Updating BCs Documentation

doc/content/source/bcs/CircuitDirichletPotential.md

@@ -10,7 +10,27 @@ documentation clear for users.
 
 ## Overview
 
-!! Replace these lines with information regarding the CircuitDirichletPotential object.
+`CircuitDirichletPotential` is a Dirichlet boundary condition for a potential based on Kirchoff's voltage law.
+
+The formulation of the potential at the wall is:
+
+\begin{equation}
+V_{source} + V_{cathode} = e \Gamma A R
+\end{equation}
+
+Where $V_{source}$ is driven the potential, $V_{cathode}$ is the potential at cathode,
+$\Gamma$ is the charged flux to the boundary, $e$ is the charge elemental, $A$ is the cross-sectional area of the plasma, and
+$R$ is the ballast resistance. When converting the density to log form and applying a scaling factor of the mesh / voltage,
+`CircuitDirichletPotential` is defined as
+
+\begin{equation}
+V_{source} + V_{cathode} = e N_{A} \Gamma \frac{A}{l_{c}^2} \frac{R}{V_{c}}
+\end{equation}
+
+Where $N_{A}$ is Avogadro's number, $l_{c}$ is the scaling factor of the mesh, and $V_{c}$ is the scaling factor of the potential.
+
+
+The charged flux is supplied as a [Postprocessor](syntax/Postprocessors/index.md) (usually the [`SideCurrent`](/postprocessors/SideCurrent.md) Postprocessor).
 
 ## Example Input File Syntax
 

doc/content/source/bcs/DCIonBC.md

@@ -1,20 +1,45 @@
 # DCIonBC
 
-!alert construction title=Undocumented Class
-The DCIonBC has not been documented. The content listed below should be used as a starting point for
-documenting the class, which includes the typical automatic documentation associated with a
-MooseObject; however, what is contained is ultimately determined by what is necessary to make the
-documentation clear for users.
-
 !syntax description /BCs/DCIonBC
 
 ## Overview
 
-!! Replace these lines with information regarding the DCIonBC object.
+`DCIonBC` is an electric field driven outflow boundary condition. `DCIonBC` assumes the electrostatic approximation for the electric field.
+
+The electrostatic electric field driven outflow is defined as
+
+\begin{equation}
+a =
+\begin{cases}
+1, & \text{sign}_{j} \mu_{j} \ \text{-} \nabla (V) \cdot \textbf{n} > 0\\
+0, & \text{sign}_{j} \mu_{j} \ \text{-} \nabla (V) \cdot \textbf{n} \leq 0\\
+\end{cases} \\[10pt]
+\Gamma_{j} \cdot \textbf{n} = a \ \text{sign}_{j} \mu_{j} \ \text{-} \nabla (V) \cdot \textbf{n} \ n_{j}
+\end{equation}
+
+Where $\Gamma$ is the outflow normal to the boundary, $\textbf{n}$ is the normal of the boundary, $\text{sign}_{j}$ indicates the advection behavior ($\text{+}1$ for positively charged species and $\text{-}1$ for negatively charged species), $\mu_{j}$ is the mobility coefficient, $n_{j}$ is the density, and $V$ is
+the potential. $a$ is defined such that the outflow is only defined when the drift velocity is direct towards the wall and zero otherwise. When converting the density to logarithmic form and applying a scaling
+factor of the mesh, the strong form for `DCIonBC` is defined as
+
+\begin{equation}
+a =
+\begin{cases}
+1, & \text{sign}_{j} \mu_{j} \ \text{-} \nabla (V) \cdot \textbf{n} > 0\\
+0, & \text{sign}_{j} \mu_{j} \ \text{-} \nabla (V) \cdot \textbf{n} \leq 0\\
+\end{cases} \\[10pt]
+\Gamma_{j} \cdot \textbf{n} = a \ \text{sign}_{j} \mu_{j} \ \text{-} \nabla (V / l_{c}) \cdot \textbf{n} \ \exp(N_{j})
+\end{equation}
+
+Where $N_{j}$ is the molar density of the specie in logarithmic form and
+$l_{c}$ is the scaling factor of the mesh.
+
 
 ## Example Input File Syntax
 
-!! Describe and include an example of how to use the DCIonBC object.
+An example of how to use `DCIonBC` can be found in the
+test file `mean_en.i`.
+
+!listing test/tests/1d_dc/mean_en.i block=BCs/OHm_physical
 
 !syntax parameters /BCs/DCIonBC
 

doc/content/source/bcs/DriftDiffusionDoNothingBC.md

@@ -10,7 +10,25 @@ documentation clear for users.
 
 ## Overview
 
-!! Replace these lines with information regarding the DriftDiffusionDoNothingBC object.
+`DriftDiffusionDoNothingBC` is an outflow boundary condition where the outflow at the
+boundary is equal to the bulk dift-diffusion equations.
+`DriftDiffusionDoNothingBC` assumes the electrostatic approximation for the electric field.
+
+The outflow is defined as
+
+\begin{equation}
+\Gamma_{j} \cdot \textbf{n} = \text{sign}_{j} \mu_{j} n_{j} \ \text{-} \nabla (V) \cdot \textbf{n} - D_{j} \nabla (n_{j}) \cdot \textbf{n}
+\end{equation}
+
+Where $\Gamma$ is the outflow normal to the boundary, $\textbf{n}$ is the normal of the boundary, $\text{sign}_{j}$ indicates the advection behavior ($\text{+}1$ for positively charged species, $\text{-}1$ for negatively charged species and $\text{0}$ for neutral species), $\mu_{j}$ is the mobility coefficient, $D_{j}$ is the diffusion coefficient, $n_{j}$ is the density, and $V$ is
+the potential. When converting the density to logarithmic form and applying a scaling factor of the mesh, the strong form for `DriftDiffusionDoNothingBC` is defined as
+
+\begin{equation}
+\Gamma_{j} \cdot \textbf{n} = \text{sign}_{j} \mu_{j} \exp(N_{j}) \ \text{-} \nabla (V / l_{c}) \cdot \textbf{n} - D_{j} \exp(N_{j}) \nabla (N_{j} / l_{c}) \cdot \textbf{n}
+\end{equation}
+
+Where $N_{j}$ is the molar density of the specie in logarithmic form and
+$l_{c}$ is the scaling factor of the mesh.
 
 ## Example Input File Syntax
 

doc/content/source/bcs/EconomouDielectricBC.md

@@ -1,20 +1,33 @@
 # EconomouDielectricBC
 
-!alert construction title=Undocumented Class
-The EconomouDielectricBC has not been documented. The content listed below should be used as a starting point for
-documenting the class, which includes the typical automatic documentation associated with a
-MooseObject; however, what is contained is ultimately determined by what is necessary to make the
-documentation clear for users.
-
 !syntax description /BCs/EconomouDielectricBC
 
 ## Overview
 
-!! Replace these lines with information regarding the EconomouDielectricBC object.
+`EconomouDielectricBC` is an type of [`PenaltyDirichletBC`](/bcs/ADPenaltyDirichletBC.md) for the potential on the boundary of a grounded ideal dielectric.
+
+The potential at the boundary of a grounded ideal dielectric is defined as
+
+\begin{equation}
+\frac{\epsilon_{i}}{d_{i}}\frac{\partial V_{i}}{\partial t} = e(\Gamma_{+} \cdot \textbf{n} -\Gamma_{e} \cdot \textbf{n})+\epsilon_{0}\frac{\partial (E \cdot \textbf{n}) }{\partial t} \\[10pt]
+E = \text{-} \nabla (V)\\[10pt]
+\Gamma_{e} \cdot \textbf{n}  = \frac{1}{4}\sqrt{\frac{8 k T_{e}}{\pi m_{e}}} \ n_e - \gamma \Gamma_{+} \cdot \textbf{n} \\[10pt]
+\Gamma_{+} \cdot \textbf{n}  = a \ \mu_{+} \ \text{-} \nabla (V) \cdot \textbf{n} \ n_{+} \\[10pt]
+a =
+\begin{cases}
+1, & \mu_{j} \ \text{-} \nabla (V) \cdot \textbf{n} > 0\\
+0, & \mu_{j} \ \text{-} \nabla (V) \cdot \textbf{n} \leq 0\\
+\end{cases}
+\end{equation}
+
+Where $\epsilon_{i}$ is the permittivity of the dielectric, $d_{i}$ is the thickness of the dielectric, $V_{i}$ is the voltage on the dielectric, $\textbf{n}$ is the normal to the boundary, $e$ is the elemental charge, $\epsilon_{0}$ is the permittivity of free space, and $E$ is the E-field normal to the dielectric. $\Gamma_{e}$ and $\Gamma_{+}$ are the electron and ion outflow flux and are defined with the [`SakiyamaElectronDiffusionBC`](/bcs/SakiyamaElectronDiffusionBC.md), [`SakiyamaSecondaryElectronBC`](/bcs/SakiyamaSecondaryElectronBC.md) and [`SakiyamaIonAdvectionBC`](/bcs/SakiyamaIonAdvectionBC.md) (please refer to those BC's for more information on the fluxes).
 
 ## Example Input File Syntax
 
-!! Describe and include an example of how to use the EconomouDielectricBC object.
+An example of how to use `EconomouDielectricBC` can be found in the
+test file `2D_RF_Plasma_actions.i`.
+
+!listing test/tests/DriftDiffusionAction/2D_RF_Plasma_actions.i block=BCs/potential_Dielectric
 
 !syntax parameters /BCs/EconomouDielectricBC
 

doc/content/source/bcs/ElectronAdvectionDoNothingBC.md

@@ -10,7 +10,25 @@ documentation clear for users.
 
 ## Overview
 
-!! Replace these lines with information regarding the ElectronAdvectionDoNothingBC object.
+`ElectronAdvectionDoNothingBC` is an outflow boundary condition where the outflow at the
+boundary is equal to the bulk election advection equation.
+`ElectronAdvectionDoNothingBC` assumes the electrostatic approximation for the electric field.
+
+The outflow is defined as
+
+\begin{equation}
+\Gamma_{e} \cdot \textbf{n} = \text{-} \mu_{e} n_{e} \ \text{-} \nabla (V) \cdot \textbf{n}
+\end{equation}
+
+Where $\Gamma$ is the outflow normal to the boundary, $\textbf{n}$ is the normal of the boundary, $\mu_{e}$ is the mobility coefficient, $n_{e}$ is the electron density, and $V$ is the potential. When converting the density to logarithmic form and applying a scaling
+factor of the mesh, the strong form for `ElectronAdvectionDoNothingBC` is defined as
+
+\begin{equation}
+\Gamma_{e} \cdot \textbf{n} = \text{-} \mu_{e} \exp(N_{e}) \ \text{-} \nabla (V / l_{c}) \cdot \textbf{n}
+\end{equation}
+
+Where $N_{j}$ is the molar density of the specie in logarithmic form and
+$l_{c}$ is the scaling factor of the mesh.
 
 ## Example Input File Syntax
 

doc/content/source/bcs/ElectronDiffusionDoNothingBC.md

@@ -10,7 +10,24 @@ documentation clear for users.
 
 ## Overview
 
-!! Replace these lines with information regarding the ElectronDiffusionDoNothingBC object.
+`ElectronDiffusionDoNothingBC` is an outflow boundary condition where the outflow at the
+boundary is equal to the bulk election diffusion equation.
+
+The outflow is defined as
+
+\begin{equation}
+\Gamma_{e} \cdot \textbf{n} = - D_{e} \nabla (n_{e}) \cdot \textbf{n}
+\end{equation}
+
+Where $\Gamma$ is the outflow normal to the boundary, $\textbf{n}$ is the normal of the boundary, $D_{e}$ is the diffusion coefficient, and $n_{e}$ is the electron density. When converting the density to logarithmic form and applying a scaling
+factor of the mesh, the strong form for `ElectronDiffusionDoNothingBC` is defined as
+
+\begin{equation}
+\Gamma_{e} \cdot \textbf{n} = - D_{e} \exp(N_{e}) \nabla (N_{e} / l_{c}) \cdot \textbf{n}
+\end{equation}
+
+Where $N_{e}$ is the molar density of the specie in logarithmic form and
+$l_{c}$ is the scaling factor of the mesh.
 
 ## Example Input File Syntax
 

doc/content/source/bcs/ElectronTemperatureDirichletBC.md

@@ -1,20 +1,32 @@
 # ElectronTemperatureDirichletBC
 
-!alert construction title=Undocumented Class
-The ElectronTemperatureDirichletBC has not been documented. The content listed below should be used as a starting point for
-documenting the class, which includes the typical automatic documentation associated with a
-MooseObject; however, what is contained is ultimately determined by what is necessary to make the
-documentation clear for users.
-
 !syntax description /BCs/ElectronTemperatureDirichletBC
 
 ## Overview
 
-!! Replace these lines with information regarding the ElectronTemperatureDirichletBC object.
+`ElectronTemperatureDirichletBC` is an type of [`PenaltyDirichletBC`](/bcs/ADPenaltyDirichletBC.md) for the electron temperature on the boundary.
+
+The electron temperature at the boundary is defined as
+
+\begin{equation}
+T_{e} = \frac{2}{3} \frac{n_{\varepsilon}}{n_{e}}
+\end{equation}
+
+Where $T_{e}$ is the electron temperature, $n_{\varepsilon}$ is the electron mean energy density, and $n_{e}$ is the electron density. When converting the density to logarithmic form,
+`ElectronTemperatureDirichletBC` is defined as
+
+\begin{equation}
+T_{e} = \frac{2}{3} \exp (N_{\varepsilon} - N_{e})
+\end{equation}
+
+Where $N$ is the molar density of the species in logarithmic form.
 
 ## Example Input File Syntax
 
-!! Describe and include an example of how to use the ElectronTemperatureDirichletBC object.
+An example of how to use `ElectronTemperatureDirichletBC` can be found in the
+test file `2D_RF_Plasma_actions.i`.
+
+!listing test/tests/DriftDiffusionAction/RF_Plasma_actions.i block=BCs/mean_en_physical_right
 
 !syntax parameters /BCs/ElectronTemperatureDirichletBC
 

doc/content/source/bcs/FieldEmissionBC.md

@@ -1,20 +1,60 @@
 # FieldEmissionBC
 
-!alert construction title=Undocumented Class
-The FieldEmissionBC has not been documented. The content listed below should be used as a starting point for
-documenting the class, which includes the typical automatic documentation associated with a
-MooseObject; however, what is contained is ultimately determined by what is necessary to make the
-documentation clear for users.
-
 !syntax description /BCs/FieldEmissionBC
 
 ## Overview
 
-!! Replace these lines with information regarding the FieldEmissionBC object.
+`FieldEmissionBC` is the outflow boundary condition assuming the the electron current density is defined by field emission.
+
+Using a Fowler-Nordheim calculation for the field emission, the electron current density is defined as
+
+\begin{equation}
+a =
+\begin{cases}
+1, & \nabla (V) \cdot \textbf{n} > 0\\
+0, & \nabla (V) \cdot \textbf{n} \leq 0\\
+\end{cases} \\[10pt]
+\textbf{J}_{\textbf{e}} \cdot \textbf{n} = a \ \phi^{-1} \ F^{2} \exp \left[\text{-}v(f) \ b \ \phi^{3/2} / F \right] \\[10pt]
+F =  \left( 1-a \right) \gamma \left( \text{-} \nabla V \right) \cdot \textbf{n}  \\[10pt]
+a = 1.541434e\text{-}6 A \ eV \ V^{-2} \\[10pt]
+b = 6.830890e9 eV^{-3/2} \ V \ m^{-1} \\[10pt]
+v(f) = 1 - f + \frac{1}{6}f\ln f \\[10pt]
+f = c \frac{F}{\phi^{2}} \\[10pt]
+c = 1.439964e\text{-}9 \ eV^{2} \ V^{-1} \ m
+\end{equation}
+
+Where $\textbf{J}_{\textbf{e}}$ is the electron current density, $a$ is the first Fowler–Nordheim constant, $\phi$ is the local work function, $F$ is the local field, $b$ is the second Fowler–Nordheim constant, $v(f)$ is a correction factor that depends on the scaled barrier field ($f$), $\textbf{n}$ is the normal of the boundary, $\gamma$ is the field enhancement factor, and $V$ is the potential. $a$ is defined such that the outflow is only defined when the drift velocity is direct towards the wall and zero otherwise. With the electron current density, the outward electron flux is defined as
+
+\begin{equation}
+\Gamma_{i} \cdot \textbf{n}  = \text{sign}_{i} \mu_{i} \ \text{-} \nabla (V) n_{i} - D_{i} \nabla (n_{i}) \\[10pt]
+\Gamma_{e} \cdot \textbf{n}  = \frac{2 (1 - a)}{1 + r} (\text{-} (\textbf{J}_{\textbf{e}} \cdot \textbf{n})/e - \gamma_{se} \Gamma_{i} \cdot \textbf{n})
+\end{equation}
+
+Where $\Gamma$ is the outflow normal to the boundary, $\mu_{i}$ is the mobility coefficient of the ions, $\text{sign}_{i}$ indicates the advection behavior ($\text{+}1$ for positively charged species and $\text{-}1$ for negatively charged species), $n_{i}$ is the ion density, $D_{i}$ is the diffusion coefficient of ions, $e$ is the elemental charge, and $\gamma_{se}$ is the ion induced secondary electron coefficient. $r$ is defined as the fraction of particles reflected by the surface.
+
+When converting the density to log form and applying a scaling factor of the mesh and voltage, the changes to `FieldEmissionBC` are defined as
+
+
+\begin{equation}
+F =  \left( 1-a \right) \gamma \left( \text{-} \nabla V / l_{c} \right) \cdot \textbf{n}  \\[10pt]
+a = 1.541434e\text{-}6 * V_{c}^{2} \\[10pt]
+b = 6.830890e9 eV^{-3/2} / V_{c} \\[10pt]
+c = 1.439964e\text{-}9 * V_{c}\\[10pt]
+\Gamma_{i} \cdot \textbf{n}  = \text{sign}_{i} \mu_{i} \ \text{-} \nabla (V / l_{c}) \exp(N_{i}) - D_{i} \exp(N_{i}) \nabla (N_{i} / l_{c}) \\[10pt]
+\Gamma_{e} \cdot \textbf{n}  = \frac{2 (1 - a)}{1 + r} (\text{-} (\textbf{J}_{\textbf{e}} \cdot \textbf{n})/ (e / N_{A}) - \gamma_{se} \Gamma_{i} \cdot \textbf{n})
+\end{equation}
+
+Where $N_{i}$ is the molar density of the specie in log form, $N_{A}$ is Avogadro's number, $V_{c}$ is the scaling factor of the potential, and $l_{c}$ is the scaling factor of the mesh.
+
+
 
 ## Example Input File Syntax
 
-!! Describe and include an example of how to use the FieldEmissionBC object.
+An example of how to use `FieldEmissionBC` can be found in the
+test file `field_emission.i`.
+
+!listing test/tests/field_emission/field_emission.i block=BCs/FieldEmission_left
+
 
 !syntax parameters /BCs/FieldEmissionBC
 

doc/content/source/bcs/HagelaarElectronAdvectionBC.md

@@ -1,20 +1,46 @@
 # HagelaarElectronAdvectionBC
 
-!alert construction title=Undocumented Class
-The HagelaarElectronAdvectionBC has not been documented. The content listed below should be used as a starting point for
-documenting the class, which includes the typical automatic documentation associated with a
-MooseObject; however, what is contained is ultimately determined by what is necessary to make the
-documentation clear for users.
-
 !syntax description /BCs/HagelaarElectronAdvectionBC
 
 ## Overview
 
-!! Replace these lines with information regarding the HagelaarElectronAdvectionBC object.
+`HagelaarElectronAdvectionBC` is an electric field driven outflow boundary condition.
+`HagelaarElectronAdvectionBC` assumes the electrostatic approximation for the electric field.
+
+The electrostatic electric field driven outflow is defined as
+
+\begin{equation}
+a =
+\begin{cases}
+1, & \mu_{e} \ \nabla (V) \cdot \textbf{n} > 0\\
+0, & \mu_{e} \ \nabla (V) \cdot \textbf{n} \leq 0\\
+\end{cases} \\[10pt]
+\Gamma_{e} \cdot \textbf{n} = \frac{1-r_{e}}{1+r_{e}} \left[ -(2 a_{e}-1) \ \mu_{e} \text{-} \nabla (V) \cdot \textbf{n} \ n_{e} \right]
+\end{equation}
+
+Where $\Gamma$ is the outflow normal to the boundary, $n$ is the normal of the boundary,
+$\mu_{e}$ is the mobility coefficient, $n_{e}$ is the electron density, and $V$ is
+the potential. $a$ is defined such that the outflow is only defined when the drift velocity is direct towards the wall and zero otherwise. $r$ is defined as the fraction of particles reflected by the surface. When converting the density to log form and applying a scaling
+factor of the mesh, the strong form for `HagelaarElectronAdvectionBC` is defined as
+
+\begin{equation}
+a =
+\begin{cases}
+1, & \mu_{e} \ \nabla (V) \cdot \textbf{n} > 0\\
+0, & \mu_{e} \ \nabla (V) \cdot \textbf{n} \leq 0\\
+\end{cases} \\[10pt]
+\Gamma_{e} \cdot \textbf{n} = \frac{1-r_{e}}{1+r_{e}} \left[ -(2 a_{e}-1) \ \mu_{e} \text{-} \nabla (V / l_{c}) \cdot \textbf{n} \ \exp(N_{e}) \right]
+\end{equation}
+
+Where $N_{e}$ is the molar density of the specie in log form and
+$l_{c}$ is the scaling factor of the mesh.
 
 ## Example Input File Syntax
 
-!! Describe and include an example of how to use the HagelaarElectronAdvectionBC object.
+An example of how to use `HagelaarElectronAdvectionBC` can be found in the
+test file `NonlocalPotentialBCWithSchottky.i`.
+
+!listing test/tests/1d_dc/NonlocalPotentialBCWithSchottky.i block=BCs/em_physical_right
 
 !syntax parameters /BCs/HagelaarElectronAdvectionBC