When solving physical equations on a real, 3D geometry, numerical methods require you to discretize the geometry. This involves breaking up the geometry into elements. At certain points in each element, the equations are solved and the relationship between the elements is described so that a finite set of equations is created.
If you choose to use a finite element method (FEM), you have to discretize the volume between the boundaries of your geometry, since the finite elements describe the relationship between the neighboring elements only. If you choose to use a boundary element method (BEM), the equation system is set up so that you only need to discretize your boundaries in the geometry. There are then many fewer elements; however, the physical equations connect every boundary element with every other boundary element, unlike finite elements on a volume, which are connected by equations only to their neighbors. Both methods have advantages and disadvantages. This blog post discusses an example where the BEM demonstrates its efficiency.
We wrote about the importance of corrosion management in oil platforms in a previous blog post. Such structures usually stand on a seabed with a jacket structure or float above the oil field as a floating production unit. Such jacket structures are generally quite large and complex and include a lot of pipes, steel legs, and hopefully a number of sacrificial anodes, which oxidize to protect the structure from corrosion.
When working with large structures, such as a jacket structure, there may be several hundred or even thousands of anodes that are placed quite close to the structural members. If you want to use a volumetric (3D) mesh on such structures, you will end up with a lot of surfaces and small mesh elements due to the curved surfaces and the thin gaps between the anodes and the structural parts. In order to quickly assess the cathodic protection scheme of a large jacket structure, it can be useful to utilize the boundary element method with the edge elements provided in the Corrosion Module.
Electric potential of steel versus the Ag/AgCl electrochemical potential on a jacket structure submerged in seawater, calculated with the Current Distribution on Edges, BEM interface.
The figure above demonstrates results from a model containing more than 3000 different edges. Each edge is given a radius and an appropriate current density is specified depending on whether it is made from steel (protected cathode) or represents a sacrificial anode. It is easy to combine several different dimensions and current densities using the selection capabilities in COMSOL Multiphysics.
By using the Current Distribution on Edges, BEM interface, you only need to create a discretization for the edges. The rest of the calculation is based on the boundary element method, which implies that no discretization of the surrounding volume is needed. Creating an edge mesh for this structure will take only a few seconds, but for a full-sized solid structure with a volumetric mesh, much more work and computational resources would be required.
When building a model with the Current Distribution on Edges, BEM physics interface, you need to select it from the Physics Interfaces selection list under the Electrochemistry branch.
A screenshot showing the Select Physics window, with access to the Corrosion Module.
Once you have built your geometry consisting of edges, you can add the Edge Radius of the different edges as an attribute in the BEM Geometrical Properties node.
Define the radius of your edge or tube as an Edge Radius attribute in the BEM Geometrical Properties feature.
By specifying the edge radius, you may also specify that it should compensate for the tube volume in the electrolyte. In turn, the edges are implemented as edge sources in an infinite electrolyte. However, if there are many edges or tubes and they are close together, they will reduce the effective conductivity of the electrolyte. Thus, if you check the box for compensation for the tube volume, each tube is treated as a solid tube at the cost of two more degrees of freedom at each element.
Compensation for tube volume, which displaces the electrolyte, may be enabled by the check box in the Settings window.
Now you are ready to specify the potential or current conditions on each edge. By right-clicking on the Current Distribution on Edges, BEM node, you’ll find the selection list of your available conditions.
Define boundary conditions on the edges as usual.
The default plot that appears after solving includes a Line plot along the edges, where the thickness of the tube is taken from the specified tube radius in the physics interface.
The potential on the edges with a tube plot, showing the virtual dimensions of the tubes.
In this blog post, we have shown a new way of handling large slender structures with the Current Distribution on Edges, BEM interface. By utilizing the boundary element method and specialized elements, we are able to reduce the time spent on meshing and model setup.
Responsible for costing the automotive industry billions of dollars each year, galvanic corrosion can occur when different metal surfaces come into contact with one another. Often characterized by a powdery growth on the surface of different metal parts, this type of corrosion degrades the surface of the metal when metallic ions are exchanged. Since different material combinations react differently, and joining techniques, environmental influences, and surface textures can affect the chemical reactions occurring on metal parts, developing effective corrosion protection requires an understanding of the many factors at play.
A clean rivet (left) compared to a rivet affected by galvanic corrosion (right).
Engineers at Daimler AG, the manufacturer of Mercedes-Benz, and Helmholtz-Zentrum Geesthacht (HZG), a German institute focused on medical technology, materials, and coastal research, banded together to investigate better corrosion protection methods for automotive rivets and metal sheets that are similar to those present in car paneling. Using COMSOL Multiphysics software, Dr. Daniel Höche of HZG modeled a steel punch rivet joint to gain a better understanding of the underlying electrochemistry, surface conditions, material loss, and long-term behavior of the interacting metals.
His model included the rivet, plated with an aluminum-zinc alloy; bonded sheets of aluminum and magnesium; a galvanic couple at the interface between the sheets, represented mathematically; and a 0.1% NaCl electrolyte, which represented the external environment.
The geometry of half of a steel rivet in COMSOL Multiphysics (left) and simulation results showing the current density at the rivet’s surface (right).
Höche included a corner bur in the rivet’s geometry to simulate a sharp edge. This increased the current flow and the gradients in the electrolyte potential, quickening the electrochemical reaction responsible for the corrosion. He also accounted for nonconstant growth in a magnesium hydroxide deposit, which forms on surfaces during corrosion, by treating the sheet metal and rivet like a set of electrodes.
The Batteries & Fuel Cells Module and the Chemical Reaction Engineering Module allowed Höche to analyze how the duration of exposure to the electrolyte, changes in electric current due to the magnesium hydroxide deposit, and anode/cathode area ratio affected the degradation of the metal. “Since the porosity directly affects the barrier properties, the resulting surface topology is influenced by the downward degradation velocity and the opposing growth of the deposit. Basic galvanic current density computations were modified by these layer growth aspects,” Höche commented. “This led us to study time-dependent variations in the electrochemical response of the electrodes.”
A plot of the localized current density in different areas of the rivet joint.
The simulation results show the current density and electric potential when the rivet joint is exposed to the electrolyte, indicating the proportion of the surfaces covered by the magnesium hydroxide growth.
Since other automotive components are also in danger of corroding due to environmental conditions, Höche worked with Nils Bösch of Daimler AG to study the delamination of a zinc-plated steel test sheet protected by a cathodic e-coat. Seemingly minor scratches in the coatings or paints on such sheets, which are widely used in car paneling, can give moisture and environmental electrolytes access to the electrically conductive metal underneath. In the delamination process, these coatings become unbonded from the metal sheets they protect, significantly weakening the corrosion protection.
A corrosion test on a piece of galvanized steel.
“Due to a scratch extending down to the steel surface, you can get a galvanic couple between the zinc and the steel, and the zinc corrodes,” Bösch explained. “This results in a crevice that grows continuously between the e-coat and the steel in the horizontal direction, rather than vertically through the layers.” Even though the obvious damage may appear small, because the crevice spreads underneath the e-coat, this type of corrosion is insidious and subtle. The damage may not be immediately noticed and can eventually lead to part failure.
A test sheet showing scratches on the e-coat and zinc layers.
Bösch and Höche set up a COMSOL Multiphysics model to study the electric potential in the e-coat and the electrolyte, using parametric sweeps to test results for different e-coat properties. The simulation predicted the horizontal growth of a crevice consuming the zinc in the test sheet, and showed that the width of such scratches is actually more influential than their depth in the metal. Further work will continue to investigate how e-coat flaws influence corrosion protection, using the results from the researchers’ model.
Simulation results showing the electric potential in the e-coat and electrolyte.
Thanks to their modeling work in COMSOL Multiphysics, Bösch and Höche were able to analyze and predict aspects of the galvanic corrosion process in automotive parts. Their simulation results offered insight into the electrochemical behavior of the different systems and have enabled them to recommend e-coats that exhibit the least decay in car paneling. Their research continues to bring galvanic corrosion control within reach through careful geometric design and knowledge-based processing.
Mercedes-Benz is a registered trademark of Daimler AG Corporation.
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Diabetes management that involves drawing blood from the patient several times a day is invasive and uncomfortable, to say the least. Continuing research in electroanalysis allows measurement techniques for glucose concentration that are not only becoming faster and more accurate, but also operate on much lower volumes of blood. With the development of these methods, it is possible to decrease the discomfort of diabetes management as we know it today.
“Blood Glucose Testing” by David-i98 (talk) (Uploads) – Own work. Via Wikipedia.
Electrochemical methods for glucose sensing are based on glucose oxidase (GOx), a biologically occurring enzyme that oxidizes the glucose molecule to gluconic acid. In nature, the oxidation is achieved by a biological oxidizing agent, such as the co-enzyme FAD^{+} (that co-enzyme is actually a catalyst, since it is in turn oxidized back to its starting condition by dissolved oxygen).
In an electrochemical cell, however, this oxidation can instead take place at the surface of an electrode, releasing electrons into an electrical circuit, which can be measured as a current. In a well-constructed sensor, that current will be proportional to the quantity of glucose reacted with the enzyme, and, so, to the present concentration of glucose. This concept stems from the pioneering research of the American medical scientist Leland Clark.
Because of the specificity of the biological GOx enzyme, the sensor can measure glucose concentration uniquely, even in a mixture like blood, which contains a large number of chemical compounds. Finally, this technique has been developed by the use of redox mediators that react more efficiently than oxygen itself, thus allowing a more accurate and unperturbed measurement.
Many companies around the world manufacture glucose sensors for patient use. Product development at the present time has moved beyond the proof-of-concept stage through to an effort to create a state-of-the-art product: the optimal glucose sensor, within the constraints of cost and manufacturing.
Although the principle behind an electrochemical glucose sensor is simple, its implementation can be complex. Because the sensor must work under a range of conditions, clever design is required to ensure that a measured current can still be directly correlated to the glucose concentration in the sample. Practical difficulties may include varying oxygen concentration in blood, the presence of other chemical species in the blood that react similarly to glucose, and the change in temperature of the sensor due to blood being supplied at body temperature. All of these can influence the measured electrical current in a sensor.
COMSOL Multiphysics is an ideal tool for real-world 2D and 3D problems in electroanalysis. Why? Because it’s easy to couple the electrochemistry to models of other physical effects, such as heat and mass transfer. What’s more, the ability to add user-defined variables and equations means that a detailed description of non-standard or complex phenomena, such as enzyme kinetics, can be incorporated into the model. Often, the theoretical model is developed by experimental corroboration of the behavior of the electrochemical system. Therefore, the numerical modeling approach must remain flexible.
You can find a simple model of a glucose sensor in the COMSOL Multiphysics Model Library. This particular model is of a sensor using an “interdigitated electrode”, in which the anode and cathode of the sensor cell are divided into a large number of “fingers” running parallel to each other, maximizing the current density and sensitivity of the system. Because the “digits” are usually much longer than they are wide, their length can be ignored and the problem can be simplified to a 2D unit cell.
This model includes the mass transport of the glucose and the redox mediators by diffusion and the fluxes of these species into and out of the system. Additionally, the enzyme-catalyzed reaction between the redox mediators and the glucose is included in the model using a user-defined Michaelis-Menten kinetic rate law (my colleague Eyal Spier explains this in his previous blog post “Enzyme Kinetics, Michaelis-Menten Mechanism“).
The figure below shows the concentration of the ferrocyanide redox mediator species when a steady-state current is drawn in the sensor.
At the anode (working electrode) in the bottom center, this species is oxidized to reduce the other redox species, ferricyanide, so the concentration is depleted here. At the cathode (counter electrode) at the bottom left and right corners of the unit cell, ferrocyanide is regenerated and diffuses back towards the working electrode in a “redox cycle” that delivers high current densities.
In the next plot, you can see the current against glucose concentration for a working sensor (blue line), demonstrating that the response of the idealized sensor is linear in the studied range of glucose concentrations.
However, it’s not hard to break the sensor (green line). This is done by imagining the enzyme to be a factor of 10 slower at oxidizing the glucose itself, by lowering the v_\mathrm{max} coefficient for the enzyme kinetics. Then, the linear response fails over the same concentration range. This demonstrates the importance of careful design to ensuring that the current measured in a glucose sensor really reflects the concentration present. You may wish to try your own variations on the model to see the influence of other effects.
Tip: Check out the Glucose Sensor model in the Model Gallery.
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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)
Current density expression with the Nernst-Planck equation:
(2)
General electrolyte current conservation:
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)
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:
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:
Simulating current distribution enables better understanding to avoid such problems.
The current distribution depends on several factors:
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:
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)
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)
with conservation of current including a general electrolyte current source term Q_l (A/m^{3}):
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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Lithium is an attractive material because of its tremendous specific energy capacity. You may remember from your school days the violent reaction of lithium with water. If you translate this willingness to react into electrochemical potential, you’ll see that a lithium-based battery gives you a higher voltage due to the electrochemical reaction than just about any other chemical substance. And at atomic number 3 it’s very light too, so there’s a lot of chemical energy available per unit mass. But rather than having the highly-reactive lithium metal available in the battery, the design ensures safety by using cathode materials in which the lithium ions are “intercalated”, such that the lithium ions exist in the chemical structure of another material.
In order to study and optimize the chemistry of his battery, Cugnet focused on an experimental technique called electrochemical impedance spectroscopy (EIS). This common and versatile technique is used to resolve the resistive and capacitive properties of an electrochemical system, by applying a fixed direct current and then superimposing a small sinusoidal voltage across the cell. As long as this applied oscillating voltage is small enough, the system properties remain linear, and so the current response oscillates harmonically at the same frequency as the applied voltage. It is these responses, evaluated through electrical impedance, that scientists use to measure many of the characteristics of the system, such as the electrochemical kinetics or the transfer of species to and from the electrode.
As Cugnet explains in his article in COMSOL News 2013, one disadvantage of the EIS experiment is that the results are not clearly correlated with the underlying chemistry. Normally, the measured behavior of the cell is described with an “equivalent circuit”, where an analogy is drawn to the impedance of an electrical circuit composed of ideal resistors and capacitors. However, the resistances and capacitances need to be fine-tuned and they do not fully represent the real chemical effects, like mass transfer and electrochemical reactions, that cause these phenomena.
By using a COMSOL Multiphysics model to predict impedances by solving a system of differential equations describing the various physical and chemical effects at work inside the battery, Cugnet was better able to understand the results of his experiments. What’s more, the model could be easily adapted to different frequencies or center voltages, revealing the chemical response of the lithium-ion battery under different power cycles. Visualizing concentration and current density profiles under diverse conditions makes clear the reasons why impedance spectra look the way they do, and so helps to make sense of the available measurements of a real battery system. Such a modeling approach allowed Cugnet to understand how well the different terms in his equations were able to mimic the behavior of the battery under investigation.
Using electrochemical impedance spectroscopy (EIS), battery impedance is measured at a range of frequencies in the milliHertz to kiloHertz range and displayed on a impedance plot.
Another interesting challenge in the work at INES-CEA is the complexity of the chemical interactions at the iron-phosphate cathode. This material has a microparticulate structure, so to understand the effects of diffusion of lithium into the individual particles, Cugnet used COMSOL Multiphysics to build a microscopic model of this process. He then coupled this small-scale model to a macroscopic model of the cathode and electrolyte. In this way, the influence of the local chemistry can be correlated to the overall distribution of current density. This type of chemical detail is critical to developing electrochemical devices, because the power extracted depends on it.
Model of the iron-phosphate cathode at the macroscopic (left) and microscopic (right) level.
It’s an interesting peculiarity of analytical techniques in electrochemistry that because you can only measure the total current or total voltage, a side-by-side simulation is often required in order to learn anything from your experiments! Experimental measurements are vital, though, to validate the simulation. In this case, the versatility of multiphysics modeling allowed Mikael Cugnet and INES-CEA to make progress in identifying the key points of a complex, multiphase chemical system. You can read about the full detail of the lithium-ion battery model in COMSOL News 2013, starting on page 44.
The Electrochemistry Module enables you to model applications within electroanalysis, electrolysis, and electrodialysis. All these applications are influenced by electrochemical reactions, material transport, and the transfer current, but depending on the application, these participating processes may vary in their importance or effects. All possible ways to describe an electrochemical process are available within the Electrochemistry Module, and you can pick and choose which ones you would like to access and utilize, or include them all.
Take electrochemical reaction kinetics, for example. This is a property that is of utmost importance to the electroanalyst, like Edmund. Being able to use a model to describe the kinetics for all participating reactions at an electrode surface is imperative to designers of gas and glucose sensors. There are so many parameters to consider about these kinetics: the effects of electrode potential (the driving force for electron transfer); the concentration of the reactants and products at the electrode surface; the shielding effects of non-participating species, like water, on this electron transfer; the ability for the electrocatalyst to receive or release these electrons; the active surface area of the electrode available for reactions to take place; temperature; and other factors.
Figure 1: AC impedance is one of the tools electroanalysts use to extract quantitative information about electrochemical reactions. This image is of a Nyquist plot, and the Electrochemistry Module also allows for Bode plots to be made.
These are encapsulated in terms known as exchange current densities, transfer coefficients (or symmetry factors), and overpotential. Together with other terms, these are all baked into variations of the Butler-Volmer equation. This is one of the features within the Electrochemistry Module that Edmund would have played with, comparing his models to experiments, and estimating the values of these terms. He might also have had to consider transport effects of the reacting and inert species as they move to and from the electrode, so that analytical measurements, as made by a sensor, accurately reflect the concentrations in the bulk solution.
I, on the other hand, was more concerned with the mass and charge transfer occurring within an electrochemical cell as a whole. At the chlorate plant where I worked, a 10mV decrease in required operating voltage meant a two million dollar reduction in annual operating costs. Part of reducing these costs came from optimizing the amount of energy required to force the electrochemical reactions to occur. Usually, this involved finding the perfect electrocatalyst to assist in charge transfer, and surface conditions to maximize reacting area. This was invariably done by the electroanalyst in the lab, and appropriate electrodes were manufactured for the electrochemical engineer to then work with.
Figure 2: Model of the charge transfer and potential drop in a secondary current distribution model of a unit cell from the chlor-alkali process. The Electrochemical Module also supports the modeling of processes assuming primary and tertiary current distribution.
The factors in an electrolysis cell that I worked with involved making the current distribution over the length, breadth, and width of the electrodes as evenly as possible, while avoiding the unnecessary onset of electron-robbing bireactions. Optimizing the transport of species to and from the electrode was also very important, as the greater the transfer, the greater the operating current density (and material production) before reaching the limiting current density where other reactions and even arcing may occur. The structural integrity and operating lives of the electrodes, current feeders, and separating membranes were also important.
Here, I would have utilized the features describing the electrochemical reactions, but would have coupled them to other equations describing fluid dynamics and homogeneous, non-electrochemical based reactions that were occurring in the electrolyte. A robust description of the material transport that occurs through convection, diffusion, and ionic migration (Nernst-Planck equation), would have also been included to result in a full description of the tertiary current density occurring at the electrodes.
Fifteen years ago, I and other colleagues at our chlorate company, used many assumptions and computer programming to try to describe all of these effects, both in the lab and in the plant. If we’d had the Electrochemistry Module back then, we would have improved the process greatly and life would have been a lot easier.
The geometry of a battery made of an orange.
It’s obviously not magic that enables orange batteries to generate electricity — it’s electrochemistry. A citrus fruit, oranges contain citric acid that, together with other ions, serves as an electrolyte. The citric acid electrochemically reacts with the two metal nails, which need to be of different metals to both release and gain electrons. The circuit needs to be completed by allowing the two nails to be electrically connected through a metal conductor, such as a small light bulb (part of the wow-factor in the school laboratory). In this process, our nails serve as electrodes, and there is a galvanic potential over the battery cell (the orange, in this case) that encourages the electron transfer. Since we’re leveraging chemical reactions to turn chemical energy into electrical energy, we can also refer to the orange battery as a voltaic battery, just as any other battery where the energy has been stored in the chemicals it contains.
Fun fact: an Italian physicist, Luigi Galvani, discovered “animal electricity” — when he connected two different metals in series in a frog’s leg, it began to twitch, which was actually caused by the movements of ions. Intrigued by this, another Italian physicist, Alessandro Volta, ran his own experiments and concluded that the frog’s leg was simultaneously a conductor and a detector of electricity. Building on this research, he eventually came up with what we call Volta’s Law of the electrochemical series, and later in the 1800s, the first battery.
If you are new to modeling electrochemistry applications, you may want to start off with our orange battery model tutorial. If we follow the step-by-step instructions in the PDF, we can model the currents and dissolved metal ion concentration in the orange battery.
One of our nails is made of zinc, and the other copper. The zinc nail loses the electrons in an electrode reaction as such:
Zn(s) → Zn^{2+} 2e^{-} E_{0} = –0.82 V
so that zinc ions then join the electrolyte in the battery. Meanwhile, the copper nail acts as an electrocatalyst to encourage the following hydrogen evolution reaction, from the hydrogen ions in the citric acid:
2H^{+} + 2e^{-} → H_{2}(g) E_{0} = 0 V
The Secondary Current Distribution interface is used for setting up the model for the currents in the orange and electrodes. Secondary current distribution assumes that electricity moves through the electrolyte only as ionic migration, while the reactions at the electrodes are a function of the electric potential and local concentration of reacting species. Ohm’s law is therefore applied, together with a charge balance, to solve for the electric currents in the nails and the orange’s electrolyte, which are also coupled to Butler-Volmer expressions that describe the electrochemical reactions. In this example, one of our nails is set to a cell potential of 0.5 V, while the other is grounded, and the current distribution is calculated.
Once we have set up and solved the model, we can evaluate the performance of the orange battery.
The electrolyte’s potential field. | The nails’ electric currents. |
As we can see in the image above to the left, the electric potential decreases as the current flows from the zinc electrode (left nail) to the copper electrode (right nail). The cell voltage loss is mainly due to Ohmic losses in the electrolyte. If we wanted a better-performing battery, we could swap the orange for fruit with higher conductivity (i.e., more acid content), like a lemon. Alternatively, we could put the nails closer together. In the model above to the right, we can see that the electric current increases along the z-axis. This happens as the tips of the electrodes have access to a greater volume of the orange, and therefore more reacting species, so that they are be able to gain and lose their electrons easier.
We can also study the ion concentration level after having run the battery for a while, and can visualize how the cell current changes over time. As the zinc ions build up, they impede the reacting ability of the anode, so the battery current decreases until it reaches a constant current density.
Isosurface for a 0.2 mol/m^{3} concentration level of zinc ions after running the battery for 5 minutes. | Cell current vs. time. |
The “double-peaked” graph looks like this:
This is a cyclic voltammogram, in which the current (“ammetry”) is plotted against the voltage applied to an electrochemical cell. The voltage is swept up and down across a range of values to successively drive the opposite directions of an electrolysis reaction:
Cyclic voltammetry is a very widely used technique for the interrogation of physics and chemistry at the interface between an electrode and an electrolyte, such as a saline solution. Electroactive surfaces are common to all electrochemical devices, including common energy extraction devices like batteries and fuel cells, as well as electrochemical sensors, such as those used by diabetics to monitor blood glucose concentration. Furthermore, the chemistry of the electrode-electrolyte interface is still not fully understood and is an active field of academic research.
Voltammetry can be valuable for device verification and design because a single scan contains a great deal of information about the chemical and physical behavior of a system. Also, voltammetry can be the fundamental mode of operation for a sensor, since the measured current will be linearly dependent on the concentration of the analyte in a well-designed system. Chemical modification of electrode materials can make voltammetry specific to individual biological compounds or toxic gases in a mixture, and it is a cheap technique where advances such as screen-printed electrodes allow for “disposable electrochemistry”.
For design and research, the great virtue of voltammetry is the diversity of information that it provides. It illustrates the competition between the rate of electrolysis at the electrode surface and the rate of transport of the reacting chemical species to that surface by diffusion, and it can also yield valuable information of the mechanism and rates of chemical reactions in solution. By performing voltammetry at different scan rates, where the rate of change of the voltage in time is altered, we can observe different system timescales and so different physical phenomena.
For all its importance, voltammetry is a difficult technique to understand. All the real physical effects in the system are lumped into rather arcane current-voltage curves. Although experienced electroanalytical chemists may be able to intuitively “see” the chemistry in a voltammogram, it is essential to compare voltammetry to a theoretical prediction to get quantitative information from the experiments. Since electrochemical kinetics are frequently nonlinear and voltammetry is a transient problem, the theory is not admissible to analytical solutions, except in a few limiting cases, and so computer simulation is necessary.
COMSOL’s new Electrochemistry Module includes an Electroanalysis interface tailored to modeling electroanalytical techniques like voltammetry. This interface assumes the presence of a large amount of supporting electrolyte; inert salt, such as potassium chloride, that is artificially added to the electrolyte in an electrochemical cell to increase its conductivity. Supporting electrolyte mitigates electric fields, which is advantageous because it simplifies both the experimental analysis and the underlying theory. We assume that only diffusion contributes to chemical species transport, because the solution is unstirred and the timescales are short enough that natural convection in the solution is not important. Under these conditions, the chemical species transport equations are linear and so much easier to solve.
Diffusion length scales over the typical duration of a voltammetry experiment are very short — often much less than 1 mm. For a traditional “macroelectrode”, shaped like a disk with a radius in excess of 1 mm, it is accurate to assume that diffusion is only significant in the direction normal to the electrode surface, and that the influence of the electrode edge can be neglected, so reactions and transport are uniform across the electrode surface. This makes voltammetric analysis a 1D time-dependent problem.
To facilitate the definition of the transient applied voltage and its influence on the rate of the electrolysis reaction, the Electroanalysis interface contains a pre-built “Electrode Surface” feature that allows the potential window and scan rate of the voltammogram to be set directly. This feature also automatically implements the Butler-Volmer equation for the electrode kinetics, but, as for any feature in COMSOL Multiphysics, a user-defined expression for these kinetics can be substituted. The associated Cyclic Voltammetry study then automatically solves the corresponding time-dependent problem using suitable numerical methods to integrate the time-dependent diffusion equation. A range of scan rates can be studied in a single computation using the “Parametric Sweep” feature.
In the figure above we can see four predicted voltammograms recorded at four successive scan rates, from 1 mV/s to 1 V/s. These correspond to experimental durations from almost half-an-hour down to just over one second. The currents can be seen to increase with the scan rate, but the voltammograms have the same qualitative “double-peaked” appearance. The latter can be explained because, to begin with, the voltage is in a range where the forward reaction of the reactant is not driven, so there is negligible current. As the voltage is swept up, the reaction is accelerated, so the current increases. After some time, however, the reactant concentration is depleted by the reaction at the electrode surface. There is then a change of rate-determining process, so that the current becomes controlled by diffusion of the reactant to the surface, and thereby falls off again. A similar process occurs in reverse for the back reaction in which the product is reconverted to the original reactant as the voltage scans back to its starting point.
The increase in current density with scan rate occurs because a faster scan causes the diffusion layer to be established over a shorter distance. Because the concentration varies from bulk to zero over a narrower length, the diffusive flux is larger, hence so too is the current. The peak current should actually scale with the square-root of scan rate: checking this relation is a common verification method for experimental data to check that the measurement has not been corrupted by physical effects other than diffusion.
For sensor design we always want to maximize our current in order to maximize sensitivity, so this analysis can help in practical design of the electrochemical cell and chemical environment. Comparison of predicted and measured voltammograms enables determination of material properties and other system parameters that may be unknown, such as diffusion coefficients and reaction rates.
Try comparing the above animations to understand the relation between the current and the evolving concentration profile. Note how the concentration at the electrode surface (x = 0) is driven to zero as the current increases, then, once the surface concentration is zero, the concentration gradient relaxes under diffusion and the current relaxes as well. The concentration returns to its bulk value at the electrode surface as the current is inverted by the reverse reaction during the second part of the sweep.
Since the Electroanalysis interface embeds electoanalytical modeling into the COMSOL Multiphysics environment, a powerful and flexible user interface for finite element methods, it is directly possible to extend the scope of this model. Follow-up chemical reactions of the electrochemically generated species can be included by adding a Reactions domain condition. A 2D or 3D model of the same process can be set up to study diffusion in a real system geometry. Multiple voltammetric cycles can be performed, or nonstandard voltage waveforms can be applied. Coupling to a fluid flow calculation with convection of the reacting species allows the investigation of hydrodynamic electrochemistry. We can also consider a range of related techniques like potential step chronoamperometry and electrochemical impedance spectroscopy via the same user interface.
Tip: Check out the Cyclic Voltammetry at an Electrode model in the Model Gallery.
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Electrochemistry is integral to the workings of batteries and fuel cells, which belong to the power and automotive industries, while electrochemical processes are responsible for most of the corrosion in the world, something that is often studied by material scientists and civil engineers. Electrochemical machining, etching, and deposition are prevalent in the electronics industry, but may be studied by physical chemists. Yet, in all of these situations, the underlying chemistry and physics of what goes on in an electrochemical process is often difficult to study experimentally — the electrochemical reactions sometimes occur hidden away in a porous media or, as is often the case with corrosion, over the span of many years. In many situations you can measure the global currents and voltages that are lumped over the system, but it’s almost impossible to measure the activation overpotential or the current density at the level of the electrode. In order to fully understand and optimize your processes, equipment, and surrounding environments, it is becoming necessary to also model these electrochemical applications.
COMSOL caters to a number of the industries that need to consider electrochemical applications. Our Batteries and Fuel Cells Module contains the relevant features for this type of modeling, such as optimizing the material properties and configurations of electrodes, separators, current collectors and feeders, feed channels, membranes, and electrolytes. The Electrodeposition Module is ideal for not only modeling the electrochemical reactions occurring at electrode surfaces, but also for including the effects of the surrounding system, such as the convection transport of the electrolyte, and shielding and masking that may be used to augment the deposition or etching processes. While the Corrosion Module obviously simulates the underlying corrosion processes, such as galvanic, pitting, and crevice corrosion, it is also just as useful for modeling corrosion protection processes, like ICCP and sacrificial anodes, and even to control underwater electric potentials that can be used to give away the positions of submarines.
Then again, when I’ve previously worked with modeling electrochemical applications, it was not within these fields. In fact, I studied some of the oldest electrochemical applications; the mundane production of caustic soda, chlorine, and chlorate. Industrial electrolysis also answers for the production of aluminum and even hydrogen and oxygen. These account for some of the biggest production methods and quantities of materials within the world’s process industries.
On another level, a burgeoning science is the study of biological systems and their dependence on electrochemistry. We have an example of the electrochemical treatment of a tumor that uses the Nernst-Planck Equations interface to define the competing electrochemical reaction kinetics of the chlorine and oxygen evolution processes at the anode. The production of both of these chemicals in a tumor significantly lowers the pH values at each of the respective electrodes, which leads to the destruction of the tumor. This is yet another example of electrochemistry showing up in a non-traditional field, maybe this time within biomedical engineering.
pH-profiles within the tumor at different time steps during the treatment over six minutes.
Total current density (blue line) and the current density for the chlorine (red line) and oxygen (green line) evolution reactions over six minutes.
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.