Development of Adaptive Kinetic-Fluid
Computational Tools for
Low-Temperature Plasmas
Vladimir Kolobov
Quantemol Workshop
April 4, 2014
Agenda
• Introduction: Plasma Technologies & Physical Models
• Commercial Software for Simulations of Low
Temperature Plasma Devices and Processes
• Adaptive Mesh and Algorithm Refinement (AMAR)
and Unified Flow Solver
• Towards Multi-Physics AMAR: Fluid Plasma Models
with Adaptive Cartesian Mesh
• Kinetic Solvers with Adaptive Mesh in Phase Space
• Conclusions
• Acknowledgements
• References
Plasma Technologies
•Thrusters
"The XXI century will be not only the century of
informatics, biology and space exploration, but also
the century of plasma technologies" A.I.Morozov 2006
Propulsion
Lighting
•Spray Coatings
Materials Processing
Displays
Kinetic and Hydrodynamic Approach
Kinetic
Hydrodynamic
Particles are described by five
characteristics:
1. Density
n ( r, t )
2. Mean directed velocity,
3. Temperature,
T ( r, t )
v ( r,t )
They depend on 4 scalar arguments –
3 spatial coordinates and time.
The only characteristic is the
velocity distribution function
(VDF) – the particle density in
phase space
f ( ξ, r, t )
It depends on 7 scalar arguments
– 3 spatial coordinates, 3 velocity
components and time.
• Kinetic description is much more detailed and much
more expensive
• Many problems in gas discharge physics can only be
understood at kinetic level
Commercial Software
for
Simulations of Plasma Devices and
Processes
Commercial Software Development
CFDRC developed the first
commercial tool for simulation of
industrial plasma reactors and
processes
In 2004, CFD-ACE+ software
package was sold to ESI group for
further commercialization
CFDRC retains rights for software
customization and consulting
CFD-ACE+ Software Architecture
Plasma Models
Spatial gradients can be so steep that a continuum description may break down even
at pressures of many atmospheres
High Pressure
Thermal Plasmas
• Non – Equilibrium Chemistry
• Two Temperature Te≥ T
• Te = T, LTE
• Gas Flow (turbulence)
• Heat Transfer
• Radiation (HF
EMAG)
Key Features
• Radiation Transport
• Gas Heating T > Troom
Low Pressure
Non-Equilibrium
Plasmas
• Magnetic Field Effects
• Kinetics of Ions & Fast Neutra
• Kinetics of Electrons
• Non-Local Electrodynamics
The need to develop new modeling techniques that address vastly different scales having
different physics at both low and high pressures speaks to the convergence of the discipline
Plasma Simulations with CFD-ACE+
2D Cartesian or
Cylindrical
3D Cartesian
1D Positive
column
0D Well stirred
Global
Plasma Model
Non - Equilibrium
Flow
Heat
Chemistry
Magnetic
Local Thermodynamic Equilibrium (LTE)
Plasma
Electric
ICP
Flow
Kinetic
CCP
PC
Heat
Electric
Global
Radiation
Magnetic
Turbulence
User Subroutine
Reaction Manager
Arrh. Rate
Cross-Sections
Maxw EEDF
Non-Maxw EEDF
Dashed lines indicate an
optional or alternative choice
Plasma Aided Manufacturing
Electron Kinetics in ICP
Calculated electron temperature (left) and electrostatic
potential (right) for Ar ICP at 10 mT, 6.8 MHz, 100W.
∂f 0 v
1 ∂  2 eE 
+ div(f1 ) + 2
⋅ f1  =
S0
v
∂t 3
3v ∂v  m

EEPFs at different points along the discharge axis at radial position r=4
cm: a) measured, b) calculated.
∂f1
eE ∂f 0
+ν f1 =− v∇f 0 −
m ∂v
∂t
Two-term SHE reduces 6D Boltzmann equation to a 4D Fokker-Planck equation for electrons in
v
f (r=
, v.t ) f 0 (r, v,t ) + ⋅ f1 (r, v, t )
collisional plasmas
∂f 0 ∂φ ∂f 0
v
−
− ∇ ⋅ Dr ∇f 0 −
∂t ∂t ∂ε
Using total energy (kinetic + potential) simplifies analysis of
1 ∂  
∂f

χ  Dε (r, ε ) 0 + Vε (r, ε ) f 0   =
S
electron kinetics for many gas discharge problems

χ ∂ε  
∂ε

V. Kolobov & R. Arslanbekov, Simulation of Electron Kinetics in Gas Discharges, IEEE TPS 34 (2006), 895
Ionization Waves - Striations
Argon 2 Torr 100 mA L = 20 cm R = 1 cm
Argon
1 Torr
Atmospheric Pressure Discharges
Photo by M. Laroussi
CFD-ACE+ simulations of atmospheric
pressure Nitrogen discharge between parallel
plates covered by dielectrics at driving
frequency 10 KHz.
Electrostatic potential
Pattern Formation in DBD
CFD-ACE+ simulations: fluid model with LUTs
Bhoj & Kolobov, IEEE TPS 39,(2011) 2152
Experimental measurements of pattern formation
dynamics in He DBD 200kHz, 30kPa, 690 V.
Stollenwerk et al. Phys. Rev. Letters 96 (2006) 255001
• Base case conditions:1000 V, 200 kHz, perfect
dielectric with permittivity ε =7.6.
• Patterns disappear when driving frequency increased
to 300 kHz
• Increasing ε increased peak electron density and
pattern density
Plasma Solvers with “Traditional” Mesh
• Hybrid models have been widely used for
simulations of weakly ionized plasmas using
traditional body-fitted mesh techniques
• We have performed simulations of numerous
plasma reactors and processes for semiconductor
manufacturing, lighting and other applications,
and accumulated considerable experience in
developing chemical reaction mechanisms:
• rare gases (lighting, PDP)
• O2, N2, Cl2, H2
• Fluorocarbon plasmas (SF6, C2F6, ..)
• SiH4 for deposition of SiO2
• Hydrocarbon plasmas (DLC films, CNT,
etc)
An example of 3D fluid simulations of an
industrial ICP source performed with CFDACE+ software using body-fitted mesh
techniques
• Traditional methods lack efficiency for
simulations of dynamically evolving features
requiring high resolution in selected areas of
computational domain (shock waves, ionization
fronts, streamers, etc).
Adaptive Mesh & Algorithm
Refinement
and
Unified Flow Solver
Third Wave of CFD
Evolution of Mesh Types
Structured body-fitted
Hybrid body-fitted
Unstructured body-fitted
Cartesian with embedded
boundary
Embedding CFD inside an MCAD tool,
Mesh generation becomes a part of the
solution process (Mentor Graphics)
Unified Flow Solver (UFS)
Developed by CFDRC for U.S. Air
Force
Supported by NASA, MDA, and
U.S. Navy
Direct Numerical
Solution
Boltzmann
Solvers
Domain
Decomposition
Direct Simulation
Monte-Carlo
(DSMC)
Next Generation
Computational Tool for
High-Altitude
Aerothermodynamics and
Low-Speed Micro-Flows
Adaptive Mesh & Algorithm Refinement
Coupling
Algorithm
Continuum Flow
Solvers
• Direct Boltzmann Solver by Discrete Velocity Method
• Dynamic adaptation of computational (Cartesian)
mesh to solution and geometry
• Auto switching between kinetic and fluid models
based on continuum breakdown criteria
V. Kolobov et al, J. Comp. Phys 223, (2007) 589
Illustration of UFS methodology
Steady Gas Flow Around Cylinder for different Kn numbers: M=3
Kn=0.01
Kn=0.1
Kn=0.3
Kn=1
Kinetic (Red) and Fluid (Blue)
Domains
Mesh Refinement Criterion:
∇ρ < ε =
0.1
Density Profile
Continuum Breakdown Criterion
2
2
2
 ∇p  1  ∂u   ∂υ  
+   + 
Kn 
0.1
 <δ =


 p  T  ∂x   ∂y  
UFS Simulations of Transient Flows
Computational grid
Shock Wave Interaction
with Boundary Layer in
Micro Channel
Kinetic &
Fluid Domains
Ma = 3
Kn = 0.2
Gas Density
V. I. Kolobov, et al, Unified Flow Solver for Transient
Rarefied-Continuum Flows, AIP Conf. Proc.1501
(2012) 414; V. V. Aristov, et al, Vacuum 86 (2012)
Space Filling Curves & Forest of Octrees
The procedure of domain decomposition is performed using SFC. Using SFC, 2D
or 3D spatial data can be stored in a single dimensional array and domain
decomposition is easily achieved by chopping this array into even pieces.
• During sequential traversing of cells by natural order, the physical space is filled with curves
in N-order (Morton ordering). After this ordering of cells, all cells can be considered as a onedimensional array.
• A weight is assigned to each cell, proportional to CPU time required for computations in this
cell. The array modified with corresponding weights, is subdivided into sub-arrays equal to
the number of processors.
• This method allows most efficient domain decomposition between different processors.
Towards Multi-Physics AMAR
Kinetic Module:
•
DVM Boltzmann & Vlasov
Solvers
•
Fokker Planck solver for
electrons
•
Lagrangian Particle Solvers
(DSMC, Photon Monte Carlo)
Mesoscopic:
• Lattice Boltzmann Method
Fluid:
• Plasma & Chemistry
Hybrid Model of Radiation Transport:
selection of PMC or diffusion solver base
on local photon MFP
Fluid Plasma Models
with
Adaptive Cartesian Mesh
Minimal Plasma Model
A simplest set of equations for the electron
and ion densities, and Poisson equation for
electrostatic potential.
Electron diffusion and ion transport are
neglected

∂ne
+ div Γ e =
ν ne
∂t
( )


Γ e =µ Ene
∂ni
= ν ne
 E 
∂t
=
ν ν 0 exp  − 0 

∆
=
ϕ 4π ( ne − ni )
Mesh Adaptation criteria
 ne
ne 
=
+
α 20 
 + 3 ( log10 (ne ) + log10 (ni ) + log10 (ν i ) + log10 ( E ) )
max(
)
max(
)
n
n
e
e 

When gradient of α exceeded a fixed value
this particular cell was refined
∇α > 1
in a cell,
When ∇α < 1/ 4 in a cell, this cell was coarsened
The procedure of mesh adaptation performed every 10 time steps
E
Streamer development in positive corona
2D axisymmetric simulations
3D parallel simulations on 8 cores with
~1.5 million cells: axi-symmetric channels
break into separate streamers.
electron density
Computational grid, electron density contours at 107 cm -3, and the
electric field strength (color, maximum value 7 106 V/m )
Extended Plasma Model
• A set of equations for the electron and ion
densities, and Poisson equation for
electrostatic potential
• Ion transport is included
• Multiple species ions are allowed by
introducing species specific mass,
diffusion coefficients, and ionization rates
• Local field approximation for ionization
rate
• Electron Energy Transport with account
for electron thermal conductivity
• CANTERA link implementation is
underway for complex reaction
mechanisms

∂ne
+ div Γ e =
ν ne
∂t
( )


Γ e =− µe Ene − De∇ne
∆
=
ϕ 4π ( ne − ni )
∂ni
+ div(Γi ) =
ν ne
∂t


=
Γi µi Eni − Di ∇ni
 E 
=
ν ν 0 exp  − 0 
 E

∂nε
+ ∇ ⋅ Γ ε = Sε
∂t
nε = neε

5 
5
Γε =− µe Enε − De∇nε
3
3
Effect of Electron Energy Transport
2D axi-symmetric simulations
The model reproduces
basic structure of
classical DC glow
discharges
Spatial distribution of electron density
Axial distributions of electron and ion densities (left), electrostatic potential and
electron temperature (right) with account of electron thermal conductivity
Kinetic Solvers with Adaptive
Mesh in Phase Space
Adaptive Mesh in Phase Space
spatial mesh
VDF
Tree-of-Trees Structure: Tree-based
velocity mesh created in each spatial cell.
Hypersonic flow over a square at M=30,Kn=0.1:
VDFs at different locations
velocity mesh
Electron Kinetics
electron
diffusion in
energy
thermal
electron
runaway
• Kinetic solvers for electrons in different
regimes of EDF formation:
• Fokker Planck (collisional, slow)
• Vlasov (nearly collisionless regimes)
• Boltzmann (fast runaway, anisotropic)
genuine
electron
runaway
Different regimes of EDF formation:
Kolobov, Phys. Plasmas 20, 101610
(2013)
Computational mesh & EDF
contours in velocity space
a : electron streaming;
b : isotropic “body”, anisotropic “tail”
Adapted mesh in velocity space
Linking Collision Theory & Physical Kinetics
Computational mesh and VDF contours for different values of electric field strength, elastic
and inelastic collisions. White circle corresponds to inelastic threshold.
In semiconductors, phased motion of hot electrons in configuration and momentum
spaces causes spatial modulation of electron concentration and mean electron
velocity with spatial period
λE = ε 0 / eE
In gas discharges, similar phenomena result in plasma stratification and
propagation of ionization waves (moving striations).
Precise calculations of EDF require detailed knowledge of differential
collision cross sections for elastic and inelastic processes
Conclusions
• Commercial software for non-equilibrium low temperature
plasmas (CFD-ACE+, 10+ years old)
• Third Wave of CFD: embedding CFD inside MCAD tools,
mesh generation is part of the solution process, AMR
• Unified Flow Solver (4th Wave of CFD ?):
• AMAR methodology (AMR + Adaptive Physics)
• massively parallel, GPU
• AMAR extensions to Multi-Physics:
• Hybrid Model of Radiation transport
• Hybrid Plasma Models
• Electron Kinetics
• Kinetic Solvers with Adaptive Mesh in Phase Space
• Linking Scattering Theory & Physical Kinetics
Acknowledgements
Collaborators and co-authors:
• Dr. Robert Arslanbekov (CFDRC)
• Dr. Anna Frolova (RAS, Moscow, Russia)
• Dr. Vladimir Aristov (RAS, Moscow, Russia)
• Dr. Sergey Zabelok (RAS, Moscow, Russia)
• Dr. Felix Tcheremissine (RAS, Moscow, Russia)
• Mr. Eswar Josyula (AFRL)
• Dr. Jonathan Burt (AFRL)
• Dr. Ranjan Mehta (CFDRC)
Financial support provided by DoD and NASA through
SBIR/STTR Projects
References
•
V.I. Kolobov, "Fokker Planck modeling of electron kinetics in plasmas and
v
semiconductors",
Computational Materials Science 28, 302 (2003).
• V.I. Kolobov and R.R. Arslanbekov, "Simulation of Electron Kinetics in Gas
Discharges", IEEE Trans. Plasma. Sci. 34, 895 (2006)
• V.I.Kolobov, R.R.Arslanbekov, V.V.Aristov, A.A.Frolova, S.A.Zabelok, “Unified
solver for rarefied and continuum flows with adaptive mesh and algorithm
refinement”, J. Comput. Phys. 223 (2007) 589
• V.I.Kolobov and R.R.Arslanbekov, “Towards Adaptive Kinetic-Fluid
Simulations of Weakly Ionized Plasmas”, J. Comput. Phys. 231 (2012) 839
• R.R.Arslanbekov, V.I.Kolobov, and A.A.Frolova, “Kinetic Solvers with Adaptive
Mesh in Phase Space”, Phys. Rev. E 88 (2013) 063301
• V. I. Kolobov, “Advances in Electron Kinetics and Theory of Gas Discharges”,
Phys. Plasmas 20, 101610 (2013)