def buildMetalIonDiffusionEquation(ionVar = None,
distanceVar = None,
depositionRate = 1,
transientCoeff = 1,
diffusionCoeff = 1,
metalIonMolarVolume = 1):
r"""
The `MetalIonDiffusionEquation` solves the diffusion of the metal
species with a source term at the electrolyte interface. The governing
equation is given by,
.. math::
\frac{\partial c}{\partial t} = \nabla \cdot D \nabla c
where,
.. math::
D = \begin{cases}
D_c & \text{when $\phi > 0$} \\
0 & \text{when $\phi \le 0$}
\end{cases}
The velocity of the interface generally has a linear dependence on ion
concentration. The following boundary condition applies at the zero
level set,
.. math::
D \hat{n} \cdot \nabla c = \frac{v(c)}{\Omega} \qquad \text{at $phi = 0$}
where
.. math::
v(c) = c V_0
The test case below is for a 1D steady state problem. The solution is
given by:
.. math::
c(x) = \frac{c^{\infty}}{\Omega D / V_0 + L}\left(x - L\right) + c^{\infty}
This is the test case,
>>> from fipy.meshes.grid1D import Grid1D
>>> nx = 11
>>> dx = 1.
>>> from fipy.tools import serial
>>> mesh = Grid1D(nx = nx, dx = dx, parallelModule=serial)
>>> x, = mesh.getCellCenters()
>>> from fipy.variables.cellVariable import CellVariable
>>> ionVar = CellVariable(mesh = mesh, value = 1.)
>>> from fipy.models.levelSet.distanceFunction.distanceVariable \
... import DistanceVariable
>>> disVar = DistanceVariable(mesh = mesh,
... value = (x - 0.5) - 0.99,
... hasOld = 1)
>>> v = 1.
>>> diffusion = 1.
>>> omega = 1.
>>> cinf = 1.
>>> from fipy.boundaryConditions.fixedValue import FixedValue
>>> eqn = buildMetalIonDiffusionEquation(ionVar = ionVar,
... distanceVar = disVar,
... depositionRate = v * ionVar,
... diffusionCoeff = diffusion,
... metalIonMolarVolume = omega)
>>> bc = (FixedValue(mesh.getFacesRight(), cinf),)
>>> for i in range(10):
... eqn.solve(ionVar, dt = 1000, boundaryConditions = bc)
>>> L = (nx - 1) * dx - dx / 2
>>> gradient = cinf / (omega * diffusion / v + L)
>>> answer = gradient * (x - L - dx * 3 / 2) + cinf
>>> answer[x < dx] = 1
>>> print ionVar.allclose(answer)
1
:Parameters:
- `ionVar`: The metal ion concentration variable.
- `distanceVar`: A `DistanceVariable` object.
- `depositionRate`: A float or a `CellVariable` representing the interface deposition rate.
- `transientCoeff`: The transient coefficient.
- `diffusionCoeff`: The diffusion coefficient
- `metalIonMolarVolume`: Molar volume of the metal ions.
"""
eq = _buildLevelSetDiffusionEquation(ionVar = ionVar,
distanceVar = distanceVar,
transientCoeff = transientCoeff,
diffusionCoeff = diffusionCoeff)
coeff = _MetalIonSourceVariable(ionVar = ionVar,
distanceVar = distanceVar,
depositionRate = depositionRate,
metalIonMolarVolume = metalIonMolarVolume)
return eq + ImplicitSourceTerm(coeff)
def buildSurfactantBulkDiffusionEquation(bulkVar = None,
distanceVar = None,
surfactantVar = None,
otherSurfactantVar = None,
diffusionCoeff = None,
transientCoeff = 1.,
rateConstant = None):
r"""
The `buildSurfactantBulkDiffusionEquation` function returns a bulk diffusion of a
species with a source term for the jump from the bulk to an interface.
The governing equation is given by,
.. math::
\frac{\partial c}{\partial t} = \nabla \cdot D \nabla c
where,
.. math::
D = \begin{cases}
D_c & \text{when $\phi > 0$} \\
0 & \text{when $\phi \le 0$}
\end{cases}
The jump condition at the interface is defined by Langmuir
adsorption. Langmuir adsorption essentially states that the ability for
a species to jump from an electrolyte to an interface is proportional to
the concentration in the electrolyte, available site density and a
jump coefficient. The boundary condition at the interface is given by
.. math::
D \hat{n} \cdot \nabla c = -k c (1 - \theta) \qquad \text{at $\phi = 0$}.
:Parameters:
- `bulkVar`: The bulk surfactant concentration variable.
- `distanceVar`: A `DistanceVariable` object
- `surfactantVar`: A `SurfactantVariable` object
- `otherSurfactantVar`: Any other surfactants that may remove this one.
- `diffusionCoeff`: A float or a `FaceVariable`.
- `transientCoeff`: In general 1 is used.
- `rateConstant`: The adsorption coefficient.
"""
spSourceTerm = ImplicitSourceTerm(_AdsorptionCoeff(rateConstant = rateConstant,
distanceVar = distanceVar))
coeff = _ScAdsorptionCoeff(bulkVar = bulkVar,
surfactantVar = surfactantVar,
rateConstant = rateConstant,
distanceVar = distanceVar)
eq = _buildLevelSetDiffusionEquation(ionVar = bulkVar,
distanceVar = distanceVar,
diffusionCoeff = diffusionCoeff,
transientCoeff = transientCoeff)
if otherSurfactantVar is not None:
otherCoeff = _ScAdsorptionCoeff(bulkVar = bulkVar,
surfactantVar = otherSurfactantVar,
rateConstant = rateConstant,
distanceVar = distanceVar)
else:
otherCoeff = 0
return eq - coeff + spSourceTerm - otherCoeff
