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 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
datamax=initialRadius)
surfactantViewer = Viewer(vars=surfactantVariable,
datamin=0.,
datamax=100.)
velocityViewer = Viewer(vars=velocity, datamin=0., datamax=200.)
distanceViewer.plot()
surfactantViewer.plot()
velocityViewer.plot()
totalTime = 0
for step in range(steps):
print('step', step)
velocity.setValue(surfactantVariable.interfaceVar * k)
distanceVariable.extendVariable(velocity)
timeStepDuration = cfl * dx / velocity.max()
distanceVariable.updateOld()
advectionEquation.solve(distanceVariable, dt=timeStepDuration)
surfactantEquation.solve(surfactantVariable, dt=1)
totalTime += timeStepDuration
velocityViewer.plot()
distanceViewer.plot()
surfactantViewer.plot()
finalRadius = numerix.sqrt(2 * k * initialRadius *
initialSurfactantValue * totalTime +
initialRadius**2)
answer = initialSurfactantValue * initialRadius / finalRadius
coverage = surfactantVariable.interfaceVar
