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Pitting Corrosion
Introduction
Pitting corrosion is a type of localized corrosion by which local cavities, pits, are formed on an initially smooth metal surface. A pit may be initiated due to surface defects, such as inhomogeneities in composition or shape, or mechanical abuse resulting in a small scratch or dent. How and if the pit grows depends on a number of factors such as the type of metal, salinity, pH, and temperature of the aqueous electrolyte. A fundamental understanding of the pitting process is paramount for proper material selection in environments susceptible to this type of corrosion.
This tutorial investigates the fundamental mechanisms of pit propagation by simulating electrode kinetics, mass transfer, and the resulting geometry deformation. In the tutorial, the following overall mechanism for the pitting corrosion is assumed:
Oxygen reduction occurs, mainly outside the pit, on the metal surface
(1)
Iron is oxidized, inside and outside the pit, to counterbalance the oxygen reduction reaction
(2)
The combination of the two above reactions gives rise to a mixed electrode potential of the metal surface. Since the electrode area inside the pit is small in comparison to the total surface area of the metal, the oxygen-iron mixed potential can be assumed to be constant; that is, to not be affected by the pitting processes.
Dissolved iron then forms iron hydroxide in the electrolyte close to the electrode surface
(3)
It should be noted that, depending on pH and potentials, other iron containing oxides and hydroxide could also be considered. In this tutorial we will assume a pH ranging from about 10 to 8 in combination with fairly low values for the mixed iron-oxygen potential, which are conditions for which iron hydroxide is the most likely oxidation product.
A result of the iron hydroxide production is that protons are formed by water autoprotolysis to counterbalance the hydroxide ions consumed
(4)
Ions such as Cl- may also be transported in order to maintain electroneutrality. The transport of ions, in combination with the pit shape, determine the local pH.
As a result of the consumption of the hydroxide ions in Equation 3, the pH is reduced close to the electrode surface. If the iron oxidation reaction is catalyzed (depassivated) by H+, a lower pH within the pit will result in faster metal dissolution compared to the metal surface outside the pit.
Model Definition
Figure 1 shows the initial model geometry, defining an electrolyte film covering an iron metal electrode surface and an initial pit. The geometry is defined in 2D with axial symmetry.
Figure 1: Initial model geometry.
The model is defined using the Tertiary Current Distribution, Nernst–Planck interface, solving for the electrolyte phase potential and the concentrations of the electrolyte species H+, OH-, Cl-, Na+ and Fe2+. The Nernst–Planck equations are used assuming electroneutrality and the water autoprotolysis reaction (Equation 4) to be in equilibrium.
Fixed concentrations and electrolyte phase potential are set at the top horizontal electrolyte boundary facing the bulk of the electrolyte.
Iron dissolves at the electrode surface according to Equation 2, with the kinetics being based on a concentration-dependent Butler–Volmer expression, with the exchange current density set to be proportional to the concentration of H+. In this way the iron dissolution reaction gets activated by a lower pH.
The iron metal phase potential is set to a fixed value resulting from the mixed iron-oxygen potential, and as discussed above, this is assumed not to be affected by the local pit corrosion.
The iron hydroxide forming homogeneous reaction (Equation 3) is assumed to be irreversible.
Due to the formation of iron hydroxide (and other oxidation products), the pit is assumed to be a porous structure with the electrolyte volume fraction in the pit (z < 0) fixed to 2.5%. The transport properties (the ion diffusivities and mobilities) depend on the porosity using a Bruggeman relation so that the effective diffusivity in the pit is about of that of the bulk of the electrolyte.
That the amount of produced Fe(OH)2(s) does not affect the porosity is a crude simplification of the processes; a possible extension of the model would be to make porosity a function of Fe(OH)2(s) as it forms dynamically over time.
The geometry of the model deforms as a result of the iron dissolution, with the boundary velocity being proportional to the iron dissolution current density.
The model is solved in a time-dependent solver, simulating the propagation of the pit during 30 days.
Results and Discussion
Figure 2 shows the iron dissolution current density after 30 days. The current density is larger inside the pit than outside. After 30 days, a pronounced convex-shaped pit has formed below the metal surface.
Figure 2: Iron dissolution current density after 30 days.
Figure 3 shows the corrosion rate along the electrode surface (one line per day). The corrosion rate within the pit increases with time.
Figure 3: Electrode dissolution rate along the iron electrode surface at various times.
Figure 4 shows the initial pH. For the initial pit shape, the lowest pH is around 9.5 in the pit.
Figure 5 shows the pH at the end of the simulation. The lowest pH is now about 8.7 at the bottom of the pit. The increased pitting rate over time observed in Figure 3 is a result of the lowered pH.
Figure 4: pH in the pit for the initial conditions.
Figure 5: pH of the pit after 30 days.
Notes About the COMSOL Implementation
The Separator node is used to define the pit as a porous structure.
Linear elements are used to reduce memory usage and reduce the computation time.
The model is solved in two steps. The first Stationary study step solves for the stationary concentration and potential fields for the initial pit shape. The Time Dependent study step solves for the pit growth during 30 days, using the stationary solution for initial values.
Application Library path: Corrosion_Module/Crevice_and_Pitting_Corrosion/pitting_corrosion
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 Axisymmetric.
2
In the Select Physics tree, select Electrochemistry > Corrosion, Deformed Geometry > Corrosion, Tertiary with Electroneutrality.
3
Click Add.
4
In the Number of species text field, type 3.
5
In the Concentrations (mol/m³) table, enter the following settings:
cNa
cCl
cFe
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
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file pitting_corrosion_parameters.txt.
Geometry 1
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 R_pit*20.
4
In the Height text field, type R_pit*10.
5
Click  Build Selected.
6
Click the  Zoom Extents button in the Graphics toolbar.
Rectangle 2 (r2)
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 R_pit.
4
In the Height text field, type H_pit.
5
Locate the Position section. In the z text field, type -H_pit.
6
Click  Build Selected.
Union 1 (uni1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Union.
2
Click in the Graphics window and then press Ctrl+A to select both objects.
3
In the Settings window for Union, locate the Union section.
4
Clear the Keep interior boundaries checkbox.
5
Click  Build Selected.
Fillet 1 (fil1)
1
In the Geometry toolbar, click  Fillet.
2
Click the  Zoom Extents button in the Graphics toolbar.
3
On the object uni1, select Points 4 and 5 only.
4
In the Settings window for Fillet, locate the Radius section.
5
In the Radius text field, type R_pit.
6
Click  Build Selected.
Tertiary Current Distribution, Nernst–Planck (tcd)
Change the charge transport model to Water-based with electroneutrality. This will add the water auto protolysis as an equilibrium reaction as well as the proton and hydroxide ion concentrations to the equation system, and save for the two concentrations algebraically.
1
In the Model Builder window, under Component 1 (comp1) click Tertiary Current Distribution, Nernst–Planck (tcd).
2
In the Settings window for Tertiary Current Distribution, Nernst–Planck, locate the Electrolyte Charge Conservation section.
3
From the Charge conservation model list, choose Water-based with electroneutrality.
Definitions
Variables 1
1
In the Model Builder window, under Component 1 (comp1) right-click Definitions and choose Variables.
2
In the Settings window for Variables, locate the Variables section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file pitting_corrosion_variables.txt.
Tertiary Current Distribution, Nernst–Planck (tcd)
Species Charges 1
1
In the Model Builder window, under Component 1 (comp1) > Tertiary Current Distribution, Nernst–Planck (tcd) click Species Charges 1.
2
In the Settings window for Species Charges, locate the Charge section.
3
In the zcNa text field, type 1.
4
In the zcCl text field, type -1.
5
In the zcFe text field, type 2.
The pit is assumed to be porous, with a nonunit volume fraction of electrolyte. The Separator node may be used to specify the porosity.
Separator 1
1
In the Physics toolbar, click  Domains and choose Separator.
2
In the Settings window for Separator, locate the Domain Selection section.
3
From the Selection list, choose All domains.
4
Locate the Diffusion section. In the DcNa text field, type D_Na.
5
In the DcCl text field, type D_Cl.
6
In the DcFe text field, type D_Fe.
7
In the DcH text field, type D_H.
8
In the DcOH text field, type D_OH.
9
Locate the Porous Matrix Properties section. In the εl text field, type epsl.
Reactions 1
1
In the Physics toolbar, click  Domains and choose Reactions.
2
In the Settings window for Reactions, locate the Domain Selection section.
3
From the Selection list, choose All domains.
4
Locate the Reaction Rates section. In the RcFe text field, type -R_FeOH2.
5
In the RcOH text field, type -2*R_FeOH2.
Use a Concentration and an Electrolyte Potential node to define the boundary toward the bulk of the electrolyte, at the top of the geometry.
Concentration 1
1
In the Physics toolbar, click  Boundaries and choose Concentration.
2
Select Boundary 3 only.
3
In the Settings window for Concentration, locate the Concentration section.
4
Select the Species cNa checkbox.
5
In the c0,cNa text field, type cNa_0.
6
Select the Species cCl checkbox.
7
In the c0,cCl text field, type cCl_0.
8
Select the Species cFe checkbox.
9
In the c0,cFe text field, type cFe_0.
Electrolyte Potential 1
1
In the Physics toolbar, click  Boundaries and choose Electrolyte Potential.
2
Select Boundary 3 only.
Electrode Surface 1
1
In the Physics toolbar, click  Boundaries and choose Electrode Surface.
2
Select Boundaries 4, 5, 7, and 8 only.
3
In the Settings window for Electrode Surface, click to expand the Dissolving–Depositing Species section.
4
Click  Add.
5
In the table, enter the following settings:
Species
Density (kg/m^3)
Molar mass (kg/mol)
Fe
rho_Fe
M_Fe
6
Clear the Solve for surface concentration variables checkbox.
7
Locate the Electrode Phase Potential Condition section. In the ϕs,ext text field, type E_metal.
Electrode Reaction 1
1
In the Model Builder window, click Electrode Reaction 1.
2
In the Settings window for Electrode Reaction, locate the Stoichiometric Coefficients section.
3
In the n text field, type 2.
4
In the νcFe text field, type -1.
5
In the Stoichiometric coefficients for dissolving–depositing species: table, enter the following settings:
Species
Stoichiometric coefficient (1)
Fe
1
6
Locate the Equilibrium Potential section. In the Eeq,ref(T) text field, type Eeq_Fe.
7
Locate the Electrode Kinetics section. In the i0,ref(T) text field, type i0_ref_Fe*(tcd.cH/cH_ref).
8
In the αa text field, type 0.25.
Initial Values 1
1
In the Model Builder window, under Component 1 (comp1) > Tertiary Current Distribution, Nernst–Planck (tcd) click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values section.
3
In the cNa text field, type cNa_0.
4
In the cCl text field, type cCl_0.
5
In the cFe text field, type cFe_0.
Multiphysics
Nondeforming Boundary 1 (ndbdg1)
Switch to zero normal displacement for the nondeforming boundaries. This is a more stable boundary condition for the deformation, but it can only be used for straight boundaries.
1
In the Model Builder window, under Component 1 (comp1) > Multiphysics click Nondeforming Boundary 1 (ndbdg1).
2
In the Settings window for Nondeforming Boundary, locate the Nondeforming Boundary section.
3
From the Boundary condition list, choose Zero normal displacement.
Tertiary Current Distribution, Nernst–Planck (tcd)
1
In the Model Builder window, under Component 1 (comp1) click Tertiary Current Distribution, Nernst–Planck (tcd).
2
In the Settings window for Tertiary Current Distribution, Nernst–Planck, click to expand the Discretization section.
Using linear elements instead of quadratic saves memory and speeds up the computation for this model.
3
From the Concentration list, choose Linear.
4
From the Electrolyte potential list, choose Linear.
5
From the Electric potential list, choose Linear.
Mesh 1
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Sequence Type section.
3
From the list, choose User-controlled mesh.
Size 1
1
In the Model Builder window, right-click Free Triangular 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundaries 4, 7, and 8 only.
5
Locate the Element Size section. From the Predefined list, choose Extremely fine.
Size 2
1
Right-click Free Triangular 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Point.
4
Select Point 2 only.
5
Locate the Element Size section. From the Predefined list, choose Extremely fine.
6
Click  Build All.
Root
For solving the model we will first compute a stationary current and concentration distribution for this initial shape of the pit. The we will use this solution as initial values for a 30-day time-dependent simulation.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies > Stationary.
4
Right-click and choose Add Study.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 1
Step 1: Stationary
1
In the Settings window for Stationary, locate the Physics and Variables Selection section.
2
In the Solve for column of the table, under Component 1 (comp1), clear the checkbox for Deformed Geometry.
3
In the Solve for column of the table, under Component 1 (comp1) > Multiphysics, clear the checkboxes for Nondeforming Boundary 1 (ndbdg1) and Deforming Electrode Surface 1 (desdg1).
Step 2: Time Dependent
1
In the Study toolbar, click  Time Dependent.
2
In the Settings window for Time Dependent, locate the Study Settings section.
3
From the Time unit list, choose d.
4
In the Output times text field, type range(0,1,30).
5
Click to expand the Study Extensions section. Use automatic remeshing to stop and remesh the model if the mesh gets too distorted due to the geometry deformation.
6
Select the Automatic remeshing checkbox.
7
In the Study toolbar, click  Compute.
Results
Electrolyte Current Density, 3D (tcd)
1
In the Model Builder window, expand the Results > Electrolyte Current Density, 3D (tcd) node, then click Electrolyte Current Density, 3D (tcd).
2
In the Settings window for 3D Plot Group, locate the Plot Settings section.
3
Clear the Plot dataset edges checkbox.
Streamline 1
1
In the Model Builder window, click Streamline 1.
2
In the Settings window for Streamline, locate the Coloring and Style section.
3
Find the Point style subsection. From the Arrow length list, choose Normalized.
Color Expression 1
1
In the Model Builder window, expand the Streamline 1 node.
2
Right-click Color Expression 1 and choose Disable.
Surface 1
1
In the Model Builder window, under Results > Electrolyte Current Density, 3D (tcd) click Surface 1.
2
In the Settings window for Surface, locate the Coloring and Style section.
3
From the Color table list, choose AuroraBorealis.
4
From the Color table transformation list, choose Reverse.
5
Click the  Show Grid button in the Graphics toolbar.
6
In the Electrolyte Current Density, 3D (tcd) toolbar, click  Plot.
7
Click the  Zoom Extents button in the Graphics toolbar.
pH
1
In the Results toolbar, click  2D Plot Group.
2
In the Settings window for 2D Plot Group, type pH in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 1/Remeshed Solution 1 (sol3).
4
From the Time (d) list, choose 0.
Surface 1
1
Right-click pH and choose Surface.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type tcd.pH.
4
Locate the Coloring and Style section. From the Color table list, choose Cynanthus.
5
In the pH toolbar, click  Plot.
pH, 3D
1
In the Results toolbar, click  3D Plot Group.
2
In the Settings window for 3D Plot Group, type pH, 3D in the Label text field.
Surface 1
1
Right-click pH, 3D and choose Surface.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type tcd.pH.
4
Locate the Coloring and Style section. From the Color table list, choose Cynanthus.
pH, 3D
1
Click the  Show Grid button in the Graphics toolbar.
2
In the Model Builder window, click pH, 3D.
3
In the pH, 3D toolbar, click  Plot.
Corrosion rate
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Corrosion rate in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 1/Remeshed Solution 1 (sol3).
Line Graph 1
1
Right-click Corrosion rate and choose Line Graph.
2
Select Boundaries 4, 5, 7, and 8 only.
3
In the Settings window for Line Graph, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Tertiary Current Distribution, Nernst–Planck > Dissolving–depositing species > tcd.vbtot - Total electrode growth velocity - m/s.
4
Locate the y-Axis Data section. From the Unit list, choose mm/yr.
5
Locate the x-Axis Data section. From the Parameter list, choose Expression.
6
In the Expression text field, type z.
Color Expression 1
1
Right-click Line Graph 1 and choose Color Expression.
2
In the Settings window for Color Expression, locate the Coloring and Style section.
3
From the Color table list, choose Viridis.
4
Locate the Expression section. In the Expression text field, type t.
5
From the Unit list, choose d.
6
In the Corrosion rate toolbar, click  Plot.
Animation 1
1
In the Results toolbar, click  Animation and choose Player.
2
In the Settings window for Animation, locate the Scene section.
3
From the Subject list, choose pH.
4
Locate the Frames section. From the Frame selection list, choose All.
5
Click the  Play button in the Graphics toolbar.