4.4. Boundary Conditions - Particle Solver
Within the parameter file it is possible to define different particle boundary conditions. The number of boundaries is defined by
Part-nBounds=2
Part-Boundary1-SourceName=BC_OPEN
Part-Boundary1-Condition=open
Part-Boundary2-SourceName=BC_WALL
Part-Boundary2-Condition=reflective
Part-Boundary2-SurfaceModel=2
The Part-Boundary1-SourceName= corresponds to the name given during the preprocessing step with HOPR. The available conditions
(Part-Boundary1-Condition=) are described in the table below.
Condition |
Description |
|---|---|
|
Every particle crossing the boundary will be deleted. |
|
A perfect specular reflection, without sampling of particle impacts. |
|
Allows the definition of specular and diffuse reflection, Section Reflective Wall |
|
Allows the definition of rotational periodicity, Section Rotational Periodicity |
For rot_periodic exactly two corresponding boundaries must be defined. Every particle crossing one of these boundaries will be
inserted at the corresponding other boundary that is rotationally shifted.
4.4.1. Reflective Wall
A reflective boundary can be defined with
Part-Boundary2-SourceName=BC_WALL
Part-Boundary2-Condition=reflective
A perfect specular reflection is performed, if no other parameters are given. Gas-surface interactions can be modelled with the extended Maxwellian model [11], using accommodation coefficients of the form
$$\alpha = \frac{E_i-E_r}{E_i - E_w}$$
where $i$, $r$ and $w$ denote the incident, reflected and wall energy, respectively. The coefficient MomentumACC is utilized to
decide whether a diffuse (MomentumACC $>R$) or specular reflection (MomentumACC $<R$) occurs upon particle impact, where
$R=[0,1)$ is a random number. Separate accommodation coefficients can be defined for the translation (TransACC), rotational
(RotACC), vibrational (VibACC) and electronic energy (ElecACC) accommodation at a constant wall temperature [K].
Part-Boundary2-MomentumACC=1.
Part-Boundary2-WallTemp=300.
Part-Boundary2-TransACC=1.
Part-Boundary2-VibACC=1.
Part-Boundary2-RotACC=1.
Part-Boundary2-ElecACC=1.
An additional option Part-Boundary2-SurfaceModel is available, that is used for heterogeneous reactions (reactions that have reactants
in two or more phases) or secondary electron emission models. These models are described in detail in Section Surface Chemistry.
4.4.1.1. Wall movement (Linear & rotational)
Additionally, a linear wall velocity [m/s] can be given
Part-Boundary2-WallVelo=(/0,0,100/)
In the case of rotating walls the -RotVelo flag, a rotation frequency [Hz], and the rotation axis (x=1, y=2, z=3) must be set.
Note that the definition of the rotational direction is defined by the sign of the frequency using the right-hand rule.
Part-Boundary2-RotVelo = T
Part-Boundary2-RotFreq = 100
Part-Boundary2-RotAxis = 3
The wall velocity will then be superimposed onto the particle velocity.
4.4.1.2. Linear temperature gradient
A linear temperature gradient across a boundary can be defined by supplying a second wall temperature and the start and end vector as well as an optional direction to which the gradient shall be limited (default: 0, x = 1, y = 2, z = 3)
Part-Boundary2-WallTemp2 = 500.
Part-Boundary2-TempGradStart = (/0.,0.,0./)
Part-Boundary2-TempGradEnd = (/1.,0.,1./)
Part-Boundary2-TempGradDir = 0
In the default case of the TempGradDir = 0, the temperature will be interpolated between the start and end vector, where the
start vector corresponds to the first wall temperature WallTemp, and the end vector to the second wall temperature WallTemp2.
Position values (which are projected onto the temperature gradient vector) beyond the gradient vector utilize the first (Start)
and second temperature (End) as the constant wall temperature, respectively. In the special case of TempGradDir = 1/2/3, the
temperature gradient will only be applied along the chosen the direction. As oppposed to the default case, the positions of the
surfaces are not projected onto the gradient vector before checking wether they are inside the box spanned by TempGradStart and
TempGradEnd. The applied surface temperature is output in the DSMCSurfState as Wall_Temperature for verification.
4.4.1.3. Radiative equilibrium
Another option is to adapt the wall temperature based on the heat flux assuming that the wall is in radiative equilibrium. The temperature is then calculated from
$$ q_w = \varepsilon \sigma T_w^4,$$
where $\varepsilon$ is the radiative emissivity of the wall (default = 1) and $\sigma = \pu{5.67E-8 Wm^{-2}K^{-4}}$ is the Stefan-Boltzmann constant. The adaptive boundary is enabled by
Part-AdaptWallTemp = T
Part-Boundary1-UseAdaptedWallTemp = T
Part-Boundary1-RadiativeEmissivity = 0.8
If provided, the wall temperature will be adapted during the next output of macroscopic variables, where the heat flux calculated
during the preceding sampling period is utilized to determine the side-local temperature. The temperature is included in the
State file and thus available during a restart of the simulation. The surface output (in DSMCSurfState) will additionally
include the temperature distribution in the Wall_Temperature variable (see Section Particle Flow and Surface Sampling).
To continue the simulation without further adapting the temperature, the first flag has to be disabled (Part-AdaptWallTemp = F).
It should be noted that the the adaptation should be performed multiple times to achieve a converged temperature distribution.
4.4.2. Rotational Periodicity
The rotational periodic boundary condition can be used in order to reduce the computational effort in case of an existing
rotational periodicity. In contrast to symmetric boundary conditions, a macroscopic flow velocity in azimuthal direction can be
simulated (e.g. circular flow around a rotating cylinder). Exactly two corresponding boundaries must be defined by setting
rot_periodic as the BC condition and the rotating angle for each BC. Multiple pairs of boundary conditions with different angles
can be defined.
Part-Boundary1-SourceName = BC_Rot_Peri_plus
Part-Boundary1-Condition = rot_periodic
Part-Boundary1-RotPeriodicAngle = 90.
Part-Boundary2-SourceName = BC_Rot_Peri_minus
Part-Boundary2-Condition = rot_periodic
Part-Boundary2-RotPeriodicAngle = -90.
CAUTION! The correct sign for the rotating angle must be determined. The position of particles that cross one rotational periodic boundary is tranformed according to this angle, which is defined by the right-hand rule and the rotation axis:
Part-RotPeriodicAxi = 1 ! (x = 1, y = 2, z = 3)
The usage of rotational periodic boundary conditions is limited to cases, where the rotational periodic axis is one of the three Cartesian coordinate axis (x, y, z) with its origin at (0, 0, 0). It is also possible to define several segments with different rotation angles. Exactly two corresponding BCs must be defined for each segment. In addition, the minimum and maximum coordinates along the rotation axis must be chosen for each BC to define the position of the segment. In the following example we have two segments. One between BCs 1 and 2 and one between BCs 4 and 5
Part-Boundary1-RotPeriodicMin=-1.
Part-Boundary1-RotPeriodicMax=1.
Part-Boundary2-RotPeriodicMin=-1.
Part-Boundary2-RotPeriodicMax=1.
Part-Boundary4-RotPeriodicMin=1.
Part-Boundary4-RotPeriodicMax=3.
Part-Boundary5-RotPeriodicMin=1.
Part-Boundary5-RotPeriodicMax=3.
4.4.3. Porous Wall / Pump
The porous boundary condition uses a removal probability to determine whether a particle is deleted or reflected at the boundary.
The main application of the implemented condition is to model a pump, according to [12]. It is defined by giving the
number of porous boundaries and the respective boundary number (BC=2 corresponds to the BC_WALL boundary defined in the
previous section) on which the porous condition is.
Surf-nPorousBC=1
Surf-PorousBC1-BC=2
Surf-PorousBC1-Type=pump
Surf-PorousBC1-Pressure=5.
Surf-PorousBC1-PumpingSpeed=2e-9
Surf-PorousBC1-DeltaPumpingSpeed-Kp=0.1
Surf-PorousBC1-DeltaPumpingSpeed-Ki=0.0
Currently, two porous BC types are available, pump and sensor. For the former, the removal probability is determined through
the given pressure [Pa] at the boundary. A pumping speed can be given as a first guess, however, the pumping speed $S$ [$m^3/s$]
will be adapted if the proportional factor ($K_{\mathrm{p}}$, DeltaPumpingSpeed-Kp) is greater than zero
$$ S^{n+1}(t) = S^{n}(t) + K_{\mathrm{p}} \Delta p(t) + K_{\mathrm{i}} \int_0^t \Delta p(t’) dt’,$$
where $\Delta p$ is the pressure difference between the given pressure and the actual pressure at the pump. An integral factor
($K_{\mathrm{i}}$, DeltaPumpingSpeed-Ki) can be utilized to mimic a PI controller. The proportional and integral factors are
relative to the given pressure. However, the integral factor has not yet been thoroughly tested. The removal probability $\alpha$
is then calculated by
$$\alpha = \frac{S n \Delta t}{N_{\mathrm{pump}} w} $$
where $n$ is the sampled, cell-local number density and $N_{\mathrm{pump}}$ is the total number of impinged particle at the pump during the previous time step. $\Delta t$ is the time step and $w$ the weighting factor. The pumping speed $S$ is only adapted if the resulting removal probability $\alpha$ is between zero and unity. The removal probability is not species-specific.
To reduce the influence of statistical fluctuations, the relevant macroscopic values (pressure difference $\Delta p$ and number density $n$) can be sampled for $N$ iterations by defining (for all porous boundaries)
AdaptiveBC-SamplingIteration=10
A porous region on the specified boundary can be defined. At the moment, only the circular option is implemented. The origin of
the circle/ring on the surface and the radius have to be given. In the case of a ring, a maximal and minimal radius is required
(-rmax and -rmin, respectively), whereas for a circle only the input of maximal radius is sufficient.
Surf-PorousBC1-Region=circular
Surf-PorousBC1-normalDir=1
Surf-PorousBC1-origin=(/5e-6,5e-6/)
Surf-PorousBC1-rmax=2.5e-6
The absolute coordinates are defined as follows for the respective normal direction.
Normal Direction |
Coordinates |
|---|---|
x (=1) |
(y,z) |
y (=2) |
(z,x) |
z (=3) |
(x,y) |
Using the regions, multiple pumps can be defined on a single boundary. Additionally, the BC can be used as a sensor by defining the respective type:
Surf-PorousBC1-BC=3
Surf-PorousBC1-Type=sensor
Together with a region definition, a pump as well as a sensor can be defined on a single and/or multiple boundaries, allowing e.g. to determine the pressure difference between the pump and a remote area of interest.
4.4.4. Surface Chemistry
Modelling of reactive surfaces is enabled by setting Part-BoundaryX-Condition=reflective and an
appropriate particle boundary surface model Part-BoundaryX-SurfaceModel.
The available conditions (Part-BoundaryX-SurfaceModel=) are described in the table below.
Model |
Description |
|---|---|
0 (default) |
Standard extended Maxwellian scattering |
1 |
Empirical modelling of sticking coefficient/probability |
5 |
Secondary electron emission as given by Ref. [13]. |
7 |
Secondary electron emission due to ion impact (SEE-I with $Ar^{+}$ on different metals) as used in Ref. [14] and given by Ref. [15] with a default yield of 13 %. |
8 |
Secondary electron emission due to ion impact (SEE-E with $e^{-}$ on dielectric surfaces) as used in Ref. [16] and given by Ref. [17]. |
9 |
Secondary electron emission due to ion impact (SEE-I with $Ar^{+}$) with a constant yield of 1 %. Emitted electrons have an energy of 6.8 eV upon emission. |
10 |
Secondary electron emission due to ion impact (SEE-I with $Ar^{+}$ on copper) as used in Ref. [18] originating from [19] |
11 |
Secondary electron emission due to electron impact (SEE-E with $e^{-}$ on quartz (SiO$_{2}$)) as described in Ref. [20] originating from [21] |
For surface sampling output, where the surface is split into, e.g., $3\times3$ sub-surfaces, the following parameters mus be set
BezierSampleN = 3
DSMC-nSurfSample = 3
Part-WriteMacroSurfaceValues = T
Particles-DSMC-CalcSurfaceVal = T
Part-IterationForMacroVal = 200
where BezierSampleN=DSMC-nSurfSample. In this example, sampling is performed over and every 200 iterations.
4.4.4.1. Empirical model for a sticking coefficient
To model the sticking of gas particles on cold surfaces, an empirical model is available, which is based on experimental measurements. The sticking coefficient is modelled through the product of a non-bounce probability $B(\alpha)$ and a condensation probability $C(\alpha,T)$
$$ p_s (\alpha,T) = B(\alpha) C(\alpha,T) $$
The non-bounce probability introduces a linear dependency on the impact angle $\alpha$
$$ B(\alpha) = \begin{cases} 1 , & |\alpha| < \alpha_{\mathrm{B}} \ \dfrac{90^{\circ}-|\alpha|}{90^{\circ}-|\alpha_{\mathrm{B}}|} , & \alpha_{\mathrm{B}} \leq |\alpha| \leq 90^{\circ} \ \end{cases} $$
$\alpha_{\mathrm{B}}$ is a model-dependent cut-off angle. The condensation probability introduces a linear dependency on the surface temperature $T$
$$ C(\alpha, T) = \begin{cases} 1 , & T < T_1 \ \dfrac{T_2(\alpha)-T}{T_2(\alpha)-T_1(\alpha)} , & T_1 \leq T \leq T_2 \ 0 , & T > T_2 \ \end{cases} $$
The temperature limits $T_1$ and $T_2$ are model parameters and can be given for different impact angle ranges defined by the maximum impact angle $\alpha_{\mathrm{max}}$. These model parameters are read-in through the species database and have to be provided in the /Surface-Chemistry/StickingCoefficient dataset in the following format (example values):
$\alpha_{\mathrm{max}}$ [deg] |
$\alpha_{\mathrm{B}}$ [deg] |
$T_1$ [K] |
$T_2$ [K] |
|---|---|---|---|
45 |
80 |
50 |
100 |
90 |
70 |
20 |
50 |
In this example, within impact angles of $0°\leq\alpha\leq45°$, the model parameters of the first row will be used and for $45°<\alpha\leq90°$ the second row. The number of rows is not limited. The species database is read-in by
Particles-Species-Database = Species_Database.h5
As additional output, the cell-local sticking coefficient will be added to the sampled surface output. A particle sticking to the surface will be deleted and its energy added to the heat flux sampling. This model can be combined with the linear temperature gradient and radiative equilibrium modelling as described in Section Reflective Wall.
4.4.4.2. Secondary Electron Emission (SEE)
Different models are implemented for secondary electron emission that are based on either electron or ion bombardment, depending on the surface material. All models require the specification of the electron species that is emitted from the surface via
Part-SpeciesA-PartBoundB-ResultSpec = C
where electrons of species C are emitted from boundary B on the impact of species A.
4.4.4.2.1. Model 5
The model by Levko [13] can be applied for copper electrodes for electron and ion bombardment and is activated via
Part-BoundaryX-SurfaceModel=5. For ions, a fixed emission yield of 0.02 is used and for electrons an energy-dependent function is
employed.
4.4.4.2.2. Model 7
The model by Depla [15] can be used for various metal surfaces and features a default emission yield of 13 % and is
activated via Part-BoundaryX-SurfaceModel=7 and is intended for the impact of $Ar^{+}$ ions. For more details, see the original
publication.
The emission yield and energy can be varied for this model by setting
SurfModEmissionYield = 1.45 ! ratio of emitted electron flux vs. impacting ion flux [-]
SurfModEmissionEnergy = 6.8 ! [eV]
respectively. The emission yield represents the ratio of emitted electrons vs. impacting ions and the emission energy is given in electronvolts. If the energy is not set, the emitted electron will have the same velocity as the impacting ion.
Additionally, a uniform energy distribution function for the emitted electrons can be set via
SurfModEnergyDistribution = uniform-energy
which will scale the energy of the emitted electron to fit a uniform distribution function.
4.4.4.2.3. Model 8
The model by Morozov [17] can be applied for dielectric surfaces and is activated via
Part-BoundaryX-SurfaceModel=8 and has an additional parameter for setting the reference electron temperature (see model for
details) via Part-SurfaceModel-SEE-Te, which takes the electron temperature in Kelvin as input (default is 50 eV, which
corresponds to 11604 K).
The emission yield is determined from an energy-dependent function.
The model can be switched to an automatic determination of the bulk electron temperature via
Part-SurfaceModel-SEE-Te-automatic = T ! Activate automatic bulk temperature calculation
Part-SurfaceModel-SEE-Te-Spec = 2 ! Species ID used for automatic temperature calculation (must correspond to electrons)
where the species ID must be supplied, which corresponds to the electron species for which, during Part-AnalyzeStep, the global
translational temperature is determined and subsequently used to adjust the energy dependence of the SEE model. The global (bulk)
electron temperature is written to PartAnalyze.csv as XXX-BulkElectronTemp-[K].
4.4.4.2.4. Model 10
An energy-dependent model of secondary electron emission due to $Ar^{+}$ ion impact on a copper cathode as used in
Ref. [18] originating from [19] is
activated via Part-BoundaryX-SurfaceModel=10. For more details, see the original publications.
4.4.4.2.5. Model 11
An energy-dependent model (linear and power fit of measured SEE yields) of secondary electron emission due to $e^{-}$ impact on a
quartz (SiO$_{2}$) surface as described in Ref. [20] originating from [21] is
activated via Part-BoundaryX-SurfaceModel=11. For more details, see the original publications.
4.4.5. Deposition of Charges on Dielectric Surfaces
Charged particles can be absorbed (or reflected and leave their charge behind) at dielectric surfaces
when using the deposition method cell_volweight_mean. The boundary can be used by specifying
```
Part-Boundary1-Condition = reflective
Part-Boundary1-Dielectric = T
Part-Boundary1-NbrOfSpeciesSwaps = 3
Part-Boundary1-SpeciesSwaps1 = (/1,0/) ! e-
Part-Boundary1-SpeciesSwaps2 = (/2,2/) ! Ar
Part-Boundary1-SpeciesSwaps3 = (/3,2/) ! Ar+
```
which sets the boundary dielectric and the given species swap parameters effectively remove electrons ($e^{-}$) on impact, reflect $Ar$ atoms and neutralize $Ar^{+}$ ions by swapping these to $Ar$ atoms. Note that currently only singly charged particles can be handled this way. When multiple charged particles would be swapped, their complete charge mus be deposited at the moment.
The boundary must also be specified as an inner boundary via
BoundaryName = BC_INNER
BoundaryType = (/100,0/)
or directly in the hopr.ini file that is used for creating the mesh.