The Nernst-Planck-Poisson Equations

An ion-exchange material is typically modeled as a porous medium consisting of a fixed matrix with the pores filled up with water and additional mobile ions. This may sound completely wrong to anyone who has seen, for instance, a Nafion® membrane, one of the most common polymer electrolyte materials. This material looks perfectly transparent and homogeneous, but the matrix is made of a transparent polymer backbone. The pores, swelling when in contact with water, are in the range of nanometers.

The key feature of the ion-exchange membrane is the immobilized ions that are fixed to the backbone and located on the internal pore walls. In the case of Nafion®, the immobilized ions are groups located at the end of polymer tails extending from the polymer backbone. As we will see in the following discussion, the concentration and sign of the fixed charge in the ion-exchange membrane is crucial for the ion transport of mobile ions in the membrane.

The Poisson equation relates the sum of all charges to the potential according to

(1)

where is the electric potential of the electrolyte phase, is the permittivity, and is the space charge density.

In our case, we can split the space charge into the contributions from the mobile and immobile ions

(2)

where is Faraday’s constant; is the charge; is the concentration of the mobile ions, where is a species index and the summation is made over all ions; and is the charge density of the immobilized ions in the matrix.

In a free electrolyte outside the ion-exchange media, the immobile ion concentration is zero so that .

To model the transport of ions, we first define the electrochemical potential for each ion as

(3)

where is the molar gas constant, is the temperature, and is some (arbitrary) reference concentration.

Assuming a dilute solution (i.e., each ion interacts with the surrounding water molecules only) and that ion transport only occurs due to diffusion and migration, we define the flux of the mobile ions based on the gradient of the electrochemical potential according to

(4)

where is the mobility.

The flux of the immobilized ion is zero. Note that we would typically expect lower mobilities for the ions in the membrane compared to the free electrolyte due to porosity and tortuosity effects. However, we will ignore that in the following example.

If you are used to modeling diffusion by Fick’s law, it is also worth mentioning that the mobility and the diffusion coefficient are related by the Nernst-Einstein relation

(5)

Since there is neither production nor consumption of ions, and assuming a stationary solution, the divergence of the ion flux is zero.

(6)

Eq. (1) together with Eq. (6) (one for each species) is commonly referred to as the Nernst-Planck-Poisson (NPP) set of equations.

Modeling an Ion-Exchange Membrane with the NPP Equations

Let’s apply the NPP equations to a simple problem and investigate how the ion-selective capabilities vary as we vary the concentration of the fixed ion in the membrane.

As a modeling problem, we play around with a 100-μm-thick membrane surrounded by two free electrolyte domains of the same length and located in a 1D geometry. We model the transport of three ions — A⁺, B^-, and C^- — and we will also add a fixed charge to the middle ion-exchange membrane domain. At the leftmost and rightmost external boundaries, the concentrations and potentials are fixed. This model could, for instance, represent one of the membranes in an electrodialysis cell, where turbulence ensures good mixing, which allows us to assume constant concentrations outside the diffusion boundary layers on each side of the membrane (see Figure 1).

Figure 1. Schematic of an electrodialysis cell used for desalination of water. An ion-exchange membrane is located between each fluid compartment. The sign of the immobilized charge of the membrane, which alternates, will govern whether the membranes will predominantly allow passage for positive or negative ions.

Figure 2. Geometry and boundary conditions.

On the left boundary, we set = 0 V, = 0.1 M, = 0.1 M, and = 0. On the right boundary, we set = 0.1 V, = 0.1 M, = 0 M, and = 0.1 M. The mobilities are set to be equal for all ions and we set the charge of the immobilized ion in the membrane to -1.

As we will see, using the NPP set of equations for modeling ion-exchange membranes results in increasingly steeper gradients for the dependent variables when increasing the fixed membrane charge. Therefore, we solve the problem, using a stationary solver, by ramping up the concentration of the immobilized ion from 0 to 1 M (using an auxiliary sweep in the COMSOL® software).

Figure 3 shows the molar flux for each ion through the cell (in the direction from left to right) when varying the immobilized ion concentration from 0 to 1 M. Regardless of the membrane charge, the direction of the electric field results in A⁺ being transported to the left (the flux having a negative sign).

The boundary condition for the concentrations of B^- and C^- determine the overall flux direction for these species, but the direction of the electric field explains why B^- is transported to the right at a higher rate than C^-, which is transported to the left. When increasing the membrane charge, we can see that the flux of B^- decreases (and at close examination, C^- as well); i.e., when increasing the membrane charge, this results in an increased blocking of both B^- and C^- for membrane transport. It should also be noted that the blocking is not complete.

Figure 3. Molar fluxes through the cell for the mobile ions in the NPP model when varying the concentration of the immobilized ion in the membrane.

Plotting the concentrations for the NPP problem in Figure 4 reveals a crucial part in the explanation of the blocking effect. The concentration of A⁺ is higher in the membrane than in the surrounding free electrolyte, whereas the concentrations of B^- and C^- are suppressed. Returning to the flux definition in Eq. (4), we see that low ion concentrations of B^- and C^- have a negative impact on the flux; i.e., a low concentration results in blocking.

Figure 4. Molar concentrations of the mobile ions for the NPP model.

Why do we get an increase of A⁺ and a decrease of B^- and C^- in the membrane, with very steep gradients at the boundaries between the free electrolyte and the ion-exchange membrane? To find the answer, we return to Eq. (1). Inspecting this equation and keeping in mind that the permittivity is typically in the order of 10^-12 (F/m), we see that any net space charge deviating from zero has a huge impact on the potential, unless the nonzero net charge is confined to a very small region in space. As a result, nonzero space charges are usually only found in very thin regions close to phase boundaries, such as an electrolyte-electrode interface or, as in this case, a membrane-free electrolyte interface. Assuming a zero space charge (electroneutrality) is hence usually a very good approximation for any electrolyte solution outside a nm-range of a phase boundary. Since A⁺ is the only positive ion in our system, and as a result of electroneutrality, the concentration of A⁺ increases to approximately match the charge of the immobilized negative ion in the ion-exchange domain.

Figure 5 shows the plot of the electrolyte potential in the NPP model. Similarly to the concentration shifts in Figure 4, we see significant potential shifts at the boundaries between the ion-exchange and the free electrolyte domains. Since the flux of all species is constant through the cell (Eq. (6)), the steep gradients of the potential are needed to balance the steep gradients in the concentrations. Since the sign is opposite for B^- and C^- compared to A⁺, the concentrations of B^- and C^- are suppressed in the membrane.

Figure 5. The electric potential of the electrolyte phase for the NPP model.

In both Figure 4 and 5, the concentration and potential gradients at the phase boundaries are extremely high: In the plots, they show up as vertical lines. This results in numerical difficulties, since the mesh in these transition regions needs to be well resolved. Zooming into either Figure 4 or 5 would reveal that the transition region has a thickness of around 1 nanometer, so typically, a mesh size in the subnanometer range is needed in order to resolve the gradients. The mesh used in this example consists of approximately 500 elements. For 1D simulations, this is usually not a problem. However, when modeling in higher dimensions, the requirement for a fine mesh may result in memory issues. Is there any way to circumvent these issues related to the transition region?

Introducing Donnan Potential Conditions and Electroneutrality

The answer is yes and comes from returning to the definition of the electrochemical potential in Eq. (3). Plotting the electrochemical potential of A⁺ in Figure 6 (using the concentration at the leftmost boundary as reference) reveals that this potential varies continuously through the cell, with no sharp gradients at the membrane-free electrolyte boundaries. (Plotting the electrochemical potential of B^- and C^- would also render fairly smooth curves).

Figure 6. The electrochemical potential of A⁺ for the NPP model.

By assuming the electrochemical potential to be the same on each side outside of the transition region, it is possible to derive a relation between the ion concentrations and the potentials on each side of the interface according to

where we are using the arbitrary indices u and d to define the value on each side of the internal boundary.

This potential shift is called a Donnan potential. Donnan potentials provide one constitutive relation per mobile ion. By requiring continuity of the ion fluxes and the condition of electroneutrality, which is usually fulfilled outside a nm-length from a phase boundary, we can formulate a complete set of internal boundary conditions for all concentration variables and the electric potential on the membrane-free electrolyte boundaries. It should be stressed here that when using this formulation, we need dual instances of the dependent concentration and potential variables on the internal boundary, each representing the variable value when approaching the boundary from either the right or left, respectively. (This is called a slit condition in COMSOL Multiphysics®).

While we are at it, we can also replace the use of Poisson’s equation (Eq. (1)) in the domains by assuming electroneutrality everywhere, and instead derive a potential equation based on the sum of all species fluxes times their respective charges

(For convenience, we have also multiplied the sum by F so that we get an expression with the unit of A/m²; i.e., the total current density. In this way, Neumann boundary conditions may be expressed in this unit.)

This reduces the number of dependent concentration variables by 1. In this new equation system, we solve for N-1 concentration variables and the electric potential variable, keeping in mind that we can always derive the N^th concentration from the other concentrations and the condition of electroneutrality.

The reformulated model problem, using the condition of electroneutrality everywhere and Donnan potentials at the internal boundary, can be solved using far fewer mesh elements. Figure 7, plotting the same concentrations as Figure 4, shows the result when using only 15 mesh elements. As we can see, the results are visually identical, except for the absence of the steep gradients in Figure 7 (the seemingly vertical lines in Figure 4), which now no longer need to be resolved. By using Donnan potentials and the assumption of electroneutrality, we can hence reduce the number of degrees of freedom in our model by more than an order of magnitude, with no loss of solution accuracy.

Figure 7. Concentrations when using Donnan potential conditions and electroneutrality.

Further Boosting Model Convergence

There is actually one more simplification of the model problem that can be done: We can assume that the ion-exchange membrane completely blocks all ions but A⁺. In this case, the concentrations of B^- and C^- in the membrane are 0, and due to electroneutrality, the concentration of A⁺ is always constant and given by the immobilized charge. As a result, we do not need to solve for any concentration variable in the membrane. Since there are no concentration gradients of A⁺ in the membrane, the sole domain equation simplifies to the Laplace equation

where we note that, due to electroneutrality, we have

and that the constant electrolyte conductivity can be calculated from

Although this equation is a simplification for multiple mobile ion systems, it should be noted that it is analytically correct for single-ion conductors, such as Nafion in polymer electrolyte fuel cells, where protons are the sole mobile ions present in the membrane.

The Laplace equation is especially well suited for solving with the finite element or boundary element methods that the COMSOL® software uses in the electrochemistry interfaces. Aside from reducing the degrees of freedom and the corresponding memory requirements of the solver further, the fully blocking membrane assumption facilitates the solver convergence. For example, there is no longer a need for ramping up the fixed ion concentration by using an auxiliary sweep in the stationary solver as I described earlier.

Figure 8 compares the concentration of A⁺ and potential, respectively, between the NPP (the results when using electroneutrality and Donnan conditions are identical) and the model using a fixed membrane charge. For our example problem, the fully blocking model approximates the original model fairly well.

Figure 8. Comparison of the A⁺ concentration (left) and potential (right) between the NPP model and the simplified model, assuming a fully blocking membrane.

How to Build an Ion-Exchange Membrane Model in COMSOL Multiphysics®

The Ion-Exchange Membrane domain node in the Tertiary Current Distribution interface can be used to set up the correct domain equations based on the choice of charge conservation model. In the case of electroneutrality, it also sets up the Donnan conditions automatically on boundaries to neighboring electrolyte domains. There is also an Ion Exchange Membrane Boundary node available for setting up Donnan conditions on boundaries between different physics interfaces.

To build the NPP model, we can use one instance of the Tertiary Current Distribution interface with the charge conservation model set to Poisson. We can then use the Electrolyte domains to define the free electrolyte domains, and the Ion Exchange Membrane node to define the membrane domain.

To build the model based on electroneutrality and Donnan conditions, we can proceed as above but switch the charge conservation model to Electroneutrality. This applies the Donnan conditions automatically to the internal boundaries.

Building the fully blocking membrane model requires a few more steps. Since separate concentration variables are solved for on each side of the membrane (A+ and B- on the left; A+ and C- on the right), we have to use two separate instances of the Tertiary Current Distribution interface (with the charge conservation model set to Electroneutrality). We can use a Secondary Current Distribution interface for the membrane to solve for the potential using the Laplace equation and the Ion-Exchange Membrane Boundary nodes in the Tertiary Current Distribution interfaces to set up the Donnan potential conditions.

Next Steps

If you have a COMSOL Access account and valid software license, you can download the MPH-file for this model from the Application Gallery.

Read more about modeling electrochemical applications in this blog post: Advancing Vanadium Redox Flow Batteries with Modeling.

Editor’s note: This blog post was updated on 10/25/2018 to include information about new features available as of version 5.4 of COMSOL Multiphysics.

Nafion is a registered trademark of The Chemours Company FC, LLC.

Should Autonomous Vehicles Be Hybrid or Electric?

Imagine a future where everyone on the road is using an AV. It seems straight out of the futuristic Netflix® television series Black Mirror, right? Now, imagine if these autonomous vehicles emit dirty exhaust and need regular visits to the gas stations to fill up. It seems anachronistic, like fully outfitting a smart home with the latest IoT gadgets…only to connect it all with dial-up internet.

Autonomous vehicles aren’t, and won’t be, powered by solely gasoline. Instead, the central question is whether only completely electric AVs should be made or if AVs with hybrid engines can be produced as well. There are arguments to be made for each side.

An electric vehicle powers up at a charging station. Image by Santeri Viinamäki — Own work. Licensed under CC BY-SA 4.0, via Wikimedia Commons.

On one side of the coin, many of the automotive companies developing self-driving cars expect their primary use — at least at first — to center on ride-sharing, like taxis without drivers. A hybrid engine that combines gas and electric power enables cars to spend more time on the road (and making money) than charging in a garage.

On the other side, some car companies, along with environmental organizations, are concerned that since these vehicles would likely be driving passengers and making deliveries nonstop, the level of pollution would multiply exponentially, wreaking havoc on the environment.

Either way, electric vehicle batteries need to be optimized for autonomous driving applications, with factors like battery power output and degradation taken into account. Using the COMSOL Multiphysics® software and add-on Batteries & Fuel Cells Module, scientists and engineers can study and design battery systems for both hybrid and all-electric autonomous vehicles.

Battery Designs with Optimal Power Output

By nature, autonomous vehicles involve more electronic components than regular cars. Going beyond the car itself (as well as its lights, alarms, and radio), AVs contain navigation systems and detection and ranging equipment. Large amounts of power consumption means that batteries will get drained faster than normal. Batteries for AVs need to be designed to last longer and output more power so they can keep up with energy demands.

Battery Management Systems

In both hybrid and all-electric vehicles, the battery management system (BMS) is an extremely important design factor. By accurately monitoring the battery activity, the BMS maximizes the energy output, lifetime, and safety. Modeling a lithium-ion battery under isothermal conditions can help you analyze factors that are important in a BMS design, including:

• Voltage
• Polarization (voltage drop)
• Internal resistance
• State-of-charge (SOC)
• Rate capability

Consider a model of a 1D lithium-ion battery made of graphite and lithium-ion manganese oxide (LMO), a cost-efficient and thermally stable fuel cell material, from the default settings in the material library in the Batteries & Fuel Cells Module.

A schematic of the key components of a BMS for electric vehicles.

The battery model is made up of four domains:

1. Negative porous electrode
2. Separator
3. Positive porous electrode
4. Electrolyte

The model enables you to test inputs to see how they affect the overall performance of the battery. These factors can include the initial cell voltage; battery capacity; thickness of the separator and electrodes; and cell SOC, which is the percentage of charge remaining in an electric or hybrid car’s battery pack, analogous to the level of the fuel gauge in a gas-powered vehicle.

Drive Cycle

Vehicles operate according to a specific drive cycle, during which the varying temperature and voltage of the battery are monitored. The drive cycle tells the BMS what the SOC of the battery is: in effect, whether the battery is empty or full. Then, a control unit stops the discharge (if the battery is empty) or charge (if the battery is full).

The 1D model can be expanded to include a thermal analysis in order to perform drive-cycle monitoring. Consider a battery cell that is subjected to a drive cycle for a hybrid vehicle.

Schematic of the key components of a BMS and drive cycle.

Engineers can simulate the drive cycle of the lithium-ion battery to predict its performance, analyze parameters that are hard to measure, or validate experimental results. A few of the factors that affect a battery’s drive cycle include:

• Internal resistance and polarization in each part of the battery cell
• Cell SOC
• SOC of each electrode material
• Local temperature
• Materials

The current load input can be imported into the model from external drive cycle data, such as time versus C-rate (a battery’s discharge rate relative to its maximum capacity). In this case, the imported data corresponds to values that are typical of a hybrid electric vehicle. The analysis can tell you a lot about a battery’s drive cycle, including the cell voltage, electric potential, and total polarization. It is also possible to determine the SOC of the cell and electrodes at load (as well as the temperature) during the drive cycle.

The drive cycle (left) and simulation results showing the cell voltage over the course of the drive cycle (right).

The results of this example show that the drive cycle is suited for this type of battery design. They also show that the heat management could be improved in order for the battery to handle longer drive cycles. As we’ll discuss in the next section, optimizing the drive cycle of AVs will affect their level of success in the consumer transportation market.

Power vs. Energy Evaluation

Rate capability is used to determine whether a battery is designed for its intended purpose. There are two options: energy optimized and power optimized. Energy-optimized batteries have a large capacity or energy supply but relatively low current loads, which makes them a good choice for use in portable electronic devices. For hybrid and electric vehicles, power-optimized batteries are the better option. These batteries have a relatively low capacity but high current loads; for example, they can be recharged at very high currents.

Going back to the 1D lithium-ion battery model, you can perform a power versus energy evaluation to determine the battery’s rate capability. The simulation investigates, under different current loads, the discharge of the battery from its fully charged state and the charge of the battery from its fully discharged state.

Cell voltage during different discharge current loads (left) and a Ragone plot for the two different types of battery cells (right).

The results show the cell voltage during the different current loads and can be used to compare the energy and power outputs of the battery design. The Ragone plot (above, right) demonstrates the effect of battery chemistry and discharge rate on the capacity of the battery.

Modeling Battery Degradation in the COMSOL® Software

The transition to autonomous driving is not going to happen overnight. Many innovators think that when AVs first hit the market, it will be in the form of ride-sharing, not individual cars for a single person or family. Logically, this means that each AV in a ride-sharing company’s fleet will be accessed by about ten riders per day instead of one and operate around the clock instead of during one person’s schedule.

In effect, the use of AVs primarily for ride-sharing will cause car batteries to wear down much faster than those in regular, single-household vehicles. This is where capacity fade analysis comes into play.

Capacity Fade

Batteries undergo both capacity fade and power fade, but there’s a difference. Power fade is the reduced battery voltage for the given discharge rate. Capacity fade is the loss of battery capacity, regardless of the current rate.

Cell voltage during the discharge cycle (left) and battery capacity over the total cycle (right).

The different cell materials that make up a battery, as well as the various combinations between them, cause different rates of aging and can even accelerate the process, and therefore the loss of battery capacity. Certain factors that affect battery cell aging and degradation include:

• Stage of the load cycle
• Potential
• Local concentration
• Temperature
• Direction of the current

By performing a time-dependent analysis of a battery during the cycle, it is possible to find the voltage during discharge and compare the capacity to both the total accumulated cycling time and total number of cycles. It is also possible to analyze the electrolyte volume fraction and solid electrolyte interphase (SEI) film potential drop versus the cycle number and the local SOC on the separator-electrode boundaries. (The SEI provides the electrolytes with insulation and conductivity.) These factors can aid in the design of batteries that are optimized for long-term and constant use in AVs.

Next Steps

Get details on the specialized features and functionality for modeling batteries and fuel cells in the Batteries & Fuel Cells Module by clicking the button below.

Additional Resources

• Read more about designing energy-efficient autonomous vehicles:
  • Do Hybrid and Electric Vehicles Have the Pulling Power?
  • Bloomberg article: “Autonomous Cars Need Tougher Batteries, Lithium-Ion Pioneer Says“
  • The Verge article: “Not all of our self-driving cars will be electrically powered — here’s why“
• Try the tutorial models featured in this blog post (available with a COMSOL Access account and valid software license):
  • 1D Lithium-Ion Battery Model for Determination of Optimal Battery Usage and Design
  • 1D Lithium-Ion Battery for Drive-Cycle Monitoring
  • 1D Lithium-Ion Battery Model for Power vs. Energy Evaluation
  • Capacity Fade of a Lithium-Ion Battery

Netflix is a registered trademark of Netflix, Inc.

Solid-State Lithium-Ion Batteries: Power for the Future

Thanks to their lightweight design and high energy densities, lithium-ion batteries have become the dominant rechargeable battery on the market. Over the years, we’ve watched the use of these batteries evolve from portable electronics, like phones and laptops, to higher-power applications for electric vehicles and aerospace technology. Along the way, certain design challenges have emerged. A major concern, which we talked about in a previous blog post, relates to the safety of these devices, which overheat and catch fire in some cases.

An example of a lithium-ion battery that has caught fire. Image in the public domain, via Wikimedia Commons.

A 2015 article from the U.S. Department of Energy discusses one way to combat such hazardous effects: replacing the traditional lithium salt-containing liquid electrolyte with a solid electrolyte that conducts lithium ions. Making this design adjustment removes the flammable solvents typically found within the liquid electrolyte. And without the liquid electrolyte, there is greater freedom in the battery’s design, which opens new doors for advancing modern battery technology to meet growing demands.

With continued development, solid-state lithium-ion batteries could be used as power sources for portable electronic devices and electric vehicles. Left: A touchscreen laptop. Image by Intel Free Press. Licensed under CC BY-SA 2.0, via Wikimedia Commons. Right: Electric vehicles. Image by Mariordo. Licensed under CC BY-SA 2.0, via Wikimedia Commons.

Strides are already being made to optimize the design of solid-state lithium-ion batteries for industrial applications. For instance, several studies have been conducted to analyze the material properties of a solid electrolyte. The main disadvantage of a solid electrolyte is that its electrical conductivity is vastly lower than that of a liquid electrolyte. Fabricating solid-state lithium-ion batteries through thin-film methods has been shown to help combat this issue. Another important step is to gain a better understanding of the electrochemical processes that occur within these devices. To meet this need, Lizhu Tong of Keisoku Engineering System Co., LTD designed and analyzed a solid-state lithium-ion battery using the COMSOL Multiphysics® software.

Analyzing the Electrochemical Processes in a Solid-State Lithium-Ion Battery

For his simulation study, Tong created a 2D model of a solid-state lithium-ion battery. The model features a negative electrode that is comprised of metallic lithium (Li) and a positive electrode that is comprised of lithium-cobalt-oxide (LiCoO₂) film. For the electrolyte, a solid-state lithium-phosphate (Li₃PO₄) film is used. A cross-section schematic of the model and a figure depicting the lithium ion (Li⁺) transport within the device are shown below.

A cross-section schematic of the battery model (left) and a diagram of the Li⁺ transport in the solid electrolyte (right). Images by Lizhu Tong and taken from his COMSOL Conference 2016 Boston paper.

Note that in solid-state lithium-ion batteries, all of the electrochemical reactions occur at the interface between the solid electrolyte and the solid electrodes. No liquids or porous electrodes are included in the design. During the charge process, oxidation reactions take place at the positive electrode’s surface and the generated lithium ions travel to the negative electrode. During discharge, reduction reactions take place at the positive electrode’s surface that consume the lithium ions obtained from oxidation at the negative electrode.

A tertiary current distribution was calculated for the battery, considering the lithium species mass transport as contributing to the overall current-voltage relation for the battery. In the solid electrolyte, the combined diffusion of ions and migration of ions in the electric field were described with the Nernst-Planck equations. In the positive electrode, Fick’s law was used to describe the diffusion of intercalated lithium atoms. Butler-Volmer kinetics were used to describe the electrode reactions at the two electrode-electrolyte interfaces. The solid lithium (negative electrode) domain itself was not included in the calculations, because the metallic lithium is highly electrically conductive compared to the other solid-state materials and there is no chemical mass transport inside the lithium metal.

Charge Characteristics

Let’s begin by looking at the charge characteristics for the battery model. Here, we can see the lithium-ion concentrations in the electrolyte at the end of charging steps at rates 1.2 C and 3.2 C. The results indicate that for higher charge rates, a higher gradient of lithium ion concentration is established across the solid electrolyte, so there is greater concentration deviation from the initial uniform ion concentration.

Lithium ion concentrations in the electrolyte at the end of charge. The plots represent charge rates of 1.2 and 3.2 C, respectively. Images by Lizhu Tong and taken from his COMSOL Conference 2016 Boston paper.

The oxidation reactions during charging cause the lithium concentration in the positive electrode material to decrease near its interface with the solid electrolyte.

Lithium concentrations in the positive electrode at the end of charge. Images by Lizhu Tong and taken from his COMSOL Conference 2016 Boston paper.

The plot below compares the charge curves — in other words, cell voltage vs. time — for a range of charge rates. It is shown that after charging begins, the cell voltage initially experiences a rapid increase until it stabilizes at about 3.9 V. After this point, further increases in the voltage occur in proportion to the state-of-charge, so that the cell voltage increases more rapidly for higher charge rates. These charge curves could be compared to experimental data to validate the simulation model and further investigate the relation of the battery design to its performance.

Plot comparing charge curves for various charge rates. Image by Lizhu Tong and taken from his COMSOL Conference 2016 Boston paper.

Discharge Characteristics

Shifting gears to the discharge process, we can now look at the results for the same corresponding discharge rates. At the end of discharge, the concentration profile across the solid electrolyte appears similar to that at the end of charge, but with the regions of high and low concentration reversed, since the lithium ions diffuse in the opposite direction.

Lithium-ion concentrations in the electrolyte at the end of discharge. The plots represent charge rates of 1.2 and 3.2 C, respectively. Images by Lizhu Tong and taken from his COMSOL Conference 2016 Boston paper.

Lithium ions are now reduced from the solid electrolyte at the positive electrode’s surface, resulting in an elevated concentration of lithium close to the interface of the positive electrode with the solid electrolyte, as illustrated below.

Lithium concentrations in the positive electrode at the end of discharge. Images by Lizhu Tong and taken from his COMSOL Conference 2016 Boston paper.

The cell voltage drops from over 4 V to about 3.8 V during discharge. Once the battery approaches a low state-of-charge, internal losses mean the discharge current can no longer be maintained and the cell voltage falls rapidly. Of course, this occurs more rapidly for higher discharge rates.

Plot comparing discharge curves for various discharge rates. Image by Lizhu Tong and taken from his COMSOL Conference 2016 Boston paper.

Building Safer and More Flexible Lithium-Ion Batteries with Simulation

Solid-state lithium-ion batteries have the potential to address many of the safety concerns surrounding traditional lithium-ion battery designs. But realizing their application on an industrial scale requires a deeper understanding of the underlying electrochemical processes that occur within the device. As we’ve showcased here, COMSOL Multiphysics provides you with the features and functionality to model such processes and delivers useful results that can further the development of solid-state lithium-ion batteries.

Learn More About Modeling Lithium-Ion Batteries Using COMSOL Multiphysics®

• Read the COMSOL Conference paper: “Two-dimensional Simulation of All-solid-state Lithium-ion Batteries“
• Download a related tutorial from our Application Gallery: All-Solid-State Lithium-Ion Battery
• See how you can further advance your analysis of lithium-ion batteries with simulation:
  • How to Model Short Circuits in Lithium-Ion Batteries
  • Li-Ion Battery: Heterogeneous Alternative to the Newman Model
  • Studying Impedance to Analyze the Li-Ion Battery with an App
]]> https://www.comsol.de/blogs/modeling-electrochemical-processes-in-a-solid-state-lithium-ion-battery/feed/ 2 https://www.comsol.de/blogs/how-to-model-electrochemical-resistance-and-capacitance/ https://www.comsol.de/blogs/how-to-model-electrochemical-resistance-and-capacitance/#comments Wed, 24 Aug 2016 21:20:15 +0000 http://com.staging.comsol.com/blogs?p=178101 Resistive and capacitive effects are fundamental to the understanding of electrochemical systems. The resistances and capacitances due to mass transfer can be represented through physical equations describing the corresponding fundamental phenomena, like diffusion. Further, when considering the resistive or capacitive behavior of double layers, thin films, and reaction kinetics, such effects can be treated simply through physical conditions relating electrochemical currents and voltages. Lastly, resistances and capacitances from external loading circuits can easily be represented in the COMSOL Multiphysics® software.

What Are Resistive and Capacitive Currents?

When interpreting the behavior of an electrical circuit, we often speak of resistances and capacitances. Current passing through a resistor with a resistance R (Ω) scales with the applied voltage V, while current across a capacitor with a capacitance C (F) scales with the rate of change of voltage:

I_\mathrm{res} = \frac{V}{R}
I_\mathrm{cap} = C\,\frac{\partial V}{\partial t}

In an electrochemical cell, we can apply voltage and measure current, or vice versa. Therefore, the response of the cell is often considered in terms of resistances and capacitances. In AC impedance analysis, the resistive current is in phase with the applied voltage, while the capacitive current is out of phase. While we can recover resistances and capacitances from the measured data, a wide variety of physical effects contribute to the measured currents. Considering the solution to the equations describing the physical processes that cause resistance and capacitance can provide greater physical insight. In this way, we use simulation as a means of fitting measured data, interpreting it in terms of the underlying physical causes.


A plot of electrolyte potential (color plot) illustrates the voltage associated with resistance to current flow (black arrows) in a plating cell for electrodeposition on a patterned wafer. The changing resistance of the deposited layer is also included in the model.

Electrochemical Effects that Cause Resistance and Capacitance

Effects interpreted as resistance in an electrochemical cell most often arise due to the finite conductivity of the electrolyte. Passing current through the electrode, between the anode and cathode, requires a certain applied voltage, just like in a conventional resistor in a circuit. The Primary Current Distribution and Secondary Current Distribution physics interfaces in COMSOL Multiphysics solve Ohm’s law to predict the magnitude of this voltage, usually called the ohmic drop, as a function of electrolyte conductivity and electrode geometry.

When electrode reaction kinetics are important, an additional voltage, called overpotential, is required to overcome the activation energy for the electrochemical reaction occurring at an electrode surface. Common electrode reaction kinetic rate laws, such as the Tafel law or Butler-Volmer equations, define the current density in terms of an overpotential — this current-voltage relation can be interpreted as a resistance. However, unlike a simple resistor in an electrical circuit, the impedance due to electrode kinetics is usually nonlinear. This is shown in the Tafel law, which sets the overpotential proportional to the logarithm of current density. The relation is approximately linear only in the limit of small amplitudes of applied voltage, as typically used in electrochemical impedance spectroscopy.

In electroanalysis and impedance spectroscopy, mass transport by diffusion is also often discussed in terms of the measured resistance and capacitance. In traditional equivalent circuit analysis, the Warburg element is a lumped impedance that accounts for the observed effects due to diffusion. Similarly, the Randles circuit is typically used to describe the combination of all of the above effects: diffusion, electrode kinetics, and solution resistance.

However, the influence of kinetics and diffusion on an electrochemical cell’s behavior can be described directly by the corresponding physical equations. In the Electroanalysis interface, Fick’s laws of diffusion are solved together with electrode kinetic relations, such as the Tafel or Butler-Volmer equations, yielding impedance spectra directly in terms of the fundamental electrochemistry. In the tertiary current density interfaces, mass transport resistance may also include the contribution from migration and advection, in addition to diffusion.

To see an example of predicting an electrochemical impedance spectrum from physical equations, without using an equivalent circuit, check out our Electrochemical Impedance Spectroscopy tutorial and the lithium-ion battery examples.

Film Resistance

Another important situation that introduces resistive and capacitive effects is the behavior of thin layers or surface films at the electrode-electrolyte interface. In these cases, it is often more straightforward to express the behavior directly in terms of a resistance or capacitance.

Film resistance occurs whenever a thin layer of material is created at an electrode surface, with significantly different electrical conductivity from the bulk electrode material. This additional resistance can lower the system’s efficiency. In the case of a battery, for example, less of the generated cell voltage from the electrochemical reactions is available to the external circuit as power.

When exposed to air, most metals quickly become oxidized at the surface and therefore are covered with a layer of metal oxide. The metal oxides tend to conduct electricity far more poorly than the original metal, thus introducing an additional film resistance into the system. This can also occur in subsea corrosion, as demonstrated in our anode film resistance example.

Another example of film resistance is in electrodeposition, where the bulk electrode material and depositing species are not the same. A layer of increasing thickness with altered conductivity is then deposited on the surface, with different properties than the bulk electrode.

Film resistance can be defined at an electrode surface in COMSOL Multiphysics. The most straightforward option is to set a given film resistance in Ω·m². The units of m² are required to express the local resistance for a current density in A m^-2 normal to the film, with a potential difference in V across the film that drives the current flow. Alternatively, we can provide a reference thickness, change in thickness, and thin film conductivity. This is useful when the film’s thickness changes with time.


Film resistance settings. Left: A constant film resistance for an electrode with a surface shape and composition that doesn’t change. Right: A more general case where the film thickness can change during the simulation, under ongoing electrodeposition.

Double-Layer Capacitance

At the interface between a charged electrode and the electrolyte, a region known as the electrical double layer (EDL) arises. Here, electrical attraction and repulsion cause a highly nonuniform distribution of ion concentrations. (In a future blog post, we will discuss the physics and modeling of the EDL in greater detail. In the meantime, check out this previous blog post describing the flow inside a battery.)

Changing the electrode potential through a time-varying applied voltage generates an accumulation or release of charge in the double layer. The amount of charge accumulated in the double layer is described by a double-layer capacitance. The scale of the double layer is given by the Debye length for the system, which is typically in the order of nanometers. As such, in comparison to the typical size of a diffusion layer or electrochemical cell on the micrometer-to-meter scale, this capacitance can be considered as a surface effect at the electrode-electrolyte interface.

In COMSOL Multiphysics, you will find several tools for modeling the EDL in detail (see our Diffuse Double Layer with Charge Transfer tutorial model). This is now easier than ever before with the new Nernst-Planck-Poisson Equations multiphysics interface available in COMSOL Multiphysics® version 5.2a. However, despite great theoretical effort, it is not possible to devise a model that accurately accounts for all of the experimentally observed phenomena in the EDL. For this reason, an empirical description of the double-layer capacitance may be preferred, where the capacitance is defined empirically by fitting data from experiments.

Double-layer capacitance is especially important in impedance spectroscopy, where the voltage is varied rapidly at high frequencies. Together with the finite electrode reaction rate and a finite rate of mass transport, it is a key factor in the characteristic “semicircle with a tail” shape of the Nyquist plot for many electrochemical systems.


An experimentally measured Nyquist plot for a lithium-ion battery, derived from our Lithium-Ion Battery Impedance demo app.

When defining an electrode surface, we can add a Double Layer Capacitance subnode. This electrode feature always requires a surface capacitance per area (in F m^-2), though certain circumstances may require additional inputs. For example, for a porous electrode, the Porous Matrix Double Layer Capacitance feature requires the input of a specific surface area to determine the true surface area of the electrode-electrolyte interface, where a double layer exists and at which capacitive charging occurs. You can also use the built-in tools to calculate the double layer area of a porous electrode based on the particle properties of spheres, cylinders, or flakes. In this case, there is no need to explicitly know the specific surface area of the electrode.


An overview of the settings for the Double Layer Capacitance feature. The differences between nonporous and porous electrodes are shown, as well as the possibility to calculate the double layer surface area from some fundamental geometric properties instead of adding this directly as input.

Addressing More Complex Circuit Setups

There may be occasional cases where it is desirable to connect an electrode to a more complex circuit in order to see how your electrochemical system couples with a complex driving circuit. Again, thanks to the flexibility of COMSOL Multiphysics, connecting any of the electrochemical physics interfaces to the Electrical Circuit interface is possible.


A typical setup for coupling a current-driven electrochemical system to a circuit representing instrumentation that supplies the current. Note how the currents in the Electrode Current and External Device settings are set to a common variable I_couple. The value of this variable is determined by constraining the voltages associated with these two nodes to be equal.

Advance Your Electrochemical Modeling by Accurately Describing Resistances and Capacitances

In today’s blog post, we have explained the approaches for describing electrochemical resistances and capacitances, while highlighting different cases in which each can be applied. We encourage you to use these tools and the knowledge that you have gained here to maximize the accuracy and usefulness of your electrochemical models.

Additional Resources for Modeling Electrochemical Systems in COMSOL Multiphysics®

• To learn more about including resistance and capacitance features in your electrochemical models, download the following tutorials:
  • Electrochemical Impedance Spectroscopy
  • Capacity Fade of a Lithium-Ion Battery
  • Modeling Impedance in the Lithium-Ion Battery
• Watch this archived webinar for further details on simulating electrochemical systems in COMSOL Multiphysics
]]> https://www.comsol.de/blogs/how-to-model-electrochemical-resistance-and-capacitance/feed/ 1 https://www.comsol.de/blogs/building-an-app-to-optimize-the-design-of-an-sofc-stack/ https://www.comsol.de/blogs/building-an-app-to-optimize-the-design-of-an-sofc-stack/#comments Tue, 23 Aug 2016 08:09:00 +0000 http://com.staging.comsol.com/blogs?p=177421 Today, guest blogger Matteo Lualdi of resolvent ApS, a COMSOL Certified Consultant, discusses the benefits of creating a simulation app to analyze a solid oxide fuel cell stack.

For many businesses, numerical modeling and simulation are valuable tools at various stages of the design workflow, from product development to optimization. Apps further extend the reach of these tools, hiding complex multiphysics models beneath easy-to-use interfaces. Here’s a look at one such example: a solid oxide fuel cell stack app.

Solid Oxide Fuel Cells: A Complex Multiphysics System

Consider a device that can generate heat and power for your home as well as power the electric motor of your car. Now think about using such a device to produce electricity with small losses, without being limited by Carnot efficiency. The thought that should come to mind is a fuel cell.

Fuel cells, like batteries, are devices that electrochemically convert chemical energy present in the fuel into electrical energy. But unlike batteries, which have to be recharged, fuel cells are continuously fed with chemicals. Solid oxide fuel cells (SOFCs), in particular, are recognized as one of the most efficient and flexible types of fuel cells. This is due to their very small kinetic losses, despite the slightly less advantageous thermodynamics at higher temperatures. However, from an engineering point of view, these devices present more of a challenge due to their high operating temperatures (600°C to 1000°C).


A schematic of an SOFC. Image in the public domain, via Wikimedia Commons.

Because of these high temperatures and the overall complexity of the system, measuring critical parameters like temperature, composition, and current densities within the stack is quite difficult. However, addressing such elements is particularly important in optimizing the performance of the device and identifying any potential issues that occur at different operating conditions.

With the capabilities of the COMSOL Multiphysics® software, it is possible to achieve precise estimates of such parameters. The software also helps with the interpretation of experimental results and enables users to virtually perform potentially destructive tests, which can otherwise be quite expensive and difficult to fully understand.

Modeling an SOFC Stack in COMSOL Multiphysics®

Modeling an SOFC stack requires combining a rather intricate geometry with large aspect ratios and multiple physics in a 3D problem. Each SOFC features two reactant flows: air and fuel. These two elements can have different compositions and several parameters that describe inlet and environmental conditions.


Model geometry of an SOFC stack.

With regards to the multiple physics that are involved, the model needs to solve:

• Chemistry for the air and fuel sides
• Flow for the air and fuel sides
• Transport of species for the air and fuel sides
• Heat transfer
• Electrical current

Designing this complex model and modifying its parameters requires knowledge about both technology and mathematical modeling. The combined expertise needed to run such calculations often translates into a heavy workload for the model developers, who have to dedicate a great share of their time to run calculations.

Now, thanks to the Application Builder in COMSOL Multiphysics®, simulation experts have the ability to design a user-friendly tool that can be deployed to other team members and throughout an organization to study such complex systems. This can greatly improve overall productivity by delivering simulation results more quickly to users, while giving simulation experts more time to focus on model development and, of course, to continue to provide assistance with simulation strategies along the way.

Creating a User-Friendly Simulation App to Analyze an SOFC Stack

Like the initial model highlighted above, the app presented here was developed at Topsoe Fuel Cell, a leader in SOFC technology. Both the model and the corresponding app have served as an important development tool throughout the organization for advancing the design of SOFC stacks.

The app’s user interface (UI) features several tabs, as highlighted in the following series of screenshots. Under the Input tab, for example, users with little to no simulation expertise can easily modify several operating conditions for the SOFC design. In this case, such conditions range from temperatures and currents to flows and heat losses — parameters that can be updated based on the specific modeling needs.


Parameters available for modification under the Input tab.

Perhaps an app user does have knowledge of the solver and other special settings. If so, under the Solver & Special Input tab, these users can modify various elements related to the solver, including mass constraint, the oxidized species, and the different species in the solver on the fuel side. There is also the option to make adjustments to the settings that pertain to the area-specific resistance (ASR).


The Solver & Special Input tab.

After applying the appropriate changes to the settings, the next step is to compute the simulation. While doing so, app users can follow the progress of the simulation as they await the results. So what types of computations can users perform with this tool? For one, the app can be used to compute the temperature distribution in the SOFC stack, as shown below.


Temperature variation in the SOFC stack.

With the app, users can also analyze the current densities of the SOFC stack as well as the hydrogen mole fraction. You can see such simulation results highlighted in the following plots.




Left: Plot showing the current density. Right: Plot showing the H₂ mole fraction.

As the design workflow progresses and users obtain a better understanding of their device, they may request to change or include new input parameters and outputs for analysis. With the flexibility and customization available in the Application Builder, it is easy for app designers to make these modifications and thus meet the specific needs of their end users.

Experience the Benefits of Building and Utilizing Simulation Apps

Turning a complex model into an app is a powerful solution for bringing simulation capabilities to a larger number of people. Whether deploying apps to other engineers, system developers, or salespeople at your organization, these tools serve as a viable resource in helping to deliver reliable simulation results quickly to verify operating strategies or provide some insight into experimental results.

From a consultant-to-client point of view, a design workflow that includes an app based on a validated model creates a greater value for clients. Rather than receiving a report with only a few sensitivity analyses, they have the power to investigate the impact of various parameters on their design and therefore achieve faster and more reliable results.

About the Guest Author

Matteo Lualdi is the simulation manager at resolvent ApS, a COMSOL Certified Consultant. He received two master’s degrees in energy engineering from Politecnico di Milano and the Royal Institute of Technology in Stockholm within the Top Industrial Managers for Europe (T.I.M.E.) framework. Matteo later earned his PhD in chemical engineering from the Royal Institute of Technology. His experience prior to resolvent ApS includes working in the fuel cell and catalyst business, where he was responsible for system simulations and catalytic reactor sizing.

]]> https://www.comsol.de/blogs/building-an-app-to-optimize-the-design-of-an-sofc-stack/feed/ 0 https://www.comsol.de/blogs/using-simulation-in-the-race-against-corrosion/ https://www.comsol.de/blogs/using-simulation-in-the-race-against-corrosion/#comments Mon, 28 Dec 2015 13:59:40 +0000 http://com.staging.comsol.com/blogs/?p=123441 Corrosion is one of the most serious factors affecting the transportation industry. In an effort to minimize its impact, a German research institute and the manufacturers of Mercedes-Benz joined forces to investigate the corrosion occurring in automotive rivets and sheet metal. Using COMSOL Multiphysics simulation, they were able to study corrosion’s effects on car components.

Modeling Galvanic Corrosion in Rivets and Joints

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.

Predicting the Dangers of Delamination

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.

Read More About Using Simulation for Automotive Component Design

• Find the full story on page 14 of COMSOL News 2015
• On the blog: Modeling Corrosion for Automotive Applications

Mercedes-Benz is a registered trademark of Daimler AG Corporation.

]]> https://www.comsol.de/blogs/using-simulation-in-the-race-against-corrosion/feed/ 2 https://www.comsol.de/blogs/studying-impedance-to-analyze-the-li-ion-battery-with-an-app/ https://www.comsol.de/blogs/studying-impedance-to-analyze-the-li-ion-battery-with-an-app/#comments Tue, 14 Jul 2015 16:02:07 +0000 http://com.staging.comsol.com/blogs/?p=92241 Batteries generally operate through numerous processes that depend on even more parameters. How can you find out more about what’s going on within them? One approach is to look at the cell’s electrical impedance. The Lithium-Ion Battery Impedance demo app, available in the Application Gallery, can be used to interpret the impedance of a specific lithium-ion battery design with minimal effort. It can also help parameterize the system, a useful step for setting up accurate time-dependent models in the future.

Impedance Spectroscopy: An Experimental Method

Electrochemical impedance spectroscopy (EIS) is a very widely used technique in electroanalysis. It is used to study the harmonic response of an electrochemical system. For batteries, a small, sinusoidal variation is applied to the potential difference between the two electrodes, and the resulting current is analyzed in the frequency domain in terms of impedance. Normally, the perturbation is applied about the open-circuit voltage.

In electrical analysis, the impedance is a complex quantity that includes a real and an imaginary component. The former corresponds to a resistance in-phase with the applied voltage; the latter is a reactance 90° out-of-phase with the applied voltage. The real and imaginary components of the impedance give information about the kinetic and mass transport properties of the cell, as well as its capacitive properties. By measuring impedance at a range of frequencies, the relative influence of the various constituent physics of the system can be interpreted as a function of their characteristic timescales.

How to Model Impedance in a Lithium-Ion Battery

Several processes within a lithium-ion battery cell exhibit time-dependent responses that are detectable in the frequency domain. For a typical lithium-ion battery cell made up of two porous electrodes with a porous separator in between, as seen in the figure below, the following processes are accountable:

• Charge transfer reaction on the surface of the active electrode materials.
• Mass transport (diffusion and migration) in the electrolyte.
• Diffusion of lithium within the active electrode material particles.
• Change of double-layer charge on the active electrode material, electrical conductor, and other interfaces.
• Contact resistance between electrically conducting materials.


Processes and materials in a lithium-ion battery cell.

The Lithium-Ion Battery interface easily accounts for all of these phenomena. In the electrolyte, both charge and mass balances are set up. In the active material particles, mass balances are solved. For instance, the charge transfer reaction can be modeled with Butler-Volmer kinetics. At the surface of all solid phase materials, double-layer currents and a film resistance can be introduced. All equations in the interface are based on the transient descriptions.

For impedance modeling, the COMSOL Multiphysics software automatically transforms these equations into the frequency domain and linearizes them around a given voltage and current. The linearization approach is in line with the harmonic interpretation of impedance data, and it can be used because the perturbation to the cell potential is chosen to be small.

How Can Impedance Data Be Interpreted?

A common way to display the impedance of a system is with a Nyquist plot, in which the negative imaginary component of impedance is plotted against the real component of impedance. For a single porous electrode (see the figure above), the Nyquist plot often looks like that shown below.


Nyquist plot and the contributions of various properties.

The semicircles within the mid-high frequency window display the charging of double layers on the materials within the electrode and the contribution of different resistances; these can be due to the electrode materials and the presence of resistive films, for instance. One of the semicircles gives an indication of the rate of the charge transfer reaction.

At lower frequencies, a “tail” is shown. The shape of the tail is principally affected by the diffusion within the electrolyte and active electrode materials. In essence, it is controlled by the diffusion coefficients and the particle size for the electrode material. The real impedance at the leftmost point in the Nyquist plot gives a measure of the ionic and electrical conductivity within the cell.

All in all, impedance supplies a considerable amount of information that a model can help to organize. One approach is to vary model parameters to pinpoint what affects the impedance and at what frequency, as seen in the figure below. Alternatively, one can fit the model to experimental impedance data through an optimization procedure and examine the optimized properties.


Nyquist plots with the change of various parameters.

The Lithium-Ion Battery Impedance App

With a simulation app, you can decipher experimental EIS measurements in a quick, straightforward manner. It functions by using experimental data from EIS measurements as input, simulates these measurements, and then fits the model to the experimental data through parameter estimation.

The studied battery cell design consists of the following components:

• Positive porous electrode: NCA (LiNi_0.08Co_0.15Al_0.05O₂) active material, electrical conductor, and binder.
• Negative porous electrode: LTO (Li₄Ti₅O₁₂) active material, electrical conductor, and binder.
• Separator: Celgard 2325.
• Electrolyte: 1.2 M LiPF₆ in EC:EMC (3:7 by weight).

The fitting is done to the measurements of the positive electrode at frequencies ranging from 10 mHz to 1 kHz.

The thicknesses of the electrodes and separator together with the current collector area and the initial state-of-charge of the electrodes can be varied in the Cell Properties section. An Experimental Data section enables you to import any measured impedance data that you want to investigate.

In the Parameter Estimation section, the selection of the estimated control parameters is made. Available parameters include the exchange current density, the resistivity of the resistive layer on the particles, the double-layer capacitance of NCA, and the double-layer capacitance of the carbon support in the positive electrode.

Here is what the user interface looks like when the battery design has been optimized:


The lithium-ion battery impedance app.

In this case, the magnitudes of the optimized parameters tell us whether the associated processes are present. Is there any substantial charging of double layers? Is the resistance due to film resistance on the active particles significant? How fast is the charge transfer reaction? The parameters are also ready to be used for setting up, for instance, a time-dependent battery model of the system. You can also easily move on and benchmark different batteries or import impedance data from another battery cell, perhaps an aged one. With COMSOL Multiphysics, there are many possibilities!

Join the Simulation Revolution: Run the Demo App Now

• Download the Lithium-Ion Battery Impedance demo app
• Learn more about building simulation applications
]]> https://www.comsol.de/blogs/studying-impedance-to-analyze-the-li-ion-battery-with-an-app/feed/ 0 https://www.comsol.de/blogs/current-distribution-interface-use/ https://www.comsol.de/blogs/current-distribution-interface-use/#comments Mon, 10 Feb 2014 21:06:47 +0000 http://com.staging.comsol.com/blogs/?p=27095 When designing electrochemical cells, we consider the three classes of current distribution in the electrolyte and electrodes: primary, secondary, and tertiary. We recently introduced the essential theory of current distribution. Here, we illustrate the different current distributions with a wire electrode example to help you choose between the current distribution interfaces in COMSOL Multiphysics for your electrochemical cell simulation.

Introducing the Three Current Distribution Interfaces

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.

Recap of the Essential Equations

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

Primary Current Distribution

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 () and another for the electric potential in the electrodes (). With the above described assumptions for a primary current distribution, you get the following equations to be solved:

Electrode: with

Electrolyte: with

Electrode-Electrolyte-Interface:

Here, denotes the conductivity of the electrolyte, which is constant by the above assumptions. The index represents the electrode and the electrolyte. denotes the equilibrium potential for reaction .

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).

When Do I Use the Primary Current Distribution Interface?

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.

Secondary Current Distribution

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 (). 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: with

Electrolyte: with

Electrode-Electrolyte-Interface:

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 : denotes the local charge transfer current density, the exchange current density, and the anodic and the cathodic charge transfer coefficient. 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 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).

When Do I Use the Secondary Current Distribution Interface?

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.

Tertiary Current Distribution

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 (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 (), the electrode potential (), and the set of species concentrations . With the assumptions described above you get the following equations:

Electrode: with

Electrolyte: with

Electrolyte electroneutrality:

Electrode-Electrolyte-Interface:

Typical current density expression:

It is essential that the reference concentration 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).

When Do I Use the Tertiary Current Distribution Interface?

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.

What Other Options Are There?

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.

Secondary Current Distribution with Chemical Species Transport

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.

Electroanalysis

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.



Concluding Thoughts

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.

Browse the Electrochemical Add-On Modules

• Batteries & Fuel Cells Module
• Electrodeposition Module
• Corrosion Module
• Electrochemistry Module

Other Post in This Series

1. Theory of Current Distribution
]]> https://www.comsol.de/blogs/current-distribution-interface-use/feed/ 4 https://www.comsol.de/blogs/theory-current-distribution/ https://www.comsol.de/blogs/theory-current-distribution/#comments Fri, 07 Feb 2014 17:01:12 +0000 http://com.staging.comsol.com/blogs/?p=27077 In electrochemical cell design, you need to consider three current distribution classes in the electrolyte and electrodes. These are called primary, secondary, and tertiary, and refer to different approximations that apply depending on the relative significance of solution resistance, finite electrode kinetics, and mass transport. Here, we provide a general introduction to the concept of current distribution and discuss the topic from a theoretical stand-point.

General Introduction to Current Distribution

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.

Essential Theory

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:

with conservation of current

where denotes the current density vector (A/m²) in the electrode, denotes the conductivity (S/m), the electric potential in the metallic conductor (V), and denotes a general current source term (A/m³, 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 denotes the current density vector (A/m²) in the electrolyte, denotes the Faraday constant (C/mol), and the flux of species (mol/(m²·s)) with charge number . 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 represents the concentration of the ion (mol/m³), the diffusion coefficient (m²/s), its mobility (s·mol/kg), the electrolyte potential, and 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 (A/m³):

\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.

Next Up: Choosing the Right Current Distribution Interface

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!

Other Post in This Series

1. Which Current Distribution Interface Do I Use?
]]> https://www.comsol.de/blogs/theory-current-distribution/feed/ 0 https://www.comsol.de/blogs/lithium-ion-battery-analysis-ines-cea/ https://www.comsol.de/blogs/lithium-ion-battery-analysis-ines-cea/#comments Thu, 27 Jun 2013 13:58:02 +0000 http://com.dev.comsol.com/blogs/?p=13721 During my time as a PhD student, a blue “Chemical Landmark” plaque was fitted to the building a couple of hundred yards down the road from my lab. The plaque commemorates the achievements of the researchers who made the lithium-ion (Li-ion) battery viable. Whether or not you know about the electrochemistry of rechargeable lithium-ion batteries, you probably rely on one. We carry them around in our phones and laptops, and ride in cars and planes that use them for power. Improving the efficiency and safety of such devices is big business, and it rests on understanding their chemistry and correlating it to experiment. Mikael Cugnet at INES-CEA in France used COMSOL Multiphysics to advance his team’s understanding of their battery experiments, considering the real effects behind electrochemical impedance measurements.

Safe Power from Lithium

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.

Lithium-ion Battery Analysis: From Electrical Circuits to Chemistry

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.

Macroscopic vs. Microscopic

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.

Further Reading

• COMSOL News 2013
• “Modeling the Li-ion Battery” whitepaper
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