FiPyexamples.levelSet.electroChem.levelerΒΆ
Model electrochemical superfill of copper with leveler and accelerator additives.
This input file is a demonstration of the use of FiPy for modeling copper superfill with leveler and accelerator additives. The material properties and experimental parameters used are roughly those that have been previously published [31].
To run this example from the base FiPy directory type:
$ python examples/levelSet/electroChem/leveler.py
at the command line. The results of the simulation will be displayed and the word
finished in the terminal at the end of the simulation. The simulation will
only run for 200 time steps. To run with a different number of time steps change
the numberOfSteps argument as follows,
>>> runLeveler(numberOfSteps=10, displayViewers=False, cellSize=0.25e-7)
1
Change the displayViewers argument to True if you wish to see the
results displayed on the screen. This example requires gmsh to
construct the mesh.
This example models the case when suppressor, accelerator and leveler additives are present in the electrolyte. The suppressor is assumed to absorb quickly compared with the other additives. Any unoccupied surface sites are immediately covered with suppressor. The accelerator additive has more surface affinity than suppressor and is thus preferential adsorbed. The accelerator can also remove suppressor when the surface reaches full coverage. Similarly, the leveler additive has more surface affinity than both the suppressor and accelerator. This forms a simple set of assumptions for understanding the behavior of these additives.
The following is a complete description of the equations for the
model described here. Any equations that have been omitted are the
same as those given in
examples.levelSet.electroChem.simpleTrenchSystem. The
current density is governed by
![i = \frac{ c_m }{ c_m^\infty }
\sum_{ j } \left[ i_j \theta_j \left( \exp{ \frac{-\alpha_j F \eta
}{ R T }} - \exp{ \frac{ \left( 1 - \alpha_j \right) F \eta}{ R T
}} \right) \right]](../../../_images/math/ed3c98e7ad93ca98288ad67bd6a122c3b6233d12.svg)
where 
represents 
for suppressor, 
for accelerator, 
for leveler and 
for vacant. This model
assumes a linear interpolation between the three cases of complete
coverage for each additive or vacant substrate. The governing
equations for the surfactants are given by,
It has been found experimentally that 
.
If the surface reaches full coverage, the equations do not naturally prevent the
coverage rising above full coverage due to the curvature terms. Thus, when
then the equation for accelerator becomes
and when 
, the
equation for leveler becomes 
The parameters 
, 
and 
are both functions of 
given by,
The following table shows the symbols used in the governing equations
and their corresponding arguments for the
runLeveler() function.
The following images show accelerator and leveler contour plots that can be obtained by running this example.
