def runSimpleTrenchSystem(faradaysConstant=9.6e4,
gasConstant=8.314,
transferCoefficient=0.5,
rateConstant0=1.76,
rateConstant3=-245e-6,
catalystDiffusion=1e-9,
siteDensity=9.8e-6,
molarVolume=7.1e-6,
charge=2,
metalDiffusion=5.6e-10,
temperature=298.,
overpotential=-0.3,
metalConcentration=250.,
catalystConcentration=5e-3,
catalystCoverage=0.,
currentDensity0=0.26,
currentDensity1=45.,
cellSize=0.1e-7,
trenchDepth=0.5e-6,
aspectRatio=2.,
trenchSpacing=0.6e-6,
boundaryLayerDepth=0.3e-6,
numberOfSteps=5,
displayViewers=True):
cflNumber = 0.2
numberOfCellsInNarrowBand = 10
cellsBelowTrench = 10
yCells = cellsBelowTrench \
+ int((trenchDepth + boundaryLayerDepth) / cellSize)
xCells = int(trenchSpacing / 2 / cellSize)
from fipy.tools import serialComm
mesh = Grid2D(dx=cellSize,
dy=cellSize,
nx=xCells,
ny=yCells,
communicator=serialComm)
narrowBandWidth = numberOfCellsInNarrowBand * cellSize
distanceVar = DistanceVariable(name='distance variable',
mesh=mesh,
value=-1.,
hasOld=1)
bottomHeight = cellsBelowTrench * cellSize
trenchHeight = bottomHeight + trenchDepth
trenchWidth = trenchDepth / aspectRatio
sideWidth = (trenchSpacing - trenchWidth) / 2
x, y = mesh.cellCenters
distanceVar.setValue(1.,
where=(y > trenchHeight) |
((y > bottomHeight) &
(x < xCells * cellSize - sideWidth)))
distanceVar.calcDistanceFunction(order=2)
catalystVar = SurfactantVariable(name="catalyst variable",
value=catalystCoverage,
distanceVar=distanceVar)
bulkCatalystVar = CellVariable(name='bulk catalyst variable',
mesh=mesh,
value=catalystConcentration)
metalVar = CellVariable(name='metal variable',
mesh=mesh,
value=metalConcentration)
expoConstant = -transferCoefficient * faradaysConstant \
/ (gasConstant * temperature)
tmp = currentDensity1 * catalystVar.interfaceVar
exchangeCurrentDensity = currentDensity0 + tmp
expo = numerix.exp(expoConstant * overpotential)
currentDensity = expo * exchangeCurrentDensity * metalVar \
/ metalConcentration
depositionRateVariable = currentDensity * molarVolume \
/ (charge * faradaysConstant)
extensionVelocityVariable = CellVariable(name='extension velocity',
mesh=mesh,
value=depositionRateVariable)
surfactantEquation = AdsorbingSurfactantEquation(
surfactantVar=catalystVar,
distanceVar=distanceVar,
bulkVar=bulkCatalystVar,
rateConstant=rateConstant0 + rateConstant3 * overpotential**3)
advectionEquation = TransientTerm() + AdvectionTerm(
extensionVelocityVariable)
metalEquation = buildMetalIonDiffusionEquation(
ionVar=metalVar,
distanceVar=distanceVar,
depositionRate=depositionRateVariable,
diffusionCoeff=metalDiffusion,
metalIonMolarVolume=molarVolume,
)
metalVar.constrain(metalConcentration, mesh.facesTop)
from surfactantBulkDiffusionEquation import buildSurfactantBulkDiffusionEquation
bulkCatalystEquation = buildSurfactantBulkDiffusionEquation(
bulkVar=bulkCatalystVar,
distanceVar=distanceVar,
surfactantVar=catalystVar,
diffusionCoeff=catalystDiffusion,
rateConstant=rateConstant0 * siteDensity)
bulkCatalystVar.constrain(catalystConcentration, mesh.facesTop)
if displayViewers:
try:
from mayaviSurfactantViewer import MayaviSurfactantViewer
viewer = MayaviSurfactantViewer(distanceVar,
catalystVar.interfaceVar,
zoomFactor=1e6,
datamax=0.5,
datamin=0.0,
smooth=1,
title='catalyst coverage')
except:
viewer = MultiViewer(
viewers=(Viewer(distanceVar, datamin=-1e-9, datamax=1e-9),
Viewer(catalystVar.interfaceVar)))
else:
viewer = None
levelSetUpdateFrequency = int(0.8 * narrowBandWidth \
/ (cellSize * cflNumber * 2))
for step in range(numberOfSteps):
if step > 5 and step % 5 == 0 and viewer is not None:
viewer.plot()
if step % levelSetUpdateFrequency == 0:
distanceVar.calcDistanceFunction(order=2)
extensionVelocityVariable.setValue(depositionRateVariable())
distanceVar.updateOld()
distanceVar.extendVariable(extensionVelocityVariable, order=2)
dt = cflNumber * cellSize / extensionVelocityVariable.max()
advectionEquation.solve(distanceVar, dt=dt)
surfactantEquation.solve(catalystVar, dt=dt)
metalEquation.solve(metalVar, dt=dt)
bulkCatalystEquation.solve(bulkCatalystVar,
dt=dt,
solver=GeneralSolver(tolerance=1e-15,
iterations=2000))
try:
import os
filepath = os.path.splitext(__file__)[0] + '.gz'
print catalystVar.allclose(numerix.loadtxt(filepath), rtol=1e-4)
except:
return 0
def runLeveler(kLeveler=0.018,
bulkLevelerConcentration=0.02,
cellSize=0.1e-7,
rateConstant=0.00026,
initialAcceleratorCoverage=0.0,
levelerDiffusionCoefficient=5e-10,
numberOfSteps=400,
displayRate=10,
displayViewers=True):
kLevelerConsumption = 0.0005
aspectRatio = 1.5
faradaysConstant = 9.6485e4
gasConstant = 8.314
acceleratorDiffusionCoefficient = 4e-10
siteDensity = 6.35e-6
atomicVolume = 7.1e-6
charge = 2
metalDiffusionCoefficient = 4e-10
temperature = 298.
overpotential = -0.25
bulkMetalConcentration = 250.
bulkAcceleratorConcentration = 50.0e-3
initialLevelerCoverage = 0.
cflNumber = 0.2
cellsBelowTrench = 10
trenchDepth = 0.4e-6
trenchSpacing = 0.6e-6
boundaryLayerDepth = 98.7e-6
i0Suppressor = 0.3
i0Accelerator = 22.5
alphaSuppressor = 0.5
alphaAccelerator = 0.4
alphaAdsorption = 0.62
m = 4
b = 2.65
A = 0.3
Ba = -40
Bb = 60
Vd = 0.098
Bd = 0.0008
etaPrime = faradaysConstant * overpotential / gasConstant / temperature
mesh = TrenchMesh(cellSize=cellSize,
trenchSpacing=trenchSpacing,
trenchDepth=trenchDepth,
boundaryLayerDepth=boundaryLayerDepth,
aspectRatio=aspectRatio,
angle=numerix.pi * 4. / 180.)
distanceVar = GapFillDistanceVariable(name='distance variable',
mesh=mesh,
value=-1.)
distanceVar.setValue(1., where=mesh.electrolyteMask)
distanceVar.calcDistanceFunction()
levelerVar = SurfactantVariable(name="leveler variable",
value=initialLevelerCoverage,
distanceVar=distanceVar)
acceleratorVar = SurfactantVariable(name="accelerator variable",
value=initialAcceleratorCoverage,
distanceVar=distanceVar)
bulkAcceleratorVar = CellVariable(name='bulk accelerator variable',
mesh=mesh,
value=bulkAcceleratorConcentration)
bulkLevelerVar = CellVariable(name='bulk leveler variable',
mesh=mesh,
value=bulkLevelerConcentration)
metalVar = CellVariable(name='metal variable',
mesh=mesh,
value=bulkMetalConcentration)
def depositionCoeff(alpha, i0):
expo = numerix.exp(-alpha * etaPrime)
return 2 * i0 * (expo - expo * numerix.exp(etaPrime))
coeffSuppressor = depositionCoeff(alphaSuppressor, i0Suppressor)
coeffAccelerator = depositionCoeff(alphaAccelerator, i0Accelerator)
exchangeCurrentDensity = acceleratorVar.interfaceVar * (
coeffAccelerator - coeffSuppressor) + coeffSuppressor
currentDensity = metalVar / bulkMetalConcentration * exchangeCurrentDensity
depositionRateVariable = currentDensity * atomicVolume / charge / faradaysConstant
extensionVelocityVariable = CellVariable(name='extension velocity',
mesh=mesh,
value=depositionRateVariable)
kAccelerator = rateConstant * numerix.exp(-alphaAdsorption * etaPrime)
kAcceleratorConsumption = Bd + A / (numerix.exp(Ba *
(overpotential + Vd)) +
numerix.exp(Bb * (overpotential + Vd)))
q = m * overpotential + b
levelerSurfactantEquation = AdsorbingSurfactantEquation(
levelerVar,
distanceVar=distanceVar,
bulkVar=bulkLevelerVar,
rateConstant=kLeveler,
consumptionCoeff=kLevelerConsumption * depositionRateVariable)
accVar1 = acceleratorVar.interfaceVar
accVar2 = (accVar1 > 0) * accVar1
accConsumptionCoeff = kAcceleratorConsumption * (accVar2**(q - 1))
acceleratorSurfactantEquation = AdsorbingSurfactantEquation(
acceleratorVar,
distanceVar=distanceVar,
bulkVar=bulkAcceleratorVar,
rateConstant=kAccelerator,
otherVar=levelerVar,
otherBulkVar=bulkLevelerVar,
otherRateConstant=kLeveler,
consumptionCoeff=accConsumptionCoeff)
advectionEquation = TransientTerm() + FirstOrderAdvectionTerm(
extensionVelocityVariable)
metalEquation = buildMetalIonDiffusionEquation(
ionVar=metalVar,
distanceVar=distanceVar,
depositionRate=depositionRateVariable,
diffusionCoeff=metalDiffusionCoefficient,
metalIonMolarVolume=atomicVolume)
metalVar.constrain(bulkMetalConcentration, mesh.facesTop)
bulkAcceleratorEquation = buildSurfactantBulkDiffusionEquation(
bulkVar=bulkAcceleratorVar,
distanceVar=distanceVar,
surfactantVar=acceleratorVar,
otherSurfactantVar=levelerVar,
diffusionCoeff=acceleratorDiffusionCoefficient,
rateConstant=kAccelerator * siteDensity)
bulkAcceleratorVar.constrain(bulkAcceleratorConcentration, mesh.facesTop)
bulkLevelerEquation = buildSurfactantBulkDiffusionEquation(
bulkVar=bulkLevelerVar,
distanceVar=distanceVar,
surfactantVar=levelerVar,
diffusionCoeff=levelerDiffusionCoefficient,
rateConstant=kLeveler * siteDensity)
bulkLevelerVar.constrain(bulkLevelerConcentration, mesh.facesTop)
eqnTuple = ((advectionEquation, distanceVar, (),
None), (levelerSurfactantEquation, levelerVar, (), None),
(acceleratorSurfactantEquation, acceleratorVar, (),
None), (metalEquation, metalVar, (), None),
(bulkAcceleratorEquation, bulkAcceleratorVar, (),
GeneralSolver()), (bulkLevelerEquation, bulkLevelerVar, (),
GeneralSolver()))
narrowBandWidth = 20 * cellSize
levelSetUpdateFrequency = int(0.7 * narrowBandWidth / cellSize /
cflNumber / 2)
totalTime = 0.0
if displayViewers:
try:
raise Exception
from mayaviSurfactantViewer import MayaviSurfactantViewer
viewers = (MayaviSurfactantViewer(distanceVar,
acceleratorVar.interfaceVar,
zoomFactor=1e6,
datamax=0.5,
datamin=0.0,
smooth=1,
title='accelerator coverage'),
MayaviSurfactantViewer(distanceVar,
levelerVar.interfaceVar,
zoomFactor=1e6,
datamax=0.5,
datamin=0.0,
smooth=1,
title='leveler coverage'))
except:
class PlotVariable(CellVariable):
def __init__(self, var=None, name=''):
CellVariable.__init__(self, mesh=mesh.fineMesh, name=name)
self.var = self._requires(var)
def _calcValue(self):
return numerix.array(self.var(self.mesh.cellCenters))
viewers = (Viewer(PlotVariable(var=acceleratorVar.interfaceVar)),
Viewer(PlotVariable(var=levelerVar.interfaceVar)))
for step in range(numberOfSteps):
if displayViewers:
if step % displayRate == 0:
for viewer in viewers:
viewer.plot()
if step % levelSetUpdateFrequency == 0:
distanceVar.calcDistanceFunction()
extensionVelocityVariable.setValue(depositionRateVariable)
extOnInt = numerix.where(
distanceVar.globalValue > 0,
numerix.where(distanceVar.globalValue < 2 * cellSize,
extensionVelocityVariable.globalValue, 0), 0)
dt = cflNumber * cellSize / extOnInt.max()
distanceVar.extendVariable(extensionVelocityVariable)
for eqn, var, BCs, solver in eqnTuple:
eqn.solve(var, boundaryConditions=BCs, dt=dt, solver=solver)
totalTime += dt
point = ((1.25e-08, ), (3.125e-07, ))
value = 2.02815779e-08
return abs(float(distanceVar(point, order=1)) - value) < cellSize / 10.0
def runGold(faradaysConstant=9.6e4,
consumptionRateConstant=2.6e+6,
molarVolume=10.21e-6,
charge=1.0,
metalDiffusion=1.7e-9,
metalConcentration=20.0,
catalystCoverage=0.15,
currentDensity0=3e-2 * 16,
currentDensity1=6.5e-1 * 16,
cellSize=0.1e-7,
trenchDepth=0.2e-6,
aspectRatio=1.47,
trenchSpacing=0.5e-6,
boundaryLayerDepth=90.0e-6,
numberOfSteps=10,
taperAngle=6.0,
displayViewers=True):
cflNumber = 0.2
numberOfCellsInNarrowBand = 20
mesh = TrenchMesh(cellSize = cellSize,
trenchSpacing = trenchSpacing,
trenchDepth = trenchDepth,
boundaryLayerDepth = boundaryLayerDepth,
aspectRatio = aspectRatio,
angle = numerix.pi * taperAngle / 180.)
narrowBandWidth = numberOfCellsInNarrowBand * cellSize
distanceVar = GapFillDistanceVariable(
name = 'distance variable',
mesh = mesh,
value = -1.)
distanceVar.setValue(1., where=mesh.electrolyteMask)
distanceVar.calcDistanceFunction()
catalystVar = SurfactantVariable(
name = "catalyst variable",
value = catalystCoverage,
distanceVar = distanceVar)
metalVar = CellVariable(
name = 'metal variable',
mesh = mesh,
value = metalConcentration)
exchangeCurrentDensity = currentDensity0 + currentDensity1 * catalystVar.interfaceVar
currentDensity = metalVar / metalConcentration * exchangeCurrentDensity
depositionRateVariable = currentDensity * molarVolume / charge / faradaysConstant
extensionVelocityVariable = CellVariable(
name = 'extension velocity',
mesh = mesh,
value = depositionRateVariable)
catalystSurfactantEquation = AdsorbingSurfactantEquation(
catalystVar,
distanceVar = distanceVar,
bulkVar = 0,
rateConstant = 0,
consumptionCoeff = consumptionRateConstant * extensionVelocityVariable)
advectionEquation = TransientTerm() + FirstOrderAdvectionTerm(extensionVelocityVariable)
metalEquation = buildMetalIonDiffusionEquation(
ionVar = metalVar,
distanceVar = distanceVar,
depositionRate = depositionRateVariable,
diffusionCoeff = metalDiffusion,
metalIonMolarVolume = molarVolume)
metalVar.constrain(metalConcentration, mesh.facesTop)
if displayViewers:
try:
from mayaviSurfactantViewer import MayaviSurfactantViewer
viewer = MayaviSurfactantViewer(distanceVar, catalystVar.interfaceVar, zoomFactor = 1e6, datamax=1.0, datamin=0.0, smooth = 1, title = 'catalyst coverage', animate=True)
except:
class PlotVariable(CellVariable):
def __init__(self, var = None, name = ''):
CellVariable.__init__(self, mesh = mesh.fineMesh, name = name)
self.var = self._requires(var)
def _calcValue(self):
return numerix.array(self.var(self.mesh.cellCenters))
viewer = MultiViewer(viewers=(
Viewer(PlotVariable(var = distanceVar), datamax=1e-9, datamin=-1e-9),
Viewer(PlotVariable(var = catalystVar.interfaceVar))))
else:
viewer = None
levelSetUpdateFrequency = int(0.7 * narrowBandWidth / cellSize / cflNumber / 2)
step = 0
while step < numberOfSteps:
if step % 10 == 0 and viewer is not None:
viewer.plot()
if step % levelSetUpdateFrequency == 0:
distanceVar.calcDistanceFunction()
extensionVelocityVariable.setValue(numerix.array(depositionRateVariable))
dt = cflNumber * cellSize / max(extensionVelocityVariable.globalValue)
distanceVar.extendVariable(extensionVelocityVariable)
advectionEquation.solve(distanceVar, dt = dt)
catalystSurfactantEquation.solve(catalystVar, dt = dt)
metalEquation.solve(metalVar, dt = dt)
step += 1
point = ((5e-09,), (1.15e-07,))
value = 1.45346701e-09
return abs(float(distanceVar(point, order=1)) - value) < cellSize / 10.0
def runSimpleTrenchSystem(
faradaysConstant=9.6e4,
gasConstant=8.314,
transferCoefficient=0.5,
rateConstant0=1.76,
rateConstant3=-245e-6,
catalystDiffusion=1e-9,
siteDensity=9.8e-6,
molarVolume=7.1e-6,
charge=2,
metalDiffusion=5.6e-10,
temperature=298.0,
overpotential=-0.3,
metalConcentration=250.0,
catalystConcentration=5e-3,
catalystCoverage=0.0,
currentDensity0=0.26,
currentDensity1=45.0,
cellSize=0.1e-7,
trenchDepth=0.5e-6,
aspectRatio=2.0,
trenchSpacing=0.6e-6,
boundaryLayerDepth=0.3e-6,
numberOfSteps=5,
displayViewers=True,
):
cflNumber = 0.2
numberOfCellsInNarrowBand = 10
cellsBelowTrench = 10
yCells = cellsBelowTrench + int((trenchDepth + boundaryLayerDepth) / cellSize)
xCells = int(trenchSpacing / 2 / cellSize)
from fipy.tools import serialComm
mesh = Grid2D(dx=cellSize, dy=cellSize, nx=xCells, ny=yCells, communicator=serialComm)
narrowBandWidth = numberOfCellsInNarrowBand * cellSize
distanceVar = DistanceVariable(name="distance variable", mesh=mesh, value=-1.0, hasOld=1)
bottomHeight = cellsBelowTrench * cellSize
trenchHeight = bottomHeight + trenchDepth
trenchWidth = trenchDepth / aspectRatio
sideWidth = (trenchSpacing - trenchWidth) / 2
x, y = mesh.cellCenters
distanceVar.setValue(1.0, where=(y > trenchHeight) | ((y > bottomHeight) & (x < xCells * cellSize - sideWidth)))
distanceVar.calcDistanceFunction(order=2)
catalystVar = SurfactantVariable(name="catalyst variable", value=catalystCoverage, distanceVar=distanceVar)
bulkCatalystVar = CellVariable(name="bulk catalyst variable", mesh=mesh, value=catalystConcentration)
metalVar = CellVariable(name="metal variable", mesh=mesh, value=metalConcentration)
expoConstant = -transferCoefficient * faradaysConstant / (gasConstant * temperature)
tmp = currentDensity1 * catalystVar.interfaceVar
exchangeCurrentDensity = currentDensity0 + tmp
expo = numerix.exp(expoConstant * overpotential)
currentDensity = expo * exchangeCurrentDensity * metalVar / metalConcentration
depositionRateVariable = currentDensity * molarVolume / (charge * faradaysConstant)
extensionVelocityVariable = CellVariable(name="extension velocity", mesh=mesh, value=depositionRateVariable)
surfactantEquation = AdsorbingSurfactantEquation(
surfactantVar=catalystVar,
distanceVar=distanceVar,
bulkVar=bulkCatalystVar,
rateConstant=rateConstant0 + rateConstant3 * overpotential ** 3,
)
advectionEquation = TransientTerm() + AdvectionTerm(extensionVelocityVariable)
metalEquation = buildMetalIonDiffusionEquation(
ionVar=metalVar,
distanceVar=distanceVar,
depositionRate=depositionRateVariable,
diffusionCoeff=metalDiffusion,
metalIonMolarVolume=molarVolume,
)
metalVar.constrain(metalConcentration, mesh.facesTop)
from surfactantBulkDiffusionEquation import buildSurfactantBulkDiffusionEquation
bulkCatalystEquation = buildSurfactantBulkDiffusionEquation(
bulkVar=bulkCatalystVar,
distanceVar=distanceVar,
surfactantVar=catalystVar,
diffusionCoeff=catalystDiffusion,
rateConstant=rateConstant0 * siteDensity,
)
bulkCatalystVar.constrain(catalystConcentration, mesh.facesTop)
if displayViewers:
try:
from mayaviSurfactantViewer import MayaviSurfactantViewer
viewer = MayaviSurfactantViewer(
distanceVar,
catalystVar.interfaceVar,
zoomFactor=1e6,
datamax=0.5,
datamin=0.0,
smooth=1,
title="catalyst coverage",
)
except:
viewer = MultiViewer(
viewers=(Viewer(distanceVar, datamin=-1e-9, datamax=1e-9), Viewer(catalystVar.interfaceVar))
)
else:
viewer = None
levelSetUpdateFrequency = int(0.8 * narrowBandWidth / (cellSize * cflNumber * 2))
for step in range(numberOfSteps):
if step % 5 == 0 and viewer is not None:
viewer.plot()
if step % levelSetUpdateFrequency == 0:
distanceVar.calcDistanceFunction(order=2)
extensionVelocityVariable.setValue(depositionRateVariable())
distanceVar.updateOld()
distanceVar.extendVariable(extensionVelocityVariable, order=2)
dt = cflNumber * cellSize / extensionVelocityVariable.max()
advectionEquation.solve(distanceVar, dt=dt)
surfactantEquation.solve(catalystVar, dt=dt)
metalEquation.solve(metalVar, dt=dt)
bulkCatalystEquation.solve(bulkCatalystVar, dt=dt, solver=GeneralSolver(tolerance=1e-15, iterations=2000))
try:
import os
filepath = os.path.splitext(__file__)[0] + ".gz"
print catalystVar.allclose(numerix.loadtxt(filepath), rtol=1e-4)
except:
return 0
