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 import Grid1D
>>> nx = 11
>>> dx = 1.
>>> from fipy.tools import serialComm
>>> mesh = Grid1D(nx = nx, dx = dx, communicator=serialComm)
>>> x, = mesh.cellCenters
>>> from fipy.variables.cellVariable import CellVariable
>>> ionVar = CellVariable(mesh = mesh, value = 1.)
>>> from fipy.variables.distanceVariable \
... import DistanceVariable
>>> disVar = DistanceVariable(mesh = mesh,
... value = (x - 0.5) - 0.99,
... hasOld = 1)
>>> v = 1.
>>> diffusion = 1.
>>> omega = 1.
>>> cinf = 1.
>>> eqn = buildMetalIonDiffusionEquation(ionVar = ionVar,
... distanceVar = disVar,
... depositionRate = v * ionVar,
... diffusionCoeff = diffusion,
... metalIonMolarVolume = omega)
>>> ionVar.constrain(cinf, mesh.facesRight)
>>> from builtins import range
>>> for i in range(10):
... eqn.solve(ionVar, dt = 1000)
>>> 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
Testing the interface source term
>>> from fipy.meshes import Grid2D
>>> from fipy import numerix, serialComm
>>> mesh = Grid2D(dx = 1., dy = 1., nx = 2, ny = 2, communicator=serialComm)
>>> from fipy.variables.distanceVariable import DistanceVariable
>>> distance = DistanceVariable(mesh = mesh, value = (-.5, .5, .5, 1.5))
>>> ionVar = CellVariable(mesh = mesh, value = (1, 1, 1, 1))
>>> depositionRate = CellVariable(mesh=mesh, value=(1, 1, 1, 1))
>>> source = depositionRate * distance.cellInterfaceAreas / mesh.cellVolumes / ionVar
>>> sqrt = numerix.sqrt(2)
>>> ans = CellVariable(mesh=mesh, value=(0, 1 / sqrt, 1 / sqrt, 0))
>>> print(numerix.allclose(source, ans))
True
>>> distance[:] = (-1.5, -0.5, -0.5, 0.5)
>>> print(numerix.allclose(source, (0, 0, 0, sqrt)))
True
: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.
"""
diffusionCoeff = _LevelSetDiffusionVariable(distanceVar, diffusionCoeff)
eq = TransientTerm(transientCoeff) - DiffusionTermNoCorrection(
diffusionCoeff)
mesh = distanceVar.mesh
coeff = depositionRate * distanceVar.cellInterfaceAreas / (
mesh.cellVolumes * metalIonMolarVolume) / ionVar
return eq + ImplicitSourceTerm(coeff)
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 import Grid1D
>>> nx = 11
>>> dx = 1.
>>> from fipy.tools import serialComm
>>> mesh = Grid1D(nx = nx, dx = dx, communicator=serialComm)
>>> x, = mesh.cellCenters
>>> from fipy.variables.cellVariable import CellVariable
>>> ionVar = CellVariable(mesh = mesh, value = 1.)
>>> from fipy.variables.distanceVariable \
... import DistanceVariable
>>> disVar = DistanceVariable(mesh = mesh,
... value = (x - 0.5) - 0.99,
... hasOld = 1)
>>> v = 1.
>>> diffusion = 1.
>>> omega = 1.
>>> cinf = 1.
>>> eqn = buildMetalIonDiffusionEquation(ionVar = ionVar,
... distanceVar = disVar,
... depositionRate = v * ionVar,
... diffusionCoeff = diffusion,
... metalIonMolarVolume = omega)
>>> ionVar.constrain(cinf, mesh.facesRight)
>>> for i in range(10):
... eqn.solve(ionVar, dt = 1000)
>>> 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
Testing the interface source term
>>> from fipy.meshes import Grid2D
>>> from fipy import numerix, serialComm
>>> mesh = Grid2D(dx = 1., dy = 1., nx = 2, ny = 2, communicator=serialComm)
>>> from fipy.variables.distanceVariable import DistanceVariable
>>> distance = DistanceVariable(mesh = mesh, value = (-.5, .5, .5, 1.5))
>>> ionVar = CellVariable(mesh = mesh, value = (1, 1, 1, 1))
>>> depositionRate = CellVariable(mesh=mesh, value=(1, 1, 1, 1))
>>> source = depositionRate * distance.cellInterfaceAreas / mesh.cellVolumes / ionVar
>>> sqrt = numerix.sqrt(2)
>>> ans = CellVariable(mesh=mesh, value=(0, 1 / sqrt, 1 / sqrt, 0))
>>> print numerix.allclose(source, ans)
True
>>> distance[:] = (-1.5, -0.5, -0.5, 0.5)
>>> print numerix.allclose(source, (0, 0, 0, sqrt))
True
: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.
"""
diffusionCoeff = _LevelSetDiffusionVariable(distanceVar,
diffusionCoeff)
eq = TransientTerm(transientCoeff) - DiffusionTermNoCorrection(diffusionCoeff)
mesh = distanceVar.mesh
coeff = depositionRate * distanceVar.cellInterfaceAreas / (mesh.cellVolumes * metalIonMolarVolume) / ionVar
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 : ~fipy.variables.cellVariable.CellVariable
The bulk surfactant concentration variable.
distanceVar : ~fipy.variables.distanceVariable.DistanceVariable
surfactantVar : ~fipy.variables.surfactantVariable.SurfactantVariable
otherSurfactantVar : ~fipy.variables.surfactantVariable.SurfactantVariable
Any other surfactants that may remove this one.
diffusionCoeff : float or ~fipy.variables.faceVariable.FaceVariable
transientCoeff : float
In general 1 is used.
rateConstant : float
The adsorption coefficient.
"""
spCoeff = rateConstant * distanceVar.cellInterfaceAreas / bulkVar.mesh.cellVolumes
spSourceTerm = ImplicitSourceTerm(spCoeff)
bulkSpCoeff = spCoeff * bulkVar
coeff = bulkSpCoeff * surfactantVar.interfaceVar
diffusionCoeff = _LevelSetDiffusionVariable(distanceVar, diffusionCoeff)
eq = TransientTerm(transientCoeff) - DiffusionTermNoCorrection(
diffusionCoeff)
if otherSurfactantVar is not None:
otherCoeff = bulkSpCoeff * otherSurfactantVar.interfaceVar
else:
otherCoeff = 0
return eq - coeff + spSourceTerm - otherCoeff
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.
"""
spCoeff = rateConstant * distanceVar.cellInterfaceAreas / bulkVar.mesh.cellVolumes
spSourceTerm = ImplicitSourceTerm(spCoeff)
bulkSpCoeff = spCoeff * bulkVar
coeff = bulkSpCoeff * surfactantVar.interfaceVar
diffusionCoeff = _LevelSetDiffusionVariable(distanceVar,
diffusionCoeff)
eq = TransientTerm(transientCoeff) - DiffusionTermNoCorrection(diffusionCoeff)
if otherSurfactantVar is not None:
otherCoeff = bulkSpCoeff * otherSurfactantVar.interfaceVar
else:
otherCoeff = 0
return eq - coeff + spSourceTerm - otherCoeff