Advanced composites are used extensively throughout the Boeing 787 Dreamliner, as shown in the diagram below. Also known as carbon fiber reinforced plastic (CFRP), the composites are formed from a lightweight polymer binder with dispersed carbon fiber filler to produce materials with high strength-to-weight ratios. Many wing components, for example, are made of CFRP, ensuring that they can support the load imposed during flight while minimizing their overall contribution to the weight of an aircraft.

*Advanced composites are used throughout the body of the Boeing 787. Copyright © Boeing.*

Despite their remarkable strength and light weight, CFRPs are generally not conductive like their aluminum counterparts, thus making them susceptible to lightning strike damage. Therefore, electrically conductive expanded metal foil (EMF) is added to the composite structure layup, shown in the figure below, to dissipate the high current and heat generated by a lightning strike.

*The composite structure layup shown at left consists of an expanded metal foil layer shown at right. This figure is a screenshot from the COMSOL Multiphysics® software model featured in this blog post. Copyright © Boeing.*

The figure also shows the additional coatings on top of the EMF, which are in place to protect it from moisture and environmental species that cause corrosion. Corrosive damage to the EMF could result in lower conductivity, thereby reducing its ability to protect aircraft structures from lightning strike damage. Temperature variations due to the ground-to-air flight cycle can, however, lead to the formation of cracks in the surface protection scheme, reducing its effectiveness.

During takeoff and landing, aircraft structures are subjected to cooling and heating, respectively. Thermal stress manifests as the expansion and compression — or ultimately the displacement — of adjacent layers throughout the depth of the composite structure. Although a single round-trip is not likely to pose a significant risk, over time, each layer of the composite structure contributes to fatigue damage buildup. Repetitive thermal stress results in cumulative strain and higher displacements, which are, in turn, associated with an increased risk of crack formation. The stresses in a material depend on its mechanical properties quantified by measurable attributes such as yield strength, Young’s modulus, and Poisson’s ratio.

By taking the thermal and mechanical properties of materials into account, it is possible to use simulation to design and optimize a surface protection scheme for aircraft composites that minimizes stress, displacement, and the risk of crack formation.

Evaluating the thermal performance of each layer in the surface protection scheme is essential in order to reduce the risks and maintenance costs associated with damage to the protective coating and EMF. Therefore, researchers at Boeing Research & Technology (BR&T), pictured below, are using multiphysics simulation and physical measurements to investigate the effect of the EMF design parameters on stress and displacement throughout the composite structure layup.

*The research team at Boeing Research & Technology from left to right: Patrice Ackerman, Jeffrey Morgan, Robert Greegor, and Quynhgiao Le. Copyright © Boeing.*

In their work, the researchers at BR&T have developed a coefficient of thermal expansion (CTE) model in COMSOL Multiphysics® simulation software. The figure shown above that presents the composite structure layup and EMF is a screenshot acquired from the model geometry used for their simulations in COMSOL Multiphysics.

The CTE model was used to evaluate heating of the aircraft composite structure as experienced upon descent, where the final and initial temperatures used in the simulations represent the ground and altitude temperatures, respectively. The *Thermal Stress* interface, which couples heat transfer and solid mechanics, was used in the model to simulate thermal expansion and solve for the displacement throughout the structure.

The material properties of each layer in the surface protection scheme as well as of the composites are custom-defined in the CTE model. The relative values of the coefficient of thermal expansion, heat capacity, density, thermal conductivity, Young’s modulus, and Poisson’s ratio are presented in the chart below.

*This graph presents the ratio of each material parameter relative to the paint layer. Copyright © Boeing.*

From the graph, trends can be identified that provide early insight into the behavior of the materials, which aids in making design decisions. For example, the paint layer is characterized by higher values of CTE, heat capacity, and Poisson’s ratio, thus indicating that it will undergo compressive stress and tensile strain upon heating and cooling.

Multiphysics simulation takes this predictive design capability one big step forward by quantifying the resulting displacement due to thermal stress throughout the entire composite structure layup simultaneously, taking into account the properties of all materials. The following figure shows an example of BR&T’s simulation results and presents the stress distribution and displacement throughout the composite structure.

*Left: Top-down and cross-sectional views of the von Mises stress and displacement in a one-inch square sample of a composite structure layup. Right: Transparency was used to show regions of higher stress, in red. Lower stress is shown in blue. Copyright © Boeing.*

In the plots at the left above, the displacement pattern caused by the EMF is evident through the paint layer at the top of the composite structure while a magnified cross-sectional view shows the variations in displacement above the mesh and voids of the EMF. The cross section also makes it easy to see the stress distribution through the depth of the composite structure, where there is a trend toward lower stress in the topmost layers. Transparency was used in the plot shown at the right to depict the regions of high stress in the composites and EMF, which is noticeably higher at the intersection of the mesh wires. Stress was plotted through the depth of the composite structure layup along the vertical red line shown in the center of the plot. The figure below shows the relative stress in each layer of the composite structure layup for different metallic compositions of the EMF.

*Relative stress in arbitrary units was plotted through the depth of the composite structure layups containing either aluminum (left) or copper EMF (right). Copyright © Boeing.*

The samples vary by the presence of a fiberglass corrosion isolation layer when aluminum is used as the material for the EMF. The fiberglass acts as a buffer resulting in lower stress in the aluminum EMF, when compared with the copper.

From lightning strike protection to the structural integrity of the composite protection scheme, it all relies on the design of the expanded metal foil layer. The design of the EMF layer can vary by its metallic composition, height, width of the mesh wire, and the mesh aspect ratio. For any EMF design parameter, there is a trade-off between current-carrying capacity, displacement, and weight. By using the CTE model, the researchers at BR&T found that increasing the mesh width and decreasing the aspect ratio are better strategies for increasing the current-carrying capacity of the EMF that minimize its impact on displacement in the composite structure.

The metal chosen for the EMF can also have a significant effect on stress and displacement in the composite structure, which was investigated using simulation and physical testing. Two composite structures, one with aluminum and the other with copper EMF, underwent thermal cycling with prolonged exposure to moisture in an environmental test chamber. In the results, shown below, the protective layers remained intact for the composite structure with copper EMF. However, for the layup with aluminum, cracking occurred in the primer, at the edges, on surfaces, and was particularly substantial in the mesh overlap regions.

*Photo micrographs of the composite structure layup after exposure to moisture and thermal cycling. A crack in the vicinity of the aluminum EMF is contained within the red ellipse. Copyright © Boeing.*

Simulations confirm the experiment results. Shown below, displacements are noticeably higher throughout the composite structure layup when aluminum is used for the EMF layer, where higher displacements are associated with an increased risk for developing cracks. The higher displacement is easiest to observe in the bottom plots, which show displacement ratios for each EMF height.

*Effect of varying the EMF height on displacement in each layer of the surface protection scheme. Copyright © Boeing.*

The larger displacements caused by the aluminum EMF can be attributed in part to its higher CTE when compared with copper, which exemplifies how important the properties of materials are to the thermal stability of the aircraft composite structures.

In the early design stages and along with experimental testing, multiphysics simulation offers a reliable means to evaluate the relative impact of the EMF design parameters on stress and displacement throughout the composite structures. An optimized EMF design is essential to minimizing the risk of crack formation in the composite surface protection scheme, which reduces maintenance costs and allows the EMF to perform its important protective function of mitigating lightning strike damage.

Refer to page 4 of *COMSOL News* 2014 to read the original article, “Boeing Simulates Thermal Expansion in Composites with Expanded Metal Foil for Lightning Strike Protection of Aircraft Structures”.

This article was based on the following publicly available resources from Boeing:

- The Boeing Company. “787 Advanced Composite Design.” 2008-2013.
- J.D. Morgan, R.B. Greegor, P.K. Ackerman, Q.N. Le, “Thermal Simulation and Testing of Expanded Metal Foils Used for Lightning Protection of Composite Aircraft Structures,” SAE Int. J. Aerosp. 6(2):371-377, 2013, doi:10.4271/2013-01-2132.
- R.B. Greegor, J.D. Morgan, Q.N. Le, P.K. Ackerman, “Finite Element Modeling and Testing of Expanded Metal Foils Used for Lightning Protection of Composite Aircraft Structures,” Proceedings of 2013 ICOLSE Conference; Seattle, WA, September 18-20, 2013.

To learn more about adding material property data to your COMSOL Multiphysics® simulations, read the following blog post series on *Obtaining Material Data for Structural Mechanics Simulations from Measurements* by my colleague Henrik Sönnerlind:

General information about aircraft design and structures can be found in chapter 1 of this handbook on aircraft maintenance from the Federal Aviation Administration.

*BOEING, Dreamliner, and 787 Dreamliner are registered trademarks of The Boeing Company Corporation in the U.S. and other countries.*

Siddiq Qidwai, a mechanical engineer at NRL, was a keynote speaker at the COMSOL Conference 2013 Boston, where he enlightened a packed auditorium with a speech on the significant costs and hazards associated with corrosion. Qidwai also explained the important work they are doing to better understand the mechanism of corrosion in metals. They expect that this new knowledge will aid in designing corrosion-resistant materials.

“For the Navy specifically, corrosion is the number one maintenance problem,” explained Qidwai in his presentation. We can all certainly relate to that sentiment — on a much smaller scale — whether it’s a bicycle left in the rain, a car that’s endured a few too many harsh winters, or the pipes and metallic fixtures used regularly in our homes.

Inspired by his presentation, the work of Qidwai and his colleagues was featured in the 2014 edition of *COMSOL News*, in the article “Simulation-Led Strategy for Corrosion Prevention” on page 36.

*From left to right: Siddiq Qidwai (NRL), Virginia DeGiorgi (NRL), and Nithyanand Kota (Leidos Corp.), the researchers behind understanding the fundamental mechanism of corrosion in metals.*

Where corrosion starts and how it propagates depends on the microstructure of a metal. In the following figure, an electron micrograph acquired at NRL demonstrates the formation of an irregular surface due to pit growth in an aluminum alloy. If the microstructure were not important, we’d expect to see a much smoother surface in the image.

*Electron micrograph. Image courtesy of C. Feng and S. Policastro, NRL.*

Pitting corrosion in metals is caused by electrochemical reactions and mass transport in an electrolyte solution, as shown in the figure below. The diagram depicts the general corrosion process for steel submerged in seawater. As corrosion proceeds, metal ions, such as Fe^{2+}, diffuse into the electrolyte solution away from the metal surface, leading to the formation of pits. The diagram not only shows pit growth, but also the formation of an irregular *corrosion front*. The corrosion front is the surface along which electrochemical reactions occur.

*Diagram of the corrosion process.*

Along the front, the reaction rate varies due to differences in the size, shape, and crystallographic orientation of each grain. The microstructure of stainless steel is shown with the grain boundaries labeled in the diagram. The varied rate associated with each grain leads to the formation of an irregular corrosion front. Therefore, it is important to take metal microstructure into account when studying pitting corrosion in metals.

Now, consider what happens as pitting corrosion progresses. The structure of a metal is disrupted, leading to areas of increased stress surrounding a pit, which consequently compromises its mechanical strength. If you’ve ever scored a piece of glass to make it easier to break, you can already relate to what’s going on. Unlike the process of breaking glass, for corrosion in metals, the reduced strength is undesirable and may lead to material or component failure. Hence, research at NRL aims to establish the relationship between microstructure, corrosion, and mechanical strength to aid in the design of corrosion-resistant materials.

Qidwai and his colleagues are working toward developing a fully-coupled multiphysics model of pitting corrosion in metals, including the electrochemical reactions and mass transport. To overcome the limitations of previous research in the field, they have developed a method to incorporate metal microstructure into their model of pitting corrosion.

*Incorporating metal microstructure in a pitting corrosion model.*

Researchers at NRL were able to reconstruct the microstructure of steel from 3D images acquired using orientation imaging microscopy (OIM). The reconstructed metal microstructure is presented to the left in the figure above, along with a legend indicating the crystallographic orientation of each grain. To the right, you can see the model geometry implemented in COMSOL Multiphysics® simulation software.

To develop a complete model of pitting corrosion in 316 steel, Qidwai is using the *Transport of diluted species* interface for mass transport, the Laplace and Poisson’s equations for the electric potential, and the moving mesh (ALE) technology for the corrosion front. “You can use COMSOL Multiphysics with the Corrosion Module to solve this problem,” explains Qidwai. “All of the work that previously seemed so difficult, now is so easy because you have the module doing a lot of the work for you.”

Implementing the irregular corrosion front movement due to metal microstructure presented a unique challenge in their simulation work. “At every location along the corrosion front, we have to determine the crystal orientation to calculate the corresponding corrosion potential, which in turn determines the corrosion rate and movement of the front,” says Qidwai. “LiveLink™ *for* MATLAB® has been an essential feature for us in order to include the effect of metal microstructure.” The corrosion potential, in particular, is determined in MATLAB® for a particular crystallographic orientation.

Their work also includes decoupled structural mechanics analyses of pit growth in metals. You can see what that looks like to the left in the screenshot of the COMSOL Desktop® environment below. The simulation results in the right-hand plot demonstrate pit growth with an irregular corrosion front.

Future work at NRL will be directed toward creating a fully-coupled model of pitting corrosion that includes structural mechanics. This way, the effect of pit growth on the mechanical strength of the material is considered, along with the effect of mechanical loading on pitting corrosion. Once the model is completed and validated, it can serve as a valuable tool to aid material designers in creating corrosion-resistant materials, which is ultimately of enormous benefit to everyone.

- For more information about pitting corrosion research using multiphysics simulation at NRL, read the full article “Simulation-Led Strategy for Corrosion Prevention”.
- Watch the original keynote presentation on modeling pitting corrosion at the microstructural scale by Siddiq Qidwai from the COMSOL Conference 2013 Boston.
- Kota, N.; Qidwai, S.M.; and DeGiorgi, V.G.; “Modeling the Influence of Microstructure on Corrosion Pit Growth and Resulting Stress Concentration,”
*ECS Transactions*, Vol. 58, No. 31, pp. 35-44, 2014.

*MATLAB is a registered trademark of The MathWorks, Inc. All other trademarks are the property of their respective owners. For a list of such trademark owners, see http://www.comsol.com/tm. COMSOL AB and its subsidiaries and products are not affiliated with, endorsed by, sponsored by, or supported by these trademark owners.*

There are several ways you can protect structures from corrosion: Impressed Cathodic Current Protection (ICCP), anodic protection, and by using sacrificial anodes. We will prescribe the third method in this example.

The sacrificial anodes method essentially involves protecting the cathode by sacrificing the anode. In the case of oil platforms, the steel structure is what needs protecting. Therefore, we can turn the steel into the cathode by connecting it electronically to a metal that is less noble, i.e. anodic (such as aluminum, for example).

When the electrodes are then immersed in seawater, the less noble metal will be polarized anodically and the steel structure will be polarized cathodically. The metal anodes will undergo anodic dissolution. Meanwhile, oxygen reduction will occur at the steel structure’s surface.

*Simplified illustration of the anode and cathode polarization.*

The current density for oxygen reduction is commonly limited by the oxygen supply. This results in a near-constant, transport-limited current for a range of potential difference at the structure’s surface, spanning a few hundred millivolts.

*Polarization: Sacrificial anodes (blue) and steel surface (red).*

If you look at the graphs above, see where it says “Oxygen reduction”? The dashed lines on either side of that label indicate that when current flows, the sacrificial anodic dissolution (blue line) and cathodic oxygen reduction (red line) currents are equal in size. Note that the exact shape of the blue polarization curve is determined by the accessibility and number of anodes.

In order to fully protect the structure, design engineers need to make sure the oil platform’s various parts are safely within the corrosion-protected range. In other words, the voltage delivered by the anodes needs to be sufficient to drive oxygen reduction at the steel to balance the current. You also have to take care to avoid hydrogen evolution, as that could cause hydrogen embrittlement. During the process, increased pH may lead to formation of a calcareous film at the cathode surface, which changes the conductivity at the surface. The growth of these deposits can also be accounted for in high-fidelity models.

When designing your cathodic protection system, you will first need to figure out the potential difference between the steel structure and the seawater during oxygen reduction. Doing so, you can determine what part will be more vulnerable to corrosion — *before* it’s too late. This is where the Corrosion Module comes in handy.

Suppose we polarize the oil platform cathodically and place cylindrical sacrificial anodes in the structure’s proximity.

Instead of detailing exactly how to set up and solve the model, let’s skip ahead to the results.

You can learn how to build the model by downloading the documentation from the Model Gallery or opening the model from within the Model Library in the Corrosion Module (under the “Cathodic Protection” section).

First, let’s examine the electrolyte potential at the oil platform’s surface as well as on the anodes. From the surface plot, we can see that the further away the protected structure is from the sacrificial anodes, the lower the potential in the electrolyte (seawater). The lower the electrolyte potential at the interface, the more positive the potential difference is between the oil platform and seawater. The more prone a structure is to oxidation, the less protected it will be. Naturally, we need to make sure that we always distribute sacrificial anodes well, so that the entire structure is protected.

Next, we can have a look at the electrolyte potential in one of the oil platform legs. The inside bottom section of the leg has the lowest electrolyte potential (most anodic interfacial potential difference) and will therefore be more prone to corrosion.

Finally, we can plot the current densities on the anodes. This is important because the magnitude of the current densities is directly proportional to the anode metal’s consumption rate. In other words, this can help us determine how long the sacrificial anodes will last.

Now that we’ve gone through this introduction to modeling corrosion for the oil and gas industry, why don’t you try out the model yourself? Once you’ve built it, you could try moving around the anodes or even adding some more to “retrofit” the structure.

- Model download: Oil Platform Corrosion Protection Using Sacrificial Anodes
- Learn about current distribution

As you saw in the previous blog post, we can use an example model of a wire electrode to compare the three current distribution interfaces. Here is the geometry again:

*Wire electrode model solved using COMSOL Multiphysics. Electrolyte is allowed to flow in the open volume between the wire and flat surfaces.*

The same geometry is considered in all three cases presented here: a wire electrode structure is placed between two flat electrode surfaces. The electrochemical cell can be seen as a unit cell of a larger wire-mesh electrode, which is an electrochemical cell set-up common in many large-scale industrial processes.

Below are the essential equations we mentioned in detail previously:

Nernst-Planck equation:

(1)

\textbf{N}_i = -D_i\nabla c_i-z_i u_{m,i} F c_i\nabla \phi_l+c_i\textbf{u}

Current density expression with the Nernst-Planck equation:

(2)

\textbf{i}_l = -F \left(\nabla \sum_i z_iD_i c_i\right)-F^2\nabla \phi_l \sum_i z^2_i u_{m,i} c_i+\textbf{u}F\sum_i z_ic_i

General electrolyte current conservation:

\nabla\cdot\mathbf{i}_l=Q_l

The primary current distribution accounts only for losses due to solution resistance, neglecting electrode kinetic and concentration-dependent effects. The charge transfer in the electrolyte is assumed to obey Ohm’s law. We are making two assumptions here: first, that the electrolyte is electroneutral, which cancels out the convective contribution to the current density in equation (2), and second, that composition variations in the electrolyte are negligible (it is homogeneous), which cancels out the diffusive contribution to the current density in equation (2) and allows us to treat the ionic strength as a constant. Hence, the remaining term of equation (2) results in Ohm’s law for electrolyte current density.

At the electrode-electrolyte interface we assume that the electrolysis reaction is so fast that we can neglect the influence of electrode kinetics, and so the potential difference at the electrode-electrolyte boundary deviates negligibly from its equilibrium value. In other words, there is no activation overpotential and an arbitrary current density can occur through electrolysis. Therefore, the primary current distribution only depends on the geometry of the anode and cathode.

The *Primary Current Distribution* interface in COMSOL Multiphysics defines two dependent variables: one for the electric potential in the electrolyte (\phi_l\) and another for the electric potential in the electrodes (\phi_s\). With the above described assumptions for a primary current distribution, you get the following equations to be solved:

Electrode: \textbf{i}_s = -\sigma_s\nabla\phi_s\ with \nabla\cdot\textbf{i}_s = Q_s

Electrolyte: \textbf{i}_l = -\sigma_l\nabla \phi_l\ with \nabla\cdot\textbf{i}_l = Q_l

Electrode-Electrolyte-Interface: \phi_s-\phi_l = E_{\mathrm{eq},m}

Here, \sigma_l denotes the conductivity of the electrolyte, which is constant by the above assumptions. The index s represents the electrode and l the electrolyte. E_{\mathrm{eq},m} denotes the equilibrium potential for reaction m.

In the following picture, we show the primary current density distribution for our wire electrode example. As you can see, the current density distribution is highest at the corners of the wires directly facing the cathode plates, and close to zero at the central parts of the wire structure that are geometrically shielded from the cathode.

*Primary current distribution, Ecell = 1.65 V. Current density distribution on the anode (dimensionless).*

You can use this class of current distribution for modeling cells where you have a relatively high electrolyte concentration (in relation to current density) or vigorous mixing in the electrolyte, allowing the assumption of a uniform electrolyte concentration. In addition, the electrochemical reactions have to be fast enough for negligible resistance to be associated with the reaction, compared to the magnitude of the ohmic losses (solution resistance).

One application of these conditions is at the anodes in systems for imposed current cathodic protection, while a constant current corresponding to the mass transport-limited current for oxygen reduction can be set as a boundary condition at the cathode in the primary current density interface (here’s an example of this). This can also be a valuable approximation for electrochemical processes involving relatively fast reactions, such as the oxidation of chloride ions in the chlor-alkali process.

Since the *Primary Current Distribution* interface is easy to solve and involves no nonlinear kinetic expressions, it is often suitable to use in order to calculate a baseline approximation before approaching a more complex model.

The secondary current distribution accounts for the effect of the electrode kinetics in addition to solution resistance. The assumptions about the electrolyte composition and behavior are the same as for the primary current distribution, resulting in Ohm’s law for electrolyte current. The difference between the primary and secondary current distributions lies in the description of the electrochemical reaction at the interface between an electrolyte and an electrode.

Here, the influence of electrode kinetics is included; the potential difference may differ from its equilibrium value due to additional impedance associated with the finite rate of the electrolysis reaction. The difference between the actual potential difference and the equilibrium potential difference is the activation overpotential (\eta). Thus, you get the same domain equations as in the *Primary Current Distribution* interface, but the electrode-electrolyte interface equation differs according to the overpotential:

Electrode: \textbf{i}_s = -\sigma_s\nabla\phi_s\ with \nabla\cdot\textbf{i}_s = Q_s

Electrolyte: \textbf{i}_l = -\sigma_l\nabla \phi_l\ with \nabla\cdot\textbf{i}_l = Q_l

Electrode-Electrolyte-Interface: \eta_m = \phi_s-\phi_l-E_{eq,m}

In the *Secondary Current Distribution* interface, the current density due to the electrochemical reactions is described as a function of the overpotential. The physics interface can use any relation between current density and overpotential, with common examples such as the Butler-Volmer equation (3) and the Tafel equation included as built-in options.

(3)

i_{loc,m} = i_{0,m}\left(e^\frac{\alpha_{a,m} F \eta_m}{RT}-e^\frac{-\alpha_{c,m} F \eta_m }{RT}\right)

In the Butler-Volmer equation above, for reaction m:i_{loc,m} denotes the local charge transfer current density, i_{0,m} the exchange current density, and \alpha_{a,m} the anodic and \alpha_{c,m} the cathodic charge transfer coefficient. R is the universal gas constant. This equation describes the case when the charge transfer of one electron is the rate determining step in the net charge transfer reaction. The expression can be derived by analogy to the Arrhenius equation for a homogeneous chemical reaction, by assuming the free energy of the charged species to be influenced by the potential. Hence, the activation energy changes with the potential difference at the electrode-electrolyte interface.

The sum of all electrode reaction currents is implemented as a current density condition on the boundary between an electrode and an electrolyte domain according to:

-\textbf{i}_s \cdot \textbf{n} = \textbf{i}_l \cdot \textbf{n} = i_\mathrm{DL} + \sum_m{i_{loc,m}}

The additional capacitive current i_\mathrm{DL} arises from charge and discharge of the electrical double layer.

In general, accounting for the effect of electrode kinetics by means of an activation overpotential will tend to make the current distribution more uniform. You can see this in the wire electrode example in the figure below.

Compared to the primary current distribution, the secondary current distribution is smoother, with a smaller difference between the minimum and maximum values. When the activation overpotential is included, a high local current density would introduce a high local activation overpotential at the electrode surface, which causes the current to naturally take a different path. To look at this another way, you can understand the electrochemical reaction as proceeding at a finite rate. In some regions, the reaction is kinetically limited, and so the distribution of current densities over the surface is less extreme than in the case when the reaction can proceed arbitrarily quickly.

*Secondary current distribution, Ecell = 1.65 V. Current density distribution on the anode (dimensionless).*

*Secondary Current Distribution* is the workhorse interface for modeling industrial applications in electrochemistry. You can use this class of current distribution for modeling cells where you can neglect concentration overpotential, due to good mixing or relatively high electrolyte concentration, but when the electrode kinetics cause losses that are not negligible compared to the ohmic losses. In industrial applications it is usually not a problem to provide an electrolyte of high concentration with vigorous mixing. You can also use the *Secondary Current Distribution* interface as a first step in your simulation of electrochemical cells to estimate the activation losses, before you eventually introduce concentration-dependent reaction kinetics.

The tertiary current distribution accounts for the effect of variations in electrolyte composition and ionic strength on the electrochemical process, as well as solution resistance and electrode kinetics. To do this, it solves the Nernst-Planck equation (1) explicitly for each chemical species to describe its mass transport through diffusion, migration, and convection. Additionally, the species concentrations are subject to the electroneutrality approximation. The kinetic expressions for the electrochemical reactions account for both activation and concentration overpotential, meaning that the rate of an electrolysis reaction can be transport-limited by exhaustion of the reactant at the electrode-electrolyte interface. This implies that all ions and all electroactive species in the electrolyte must be included in the model.

Unlike the primary and secondary current distributions, the electrolyte current density is no longer assumed to follow Ohm’s law in the tertiary current distribution. The imposition of electroneutrality still means that convective flux does not contribute to the current density, due to equation (2), but now the influence of the concentration variations in the electrolyte cannot be neglected. Therefore, the diffusion term in equation (2) may be non-zero.

At the electrode-electrolyte interface, the current density of charge transfer reactions is expressed as a function not only of the overpotential, but also of the concentration of the electroactive species at the interface. For a reaction rate determined by a one-electron charge transfer step, the reaction kinetics is expressed using a Butler-Volmer expression for the charge transfer current density i_{loc,m} (compare with equation (3)), which in this case can contain concentration dependencies.

The *Tertiary Current Distribution* interface in the COMSOL software solves for the electrolyte potential (\phi_l\), the electrode potential (\phi_s\), and the set of species concentrations c_i. With the assumptions described above you get the following equations:

Electrode: \textbf{i}_s = -\sigma_s\nabla\phi_s\ with \nabla\cdot\textbf{i}_s = Q_s

Electrolyte: \textbf{i}_l = F\sum_{i=1}^n z_i (-D_i\nabla c_i-z_i u_{m,i} F c_i\nabla \phi_l) with \nabla\cdot\textbf{i}_l = Q_l

Electrolyte electroneutrality: \sum_i z_ic_i = 0

Electrode-Electrolyte-Interface: \eta_m = \phi_s-\phi_l-E_{eq,m}

Typical current density expression: i_{loc,m} = i_{0,m}\left(\frac{c_\mathrm{Red}}{c_\mathrm{ref}} e^\frac{\alpha_{a,m} F \eta_m}{RT}-\frac{c_\mathrm{Ox}}{c_\mathrm{ref}}e^\frac{-\alpha_{c,m} F \eta_m }{RT}\right)

It is essential that the reference concentration c_\mathrm{ref} is the same for all species involved in a reaction. This ensures that at zero current density (equilibrium) the overpotential obeys the thermodynamic Nernst equation.

In the image below, you can see the tertiary current distribution for the wire example. Due to the dependence of the concentration, the tertiary current distribution becomes influenced by the flow of the electrolyte and hence the availability of the reactant by mass transport. Where the flow velocity is small between the wires, electrolyte consumed to draw Faradaic current is not replenished, leading to a depletion zone of the reactant in these parts in the cell. This significantly lowers the local current density, which can be described as “mass-transport limited”, leading a greater amount of the current to be drawn from the outer edges of the wires. A corresponding increased voltage drop is observed due to the transport limitation of current: this is the “concentration overpotential”.

*Tertiary current distribution, Ecell = 1.65 V. Current density distribution on the anode (dimensionless).*

You can use this class of current distribution for modeling cells with poor mixing or relatively low electrolyte concentration (compared to net current density), such that the electrolyte composition varies significantly throughout the cell and the resistive losses cannot be described by Ohm’s law. Solving the Nernst-Planck equations for *all* species concentrations with concentration of current and electroneutrality makes the equation set nonlinear and very complicated for the tertiary current distribution, which results in more time and memory storage requirements for the simulation. It is good practice to predict and understand the likely behavior of an electrochemical cell with secondary current distribution before modeling the tertiary current distribution.

The primary, secondary, and tertiary current distributions distinguish successive levels of approximation in the analysis of the current-voltage relation of an electrochemical cell. There are other modeling approaches that may be suitable, however, to extract maximum information about a cell’s behavior while minimizing the complexity of the model as much as possible.

In cases where the current density may be limited by mass transport of the electroactive species, but the electrolyte composition remains near-constant, it may not be necessary to solve for the full tertiary current distribution. Instead, the constant ionic strength means that we can assume that the solution obeys Ohm’s law with a constant conductivity, and so *Secondary Current Distribution* is used to solve for the electrolyte potential. However, the kinetic rate law is made concentration-dependent by coupling to a chemical species transport model that solves for the diffusion of the chemical species (and, where necessary, their migration and convection).

In fact, this is the method used for the tertiary current distribution of the wire electrode example, since it is the depletion of the reactant rather than the bulk electrolyte species that has the dominating effect. You can see another example of this coupling in the orange battery model. Also, this partial coupling of charge transport with mass transport is a very common approach in the analysis of batteries and fuel cells.

A special case of the above occurs when the inert (supporting) electrolyte is in considerable excess compared to the quantity of reacting (electroactive) species. Hence, the ionic strength of the solution is large compared to the Faradaic current density. In this case, the electric field is small and so the electrolyte potential is almost constant — solution resistance does not contribute noticeably to the behavior of the electrochemical cell.

In cases where solution resistance is unimportant, but electrode kinetics (activation) and mass transport of the electroactive species are important, you can use the *Electroanalysis* interface. This is a chemical species transport interface solving the diffusion-convection equation for mass transport, which incorporates electrode kinetic boundary conditions to drive a flux of the chemical species at electrode-electrolyte interfaces as a function of the local overpotential.

The electroanalytical approximation of zero solution resistance applies to the standard experimental set-ups for electrochemical techniques such as cyclic voltammetry, chronoamperometry, and electrochemical impedance spectroscopy. You can see an example of a cyclic voltammetry model using this approximation in our Model Gallery.

This blog post has discussed the three current distribution interfaces available in the four electrochemical add-on modules for COMSOL Multiphysics, and when and why you should use each of them. The strength of the COMSOL Multiphysics software is that it offers you the ability to model all classes of current distributions (primary, secondary, and tertiary), and therefore provides you with the flexibility to gradually introduce and control the complexity of the theoretical model used to analyze an electrochemical cell.

If you are interested in using COMSOL Multiphysics for your electrochemical cell design, or have a question that isn’t addressed here, please contact us.

An electrochemical cell is characterized by the relation of the current it passes to the voltage across it. The current-voltage relation depends on diverse physical phenomena and is fundamental to performance. In a battery or fuel cell at zero current (*equilibrium*), a theoretical maximum voltage can be extracted, but we want to draw current in order to extract power.

When current is drawn, there are voltage losses; equally, the current density may not be uniformly distributed on the electrode surfaces. The performance and lifetime of electrochemical cells, such as electroplating cells or batteries, is often improved by a uniform current density distribution.

By contrast, bad design leads to poor performance, such as:

- Substantial losses and shortened lifetime of electrode material at practical operating currents in a battery or fuel cell
- Uneven plating thickness in electroplating
- Unprotected surfaces in a cathodic protection system

Simulating current distribution enables better understanding to avoid such problems.

The current distribution depends on several factors:

- Cell geometry
- Cell operating conditions
- Electrolyte conductivity
- Electrode kinetics (“activation overpotential”)
- Mass transport of the reactants (“concentration overpotential”)
- Mass transport of ions in the electrolyte

Because of this complexity, many applications benefit from suitable simplification when modeling. If one of these factors dominates the cell behavior, the others may not need to be taken into account. As a consequence, successive approximations are introduced by the classifications of primary, secondary, and tertiary current distribution.

Each of the three classes of current distribution is represented in COMSOL Multiphysics by its own interface: *Primary*, *Secondary*, and *Tertiary Current Distribution*. These interfaces are provided in all of the four different application-specific products available for modeling electrochemical cells: the Batteries & Fuel Cells Module, Electrodeposition Module, Corrosion Module, and Electrochemistry Module.

When modeling an electrochemical cell, you have to solve for the potential and current density in the electrodes and the electrolyte, respectively. You may also have to consider the contributing species concentrations and the involved electrolysis (Faradaic) reactions.

The electrodes in an electrochemical cell are normally metallic conductors and so their current-voltage relation obeys Ohm’s law:

\textbf{i}_s = -\sigma_s\nabla\phi_s\ with conservation of current \nabla\cdot\textbf{i}_s = Q_s

where \textbf{i}_s denotes the current density vector (A/m^{2}) in the electrode, \sigma_s denotes the conductivity (S/m), \phi_s\ the electric potential in the metallic conductor (V), and Q_s denotes a general current source term (A/m^{3}, normally zero).

In the electrolyte, which is an ionic conductor, the net current density can be described using the sum of fluxes of all ions:

\textbf{i}_l = F\sum_i{z_i\textbf{N}_i}

where \textbf{i}_l denotes the current density vector (A/m^{2}) in the electrolyte, F denotes the Faraday constant (C/mol), and N_i the flux of species i (mol/(m^{2}·s)) with charge number z_i. The flux of an ion in an ideal electrolyte solution is described by the Nernst-Planck equation and accounts for the flux of solute species by diffusion, migration, and convection in the three respective additive terms:

(1)

\textbf{N}_i = -D_i\nabla c_i-z_i u_{m,i} F c_i\nabla \phi_l+c_i\textbf{u}

where c_i represents the concentration of the ion i (mol/m^{3}), D_i the diffusion coefficient (m^{2}/s), u_{m,i} its mobility (s·mol/kg), \phi_l\ the electrolyte potential, and \textbf{u} the velocity vector (m/s).

On substituting the Nernst-Planck equation into the expression for current density, we find:

(2)

\textbf{i}_l = -F \left(\nabla \sum_i z_iD_i c_i\right)-F^2\nabla \phi_l \sum_i z^2_i u_{m,i} c_i+\textbf{u}F\sum_i z_ic_i

with conservation of current including a general electrolyte current source term Q_l (A/m^{3}):

\nabla\cdot\mathbf{i}_l=Q_l

As well as conservation of current in the electrodes and electrolyte, you also have to consider the interface between the electrode and the electrolyte. Here, the current must also be conserved. Current is transferred between the electrode and electrolyte domains either by an electrochemical reaction, also called electrolysis or Faradaic current, or by dynamic charging or discharging of the charged double layer of ions adjacent to the electrode, also called capacitive or non-Faradaic current.

This general treatment of electrochemical theory is usually too complicated to be practical. By assuming that one or more of the terms in Equation (2) are small, the equations can be simplified and linearized. The three different current distribution classes applied in electrochemical analysis are based on a range of assumptions made to these general equations, depending on the relative influence of the different factors affecting the current distribution as listed above. In the next blog post in the series we’ll discuss the detailed content of these assumptions: going from primary to secondary to tertiary, fewer assumptions are made. Therefore, the complexity increases, but so does the level of detail available from the simulation.

Below you can see the geometry from a modeling example of a wire electrode. This example models the primary, secondary, and tertiary current distributions of an electrochemical cell. In the open volume between the wire and the flat surfaces, electrolyte is allowed to flow. You can think of the electrochemical cell as a unit cell of a larger wire-mesh electrode — a common electrochemical cell set-up in many large-scale industrial processes.

*Geometry of the electrochemical cell. Wire electrode (anode) between two flat electrodes (cathodes). Flow inlet to the left, outlet to the right. The top and bottom flat surfaces are inert.*

Now, you might be wondering which of the three current distribution interfaces you should use for your particular electrochemical cell simulations. In an upcoming blog post, we will use the wire electrode example shown here for a comparison of the three current distributions. Stay tuned!

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A blast furnace is used to produce industrial metals with different compositions, shapes, and degrees of purity. The metals are processed in a molten state that is reached at extremely high temperatures. This makes it very important to keep workers and production equipment safe. When the molten metal first leaves a furnace through a taphole and travels through a runner where the slag is separated out, it could splash and cause a halt in production, and even damage equipment and harm operators.

*Molten metal exits a blast furnace (left) and travels through a runner, where slag is separated from the cast iron (right).*

A strong, reliable barrier is necessary to contain splashes and provide protection. As you can read on page 20 of COMSOL News 2013, a company called Terres Réfractaires du Boulonnais (TRB), part of Groupe CB, uses a thick concrete liner to contain the molten metal as it exits the blast furnace. This liner is subject to many effects, like thermal shock, corrosion, and erosion. A typical roof runner only lasts for one month before it needs to be relined. One blast furnace can have two, three, or even four taps, and at TRB there are hundreds of blast furnaces — so quite a lot of taps. This calls for a lot of heavy materials to handle, so TRB wants to avoid over-engineering them and find a design that is as light, robust, safe, and economical as possible.

In the past, TRB used trial and error methods to come up with the best type of concrete for liners, and to determine the ideal thickness. If a liner was damaged too quickly, they just kept trying out new combinations. They ran all of these experiments without knowing what the temperatures were in the roof runner. They could not place expensive sensors inside because extreme heat and splashing molten metal would destroy them.

*Thermograms of the inside of a roof runner (top) and a standard photograph from the same angle (bottom). This is basically the only way that TRB can measure the temperature characteristics of their process, and it was used to verify the model.*

TRB turned to COMSOL Multiphysics and the Heat Transfer Module to calculate the temperature in the roof runners and optimize their design. The roof runners are made of two different types of concrete. One is in direct contact with molten metal and must resist the scalding fluid, corrosion, and thermal shocks. The second type of concrete never comes in contact with molten metal, and it acts as a mechanical frame for the roof runner. TRB conducted a thorough thermal analysis in which they carefully validated their assumptions in order to set up an accurate model of their blast furnaces. The details of the analysis are provided in the full version of the story available in COMSOL News.

*The simulation indicates that, during the first tap, the air becomes almost stationary under the roof runner.*

TRB wanted to mimic the real operation of the blast furnace at every point of the process. They simulated pre-heating, the first time the molten metal is tapped and sent through the runner, and even a full week of pauses and tapping to see what temperature the roof runner’s outer cast iron shell would eventually reach.

Before they turned to thermal analysis, TRB could really only guess how hot the blast furnace gets. Thanks to COMSOL Multiphysics, they were able to reach an accurate calculation within 10 degrees.

Until TRB worked with COMSOL Multiphysics, their concrete runners were very thick. With thermal analysis, they learned that they had over-engineered them and that they could reduce the thickness of the concrete. Besides lowering the cost of raw materials in future roof runners by making the liner thinner, if the liners are lighter they will also be easier to maneuver, and this will improve production rates and safety.

- “When It’s Impossible to Take Actual Measurements, Multiphysics Provides the Answers” on page 20 in COMSOL News 2013
- “Ugitech optimizes steel casting process using COMSOL Multiphysics“

Galvanic, crevice and other types of corrosion are electrochemical in nature and give rise to small currents passing between the corroding and most noble regions, via the material and surrounding electrolyte. To combat this, you can impose an opposite electric current in a process known as cathodic protection. In many cases, this leads to an even larger current passing through the material and surrounding electrolyte.

This is where problems arise. Electric currents give rise to electric potentials, and these can be measured. If a submarine, for example, is imposing an electric current through its hull and propeller, then there is a pretty big (in terms of area) electric potential that could be used to detect the submarine, or even set off a mine. This is further complicated by the submarine’s rotating propeller, which modulates the signal.

What to do? One solution is to turn the cathodic protection process off when the submarine is in an area of reasonable danger. But this won’t work as the corroding process itself has its own electric potential signature that can be detected.

David Schaefer at the University of Duisburg-Essen, in Germany, is simulating these underwater electrical potential signatures of submarines, on the commission of the Technical Center for Ships and Naval Weapons. Using COMSOL Multiphysics, he and his team were able to simulate the signal and find that you could run cathodic protection at a certain amperage, resulting in a signal less noticeable than that during corrosion. Read more about it in the COMSOL News 2012 article “Submarines: Corrosion Protection or Enemy Detection?” (on page 67).

*Underwater Electric Potential Signatures below the keel, for different currents imposed through
cathodic protection. A signature is evident when corrosion is occurring (top left), which can be
optimized at 3.5 A (top middle). Overprotection results in larger signatures.*

The course will go through a number of corrosion scenarios, like galvanic and crevice corrosion, where the electrochemical kinetics at corroding surfaces have to be considered. Yet, you can also consider corrosion on a bigger scale, such as the model of an oil rig shown here.

This models the corrosion protection of a support structure for oil platforms and the like, placed in seawater. The structure is protected by placing a series of sacrificial anodes around it. By modeling the primary current density distribution, the electric potential distribution on the structure can be obtained, and areas where corrosion could still occur can be identified. Rearranging the anodes would then be done.

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The above image shows the type of modeling that can be done. A magnesium alloy and mild steel are connected in a salt water environment. As all high-school scientists know, the connection of different metals like this usually leads to corrosion. What is seen is the corrosion of the magnesium alloy over time. The model takes into account the change in the geometry as a result of this corrosion, and the corroding current density (streamlines) changes as a result of this.

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This site suggests that corrosion is a $276 billion/yr problem in the US. You would think there would be a reason to model it. Well, we’ve seen corrosion modeled in COMSOL before. This User’s Story gives a great account of how it can be done, and I suggest you look into it.

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