e
Heat conduction
qn   n  Tn
 1 Ws 
m s  m 



e 
v the
2
 ee
 Ce
5/ 2

kTe 

ne

n
Ci / Ce  me / mi  Z 2
Electrical resistance:
j  E
bzw.
j
E

me
 || 
 stoß
2
ne  e
 m2 
 
 s 