Chemistry Reference
In-Depth Information
9.1.5.3 Particle Models
An alternative approach for a kinetic description of the behavior of plasmas uses the
techniques of particle simulation. Particle simulations of plasmas can be divided into
two general categories, namely, Monte Carlo (MC) simulations and particle-in-cell
(PIC) methods [37-40]. Both approaches use very similar techniques to advance
particles under the action of external and interparticle forces.
In MC simulations, the trajectories of test particles (usually charged particles)
between collisions with the background gas particles are obtained by the numerical
integration of the equation of motion for each test particle. The dynamics during a
collision process is not resolved in the MC simulation and the collisions that result
fromshort-rangeforcesareassumedtobebinary.Thetime(orlength)ofthefreeflight
between two collisions is calculated by generating a random number and relating it
to the collision frequency (or mean free path) known in dependence on the particle
energy at each position and time. Sequences of random numbers are further used to
determine, for example, the type of collision process that occurred after a free flight,
taking into account the cross-section values of the different collision processes at
the energy of the colliding test particle, the change of the vectorial velocity during a
collision process, as well as the incident of an interaction with the walls of the plasma
reactor and the corresponding change of the velocity in the case of a reflection event.
Monte Carlo simulations can easily be applied to multidimensional problems and
complicated geometries, but this advantage is balanced by the necessity to treat a
large number of test particles in any simulation to get sufficient accuracy.
PIC simulations are most commonly used to simulate fully ionized plasmas. In
the PIC approach, the equations of motion for superparticles, representing a large
number of real electrons and ions each, are integrated for each superparticle, taking
into account the electromagnetic fields calculated on a numerical grid. Collisions are
of long range because the interparticle forces are Coulomb forces between charge
carriers. These forces can be included in the simulation by explicitly summing the
interparticle forces and adding the net force given by the applied electromagnetic
field, as it is done in the so-called particle-particle (PP) method.
An alternative procedure of including Coulomb forces between charge carriers
is performed in the so-called particle-mesh (PM) method. Here, the densities and
current densities of electrons and ions are summed on the numerical grid to obtain
the electric charge density and total current density. Knowing these quantities, the
electric and magnetic fields are then determined by solving Maxwell's equations
on the same numerical grid. The PM method is computationally faster than the PP
method at the cost of a resolution loss for the potential and fields.
The Newtonian equations of motion in the PIC are most commonly solved by the
finite-difference Leapfrog approximation scheme according to [38]
x
n
+
1
i
v
n
+
1
/
2
i
=
x
i
+
t
(9.49)
F
(
m
x
i
,
t
n
v
n
+
1
/
2
i
=
v
n
−
1
/
2
i
+
t
,
(9.50)
an explicit time-stepping scheme that is second-order accurate in both space and time.
