Boundary conditions

The boundary conditions (BCs) are essential to FESTIM simulations. They describe the mathematical problem at the boundaries of the simulated domain. If no BC is set on a boundary, it is assumed that the flux is null. This is also called a symmetry BC.

All boundary conditions in FESTIM require a surface subdomain. The subdomain can be a surface, an edge or a point and can be defined with the festim.SurfaceSubdomain class. See the Surface Subdomains section for more information on how to define subdomains.

Hydrogen transport BCs

Some BCs are specific to hydrogen transport. FESTIM provides a handful of convenience classes making things a bit easier for the users.

Imposing the concentration

The concentration of a defined species can be imposed on boundaries with festim.FixedConcentrationBC.

from festim import FixedConcentrationBC, Species, SurfaceSubdomain

boundary = SurfaceSubdomain(id=1)
H = Species(name="Hydrogen")

my_bc = FixedConcentrationBC(subdomain=boundary, value=10, species=H)

The imposed concentration can be dependent on space, time and temperature:

from festim import FixedConcentrationBC, Species, SurfaceSubdomain

boundary = SurfaceSubdomain(id=1)
H = Species(name="Hydrogen")

my_custom_value = lambda x, t, T: 10 + x[0]**2 + t + T

my_bc = FixedConcentrationBC(subdomain=boundary, value=my_custom_value, species=H)

Note

When defining custom functions for values, only the arguments x, t and T can be defined. Spatial coordinates can be referred to by their indices, such as x[0], x[1], and x[2], regardless of the coordinate system used. Time dependence must use t, and T for temperature dependence.

Imposing a particle flux

When a particle flux needs to be imposed on a boundary, use the festim.ParticleFluxBC class.

from festim import ParticleFluxBC, Species, SurfaceSubdomain

boundary = SurfaceSubdomain(id=1)
H = Species(name="Hydrogen")

my_flux_bc = ParticleFluxBC(subdomain=boundary, value=2, species=H)

As for the fixed concentration boundary condition, the flux can be dependent on space, time and temperature. But for particle fluxes, the values can also be dependent on a species’ concentration:

from festim import ParticleFluxBC, Species, SurfaceSubdomain

boundary = SurfaceSubdomain(id=1)
H = Species(name="Hydrogen")

my_custom_value = lambda t, c: 10*t**2 + 2*c

my_flux_bc = ParticleFluxBC(
    subdomain=boundary,
    value=my_custom_value,
    species=H,
    species_dependent_value={"c": H},
)

Note

The species_dependent_value argument requires a dictionary to be passed, mapping any arguments in the custom function given to value, to any species defined.

For instance with three species A, B and C, the dictionary can be defined as:

from festim import Species

A = Species(name="A")
B = Species(name="B")
C = Species(name="C")

my_custom_value = lambda c_A, c_B, c_C: 2*c_A + 3*c_B + 4*c_C

species_dependent_value = {"c_A": A, "c_B": B, "c_C": C}

Sievert’s law of solubility

Impose the concentration of a species as \(c_\mathrm{m} = S(T) \sqrt{P}\) where \(S\) is the Sievert’s solubility and \(P\) is the partial pressure of the species on this surface (see festim.SievertsBC).

from festim import SievertsBC, SurfaceSubdomain, Species

boundary = SurfaceSubdomain(id=1)
H = Species(name="Hydrogen")

custom_pressure_value = lambda t: 2 + t

my_bc = SievertsBC(subdomain=3, S_0=2, E_S=0.1, species=H, pressure=custom_pressure_value)

Henry’s law of solubility

Similarly, the the concentration of a species can be set from Henry’s law of solubility \(c_\mathrm{m} = K_H P\) where \(K_H\) is the Henry solubility (see festim.HenrysBC).

from festim import HenrysBC, SurfaceSubdomain, Species

boundary = SurfaceSubdomain(id=1)
H = Species(name="Hydrogen")

pressure_value = lambda t: 5 * t

my_bc = HenrysBC(subdomain=3, H_0=1.5, E_H=0.2, species=H, pressure=pressure_value)

Surface reactions

Surface reactions on boundary can be defined with the festim.SurfaceReactionBC class.

The surface reaction class can be used to impose dissociation and recombination reactions on the surface of the material. A reaction is defined by specifying the reactants, products, and forward/backward rate constants. For example:

from festim import Species, SurfaceReactionBC, SurfaceSubdomain

boundary = SurfaceSubdomain(id=1)
A = Species("A")
B = Species("B")
C = Species("C")

my_bc = SurfaceReactionBC(
    reactant=[A, B],
    gas_pressure=1e5,
    k_r0=1,
    E_kr=0.1,
    k_d0=1e-5,
    E_kd=0.1,
    subdomain=boundary,
)

The net reaction rate is:

\[R = K_r c_A c_B - K_d c_C\]

where \(K_r\) and \(K_d\) are the temperature-dependent forward and backward rate constants, respectively, and \(c_A\), \(c_B\), and \(c_C\) are the concentrations of species A, B, and C at the surface. From this FESTIM automatically applies the corresponding fluxes as Neumann boundary conditions (see festim.ParticleFluxBC).

Recombination and Dissociation

Hydrogen recombination/dissociation can be modelled as:

\[\mathrm{H + H} \overset{K_r}{\underset{K_d}{\rightleftharpoons}} \mathrm{H_2}\]

If the partial pressure of \(\mathrm{H}_2\) is known or assumed constant, the product species does not need to be included explicitly, and the flux simplifies to:

\[\mathbf{J}_{\mathrm{H}} \cdot \mathbf{n} = K_r c_{\mathrm{H}}^2 - K_d P_{\mathrm{H_2}}\]

Both rate coefficients can follow Arrhenius laws.

from festim import Species, SurfaceReactionBC, SurfaceSubdomain

boundary = SurfaceSubdomain(id=1)
H = Species("H")

my_bc = SurfaceReactionBC(
    reactant=[H, H],
    gas_pressure=1e5,
    k_r0=1,
    E_kr=0.1,
    k_d0=1e-5,
    E_kd=0.1,
    subdomain=boundary,
)

Multiple isoptopes

Surface reactions can involve multiple hydrogen isotopes, enabling the modelling of more complex interactions between species. For example, in a system with both mobile hydrogen and tritium, various molecular recombination pathways may occur at the surface, resulting in the formation of \(\mathrm{H_2}\), \(\mathrm{T_2}\), and \(\mathrm{HT}\):

\[\mathrm{H + H} \rightleftharpoons \mathrm{H_2}, \quad \mathrm{T + T} \rightleftharpoons \mathrm{T_2}, \quad \mathrm{H + T} \rightleftharpoons \mathrm{HT}\]

from festim import Species, SurfaceReactionBC, SurfaceSubdomain

boundary = SurfaceSubdomain(id=1)
H = Species("H")
T = Species("T")

reac1 = SurfaceReactionBC(
    reactant=[H, H],
    gas_pressure=1e6,
    k_r0=1.0,
    E_kr=0.1,
    k_d0=0.5,
    E_kd=0.1,
    subdomain=boundary,
)
reac2 = SurfaceReactionBC(
    reactant=[T, T],
    gas_pressure=1e6,
    k_r0=1.0,
    E_kr=0.1,
    k_d0=0.5,
    E_kd=0.1,
    subdomain=boundary,
)
reac3 = SurfaceReactionBC(
    reactant=[H, T],
    gas_pressure=1e6,
    k_r0=1.0,
    E_kr=0.1,
    k_d0=0.5,
    E_kd=0.1,
    subdomain=boundary,
)

Heat transfer BCs

Some BCs are specific to heat transfer. FESTIM provides a handful of convenience classes making things a bit easier for the users.

Imposing the temperature

The temperature can be imposed on boundaries with festim.FixedTemperatureBC.

from festim import FixedTemperatureBC, SurfaceSubdomain

boundary = SurfaceSubdomain(id=1)

my_bc = FixedTemperatureBC(subdomain=boundary, value=10)

To define the temperature as space or time dependent, a function can be passed to the value argument:

from festim import FixedTemperatureBC, SurfaceSubdomain

boundary = SurfaceSubdomain(id=1)

my_custom_value = lambda x, t: 10 + x[0]**2 + t

my_bc = FixedTemperatureBC(subdomain=boundary, value=my_custom_value)

Imposing a heat flux

When a heat flux needs to be imposed on a boundary, use the festim.HeatFluxBC class.

from festim import HeatFluxBC, SurfaceSubdomain

boundary = SurfaceSubdomain(id=1)

my_flux_bc = HeatFluxBC(subdomain=boundary, value=5)

As for the fixed temperature boundary condition, the flux can be dependent on space and time. But for heat fluxes, the values can also be dependent on a temperature:

from festim import HeatFluxBC, SurfaceSubdomain

boundary = SurfaceSubdomain(id=1)

my_custom_value = lambda x, t, T: 2 * x[0] + 10 * t + T

my_flux_bc = HeatFluxBC(subdomain=boundary, value=my_custom_value)