Under-Deposit Corrosion

This example models the effect of corrosion product on galvanic corrosion between a magnesium alloy (AE44) and mild steel in contact with brine solution. The present model is based upon Galvanic Corrosion with Electrode Deformation model available in the Corrosion Module Application Library. In this model, deposition of magnesium hydroxide corrosion product is modeled using the interface tracking Level Set interface, in addition to dissolution of the magnesium alloy.

The example is based on a paper by W. Sun and others (Ref. 1).

Model Definition

The Galvanic Corrosion with Electrode Deformation model considered electrochemical reactions occurring at the magnesium and steel surfaces and deformation of magnesium surface due to the dissolution reaction, neglecting the effect of corrosion product formed at the electrode surfaces. The present model extends the Galvanic Corrosion with Electrode Deformation model by accounting for mass transport of Mg2+ and OH- ions, precipitation reaction forming magnesium hydroxide corrosion product (Mg(OH)2) and its effect on transport properties. The present model also tracks the corrosion product interface using the Level Set interface, which is used to describe transport properties separately for electrolyte and corrosion product deposition regions.

Figure 1: Model geometry.

The model geometry is the same as used in Galvanic Corrosion with Electrode Deformation model and is shown in Figure 1.

Electrochemical reactions

The magnesium alloy is an anode of the galvanic couple, oxidizing magnesium according to

The mild steel acts as a cathode for this galvanic couple, wherein oxygen reduction reaction occurs according to

The same electrode kinetics as described in Galvanic Corrosion with Electrode Deformation model is used here.

Mass transport

The transport of Mg2+ ions generated at the magnesium surface and OH- ions generated at the mild steel surface is considered to be by diffusion and migration mechanisms. Assuming diluted species in a supporting electrolyte, mass transport of the ionic species is prescribed using a Transport of Diluted Species interface according to

where Ni denotes the transport vector (SI unit: mol/(m2·s)), Di,eff the effective diffusion coefficients of the ionic species, ci the concentration in the electrolyte (SI unit: mol/m3), zi the charge for the ionic species, ui,eff the effective mobility of the charged species (SI unit: m2/(s·J·mole)), F Faraday’s constant (SI unit: A·s/mole), and

the potential in the electrolyte (V).

The material balances are expressed through

one for each species, that is i = Mg2+ and OH-.

The transport properties such as effective diffusion coefficients, are defined using the electrolyte volume fraction:

where

is the electrolyte volume fraction which is defined in terms of level set variable and varies from 1 in the electrolyte domain to 0 in the corrosion product deposition region,

is the diffusion coefficients of the ionic species in the electrolyte domain and

is the diffusion coefficients of the ionic species in the corrosion product deposition region.

Due to a porous nature of corrosion product deposits, diffusion coefficients need to be updated to account for the effect of porosity. Di,cp is defined in terms of MacMullin number, NM, and porosity of corrosion product region,

, to define average characteristic of transport phenomenon in the deposited region according to (Ref. 1)

where

is considered to be 0.55 and NM is defined according to

where

is the tortuosity of corrosion product region and is considered to be 1.

The effective mobilities are calculated using the Nernst-Einstein relation:

The Mg2+ and OH- ions react with each other to form magnesium hydroxide precipitates according to

The precipitation reaction of magnesium hydroxide is assumed to be an irreversible reaction which initiates when the ion product,

, exceeds the solubility product for magnesium hydroxide,

(SI unit: mol3/m9). The rate of reaction for consumption of Mg2+ and OH- ions (SI unit: mol/(m2·s)) is described according to

where k is the precipitation reaction rate constant (SI unit: m7/(mol2·s)), H(ζ) is the Heaviside step function and ζ is the step function variable which is defined according to

The reaction source term (SI unit: (mol/(m3·s)) is defined in terms of the reaction rate for consumption of Mg2+ and OH- ions (SI unit: mol/(m2·s)) according to

where δ is the level set delta function which is used to prescribe the deposition reaction at the corrosion product interface.

The boundary condition at the magnesium surface is:

where n denotes the normal vector to the boundary.

The boundary condition at the magnesium surface is:

The initial conditions and concentration at the top boundary are set according to

All other boundaries are insulating:

Corrosion product interface tracking

The Level Set interface is used to keep track of the deformation due to corrosion product deposition. The Level Set interface automatically sets up the equations for the movement of the interface between the liquid electrolyte and the porous corrosion product. The interface is represented by the 0.5 contour of the level set variable

. The level set variable varies from 1 in the electrolyte domain to 0 in the deposited region. The level set variable can thus be thought of as the electrolyte volume fraction. The transport of the level set variable is given by:

The ε parameter determines the thickness of the interface and is defined as ε = hmax/4, where hmax is the maximum mesh element size in the domain. The γ parameter determines the amount of reinitialization. A suitable value for γ is the maximum velocity magnitude occurring in the model.

The level set delta function is approximated by:

As the magnesium hydroxide corrosion product deposits at the electrode surfaces, the magnesium surface is also deforming due to dissolution reaction. The velocity field used in the transport equation for level set variable needs to account for both of these components: velocity representing magnesium hydroxide corrosion product deposition, udep (SI unit: m/s) and velocity representing magnesium dissolution, ucorr (SI unit: m/s).

The magnesium hydroxide corrosion product deposition velocity in normal direction is defined according to

where

is the molar mass (58 g/mol) and

is the density of Mg(OH)2 (2450 kg/m3).

The magnesium dissolution velocity in normal direction is defined according to

where

is the mean molar mass (25 g/mol) and

is the density (1820 kg/m3) of the magnesium alloy. Since ucorr is available only at the deforming boundary, we use a general extrusion operator to make it available throughout the domain as well.

Assuming that deposits form parallel vertical rods type of porous structure, only y-component of dissolution velocity is used in the level set transport equation. Hence, the velocity field used in the transport equation for level set variable is defined according to

where

and

are x- and y-components of the interface normal nLS which is defined in terms of level set variable according to

is y-component of the unit normal vector n, pointing out of the electrolyte domain.

The Heaviside step function

ensures that the level set variable is convected only adjacent to the corrosion product region.

The level set variable of value 0 enters the domain from the bottommost boundaries of the domain, which are prescribed using the Inlet boundary condition. The rest of the boundaries are prescribed using the Outlet boundary condition.

Electrolyte properties

Finally, the effective electrolyte conductivity,

, is defined for the electrolyte domain and the porous corrosion product region separately using the electrolyte volume fraction defined in terms of level set variable according to

where σ is the electrolyte conductivity in the electrolyte domain and is considered to be equal to 2.5 S/m.

The electrolyte conductivity in the corrosion product deposition region,

, is described according to

where sL is the fluid saturation and is considered to be 1, m is the cementation exponent and is considered to be 1 and n is the saturation coefficient and is considered to be 2 in this model.

Solve the model in a time-dependent study, simulating the corrosion for three days of immersion in salt water.

Results and Discussion

Figure 2 shows a surface plot of volume fraction of fluid 1 at the end of the simulation (t = 72 h). The region with volume fraction of fluid 1 of value 1 represents the corrosion product deposition whereas the same of value 0 represents the electrolyte. A contour plot of volume fraction of fluid 1 of value 0.5 represents the position of corrosion product interface at the end of the simulation. It can be seen that the corrosion product deposition is maximum at the contact point of the metals and it decreases with distance away from the contact point along the magnesium and mild steel surfaces. The corrosion product thickness is also found to be thicker at the magnesium surface than the same at the mild steel surface. This behavior is attributed to the faster precipitation reaction at the magnesium surface, since OH- ions transport faster than Mg2+ ions in the electrolyte. Figure 2 represents deformed boundaries due to deposition of corrosion product as well as dissolution of magnesium.

Figure 2: The corrosion product deposition and dissolution of magnesium after 72 h.

Figure 3 shows the electrode current densities at the beginning and the end of the simulation. As expected, the highest electrode current densities are found at the contact point between the two metals. However, the electrode current density distribution along the magnesium and mild steel surfaces is considerably different when compared to a scenario where the effect corrosion product is not considered (refer to Figure 2 in Galvanic Corrosion with Electrode Deformation model). It can be seen that the electrode current density at the end of the simulation is lower adjacent to the contact point between the two metals which is attributed to the corrosion product deposited at the magnesium and mild steel surfaces

Figure 3: Electrode current densities at t = 0 and t = 72 h.

Figure 4 shows the current density and potential distribution in the electrolyte, and the model geometry, at the beginning of the simulation. The electrolyte potential and current density distribution at the beginning of simulations is the same as reported in Figure 3 of Galvanic Corrosion with Electrode Deformation model, as expected.

Figure 4: Model geometry, electrolyte potential, and current densities at t = 0.

Figure 5 shows the current density and potential distribution in the electrolyte, the changed geometry due to metal dissolution and corrosion product deposited region at the end of the simulation. Because the electrode current densities are highest at the contact point of the metals, the metal dissolution is also maximum at this point. A streamline plot of electrolyte current density vector at the end of simulations is slightly different than that reported in Figure 4 of Galvanic Corrosion with Electrode Deformation model. The difference in the electrolyte current density streamline plot is apparent in the corrosion product deposited region where streamlines are found to change the shape.

Figure 5: Model geometry, electrolyte potential, and current densities after 72 h.

Reference

1. W. Sun, G. Liu, L. Wang, T. Wu and Y. Liu, “An arbitrary Lagrangian-Eulerian model for studying the influences of corrosion product deposition on bimetallic corrosion”, J Solid State Electrochem, vol. 17, pp. 829–840, 2013.

Corrosion_Module/Galvanic_Corrosion/under_deposit_corrosion

Modeling Instructions

Root

From the menu, choose .

Browse to the model’s Application Libraries folder and double-click the file galvanic_corrosion_with_deformation.mph.

Component 1 (comp1)

Add and interfaces to solve for the mass transport of Mg2+ and OH− ions and to track the position of the Mg(OH)2 corrosion product precipitate, respectively.

Add Physics

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click to expand the section. In the text field, type 2.

In the table, enter the following settings:


cMg
cOH

Click in the window toolbar.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Global Definitions

Start by updating the parameters required for corrosion product modeling.

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file under_deposit_corrosion_parameters.txt.

Definitions

Define a step function for use when defining variables for precipitation reaction rate and corrosion product velocity.

Step 1 (step1)

In the toolbar, click

and choose .

In the window for , click to expand the section.

In the text field, type 0.001.

Variables 1

Load the model variables from a text file.

In the toolbar, click

and choose .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file under_deposit_corrosion_variables.txt.

The orange color for the variable v_corprod is caused by the undefined nonlocal general extrusion coupling.

General Extrusion 1 (genext1)

Now, define the nonlocal general extrusion coupling.

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Boundary 5 only.

Locate the section. Clear the text field.

Locate the section. From the list, choose .

Select the check box.

Clear the i text field.

Click to expand the section. From the list, choose .

Transport of Diluted Species (tds)

Next, set up mass transport physics.

In the window, under click .

In the window for , locate the section.

Clear the check box.

Select the check box.

Select Domain 1 only.

Transport Properties 1

In the window, under click .

In the window for , locate the section.

In the DcMg text field, type DMg*ls.Vf2+DMge*ls.Vf1.

In the DcOH text field, type DOH*ls.Vf2+DOHe*ls.Vf1.

Locate the section. In the V text field, type phil.

In the zcMg text field, type 2.

In the zcOH text field, type -1.

Initial Values 1

In the window, click .

In the window for , locate the section.

In the cOH text field, type cOH0.

Reactions 1

Use a node to define the sink terms for Mg2+ and OH−. The use of the variable ls.delta ensures that the sink term is applied only at the corrosion product interface.

In the toolbar, click

and choose .

Select Domain 1 only.

In the window for , locate the section.

In the RcMg text field, type -R_Mg*ls.delta.

In the RcOH text field, type -R_OH*ls.delta.

Concentration 1

Define the concentration at the top boundaries of the domain.

In the toolbar, click

and choose .

Select Boundaries 3 and 6 only.

In the window for , locate the section.

Select the check box.

In the c0,cOH text field, type cOH0.

Electrode Surface Coupling 1

Now, define the source terms for Mg2+ and OH− at the anode and cathode boundaries, respectively, making use of the local current densities obtained from the interface.

In the toolbar, click

and choose .

Select Boundaries 2 and 4 only.

Reaction Coefficients 1

In the window, expand the node, then click .

In the window for , locate the section.

From the iloc list, choose .

Locate the section. In the n text field, type 4.

In the νcOH text field, type 4.

Electrode Surface Coupling 2

In the toolbar, click

and choose .

Select Boundary 5 only.

Reaction Coefficients 1

In the window, expand the node, then click .

In the window for , locate the section.

From the iloc list, choose .

Locate the section. In the n text field, type 2.

In the νcMg text field, type -1.

Level Set (ls)

Now, set up the level set physics to track the position of the corrosion product interface.

In the window, under click .

Select Domain 1 only.

Level Set Model 1

In the window, under click .

In the window for , locate the section.

In the γ text field, type max(vn_corprod,eps).

In the εls text field, type ls.hmax/4.

Locate the section. Specify the u vector as


u_corprod	x
v_corprod	y

Note that the variables u_corprod and v_corprod use the step function variables defined in terms of the level set function to ensure that only the level sets adjacent to the corrosion product are advected.

Initial Values 1