Download DSD Code (continued)

DSD - A Particle Simulation Code
for Modeling Dusty Plasmas
Glenn Joyce
Icharus Research Inc.
Co-workers:
Martin Lampe
Gurudas Ganguly
Division of Plasma Physics
Naval Research Laboratory
Low Density Plasmas
 Many dust grain experiments are performed in low density plasmas.
 The plasma is sustained by electron ionization due to EM waves .
 There are ion drifts in the plasma with vion ~ 0 at the center and
vion ~ kT
at the edge of the plasma
m
e
i
Dust Grain Properties
 Dust grains normally float at a negative potential of ~ -2Te.
The dust charge Ze  aTe/e.
For radius a = 3.5m, and Te = 3eV, Z ~ 15,000
 For a grain separation of ~ 100 m, the potential energy is ~ 4keV.
 At high gas pressures, the grains have a kinetic energy < 1eV
 The coupling constant,  ~ P.E./K.E. >> 1.
 The grains can be strongly coupled and settle into a crystalline form.
Predicted by Ikezi in 1986.
Seen by a number of experimenters in 1990s.
 Charged dust grains in discharges form clouds at the sheath edge.
There is a balance of the sheath electric field and gravity.
Dust Grain Experiments (P>Pcrit)
 At pressures above a critical pressure Pcrit,
the dust is cold and is strongly coupled.
 There is a wake downstream of each grain caused by the streaming ions.
 The wake causes positively charged regions behind the grains
that can attract other grains.
 Experiments have observed various structures.
 Grains line up directly behind each other
in the streaming direction.
DYNAMIC SHIELDING WITH ION FLOW
FE
E
mg
• Plasma mediated (dynamically shielded) potential
eikr Z e
+
+
 (r)   d k
3
2 2 k 2 D(k,k u i i )
Downstream
attractive
Upstream and sideways
Debye-like
+
Dissipation removes
nodes
Ion Flow
Ion flow
• Discharge experiments:
- ions stream past dust grains at ~ cs
- forms wake fields
Dust Grain Experiments (P<Pcrit)
 Below Pcrit, the grains have a large random kinetic energy
and are weakly coupled.
 Dust grains lose energy due to neutral gas friction.
Despite this they reach a large random kinetic energy.
 A dust crystal formed at high pressure melts
when the pressure is decreased
Dust Simulation
 Modeling dusty plasmas is less developed than experiments.
 Kinetic models fall into two general classes:
1) Molecular dynamics models
follow the dynamics of the dust grains
greatly simplify the effects of the plasma.
2) Particle-in-cell or fluid models
follow the plasma dynamics
do not include the self-consistent dynamics
of a large number of dust grains.
 These codes provide insight into aspects of the physics.
 They do not simulate many phenomena of interest
in ground-based or microgravity experiments.
Molecular dynamics simulations:
 Force on a particle is found by summing the forces from all other particles.
 Works best if inter-particle forces are short-range.
Each particle is influenced by a small number particles in its vicinity.
 In a plasma, Coulomb interactions are long-range.
Every charged particle is influenced by every other charged particle.
Nreal lies somewhere between 1010 and 1022
 Simulations use a number of particles N ~106.
 Two problems arise:
Molecular dynamics requires N2 calculations per time step.
The reduction from Nreal to N causes an unphysical
enhancement of short-range “collisional” interactions.
The “particle-in-cell” or PIC approach:
 Each particle’s charge is distributed on the nearest grid points.
 Poisson’s equation is solved using the charge density on the grid.
The computational burden is reduced when N is large.
 Each point particle, j, located at rj, becomes a finite-sized
charge distribution with a specified shape S(r–rj).
 This eliminates the strong short-range interaction of point particles.
 One approach has been to use molecular dynamics simulations
of the dust grains, using a Debye-shielded Coulomb potential.
Ze exp(r / d )
 (r ) 
r
 The plasma does not appear except as the source of Debye shielding.
 The simulations show strongly coupled plasma crystals,
but the force is isotropic, and leads to isotropic structures (bcc & fcc).
 Experiments can show a strong anisotropy between
flow direction and the transverse plane.
 In some regimes, grains form hexagonal structures in the transverse plane,
with the lattice points directly above each other.
 Grains can also form vertical strings, with a weak interaction between strings.
The NRL Dynamically Shielded Dust (DSD) Simulation Code
 To model strongly coupled dust in plasmas, we use:
The techniques of molecular dynamics simulation
The techniques of PIC simulation
The “particle-particle/particle-mesh” (P3M)
(Hockney and Eastwood)
 We also use the dressed test particle representation:
one of the theoretical foundations of plasma physics.
 Many of the techniques are common to all PIC plasma simulation codes.
 The unique properties are the accurate representation of both
the long-range and short-range parts of the interaction between dust
grains described by the complete plasma dielectric response.
DSD Code (continued)
We assume:
 The charge density of the dust is small compared to the charge density of
electrons or ions.
 The response of the plasma to the dust is a linear response, which is
represented by the plasma dielectric function D(k,w).
 The presence of a single dust grain with charge Ze, together with the
response of the plasma, induces a potential (in k-space)
 truek  
q
2 2k 2 D k,kˆ  v D


 vD is the streaming velocity of the plasma, with respect to the dust.
 Since the plasma response is linear, the potential due to the presence of
many dust particles is the sum of terms as above.
 Each dust grain may be regarded, as a particle “dressed” by the plasma.
DSD Code (continued)
In k-space the potential is given by
 true k  
Ze
2π 2 k 2 Dk , k  ud 
k  fi 0 v  / dv
2
  k  v  i i
1 ωpi
D(k ,  )  1  2 2  2
k λ De k 1  i d 3 v fi 0 v 
i
  k  v  i i
3
d v
Zero streaming velocity
yields Debye-Huckel potential
qe r (1Te / Ti ) / D
 truer  
D r
DSD Code (continued)
 The model represents the dust as simulation particles
interacting via the dressed particle potential.
 The plasma appears only implicitly through the function D(k,w).
 It is not necessary in the code to resolve the fast plasma time scales.
However:
 If we use only the PIC technique to calculate the inter-grain forces,
we lose the short-range force.
 If we use only the molecular dynamics technique,
we need to calculate N2 interactions.
 So we use both techniques together for dressed particles,
according to the P3M procedure introduced by Hockney and Eastwood
for particles interacting via the bare Coulomb potential.
DSD Code (continued)
 If S(k) is the structure function of finite-sized particles in a PIC model,
i.e. the Fourier transform of the particle charge distribution S(r),
then in k-space the potential induced by the dressed PIC particle is
 ref k  
qS ( k )
2 2k 2 D k , kˆ  v D


 This is the reference potential.
It has the property that its interaction force becomes small
inside a cell surrounding each particle.
DSD Code (continued)
 We define a short-range potential sr  true – ref.
The potential sr is small outside a short range about the particle.
 We sum directly the short-range force on a given particle by all other
particles.
 For the short range force, only particles in nearest neighbor cells contribute.
This is not an N2 operation.
DSD Code (continued)
The method of pushing the particles, is
 Calculate the charge density on a mesh.
 Calculate the reference force from the reference potential, ref
 Change the momentum using this force.
 Calculate the momentum change due to the short-range force
using pair-wise interactions.
 The details are described in the book by Hockney and Eastwood.
Dust Cloud Simulations
 In discharge experiments,
The dust accumulates near the sheath edge at the lower electrode.
Gravity and the sheath electric field balance the grains vertically.
The grains are confined laterally by external electric fields.
 We impose these fields in the DSD code.
The vertical electric field increases linearly with z.
The lateral electric fields are weaker than the vertical field
and increase linearly from the box center.
 We load the grains randomly in the simulation box.

      
      
CHARACTERISTICS OF DSD
 Time steps can be long compared to those required for simulating ions.
 Kinetic electron and ion properties such as wake effects due to streaming
plasmas are included correctly.
 Scattering of ions and electrons off each other and off neutral atoms
are included in the dielectric function.
 Ion drag forces on the dust grains are included.
Deposition of charge on the dust grains can also be included.
CHARACTERISTICS OF DSD
 Approximations:
1) The plasma response is linear.
Good approximation for streaming ions
2) The plasma is represented as uniform
in density, temperature and flow velocity.
3) Boundary conditions are periodic because of Fourier transforms.
4) The grain velocity is neglected in the shielded potential.
Dust Cloud Simulations
Simulation parameters:
Plasma
ne = ni = 2x108cm-3
Te = 3 eV
mi = 83.5 mproton (krypton)
Te/Ti = 24
M = 1.2
i = 2x106sec-1
Dust
a = 3.5m
Zd = 104
md = 1.4x1014 mproton
d = 71sec-1
Example of DSD Structures
 Weakly coupled at P =125mT
 Strongly coupled at P=150mT
 In the plane normal to the ion streaming, there can be various structures
Application of the DSD Code to Dust heating
Dust Grain Experiments (P>Pcrit)
 At pressures above a critical pressure Pcrit,
the dust is cold and is strongly coupled.
Dust Grain Experiments (P<Pcrit)
 Below Pcrit, the grains have a large random kinetic energy
and are weakly coupled.
 Dust grains lose energy due to neutral gas friction.
Despite this they reach a large random kinetic energy.
 Dust forms a weakly coupled fluid.
 A dust crystal formed at high pressure melts
when the pressure is decreased
PHASE TRANSITION IN A DUSTY PLASMA
Typical DSD Simulation Output
Gas
Crystal
Condensation
Physics?
Melting
Physics?
Pm
Pc
PHASE TRANSITION IN A DUSTY PLASMA
DSD Simulation
PHASE TRANSITION IN A DUSTY PLASMA
DSD Simulation
SIMULATION OUTPUT SUMMARY
Condensed State: P > Pc
Gaseous State: P < Pc
Theory of Dust Heating
 Non-collective mechanisms do not explain grain heating
in the weakly coupled fluid phase.
 Grain heating can be explained by an ion-grain two-stream instability.
 The instability is stabilized by ion-neutral and dust-neutral collisions.
 Consider: A dusty plasma
1) warm electrons, De=(Te/4nee2)1/2
2) ions, i = (4nie2/mi)1/2
3) dust, d = (4nde2/md)1/2.
4) dust and ions collide with gas with frequencies d and i.
 d2
i2
1
0  1 2 2 

k De    i d    k  u   k  u  i i 
 All modes are stable if both d/d and i/i are large enough.
Dust Heating (continued)
 The two-stream instability is present in the DSD model.
A modified dispersion relation can be derived.
 d2
i2
1
0  1 2 2 

k De    i d    k  u   k  u  i i 
 This dispersion relation is quadratic in  and can be solved analytically.
 Solutions agree quite well with the full dispersion relation.
Comparison of full and partial
Dispersion Relations
LINEAR AND NONLINEAR WAVE PROPERTIES
P=100mT
P=125mT
P=150mT
PHYSICAL PICTURE (Gas to Solid Phase Transition)
• CONDENSATION: (suppression of ion-dust two stream instability)
- ions stream by the dust, at V0 ~ cs >> vti
- ion-dust two-stream instability
 d2
i2
1
1


0
(kDe ) 2  (  i d ) (  kVo cos )(  kVo cos  i i )
- convective in presence of collisions
• Instability stabilized if both species are
sufficiently collisional; at P = Pc
- P > Pc: no heat source
-Td falls; condensation to dust crystal
Theory
Simulation points:
Stable
Unstable
POSSIBLE HEAT SOURCES
1. Stochastic heating in a broad
wave spectrum
2. Dust-dust collisions
Homogeneous DSD Simulation: P=100mT Mach 1.
Conclusions
 Dust heating below Pcrit is due to a streaming instability.
 Above Pcrit the instability is stabilized.
 A low frequency dust mode is seen above Pcrit
This mode disappears at higher pressures.
 For the parameters of this study, the particles are aligned vertically.
 Various structures appear in the horizontal planes.
 DSD is in qualitative agreement with experimental observations.
DSD Code (continued)
 The two-stream instability is present in the DSD model.
A modified dispersion relation can be derived.
 d2
i2
1
0  1 2 2 

k De    i d  k  uk  u  i i 
 This dispersion relation is quadratic in  and can be solved analytically.
 Solutions agree quite well with the full dispersion relation.
DSD Code (continued)
In k-space the potential is given by
 true k  
Ze
2π 2 k 2 Dk , k  u i 
where D is the plasma dispersion function
k  f i 0 v  / dv
ω2pi
  k  v  i i
1
D(k ,  )  1  2 2  2
f i 0 v 
k λ De k 1  i d 3 v
i
  k  v  i i
3
d
 v
For zero streaming velocity, the potential is the Debye-Huckel potential
qe  r (1Te / Ti ) / D
 truer  
D r
Dust Simulation
 Particle simulation of dusty plasmas presents two severe challenges.
 Dust dynamics occurs on a slower time scale than electrons and ions
(a dust particle is ~1012 times the mass of an ion),
and on a much coarser spatial scale
(the plasma particle density np ~ 103nd).
 One must resolve short and long spatial scales.
Short: close approach of dust and plasma particles
Long: collective electrostatic interactions.
 For strongly-coupled dust, the short-range interaction is important.
 Electrons and ions are weakly coupled, and long range
collective interactions are important.
 The electron-ion plasma serves as a medium for long-range
interactions between dust particles.
The model basically represents the dust as simulation particles interacting via
the dressed particle potential.
The plasma appears only implicitly through the function D(k,w).
It is not necessary in the code to resolve the fast plasma time scales.
However:
If we use only the PIC technique to calculate the inter-grain forces,
we lose the short-range force.
If we use only the molecular dynamics technique,
we need to calculate N2 interactions.
So we use both techniques together for dressed particles,
according to the P3M procedure which Hockney and Eastwood introduced for
particles interacting via the bare Coulomb potential.
DSD Code (continued)
 To resolve the long and short-range forces,
we use the particle-particle/particle-mesh (PPPM) scheme
of Hockney and Eastwood.
 The grain density is laid down on a mesh and Fourier transformed.
The potential is calculated and transformed back to real space.
 The potential is smoothed at short range due to finite size particle effects.
 The difference between the exact potential and the smoothed potential
is used to calculate pair-wise interactions of near grains.
 The complete dynamically shielded potential is in the code.
DSD (Dynamically Shielded Dust) Simulation Code
 The DSD simulation code efficiently resolves problems
Short-long spatial scales
Disparate time scales
 Electrons and ions do not appear as simulation particles.
 The interaction of grains is the dynamically-shielded Coulomb interaction.
 This is the Coulomb interaction mediated by the response of the plasma
Includes wakefields, ion-neutral collisions and Landau damping.
DSD Code (continued)
In k-space the potential is given by
 true k  
Ze
2π 2 k 2 Dk , k  ud 
k  fi 0 v  / dv
2
  k  v  i i
1 ωpi
D(k ,  )  1  2 2  2
k λ De k 1  i d 3 v fi 0 v 
i
  k  v  i i
3
d
 v
Zero streaming velocity
yields Debye-Huckel potential
qe r (1Te / Ti ) / D
 truer  
D r