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Electrochemical Polishing
Introduction
This example illustrates the principle of electrochemical polishing. The simplified 2D model geometry consists of two electrodes and an intermediate electrolyte domain. The positive electrode has a protrusion, representing a surface defect. The purpose of the application is to examine how this protrusion and the surrounding electrode material are depleted over a period of time.
Model Definition
The potential drop over the electrodes is 30 V, and the electrolyte has a conductivity of 10 S/m.
Figure 1: Model geometry.
Modeling the depletion of the positive electrode requires a moving boundary because the geometry changes and the current density distribution with it. A simple model for the depletion is based on the assumption that the depletion rate is proportional to the normal current density at the electrode surface. The velocity, U, normal to the mesh at the electrode surface then becomes
(1)
where K is the coefficient of proportionality, and Jn is the normal current density. In this example, K = 10−11 m3/As.
The part of the electrode and electrolyte that the model includes is about 3 mm wide and the distance between the electrodes is 0.4 mm.
Results and Discussion
After a period of 10 s, the protrusion is somewhat smoothed out, and a significant portion of the positive electrode has been depleted.
Figure 2: Potential distribution and electrode depletion after 10 s.
Using Equation 1, the expected total depletion increment, d(Δt = 10 s), over the simulated time interval can be estimated as
(2)
This estimate agrees with the maximum value for the y-displacement obtained for the model, showing that the approximate formula (which does not take effects from the curved boundary into account) is in fact very accurate.
Notes About the COMSOL Implementation
This application uses the Electric Currents interface in combination with a Deformed Geometry. The variable for the normal current density defines the deforming boundary velocity. Due to good stirring of the electrolyte, the dynamics in this example is quasi static in nature with regards to the electrolyte charge transport, and the time dependence only enters in the depletion (removal of material) of the electrode.
Application Library path: COMSOL_Multiphysics/Electromagnetics/electrochemical_polishing
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  2D.
2
In the Select Physics tree, select AC/DC > Electric Fields and Currents > Electric Currents (ec).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies > Time Dependent.
6
Click  Done.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Description
K
1e-11[m^3/(A*s)]
Coefficient of proportionality
Geometry 1
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Settings window for Geometry, locate the Units section.
3
From the Length unit list, choose mm.
Rectangle 1 (r1)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 2.8.
4
In the Height text field, type 0.4.
5
Locate the Position section. In the x text field, type -1.4.
6
Click  Build Selected.
Circle 1 (c1)
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, locate the Size and Shape section.
3
In the Radius text field, type 0.3.
4
Locate the Position section. In the y text field, type 0.6.
Difference 1 (dif1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Difference.
2
Select the object r1 only to add it to the Objects to add list.
3
In the Settings window for Difference, locate the Difference section.
4
Click to select the  Activate Selection toggle button for Objects to subtract.
5
Select the object c1 only.
6
Click  Build Selected.
7
Click the  Zoom Extents button in the Graphics toolbar.
The model geometry is now complete.
Before turning to the Deformed Geometry interface settings, define variables for the local displacement components.
Definitions
Variables 1
1
In the Definitions toolbar, click  Local Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
dx
x-Xg
m
x-displacement
dy
y-Yg
m
y-displacement
Here, Xg and Yg are geometry-frame coordinates corresponding to x and y.
Electric Currents (ec)
1
In the Model Builder window, under Component 1 (comp1) click Electric Currents (ec).
2
In the Settings window for Electric Currents, click to expand the Equation section.
3
From the Equation form list, choose Stationary.
With this setting you specify that the current distribution can be regarded as stationary on the time scale determined by the depletion rate.
Current Conservation in Solids 1
1
In the Model Builder window, under Component 1 (comp1) > Electric Currents (ec) click Current Conservation in Solids 1.
2
In the Settings window for Current Conservation in Solids, locate the Constitutive Relation Jc-E section.
3
From the σ list, choose User defined. In the associated text field, type 10.
4
Locate the Constitutive Relation D-E section. From the εr list, choose User defined. In the associated text field, type 80.
Electric Potential 1
1
In the Physics toolbar, click  Boundaries and choose Electric Potential.
2
Select Boundaries 3, 4, 6, and 7 only.
3
In the Settings window for Electric Potential, locate the Electric Potential section.
4
In the V0 text field, type 30.
Ground 1
1
In the Physics toolbar, click  Boundaries and choose Ground.
2
Select Boundary 2 only.
Electric Insulation 1
For the left and right boundaries, the default boundary condition is a good approximation if you want to simulate that the electrodes are extended indefinitely in both directions.
Component 1 (comp1)
Deforming Domain 1
1
In the Physics toolbar, click  Deformed Geometry and choose Free Deformation.
2
Select Domain 1 only.
3
In the Settings window for Deforming Domain, locate the Smoothing section.
4
From the Mesh smoothing type list, choose Laplace.
Prescribed Normal Mesh Velocity 1
1
In the Deformed Geometry toolbar, click  Prescribed Normal Mesh Velocity.
2
Select Boundaries 3, 4, 6, and 7 only.
3
In the Settings window for Prescribed Normal Mesh Velocity, locate the Prescribed Normal Mesh Velocity section.
4
In the vn text field, type -K*(-ec.nJ).
Prescribed Normal Mesh Displacement 1
1
In the Deformed Geometry toolbar, click  Prescribed Normal Mesh Displacement.
2
Select Boundaries 1, 2, and 5 only.
Mesh 1
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Physics-Controlled Mesh section.
3
From the Element size list, choose Finer.
4
Click  Build All.
5
Click the  Zoom Extents button in the Graphics toolbar.
Study 1
Step 1: Time Dependent
1
In the Model Builder window, under Study 1 click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, locate the Study Settings section.
3
In the Output times text field, type range(0,10).
4
In the Study toolbar, click  Compute.
Results
Electric Potential (ec)
1
Click the  Zoom Extents button in the Graphics toolbar.
The first default plot shows the potential field at the end of the simulation interval; compare with Figure 2.
Electric Field (ec)
The second default plot shows the electric field as a combined surface and streamline plot:
1
In the Model Builder window, click Electric Field (ec).
Current Density Norm
Next, plot the current distribution.
1
In the Results toolbar, click  2D Plot Group.
2
In the Settings window for 2D Plot Group, type Current Density Norm in the Label text field.
Surface 1
1
Right-click Current Density Norm and choose Surface.
2
In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Electric Currents > Currents and charge > ec.normJ - Current density norm - A/m².
3
In the Current Density Norm toolbar, click  Plot.
4
Click the  Zoom Extents button in the Graphics toolbar.
Add a Max/Min Point plot to determine the maximum and minimum values of the current density norm.
Current Density Norm
In the Current Density Norm toolbar, click  More Plots and choose Max/Min Point.
Max/Min Point 1
1
In the Settings window for Max/Min Point, locate the Expression section.
2
In the Expression text field, type ec.normJ.
3
In the Current Density Norm toolbar, click  Plot.
The maximum current density is thus roughly 9·105 A/m2.
Depletion, y Direction
To see the magnitude of the depletion in the y direction more easily, plot the y-component of the mesh displacement and evaluate its maximum value.
1
In the Results toolbar, click  2D Plot Group.
2
In the Settings window for 2D Plot Group, type Depletion, y Direction in the Label text field.
Surface 1
1
Right-click Depletion, y Direction and choose Surface.
2
In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Definitions > Variables > dy - y-displacement - m.
3
In the Depletion, y Direction toolbar, click  Plot.
4
Click the  Zoom In button in the Graphics toolbar.
Depletion, y Direction
In the Depletion, y Direction toolbar, click  More Plots and choose Max/Min Point.
Max/Min Point 1
1
In the Settings window for Max/Min Point, locate the Expression section.
2
In the Expression text field, type dy.
3
Locate the Display section. From the Display list, choose Max.
4
In the Depletion, y Direction toolbar, click  Plot.
The maximum value for the y-displacement is approximately 0.1 mm, which agrees with the value calculated in Equation 2.