The Van Allen radiation belts consist of highly energetic charged particles that have become trapped in Earth’s magnetic field. These particles follow the shape of the field and give the belts a doughnut-like appearance. To study the behavior of the particles in the Van Allen belts, scientists can use the Particle Tracing Module, an add-on product to the COMSOL Multiphysics® software.

What Are the Van Allen Belts?

The Van Allen belts, named after James Van Allen, are two doughnut-shaped belts of radiation that surround Earth. Extending outward from about 1000 to 60,000 km, these belts fluctuate in size, sometimes expanding and shrinking in large orders of magnitude over the course of a couple of hours.

The inner (light gray) and outer (dark gray) Van Allen belts around Earth. The projection of Earth’s landmass is based on images by M.J. Brodzik and K.W. Knowles (Ref. 1).

While research into what affects the shape and size of these belts is still ongoing, scientists have made significant breakthroughs over the years. In 2012, NASA launched the Van Allen Probes, which showed just how complex the belts are. Much to the researchers’ surprise, the probes detected a temporary third belt (caused by a solar storm), which disappeared within a few weeks. The team also learned that only some of the particles within the belts affect the shape, while others are present at all times.

Due to the complexity of the belts, it’s important to clearly understand the fundamental physics. With particle tracing simulation, it’s simple to study the movement of particles trapped in Earth’s magnetic field and see how they give the belts their distinctive shape.

The Motion of Particles in Earth’s Magnetic Field

Earth’s magnetic field is extensive, surrounding the planet for thousands of kilometers. The field is similar to a dipole, although there are certain asymmetries and irregularities. Because the magnetic field is responsible for trapping the particles and they follow its shape, it’s important for scientists to have an accurate model of the field. The standard is, of course, the International Geomagnetic Reference Field (IGRF), which is regularly updated based on the latest findings.

Simulation of the Van Allen belts around Earth. The projection of Earth’s landmass is taken from images in Ref. 1. (Note that this image was created by running the model discussed below for a longer time interval.)

When particles enter this field, they begin to spiral around a field line toward one of the poles. As they approach, their pitch angle (i.e., the angle between the direction of the magnetic field and a particle’s trajectory) increases due to the increasing magnitude of the magnetic field, leading to the dip in the doughnut’s center. With the higher angle, eventually, the particle reaches a point (known as the mirror point) where it bounces, heading toward the other hemisphere. If the particle doesn’t become lost to the atmosphere, the pattern continues and the particle drifts from one field line to the next, gradually making its way around Earth.

Simulating the Motion of Particles Using the IGRF Model

This 3D example examines the path that particles, specifically protons, take while trapped in Earth’s magnetic field. The model consists of a simple sphere of radius R_e, which represents Earth, and a larger spherical simulation domain of radius 5R_e, where the particle trajectories are computed.

The magnetic field domain around Earth. Note that in the model, Earth is represented by a simple sphere. Here, the projection of Earth’s landmass is taken from images in Ref. 1.

To quickly compute the magnetic fields, it’s possible to use an external function to incorporate the data from IGRF into a simulation. Accessing this data is simple using the built-in Earth’s magnetic field option in the Magnetic Force feature, available in both the Charged Particle Tracing interface and Particle Tracing for Fluid Flow interface. Here, the Charged Particle Tracing interface is used with a time-dependent study.

The magnetic field lines are based on data from IGRF, while Earth’s landmass is based on images in Ref. 1.

Another feature, named Release from Grid, makes it easy to model the release of protons at a particular equatorial pitch angle. In this model, the equatorial pitch angle of the 10-MeV proton is set to 30°.

Examining the Results of the Particle Tracing Simulation

The simulation shows the three components of the proton’s motion:

1. Gyration
2. Bounce
3. Drift

While not displayed below, the timescales for drift motion are much longer than those for bounce motion, which are in turn much longer than the gyration period.

The trajectory of a proton in Earth’s magnetic field.

Scientists can also use models like this one to look at the effect of various equatorial pitch angles on the mirror point latitude. As expected, the mirror point latitude increases as the pitch angle of the particle decreases. At the limits of the plot, a particle with an equatorial pitch angle of 90° remains in the equatorial plane, whereas a particle with a pitch angle of 0° travels directly along the field line without bouncing. If there’s no bounce, the particle falls down toward Earth to form an aurora.

Left: The trajectories of multiple trapped protons, with the color expression corresponding with different equatorial pitch angles. Right: Mirror point latitude against the equatorial pitch angle for a particle.

To better visualize the results, you can create an animation. The one below depicts the movement of a trapped proton, showing how the particle’s pitch angle increases near the poles. You can also see the beginnings of how the particle’s movement creates the belts’ signature doughnut-like shape.

Animation of a trapped proton in the Van Allen belts. The projection of Earth’s landmass is based on images in Ref. 1.

With particle tracing simulation, scientists can gain a better understanding of the behavior of particles in the Van Allen belts, important information for more complex studies in the future.

Next Steps

If you want to try modeling the motion of protons within Earth’s magnetic field, click the button below. Doing so will take you to the Application Gallery, where you can log into your COMSOL Access account and then get the tutorial documentation and MPH-file (with a valid software license).

Additional Resources

• Read more about particle tracing simulation in the Phase Space Distributions in Beam Physics blog series
• Learn how to add custom particle-particle interaction forces in the Rotating Galaxy tutorial

References

1. Brodzik, M. J. and K. W. Knowles. 2002. EASE-Grid: A Versatile Set of Equal-Area Projections and Grids in M. Goodchild (Ed.) Discrete Global Grids. Santa Barbara, California USA: National Center for Geographic Information & Analysis.

The greenhouse effect has made it necessary for scientists to develop combustion processes that minimize the accumulation of carbon dioxide in the atmosphere. Possible fuels in these processes include biomass and other biofuels, which recycle carbon within a short timescale. But there’s a downside: Combustion of these materials produces carbon and ash particles that must be removed from the exhaust. To improve particle filtration, researchers studied electrostatic filter designs using models that were validated by comparing them to experimental data.

How Does an Electrostatic Precipitator Work?

It’s all in the name: An electrostatic precipitator filters carbon particles out of exhaust via static electricity. The electric filter component charges and accelerates particles, which accumulate and collect on plates in the precipitator that can then be removed. This device can be attached to chimneys and clean flue exhaust, for instance.

An electrostatic precipitator at an incineration plant in Poland. Image by LukaszKatlewa — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.

In a typical electrostatic precipitator, exhaust released through a flue goes through two electrodes, usually in the form of metal bars, plates, or wires inside a pipe or smokestack. One of the electrodes is charged with a high negative voltage, which is passed to the smoke particulates so that they gain a negative charge in turn. The second electrode, located further down the pipe, has a different voltage — usually grounded — which creates a strong electrical field between the two, thus leading to an acceleration of the negatively charged particles. Consequently, the particles are attracted by the grounded electrode and collected there until removal and disposal.

Seeking a way to improve and further develop the charging mechanism of an electrostatic precipitator, specifically the air ionization process in the electric filter, researchers Donato Rubinetti and Josef Wüest from the Institute of Biomass and Resource Efficiency at the University of Applied Sciences and Arts Northwestern Switzerland in Windisch, Switzerland, collaborated with emission-reduction company OekoSolve. They presented their findings at the COMSOL Conference 2017 Rotterdam.

Numerical Modeling of an Electric Filter in COMSOL Multiphysics®

The research team built an electrostatic precipitator model with the COMSOL Multiphysics® software and the add-on CFD Module. The model involves the interdependent phenomena of fluid mechanics, particle dynamics, and electrostatics. The model uses the Electrostatics interface, a fluid flow interface, PDE interfaces, and the Particle Tracing for Fluid Flow interface.

In a previous paper from the COMSOL Conference 2015, the group explained how they developed the electrostatics model. They initially determined the forces on the particles (Coulomb force vs. drag force) as well as where to place the electrode to optimize the capture and charging of particles.

In the paper from the COMSOL Conference 2017, the researchers discussed their test case based on that model. The physical and numerical modeling practices are identical to the previous paper, except for the geometry. To obtain the acceleration that triggers the removal of particles, they were able to achieve a broad range of space charge density across the geometry of the filter. This distribution was modeled using Poisson’s equation and the transport equation for electric charge.

Modeling an Electrostatic Filter in 2D

As represented below in both 3D and 2D, point number 5 is the electrode where the particles are negatively charged and numbers 1 through 4 are the positively charged rings that the particles are drawn to as the exhaust travels through the cylinder.

3D (left) and 2D (right) representations of the electric filter model. Images courtesy Rubinetti, Wüest, and OekoSolve AG and taken from their COMSOL Conference 2017 Rotterdam paper.

By setting up a numerical model, the researchers were able to analyze the electrical field strength for each axis. They performed 2D and 3D simulations, comparing the distributions of the electric field.

Because the 3D simulation amounted to a total of 1.6 million mesh elements and the 2D simulation approximately 300,000 mesh elements, the team wanted to find out if the 2D model would be sufficient in further studies for quicker computations. They confirmed that the 2D model is suitable for a qualitative understanding of the process and realistic enough to produce accurate results.

Setting Up the Experimental Test Rig Validation Model

After confirming the suitability of the 2D model for their experiments, the research team set up an experimental validation model. They verified this model using experimental data obtained from a test rig.

In the schematic below, the experimental setup for the test rig is shown, with electrode P3 and measurement ring units M1 through M4, with C as the current starting from the top. The emitting electrode was given a variable electric potential from -2 kV to -30 kV. Due to the ionization and acceleration of the plasma layer, the electrical current can be observed and measured on the ground rings.

Laboratory test rig schematic. Image courtesy Rubinetti, Wüest, and OekoSolve AG and taken from their COMSOL Conference 2017 Rotterdam paper.

Comparing the Results

Let’s take a look at the 2D electrical field strength results in the numerical model. In the image below on the left, the electric field lines indicate that there is a decrease in intensity from ring 1 to ring 4. The electrical field strength is depicted as blue in color because the current is so strong near the electrode before being pulled through the rings.

In the image on the right, we can see a close-up view of the electrode. Even in the close-up, the electrical field strength dissipates quickly, as the strongest area (red) is a thin line. These results demonstrate the feasibility of running 2D simulations, because they accurately show the electrical field strength at the most important part of the process close to the electrode.

Electrical field strength and field lines (left) and a close-up view near the electrode (right). Images courtesy Rubinetti, Wüest, and OekoSolve AG and taken from their COMSOL Conference 2017 Rotterdam paper.

While 2D simulation is effective for observing field strength, the researchers could only compare the measured electrical current to a certain extent because of a dimensional mismatch. The research team dedimensionalized the results to get a closer look.

As shown below, in the overview of all measurements (M1–M4) and simulations (S1–S4), the discrepancy between the results can be explained by the dimensional mismatch, as well as some other factors from the test rig setup. However, we can see in the simulation and measurements that the first ring differs significantly from the fellow rings that are closer to each other — meaning it’s still possible to predict the behavior of the physics when reduced to a 2D arrangement.

Comparison of the simulation and measurements. Image courtesy Rubinetti, Wüest, and OekoSolve AG and taken from their COMSOL Conference 2017 Rotterdam paper.

Since the COMSOL Conference 2017 Rotterdam, the researchers have resolved the mismatch and added the dimensions back into the model. They’ve also focused more intensely on the electric aspect of the simulation, especially on how the ionization processes change with temperatures up to 1000 K. Having built an axisymmetric test rig in order to get a quantitative agreement between the experiments and the model, they also built a 2D axisymmetric model.

The new test rig helped them test the behavior of the electrode by:

1. Placing the electrode in line with the cylinder axis
2. Changing the voltage of the electrode up to 30 kV
3. Measuring the current on the rings

As we can see below, the experiments now match the simulation results much more closely:

Comparison of the new validation model and measurements. Image courtesy Donato Rubinetti.

The team was able to validate the model results and modeling approach, since, as shown in the animation below, the current only started to flow above approximately 12 kV.

An animation of the updated validation model results. Animation courtesy Donato Rubinetti.

What’s Next?

From the test case to the industry-relevant model, the research team proved that multiphysics modeling can help accelerate further research and development of particle control technologies. As demonstrated, 2D cases are sufficiently accurate for insights into space change distribution within a domain.

Could further simulations provide more groundbreaking insights? In their paper, the team focused on the type of electric field that is given by the potential difference between electrodes. While the current model does not account for the influence of particles on the electric field, a second approach may account for this impact. Established by a “cloud” of charged particles, the new approach could account for a more dynamic field, moving through all parts of the pellet burner until the particles are deposited on the collector.

Usually, the experiments only allow researchers to see the particles that are already deposited. Given more time and computational resources, Rubinetti says that a 3D simulation for the complete pellet burner system could help visualize the particle cloud clearer than ever before: They’d be able to see the actual path and interactions of the charged particles.

Long-term, Rubinetti says he wants to “further develop the electrostatic precipitator modeling approach to understand the influence of external convection on ionization processes, as well as include temperature dependencies for the particle-charging processes and fluid properties such as density and viscosity.” Based on the researchers’ promising results, they continue to see opportunities for improvement, setting goals to optimize particle control technology designs.

Further Reading

• Check out the paper from the COMSOL Conference 2017 Rotterdam: “Innovation of Combustion Particle Control Technologies Assisted by Numerical Modeling“
• Read more about electrostatics modeling: How to Create Electrostatics Models with Wires, Surfaces, and Solids

Heart failure is a global health concern, affecting millions of people and keeping them from their everyday lives. But what if there was a device that could keep patients’ hearts pumping and even improve their quality of life? In his keynote presentation at the COMSOL Conference 2018 Boston, Freddy Hansen from Abbott Laboratories discussed a heart pump that could do just that. If you missed his talk, you can watch a recording of the video and read a summary below.

Freddy Hansen Discusses Improving a Heart Pump Design for Heart Failure Patients

Designing Better Left Ventricular Assist Devices

“The most complex machine ever implanted into a human being” — that’s how Freddy Hansen described the HeartMate 3 left ventricular assist device (LVAD), a pump developed by Abbott Laboratories that has a magnetically levitated rotor for patients with heart failure. At Abbott, Hansen’s team works to improve LVAD designs to compensate for the weakened muscle in the left side of the heart and thereby avoid the pooling of blood in the heart, which can be fatal. While other electronic implants (such as pacemakers) use only microwatts of power, LVADs must pump the entire bloodstream and therefore use a significant amount of power, around the 10-watt range. Other challenges for designing these devices are that they must be small enough to fit inside a patient’s chest and be bio- and hemocompatible. To address these issues, Hansen and his group use simulation.

Optimizing an LVAD System with COMSOL Multiphysics®

Hansen said that he uses the COMSOL Multiphysics® software to simulate all components of the LVAD system design. These components include an electric motor with magnetic bearings that levitate a rotor inside the device, as well as sensors and other electronics, such as electric motor control. Because each part of this system is life sustaining, Abbott engineers need to ensure each component functions as designed. The HeartMate 3 LVAD is the only part of the system that is inside the human body, and the power for the device comes from batteries and moves through a cable that goes through the patient’s skin. The controller is a patient-interface device that has buttons to help monitor the LVAD’s performance (it also includes emergency batteries).

From the video: An LVAD system. The LVAD is implanted within the chest, while the rest of the system components are outside of the body.

Hansen gave several examples of how he and his group use various modeling techniques to accurately analyze all of the components and multiphysics interactions of the LVAD system design. For instance, when designing the HeartMate 3 LVAD, they created a motor model that contains a magnetic field in the rotating part, with several coils around the core to spin and levitate the rotor. Let’s take a look at some more model examples below.

Modeling Heat Transfer from the Controller and Implant

While working on the LVAD design, Hansen posed an important question: How hot does the skin get close to the controller? As he pointed out, it’s important to determine the controller device’s temperature when against the skin, because the patient can’t just remove the device if it gets too hot — it’s what’s keeping him or her alive! To find answers, Abbott researchers designed a bioheat model with layers of skin, fat, muscle, and viscera on one side of the controller and a clothing layer on the other side to simulate the thermal effects. The model had long-distance blood cooling as well as custom convective cooling on the clothing layer with a heat transfer coefficient.

A second thermal model example began with the question: How hot does the tissue get next to the implant? To answer this, the research team analyzed published data by reverse engineering an experiment that evaluated the thermal connectivity of live tissue. Next, they simulated putting the LVAD in a water bath and compared the results with a benchmark test. (You can watch the video for more details about how he tracked the temperature of batteries and other thermal components.)

Simulating Wireless Power Transfer

The Abbott team also looked at ways that they could replace the percutaneous cable with wireless power transfer — an objective that Hansen called the “holy grail” in the industry. If accomplished, wireless power transfer would decrease the infection risk at the entry site. Not only would the improved device save patient lives but also greatly enhance their quality of life by allowing patients to shower and swim. To explore this possibility, the team developed a model of a 3D magnetic field and 0D electrical circuit in order to evaluate the efficiency and power loss. Adding the electrical circuit capability was important in this case, since the electrical circuit and magnetic coils are strongly interactive.

From the video: The wireless power transfer model design.

Evaluating the Flow of Blood Cells with Simulation

Hansen continued his keynote presentation by discussing how CFD modeling helped the team track the blood flow path through the device. The flow field through the LVAD was solved as a true multiphysics problem in terms of its interactive and cyclical nature. When studying the blood cell concentration, shear stress needs to be accounted for, because blood cells move away from higher shear stress locations. This movement affects the viscosity of the blood, which affects the flow field and, in turn, the shear stress.

When performing a qualitative evaluation of flow for their LVAD design, the researchers made sure that there was no blood flowing from the rotor back into the inlet, since this results in a lower efficiency and a higher risk of blood damage. Hansen also talked about using particle tracing to compute “washing”, which is how long a blood cell stays in the heart pump. They found that they could sort the particles by escape time as well as calculate advanced blood damage using simulation.

Hansen concluded by saying that he and the Abbott team used more than 300 models for this design as well as dozens of products and prototypes. Simulation has made it possible for them to create an optimized LVAD system and develop the HeartMate 3 LVAD for practical use.

So far, the HeartMate LVADs have saved more than 35,000 lives of patients with heart failure. Some patients go on to run marathons and participate in many other sports and activities, but what’s truly impressive about these systems? They keep hearts beating and patients alive and well.

Want to learn more about Abbott’s heart pump simulations and LVAD system design? Watch the video at the top of this post.

HeartMate 3 is a trademark of the Abbott group of companies.

Charge exchange cells can alter the charge of an ion beam, making them useful in fusion reactors, particle accelerators, and semiconductor fabrication equipment. Improving the design of these devices, though, can be time consuming, as many factors (like the input beam’s energy, cell geometry, and neutral number density) affect their performance and must be tested. While these analyses are typically done by simulation experts, they can make the design of such devices accessible to others by building a simulation application…

Changing the Charge of Ion Beams with Charge Exchange Cells

Charge exchange cells can convert a positively or negatively charged ion beam to a neutral beam. How a charge exchange cell works is that the ion beam enters a vacuum chamber where it passes through a rarefied gas. As the ions interact with the gas, a certain fraction undergo charge exchange with almost no loss in the beam’s energy or direction. The neutral beam continues on its original path, while the remaining ions (the ones that didn’t undergo charge exchange) are deflected away via electrically charged plates.

Simulation showing how the charged plates in a charge exchange cell allow ions that have undergone charge exchange reactions to continue forward but deflect the ones that have not.

Due to their ability to exchange charges between ions, these cells are used to produce neutral beams in devices like accelerators (e.g., synchrotrons), which is helpful for medical research. In addition, they can be used for ion implantation processes, including surface finishing (e.g., artificial joints); steel toughening (e.g., drill bits); and semiconductor fabrication (e.g., metal-oxide-semiconductor field-effect transistors or MOSFETs). A neutral beam is desirable in these applications, as it doesn’t accumulate significant amounts of charge on a target surface.

In order to achieve a high efficiency, it’s important to optimize a variety of aspects, such as the:

• Type of gas used (argon, xenon, etc.)
• Number density of the gas
• Shape and size of the chamber
• Magnitude of the bias voltage of the charged plates

That’s where simulation comes in, since it enables engineers to optimize these parameters without the expense of prototyping. Typically, testing different designs falls to a simulation expert like you, but this reduces the amount of time you have to work on other innovative projects. Plus, all of this testing can create a bottleneck in the overall development process, as there’s only a select number of people who can run the analyses.

A more efficient option is to create a simulation application. The application can include all of the physics of a model while displaying an easy-to-use interface with only the features you decide to showcase. By deploying such an application, you can enable team members who aren’t simulation experts to analyze and optimize charge exchange cell designs; for example, computing their efficiency, the path of the resulting beam, and more. Here, we take a look at an example that was created using the COMSOL Multiphysics® software and add-on Molecular Flow Module and Particle Tracing Module.

Note: While this blog post doesn’t go into the details of the underlying model, you can find this information in the Neutralization of a Proton Beam Through a Charge Exchange Cell tutorial.

Streamlining the Design of Charge Exchange Cells with Applications

The Charge Exchange Cell Simulator simulates a charge exchange cell converting a high-energy positive beam into a neutral beam. To make it simple for users to test different designs in various scenarios, the demo application includes three tabs with parameters for key parts of the cell:

1. Vacuum Parameters
   • Vacuum chamber’s dimensions
   • The flow rate into the gas cell that, in this case, contains argon
   • Pump speed
2. Beam Parameters
   • The input beam’s Twiss parameters and emittance
   • Total number of ions
   • Most probable kinetic energy of the ions
3. Deflection Parameters
   • Dimensions of the charged plates and the potential difference between them

With a variety of parameters available, remembering if a parameter has been changed or not can be difficult. For this reason, helpful status cards are included to let the user know about the status of the geometry and the solution. When one of the geometric parameters is changed, a message will appear, prompting the user to click the Update/Show Geometry button, making sure the user is looking at the correct geometry. Additionally, when any parameter is changed, a separate message will indicate that the input data has changed since the previous solution was computed, and the user will not be able to view the solution or create a report. These measures help to ensure users have matching inputs and associated solutions.

The application includes many other options for controlling the simulation. For instance, by clicking the Advanced Settings button, users can increase the number of time steps in order to improve the accuracy of the results, which is particularly helpful when there are high gas pressures and frequent collisions. They can also specify the amount of neutralized particles and argon ions that the simulation can include. Plus, similar to the Graphics window, a warning message will automatically appear if there aren’t enough particles for the simulation.

The Charge Exchange Cell Simulator also gives users a choice in how they want to format a generated report of the simulation results; i.e., in HTML or Microsoft® Word. There’s also a button that makes it simple to create a report, which details the model setup, values of the input parameters, and simulation results.

Results of the Demo Application

After adjusting the application settings as desired, users can click the Compute button to visualize various aspects of the charge exchange cell, including the:

• Gas pressure on the walls of the vacuum chamber and the inner cell that contains the argon gas
• Number density of the gas along the beam path
• Electric potential distribution around the plates
• Paths of the deflected ions and neutral beam

The application also calculates the efficiency of the design. How well it performs is determined by the percentage of ions that are neutralized, which can be seen in the Numerical Results section of the application as well as in the generated report. What’s more, the report will detail all of the reaction types in the simulation as well as the number of each kind, which provides further insight into the products of the exchange reactions.

With an application like this one, anyone can easily test and then optimize the performance of a charge exchange cell, enhancing the overall design process.

Try It Yourself

To get the Charge Exchange Cell Simulator, click the button below to go to the Application Gallery. With a valid software license, you can then download the demo application as well as see its accompanying documentation.

Microsoft is a registered trademark of Microsoft Corporation in the United States and/or other countries.

When investigating a crime, forensic scientists sometimes use DNA evidence to identify suspects. DNA contains more than identifying information, though, like clues to our genetic makeup. DNA separation is used to take a closer look at DNA strands, but traditional methods are time consuming. To speed up the DNA separation process, researchers from the Missouri University of Science and Technology turned to the COMSOL Multiphysics® software.

A Closer Look at Our Genetic Makeup

The molecular structure of DNA is complex: It’s a double-helix polymer made up of a long chain of nucleotides. Studying DNA is much easier by breaking up the sample into fragments of varying sizes.

The nucleotide, or base, pairs of DNA are guanine (G), adenine (A), thymine (T), and cytosine (C). Researchers try to make sense of the sequences of these genetic “letters” in areas like genome sequencing and medical diagnosis, for example, to locate genes and see how they work together within an organism. This work would not be easy to do without DNA separation — after all, the human genome has over 3 billion base pairs of DNA!

An illustration of DNA strands containing the base pair letters G, A, T, and C. Image in the public domain in the United States, via Wikimedia Commons.

Other examples of DNA analysis have entered the spotlight more recently. You may be familiar with mail-in DNA test kits that help you learn more about your ancestry. When a genetic testing company digitizes your DNA sample, it looks like a long strand of the nucleotide G, A, T, and C letters. These companies use algorithms to compare your piece of DNA from the genome against sets of reference data. Then, the algorithm determines how closely your DNA sample matches each reference set to see which ancestry groups you most likely belong to. The algorithm is only as good as its reference sets, so some ancestral groups may be underrepresented compared to others in the database.

In fields such as forensics, DNA profiling helps scientists compare samples of genetic material. Since it’s rare that two people have the same DNA pattern, forensic scientists can compare the patterns in slices of DNA molecules to reference databases, such as the Combined DNA Index System (CODIS) managed by the U.S. FBI. However, systems like CODIS are limited to the DNA profiles they contain. Investigators are starting to use ancestry databases like those mentioned above to expand their searches via a concept called familial DNA. For instance, in 2018, police investigating the case of the Golden State Killer ran the crime scene DNA against a genealogy site database and found a partial match to a distant relative. Ultimately, this helped them narrow down their search and identify the alleged suspect.

Investigating Fragmented Links in a Nucleotide Chain with DNA Separation

One common technique used to separate DNA molecules — primarily in forensics — is gel electrophoresis, which involves the migration of negatively charged nucleic acid molecules by means of a gel. When an electric current is applied, the smaller molecules move through the gel faster than the larger ones, thus the fragments are separated into bands based on size. To visualize this separation, radioactive dye is used.

An example of gel electrophoresis results. Image by Mnolf — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.

There is another method that can separate long strands of DNA much more efficiently without the use of a gel or electric fields: entropic trapping. In this microchip-based system, entropic trap arrays (structured microchannels) of different heights are set up so that the narrow channel gap is much smaller than the gyration diameter of a DNA molecule. The molecules can be separated depending on the chain length. When negatively charged DNA molecules are driven through these channels by electrophoretic forces, the elution times are dependent on length. The longer the DNA molecule, the more likely it is to be drawn into the small channels, because the longer molecules occupy more surface area.

Schematic of an array of entropic traps, with a DNA molecule in a wide channel flowing into a narrow channel. Image courtesy Missouri University of Science and Technology.

Although entropic trapping is faster and more efficient than other separation methods, the design and fabrication of the devices needed takes a lot of time and can be costly, since it relies on trial-and-error. Since discovering the entropic trapping method, researchers have run computational studies to optimize designs and look into the separation mechanisms within these devices, but commercial software had yet to be used to simulate these entropic trap systems…until now.

Simulating Polymer Dynamics in an Entropic Trap System with COMSOL Multiphysics®

To find out whether they could save time with a commercially available simulation software, researchers from the Missouri University of Science and Technology set up their entropic trap system and polymer dynamics simulation using COMSOL Multiphysics® and compared their results to experimental data.

The research team, comprised of Joontaek Park, James Jones, Meyyamai Palaniappan, Saman Monjezi, and Behrouz Behdani, says that “DNA dynamics in microchannel simulation is challenging because two different simulations — field calculation in a complex microfluidics geometry and polymer molecule dynamics — must be combined.” Fortunately, they add, “COMSOL® can relatively easily handle the former simulation,” and that “COMSOL® can open a new page in the DNA or single-polymer molecule simulation area.”

Using the add-on Particle Tracing Module, the team performed Brownian dynamics simulations of the DNA chain. The chain was set up as a single-polymer, bead-chain model within a Newtonian fluid with the help of the CFD Module. As for the beads themselves, they were treated as Brownian particles to account for the random movements of the chain as it moves through the surrounding solvent.

To describe the spring force between each bead, they used another well-known model, the worm-like chain (WLC), which describes the behavior of semiflexible polymers. Alongside the WLC, the research team used the Lennard–Jones potential to keep the beads from penetrating each other. After setting up the entropic array geometry (shown below) so that H_s is much smaller than the gyration diameter of a typical DNA molecule, the researchers used the AC/DC Module to create the electric field of potential across the channel.

Schematic of the channel structures used in the simulation. Image courtesy Missouri University of Science and Technology.

Evaluating the Simulation Results

The research team calculated the nonuniform electric field using the finite element method. The electrical field direction can be seen here, with the arrows also indicating the direction of movement of the DNA molecules.

The electrical field flux vectors in the right corner (a) and the left corner (b) of a wide channel. Image courtesy Missouri University of Science and Technology.

Next, the researchers simulated the center-of-mass trajectory for the DNA molecules according to length, at N_b = 2, 4, and 16 bead lengths, as they periodically flowed into the constricted channel. The trajectories for each of the molecules traveling at the same distance are shown below. As expected from the electric field vector above, the molecules move faster in the narrow channels, and the longer the molecule (the more beads it has), the faster it moves. The shorter molecules, meanwhile, have a reduced velocity along their trajectory, and the distribution of DNA molecules indicates a more diffusive pattern, which reduces the overall velocity through the channels by moving them away from regions where the electric field is strongest.

Center-of-mass trajectories of DNA molecules with N_b = 2, 4, and 16. Image courtesy Missouri University of Science and Technology.

This comparison can be seen in the animations below for the short bead length, N_b = 2, middle bead length N_b = 4, and the long bead length, N_b =16. As indicated by the units on the legend, the color in the animations shows the speed of the particles at any point in time. As expected, the larger the surface of a DNA molecule, the more likely it will be dragged into the smaller channel. (Note that the animations are ~10x slower than real time. If you prefer to slow the speed further, you can hover over the animation, then click the gear icon.)

N_b = 2 shorter DNA molecule flowing into and out of a wide channel in an entropic trap channel. Animation courtesy Missouri University of Science and Technology.

N_b = 4 middle-length DNA molecule flowing into and out of a wide channel in an entropic trap channel. Animation courtesy Missouri University of Science and Technology.

N_b = 16 longer molecule flowing into and out of a wide channel in an entropic trap channel. Animation courtesy Missouri University of Science and Technology.

The researchers were able to confirm that their simulation results were in good agreement with experimental data for the trajectory of DNA chains in an entropic trap, and these results show that the longer DNA chain, indeed, elutes faster than the shorter chain.

The use of COMSOL Multiphysics for polymer dynamics simulation has opened up possibilities for further research, as this was the first trial for this type of simulation using commercially available software. The team says that “COMSOL Multiphysics is a very popular and user-friendly simulation tool,” and further, that the extension of using the software for polymer dynamics will “enhance the related application and simulation studies.”

As for their own future research? The team adds that they could see pursuing the investigations on the inertia effect, the polymer configuration (branched polymer) effect, and DNA-carbon nanotube interaction.

Next Step

For more details about researchers’ work from the Missouri University of Science and Technology, click the button below:

Reference

• S. Monjezi, B. Behdani, M.B. Palaniappan, J.D. Jones, and J. Park, “Computational studies of DNA separations in micro-fabricated devices: Review of general approaches and recent applications”, Adv. in Chem. Eng. Sci., 7 (4), pp. 362–393, 2017.

In outer space and other harsh radiation environments, high-energy ions and protons pierce materials and affect nearby electronic systems. Known as a single-event effect (SEE), the particle radiation can lead to soft or hard errors in devices. Since just one hard error puts a space mission at risk, aerospace engineers must make sure that all critical electronic devices can withstand an SEE. To gain a better understanding of this phenomenon, they can accurately analyze the ion-material interaction using simulation.

What Is a Single-Event Effect?

An SEE is an electrical disturbance in a circuit caused by charged particles, such as high-energy protons, hitting a solid material. The impact of these particles — which come from the Sun, radiation belts, and galactic cosmic rays — creates holes, enabling electrons to travel through the material. While moving, these free-charge carriers eventually recombine before stopping on a node. There, the extra charge causes a change in voltage and, in turn, a soft or hard error. These errors are particularly problematic for aerospace applications, although they can also occur on Earth in areas with high levels of radiation, such as near nuclear testing sites.

The charged particles in the Van Allen belts, which surround Earth for thousands of miles, could cause an SEE in a spaceship. The projection of Earth’s landmass is based on images by M.J. Brodzik and K.W. Knowles (Ref. 1).

Soft vs. Hard Errors

If just one hard error happens in a critical system in space, it could spell the end of the mission. For instance, the highly energetic particles produced by the 2003 Halloween solar storm affected numerous devices aboard spacecraft, including the Martian Radiation Environment Experiment (MARIE), which malfunctioned and never recovered. The reason is that hard errors can be destructive, sometimes to the point where the entire device or system needs to be replaced. Thankfully, though, not all errors caused by SEEs are so damaging. Soft errors typically aren’t destructive and can be fixed with a power reset.

An example of a soft error is a single-event upset, which occurs in two key components of digital electronic devices: memory and logic systems. These systems are integral to microprocessors, such as those in motherboards and scientific instruments on spacecraft. An error in the microprocessor could lead to the entire device behaving incorrectly simply by causing a bit to flip from a 0 to a 1 or vice versa. With this type of soft error, the system is usually able to keep functioning, and the issue can be fixed by reversing the change.

Hard errors, which include single-event latchups, burnouts, and gate ruptures, lead to more permanent effects. For instance, a single-event latchup can result in an operating current that is too high, causing the device to stop working properly; data to be lost; and ultimately, the device’s destruction. These latchups often take place in integrated circuits built with complementary metal-oxide-semiconductor (CMOS) technology, which is commonly used for microcontrollers and microprocessors. Burnouts and gate ruptures take place in, for example, power MOSFETs, such as those for weather satellites and GPS. These errors lead to voltages exceeding limitations, making the device fail.

Hard errors in satellites can cause data to be lost and eventually lead to systems being destroyed.

Aerospace engineers must make sure that all important electronic devices avoid both soft and hard errors. This mission is becoming more of a challenge, though, due to the ever-increasing demand for technology with more functionality, smaller sizes, faster speeds, and lower voltages. While advantageous in terms of cost and performance, these factors mean that the critical charge (the minimum charge needed to upset a node) is getting smaller. As a result, lower-energy particles have a higher chance of causing SEEs, making devices more vulnerable to errors — even here on Earth.

By understanding how charged particles affect a material, aerospace engineers can design electronic equipment that can withstand, or even is invulnerable to, SEEs. To examine this interaction, they can use the COMSOL Multiphysics® software and add-on Particle Tracing Module, as demonstrated by a benchmark example.

Modeling Particle-Material Interaction with the COMSOL® Software

In this example, high-energy protons move toward a block of solid silicon for initial energy values ranging from 1 keV to 100 MeV. Once they hit the material, the protons undergo ionization losses, which slow the particles down, and nuclear stopping, which deflects them in random directions.

To easily capture the proton behavior, you can take advantage of the Charged Particle Tracing interface. Using the Particle-Matter Interaction node, you can account for both the energy loss as well as how the protons scatter. In addition, you can describe the effect of the proton on the material using one of the subnodes. For instance, the Ionization Loss subnode treats the interaction as a continuous force moving in the opposite direction of the particle’s motion, while the Nuclear Stopping subnode treats it as a discrete force slowing down the particle and deflecting it in a random direction.

Next, it’s important to determine the penetration depth of the particles (i.e., the ion range), as this influences whether or not they will ionize nearby electronic devices and thus cause an SEE. To find this depth, there are two approaches you can use:

1. Using an auxiliary dependent variable to mimic the continuous slowing down approximation (CSDA), which assumes that the protons will slow down at a steady rate
2. Computing the projected range by projecting the proton’s velocity onto its original direction of motion

Next up, you can see how these approaches compare to results from published literature.

Comparing the Simulation and Experimental Results

The protons tend to move in random directions at lower energies, as they’re more affected by nuclear stopping. Since this effect causes their energy to change discontinuously, you can see that the CSDA range doesn’t quite align with the experimental results at the lower end of the spectrum. However, as expected, the projected range closely matches the experiment.

For higher energies, ionization losses are more in control of the protons’ trajectories, making their movement more linear. Thus, as the energy increases, so too does the agreement between the CSDA and the experiment. Again, the projected range corresponds well with the experiment.

Left: The particle trajectories for initial energies ranging from 1 keV to 100 MeV. Right: Comparison of the CSDA (black asterisk) and projected range (black circle) with results from published literature (red asterisk).

The good agreement between the approaches in the model and experiment demonstrates that the COMSOL® software provides engineers with the tools needed to accurately examine ion-material interactions. They can then use this knowledge to confidently design electronic systems that resist SEEs.

Next Step

Want to try the benchmark example yourself? Clicking the button below takes you to the Application Gallery, where you can download the documentation for the Ion Range Benchmark model and, if you have a COMSOL Access account and valid software license, the MPH-files.

Reference

1. Brodzik, M. J. and K. W. Knowles. 2002. EASE-Grid: A Versatile Set of Equal-Area Projections and Grids in M. Goodchild (Ed.) Discrete Global Grids. Santa Barbara, California USA: National Center for Geographic Information & Analysis.

A question we get asked all of the time is: “Which of the COMSOL products should be used for modeling a particular electromagnetic device or application?” In addition to the capabilities of the core package of the COMSOL Multiphysics® software, there are currently six modules within the “Electromagnetic Modules” branch of our product tree, and another six modules spread throughout the remaining product structure that address various forms of Maxwell’s equations coupled to other physics. Let’s take a look through these and see what they offer. 

Note: This blog post was originally published on September 10, 2013. It has since been updated with additional information and examples.

Computational Electromagnetics: Maxwell’s Equations

Maxwell’s equations relate the electric charge density, ; electric field, ; electric displacement field, ; and current, ; as well as the magnetic field intensity, , and the magnetic flux density, :

To solve these equations, we need a set of boundary conditions, as well as material constitutive relations that relate the  to the  field, the  to the  field, and the  to the  field. Under varying assumptions, these equations are solved, and coupled to the other physics, in the different modules within the COMSOL product suite. 

Note: Most of the equations presented here are shown in an abbreviated form to convey the key concepts. To see the full form of all governing equations, and to see all of the various constitutive relationships available, please consult the Product Documentation.

Let’s start off with a few concepts…

Steady State, Time, or Frequency Domain?

When solving Maxwell’s equations, we try to make as many assumptions as are reasonable and correct, with the purpose of easing our computational burden. Although Maxwell’s equations can be solved for any arbitrary time-varying inputs, we can often reasonably assume that the inputs, and computed solutions, are either steady-state or sinusoidally time-varying. The former is also often referred to as the DC (direct current) case, and the latter is often to as the AC (alternating current), or frequency domain, case. 

The steady-state (DC) assumption holds if the fields do not vary at all in time, or vary so negligibly as to be unimportant. That is, we would say that the time-derivative terms in Maxwell’s equations are zero. For example, if your device is connected to a battery (which might take hours or longer to drain appreciably), this would be a very reasonable assumption to make. More formally, we would say that: , which immediately omits two terms from Maxwell’s equations. 

The frequency-domain assumption holds if the excitations on the system vary sinusoidally and if the response of the system also varies sinusoidally at the same frequency. Another way of saying this is that the response of the system in linear. In such cases, rather than solving the problem in the time domain, we can solve in the frequency domain using the relationship: , where  is the space- and time-varying field;  is a space-varying, complex-valued field; and  is the angular frequency. Solving Maxwell’s equations at a set of discrete frequencies is very computationally efficient as compared to the time domain, although the computational requirements grow in proportion to the number of different frequencies being solved for (with some caveats that we will discuss later).

Solving in the time domain is needed when the solution varies arbitrarily in time or when the response of the system is nonlinear (although even to this, there are exceptions, which we will discuss). Time-domain simulations are more computationally challenging than steady-state or frequency-domain simulations because their solution times increase in proportion to how long the time span of interest is, and the nonlinearities that are being considered. When solving in the time domain, it is helpful to think in terms of the frequency content of your input signal, especially the highest frequency that is present and significant. 

Electric Fields, Magnetic Fields, or Both?

Although we can solve Maxwell’s equations for both the electric and the magnetic fields, it is often sufficient to neglect one, or the other, especially in the DC case. For example, if the currents are quite small in magnitude, the magnetic fields are going to be small. Even in cases where the currents are high, we might not actually concern ourselves about the resultant magnetic fields. On the other hand, sometimes there is solely a magnetic field, but no electric field, as in the case of a device composed only of magnets and magnetic materials.

In the time- and frequency domains, though, we have to be a bit more careful. The first quantity we will want to check here is the skin depth of the materials in our model. The skin depth of a metallic material is usually approximated as , where  is the permeability and  is the conductivity. If the skin depth is much larger than the characteristic size of the object, then it is reasonable to say that skin depth effects are negligible and one can solve solely for the electric fields. However, if the skin depth is equal to, or smaller than, the size of the object, then inductive effects are important and we need to consider both the electric and magnetic fields. It’s good to do a quick check of the skin depth before starting any modeling. 

As one increases the excitation frequency, it is also important to know the first resonance of the device. At this fundamental resonant frequency, the energy in the electric fields and magnetic fields are exactly in balance, and we would say that we are in the high-frequency regime. Although it is generally difficult to estimate the resonant frequency, a good rule of thumb is to compare the characteristic object size, , to the wavelength, . If the object size approaches a significant fraction of the wavelength, , then we are approaching the high-frequency regime. In this regime, power flows primarily via radiation through dielectric media, rather than via currents within conductive materials. This leads to a slightly different form of the governing equations. Frequencies significantly lower than the first resonance are often termed the low-frequency regime. 

Let’s now look at how these various different assumptions are applied to Maxwell’s equations, and give us different sets of equations to solve, and then see which modules we would need to use for each. 

Steady-State Electric Field Modeling

Under the assumption of steady-state conditions, we can further assume that we are dealing solely with conductive materials, or perfectly insulative materials. In the former case, we can assume that current flows in all domains, and Maxwell’s equations can be rewritten as:

This equation solves for the electric potential field, , which gives us the electric field, , and the current, . This equation can be solved with the core COMSOL Multiphysics package and is solved in the introductory example to the software. The AC/DC Module and the MEMS Module extend the capabilities of the core package, for example, by offering terminal conditions that simplify model setup and boundary conditions for modeling of relatively thin conductive and insulative regions, as well as separate physics interfaces for modeling the current flow solely through geometrically thin, possibly multilayered, structures.

On the other hand, under the assumption that we are interested in the electric fields in perfectly insulating media, with material permittivity, , we can solve the equation:

This computes the electric field strength in the dielectric regions between objects at different electric potentials. This equation can also be solved with the core COMSOL Multiphysics package, and again, the AC/DC and MEMS modules extend the capabilities via, for example, terminal conditions, boundary conditions for modeling thin dielectric regions, and thin gaps in dielectric materials. Furthermore, these two products additionally offer the boundary element formulation, which solves the same governing equation, but has some advantages for models composed of just wires and surfaces, as discussed in this previous blog post.

Time- and Frequency-Domain Electric Field Modeling

As soon as you want to model time-varying electric fields, there will be both conduction and displacement currents, and you will want to use either the AC/DC Module or MEMS Module. The equations here are only slightly different from the first equation, above, and in the time-domain case, are written as:

This transient equation solves for both conduction currents, , and displacement currents, . This is appropriate to use when the source signals are nonharmonic and you wish to monitor system response over time. You can see an example of this in the Transient Modeling of a Capacitor in a Circuit model.

In the frequency domain, we can instead solve the stationary equation:

The displacement currents in this case are . An example of the usage of this equation is the Frequency Domain Modeling of a Capacitor model.

Magnetic Field Modeling with the AC/DC Module

Modeling of magnetic fields, in the steady-state, time-domain, or low-frequency regime, is addressed within the AC/DC Module. 

For models that have no current flowing anywhere, such as models of magnets and magnetic materials, it is possible to simplify Maxwell’s equations and solve for , the magnetic scalar potential:

This equation can be solved using either the finite element method or the boundary element method.

Once there are steady-state currents in the model, we must instead solve for , the magnetic vector potential. 

This magnetic vector potential is used to compute , and the current, , can be either imposed or simultaneously computed via augmenting with the previous equation for the electric scalar potential and current. A typical example of such a case is the magnetic field of a Helmholz coil. 

As we move to the time domain, we solve the following equation:

where .

This equation considers solely the conduction currents and induced currents, but not the displacement currents. This is reasonable if the power transfer is primarily via conduction and not radiation. One strong motivation behind solving this equation is if there are material nonlinearities, such as a B-H nonlinear material, as in this example of an E-core transformer. It should be noted, though, that there are alternative ways of solving B-H nonlinear materials via an effective H-B curve approach. 

As we move into the frequency domain, the governing equation becomes:

Note that this equation considers both conduction currents, , and displacement currents, , and is starting to look quite similar to a wave equation. In fact, this equation can solve up to and around the resonance of a structure under the assumption that there is negligible radiation, as demonstrated in this example: Modeling of a 3D Inductor. 

For a more complete introduction to the usage of the above sets of equations for magnetic field modeling, also see our lecture series on electromagnetic coil modeling. 

It is also possible to mix the magnetic scalar potential and vector potential equations, and this has applications for modeling of motors and generators. 

In addition to the above static, transient, and frequency-domain equations in terms of the magnetic vector potential and scalar potential, there also exists a separate formulation in terms of the magnetic field, which is appropriate for modeling of superconducting materials, as in this example of a superconducting wire. 

Wave Equation Modeling in the Frequency and Time Domains with the RF or Wave Optics Modules

As we get into the high-frequency regime, the electromagnetic fields become wave-like in nature, as in the modeling of antennas, microwave circuits, optical waveguides, microwave heating, and scattering in free space, as well as scattering from an object on a substrate, and we solve a slightly different form of Maxwell’s equations in the frequency domain:

This equation is written in terms of the electric field, , and the magnetic field is computed from: . It can be solved either at a specified set of frequencies, or as an eigenfrequency problem, which directly solves for the resonant frequency of the device. Examples of eigenfrequency analysis include several benchmarks of closed cavities, coils, and Fabry–Perot cavities, and such models compute both resonant frequencies and quality factor. 

When solving for the system response over a range of specified frequencies, one can directly solve at a set of discrete frequencies, in which case the computational cost scales linearly with number of specified frequencies. One can also instead exploit hardware parallelism on both single computers and clusters to parallelize and speed up solutions. There are also Frequency-Domain Modal and Adaptive Frequency Sweep (also called Asymptotic Waveform Evaluation) solvers that accelerate the solutions to some types of problems, as introduced in a general sense in this blog post, and demonstrated in this waveguide iris filter example.

If you’re instead solving in the time domain with the RF or Wave Optics modules, then we solve an equation that looks very similar to the earlier equation from the AC/DC Module:

This equation again solves for the magnetic vector potential, but includes both first and second derivatives in time, thus considering both conduction and displacement currents. It has applicability in modeling of optical nonlinearities, dispersive materials, and signal propagation. Time domain results can also be converted into the frequency domain via a Fast Fourier Transform solver, as demonstrated in this example. 

The computational requirements for these equations, in terms of memory, also are a concern. The device of interest, and the space around it, are discretized via the finite element mesh, and this mesh must be fine enough to resolve the wave. That is, at a minimum, the Nyquist criterion must be fulfilled. In practice, this means that a domain size of about 10 x 10 x 10 wavelengths in size (regardless of the operating frequency) represents about the upper limit of what is addressable on a desktop computer with 64 GB of RAM. As the domain size increases (or the frequency increases), the memory requirements will grow in proportion to the number of cubic wavelengths being solved for. This means that the above equation is well suited for structures that have characteristic size roughly no larger that 10 times the wavelength at the highest operating frequency of interest. There are, however, two ways to get around this limit.

One approach for solving for the wave-like fields around an object much smaller than the wavelength is the Time Explicit formulation. This solves a different form of the time-dependent Maxwell’s equations that can be solved using much less memory. It is primarily meant for linear material modeling, and is attractive in some cases such as for computing wideband scattering off an object in a background field. 

Another alternative exists for certain types of optical waveguiding structures, solved in the frequency domain, where it is known that the electric field varies quite slowly in the direction of propagation. In such cases, the beam envelopes method in the Wave Optics Module becomes quite attractive. This interface solves the equation:

Where the electric field is  and  is the envelope of the electric field.

The additional field, , is a so-called phase function that must be known, at least approximately, and specified as an input. Luckily, for many optical waveguiding problems, this is indeed the case. It is possible to solve for either just one, or two, such beam envelope fields at the same time. The advantage of this approach, when it can be used, is that the memory requirements are far lower than for the full-wave equation presented at the beginning of this section. Other examples of its usage include models of a directional coupler as well as modeling of self-focusing in optical glass. 

Deciding Between the AC/DC Module, RF Module, and Wave Optics Module

The dividing line between the AC/DC Module and the RF Module is a bit of a fuzzy line. It’s helpful to ask yourself a few questions:

1. Are the devices I’m working with radiating significant amounts of energy? Am I interested in computing resonances? If so, the RF Module is more appropriate.
2. Are the devices much smaller than the wavelength at the highest operating wavelength? Am I primarily interested int the magnetic fields? If so, the AC/DC Module is more appropriate.

If you’re right at the line between these, then it can even be reasonable to have both products in your suite of modules. 

Deciding between the RF Module and Wave Optics Module involves asking yourself about your applications. Although there is a lot of overlap in functionality in terms of the full-wave form of Maxwell’s equations in the time and frequency domains, there are some small differences in the boundary conditions. There are so-called Lumped Port and Lumped Element boundary conditions, applicable for microwave device modeling, which are solely part of the RF Module. Also keep in mind that only the Wave Optics Module contains the beam envelopes formulation. 

As far as material properties, the two products come with different libraries of materials: The RF Module offers a suite of common dielectric substrates, while the Wave Optics Module includes refractive indices of over a thousand different materials in the optical and IR band. For more details on this, and the other available libraries of materials, see this blog post. Of course, if you have specific questions about your device modeling needs, contact us. 

A summary of the approximate dividing lines between these modules is given in the figure below. 

Ray Tracing with the Ray Optics Module

If you are modeling devices many thousands of times the wavelength in size, then it is no longer possible to resolve the wavelength via a finite element mesh. In such cases, we also offer a geometrical optics approach in the Ray Optics Module. This approach does not directly solve Maxwell’s equations, but instead traces rays through the modeling space. This approach requires only that reflective surfaces and dielectric domains, but not uniform free space, be meshed. It is applicable for modeling of lenses, telescopes, large laser cavities, as well as structural-thermal-optical performance (STOP) analysis. It can even be combined with the output of a full-wave analysis, as demonstrated in this tutorial model. 

Multiphysics Modeling

In addition to solving Maxwell’s equations on their own, one of the core strengths of COMSOL Multiphysics is solving problems where there are couplings between several physics. One of the most common is the coupling between Maxwell’s equations and temperature, wherein the rise in temperature affects the electrical (as well as the thermal) properties. For an overview of the ways in which these kinds of electrothermal problems can be addressed, please see this blog post. 

It is also common to couple structural deformations to electric and magnetic fields. Sometimes, this just involves deformation, but sometimes, this also involves piezoelectric, piezoresistive, or magnetostrictive material response, or even a stress-optical response. The MEMS Module has a dedicated user interface for electrostatically actuated resonators, wherein an applied electric field biases a device. Structural contact and the flow of current between contacting parts can also be considered in the context of electric currents modeling. 

Beyond just temperature and deformation, though, you can also couple Maxwell’s equations for electric current to chemical processes, as addressed by the Electrochemistry, Batteries & Fuel Cells, Electrodeposition, and Corrosion modules. In the Plasma Module, you can even couple to plasma chemistry, and with the Particle Tracing Module, you can trace charged particles through electric and magnetic fields. Lastly (for now!) our Semiconductor Module solves for charge transport using the drift-diffusion equations. Each of these modules is a topic in and of itself, so we won’t try to address them all right here. 

Of course, if you would like to discuss any of these modules in greater depth, and find out how it is applicable to your device of interest, don’t hesitate to contact us via the button below.

Electron guns are frequently used in cathode ray tubes, electron microscopes, spectrometers, and particle accelerators. The electron gun extracts electrons from a hot cathode or a plasma, then accelerates them to a high kinetic energy. One of the main design challenges in building an electron gun is that the electrons repel each other, so the beam tends to spread out. Here, we discuss modeling one of the earliest electrode configurations that was devised to counteract this electrostatic repulsion: the Pierce electron gun.

Electron Gun Design

A good electron gun design must be able to do the following:

1. Extract a sufficiently high number of electrons; that is, get an adequate beam current
2. Accelerate the electrons to a certain kinetic energy
3. Focus the beam at a particular location

Often, the electrons are rather slow when they are first emitted from the cathode or plasma source, and then they get accelerated by external fields. One very simple means of acceleration is to send the beam through a metal grid held at a fixed electric potential.

Force Between Two Charged Particles

All of the emitted beam electrons exert repulsive forces on each other. Consider two charged particles at positions r₁ and r₂, with charges q₁ and q₂. The force exerted on particle 1 by particle 2 is given by Coulomb’s law,

where ε₀ = 8.854187817 × 10^-12 F/m is a physical constant called the permittivity of vacuum.

If q₁ and q₂ have the same sign — that is, both particles are either positively or negatively charged — then the force on particle 1 points away from particle 2. If one particle is positively charged and the other is negatively charged, then the force on particle 1 will point toward particle 2. Thus, like charges repel, while opposites attract.

The attractive or repulsive force gets stronger as the particles move closer together. If you divide the distance between two electrons in half, the repulsive force they exert on each other increases by a factor of four.

Force Between Many Charged Particles

Most real-world systems include a huge number of electrons, not just two. The total Coulomb force acting on an electron is the sum over the forces exerted by all other particles. For example, the total force acting on the first particle is

where N is the total number of electrons.

Imagine a cylindrical beam containing many electrons. For an electron near the center of the beam, there are an equal number of other electrons on any side, so if we take the vector sum of the forces acting on this electron, the Coulomb forces will mostly cancel out. On the other hand, for an electron near the edge of the beam, the net force will push it even farther away from the center. So, if the beam electrons are initially moving parallel to each other, they will begin to spread out, or diverge, as the beam propagates.

A diverging electron beam. The beam is released at a waist (left) where the electron velocities are parallel. At the right side, the electrons are spreading out in all directions.

To see some examples that demonstrate how an electron beam diverges in free space, see the tutorial models Electron Beam Divergence Due to Self Potential and Relativistic Diverging Electron Beam.

The repulsive force that causes the beam to spread out is strongest where the beam electrons are slowest, because those regions tend to have the highest charge density. Therefore, one of the key technical challenges in designing an electron gun is often keeping the beam focused in the first acceleration gap, immediately after the beam electrons are emitted (Ref. 1).

Finding the Optimal Electrode Shape

Our goal is to design an electron gun geometry so that the shape of the electrodes cancels out the Coulomb repulsion between the beam electrons, so that the beam goes in a straight line without spreading out.

To begin, consider a two-dimensional sheet beam that is uniform in the out-of-plane (z) direction. The beam will propagate in the positive y direction. The beam electrons are first emitted from a cathode (V = 0) located at y = 0 and are attracted toward an anode (V = V_a) located at some height y = d.

Let’s begin with a trivial solution, where the beam is infinitely wide in the x direction. In this case, any beam electron could be considered to be at the center of the beam, and the electric forces to the left and the right will cancel each other out.

Simple sheet beam between two flat electrodes, extending infinitely in the +x and –x directions.

There is a theoretical maximum amount of current that can be extracted from the cathode without causing electrons to be repelled backward. This is called the space charge limit, and a cathode releasing electrons at this current is space charge limited. During space-charge-limited emission between two parallel electrodes, the electric potential in the gap follows the distribution given by Child’s law (Ref. 2),

Now suppose the electrons are only flowing in the region x < 0, and that there are no charges in the region x > 0.

If the electrodes retained their flat shapes, then because of the electrostatic repulsion in the beam, some of the electrons close to the y-axis would spill into the region x > 0.

Thus, divergence or spreading of an electron beam occurs when the beam has a finite size, because electrons near the edge of the beam feel an imbalance in the Coulomb forces from other electrons. In the next section, we introduce an analytic approach to change the shape of the electrodes so that the beam goes directly upward and no electrons spill into the region x > 0.

The Pierce Method of Electron Gun Design

Suppose that the cathode and anode are still flat in the region x < 0, but now they take on different shapes in the region x > 0. The exact functional forms of these electrode shapes are not yet known.

Because there are no charges in the region x > 0, the electric potential must satisfy Laplace’s equation,

Consider the complex number u = y + ix. Pierce’s method (Ref. 1, 3) begins with the observation that any twice-differentiable function of u — let’s call it f(u) — will also satisfy Laplace’s equation. This can be proven by repeated application of the chain rule,

With this in mind, let the electric potential in the region x > 0 be defined as the real part of f,

Then V will also satisfy Laplace’s equation in this charge-free region. To ensure that the electric potential is continuous at x = 0, V must satisfy the potential distribution given by Child’s law (Ref. 2),

The cathode is simply the set of coordinates in the complex plane that satisfy this equation for V = 0. Similarly, the anode is the set of coordinates in the complex plane that satisfy this equation for V = V_a.

At this point, it is convenient to rewrite u in cylindrical polar coordinates,

Note that here, θ = 0 is the positive y direction rather than the x direction. Now, our expression for the electric potential in the region x > 0 is

Because r and d are real, this can be simplified to

and then invoking Euler’s formula yields the final result

The shapes of the cathode and anode are now just the curves that, when inserted into the last expression, give V = 0 and V = V_a, respectively.

The solution for V = 0 is the straight line

or a 67.5° angle from the beam propagation direction.

The solution for V = V_a is the curve

Therefore, the Pierce gun design algorithm predicts that a beam can be kept perfectly straight using a straight cathode at a 67.5° angle from the beam propagation direction, and a curved anode.

Creating the COMSOL Multiphysics® Geometry

The anode curve found in the previous section asymptotically approaches the cathode line but never quite meets it; the perfect Pierce electron gun extends infinitely far in the positive x direction, so we have to arbitrarily cut it off at some point.

In the following plot, the long straight line is the cathode, and the long curved line above it is the anode. These two lines could be extended infinitely far but would never meet. So, for a more practical model, we draw another line that intersects both the cathode and anode, then use the Convert to Solid operation in COMSOL Multiphysics® to form a domain bounded by these curves. The intersecting line segment is drawn perpendicular to the cathode because we expect the electric field to point in this direction.

The solid rectangle on the left side is the beam propagation region, and the line x = 0 is a symmetry axis here. So the complete Pierce gun geometry has the same curved anode shape on either side of the beam.

Pierce electron gun geometry, just before truncating the cathode and anode curves.

Particle Field Interaction Modeling

In this model, we use the dedicated Space Charge Limited Emission multiphysics coupling node to release a space-charge-limited beam of electrons in the positive y direction. Then, to include the effect of mutual electrostatic repulsion between the beam electrons, we use the dedicated Electric Particle Field Interaction multiphysics coupling. This causes the beam electrons to contribute to the space charge density in the domain, which is then included in the Electrostatics interface when solving for the electric potential.

The final algorithm for getting a self-consistent solution for the electric potential and the particle trajectories then looks like this:

1. Trace the particles, without considering the electrostatic repulsion between the beam electrons. From this solution, estimate the space charge density in the beam.
2. Using the estimated space charge density, as well as the boundary conditions at the cathode and anode surfaces, compute the stationary electric potential.
3. Use the electric potential from the previous step to define an Electric Force on the particles. Trace the particles again, and compute the space charge density in the beam.
4. Continue alternating between steps 2 and 3 for a fixed number of iterations or until the solution doesn’t noticeably change between iterations.

Results

A Mirror dataset was used to reflect the electric potential distribution across the y-axis. The following plot shows the particle trajectories, colored according to their speed with green being the fastest. The space charge density in the domain is plotted in grayscale, with darker shades indicating greater charge density. It is clear that the charge density is greatest in a narrow region adjacent to the cathode, and decreases as the particles accelerate.

Electric potential distribution and particle trajectories in the Pierce gun.

The following animation shows how the particles accelerate as they approach the anode. Here, the grayscale background shows some equipotential contours. The particles in the beam area move in straight lines, and the equipotential contours in the beam cross section are horizontal — both good indications that the inclined cathode and curved anode are correctly counterbalancing the electrostatic repulsion in the beam.

Try It Yourself

To download the Pierce Electron Gun model, click the button below.

Further Reading

An additional benchmark model, providing some helpful theoretical details about Child’s law and its derivation, is the Child’s Law Benchmark tutorial.

Child’s law is based on a simplifying approximation that neglects the thermal velocity of the released electrons. In reality, particles at room temperature can fly around at speeds of several hundred meters per second, immediately after leaving the cathode. To learn more about the thermal distribution of emitted electrons and its effect on space charge limited electron emission — sometimes called the Langmuir–Fry model (Refs. 4, 5) — see the example Thermionic Emission in a Planar Diode.

References

1. S. Humphries, Stanley, Charged Particle Beams, Dover, 2013.
2. J.R. Pierce, “Rectilinear electron flow in beams”, Journal of Applied Physics, vol. 11, no. 8 pp. 548–554, 1940.
3. C.D. Child, “Discharge from hot CaO”, Physical Review (Series I), vol. 32, no. 5, pp. 492–511, 1911.
4. T.C. Fry, “The thermionic current between parallel plane electrodes; velocities of emission distributed according to Maxwell’s law”, Physical Review, vol. 17, no. 4, pp. 441–452, 1921.
5. I. Langmuir, “The effect of space charge and initial velocities on the potential distribution and thermionic current between parallel plane electrodes”, Physical Review, vol. 21, no. 4, pp. 419–435, 1923.

When you first try to run a particle tracing simulation with very small particles in a fluid — generally tens of microns in diameter or smaller — you might notice the time-dependent solver taking much shorter time steps than usual. This is often due to the particle equations of motion exhibiting numerical stiffness. In this blog post, we’ll introduce the concept of stiffness as it pertains to particle simulation, then offer some guidelines for choosing the correct equation formulation based on particle size.

Example: Gravitational Settling of a Small Sphere

Let’s consider a small, spherical particle as it falls through a surrounding fluid of uniform velocity u (SI unit: m/s). The particle follows Newton’s second law of motion,

(1)

where

• m_p (SI unit: kg) is the particle mass
• q (SI unit: m) is the particle position vector
• F_t (SI unit: N) is the net force or total force acting on the particle

For a particle sinking in a fluid, the total force is the sum of the gravity force F_g and the drag force F_D,

(2)

where

• ρ_p (SI unit: kg/m³) is the density of the particle
• ρ (SI unit: kg/m³) is the density of the surrounding fluid
• g (SI unit: m/s²) is the acceleration due to gravity (about 9.8 m/s² downward at sea level)
• μ (SI unit: Pa s) is the dynamic viscosity of the surrounding fluid
• d_p (SI unit: m) is the particle diameter
• u (SI unit: m/s) is the velocity of the surrounding fluid
• v (SI unit: m/s) is the velocity of the particle (v ≡ dq/dt)

The term (ρ_p – ρ)/ρ_p in the expression for the gravitational force represents buoyancy. It approaches 1 when the particles are much heavier than the fluid they displace — solid particles in air, for example. It approaches zero when the particle and the surrounding fluid have equal density, in which case the particles are called neutrally buoyant.

The expression for the drag force used here is from the Stokes drag law. It is appropriate to use this drag law when the relative Reynolds number of the particle is very small,

\textrm{Re}_\textrm{r} \equiv \frac{\rho d_\textrm{p}\left|\mathbf{u}-\mathbf{v}\right|}{\mu} \ll 1

This is more likely to be valid for smaller particles.

Assume the particles are not changing size (thus d_p and m_p are constant). The mass of a sphere is

(3)

By combining Eqs. 1–3, we arrive at a simplified expression for the particle’s equation of motion,

(4)

where the constant τ_p has been introduced,

τ_p has units of time and is usually called the Lagrangian time scale or the particle velocity response time, for reasons that will soon be apparent.

As a further simplification, assume the surrounding fluid is quiescent (u = 0) and that the particle is initially at rest (q = 0 and v = 0 at time t = 0). Suppose we align our coordinate system so that the gravity vector points in the –y direction. Then, continuing from Eq. 4, the equation for the y-component of the particle position becomes

(5)

The exact or analytic solution to Eq. 5, given the initial conditions q_y = 0 and v_y = 0, is

where v_t is the terminal velocity,

Conversion to Dimensionless Variables

To get a better understanding of how the particle accelerates during the first few multiples of τ_p, we can replace the time, position, and velocity (t, q_y, v_y) with the corresponding dimensionless quantities (t‘, q_y‘, v_y‘), defined as

Substituting these dimensionless variables back into the analytic solution yields

In the figure below, the dimensionless position and velocity are plotted as functions of the dimensionless time t‘. This plot illustrates that the particle velocity asymptotically approaches the terminal velocity, with most of the acceleration happening during the first few multiples of the Lagrangian time scale τ_p. The particle position appears to change linearly after this initial acceleration period.

Plot of the dimensionless position and velocity of a particle undergoing gravitational settling, starting from rest.

Time Scales for Some Typical Particle Sizes

To get a better idea of the time scales involved in particle acceleration, suppose the particles are silica glass beads with a density of about 2200 kg/m³. The following table gives some values for the Lagrangian time scale in air and in water for different particle sizes.

The diameter-squared dependence of τ_p means that large particles have a much greater velocity response time and a much greater terminal velocity than small particles. This has two major consequences:

1. Large particles fall to the ground much faster than small particles.
2. When large particles are launched into a fluid with some initial velocity, they follow ballistic trajectories, capable of traveling a considerable distance before the drag force slows them down. In contrast, smaller particles will match the fluid velocity much sooner; when they spread out, it is more likely due to turbulent diffusion of the surrounding fluid.

Numerical Particle Tracking Simulation

In the previous section, we were quite lucky that Eq. 4 had an exact analytic solution. It was only possible to obtain an exact solution because of all of the simplifying assumptions involved, most notably that the fluid velocity u was zero everywhere. In most real-world systems, the velocity of the surrounding fluid is not only nonzero but also spatially nonuniform, and then it is very unlikely that an exact solution can be found with pen and paper alone.

For more general problems, we can turn to numerical simulation to get an approximate answer. The main idea is that, given the initial particle position q₀ and velocity v₀ at the initial time t = 0, we can use numerical time stepping algorithms to estimate the solution at a set of discrete time steps t₁, t₂, t₃, etc. A wide variety of different time stepping algorithms have been devised for this purpose, many of which are available in the COMSOL Multiphysics® software.

Solving a set of differential equations numerically introduces some amount of error — the difference between the real-world particle motion and the computed numerical solution. While one usually cannot hope to get a perfect solution from a numerical simulation, a more realistic goal is that the simulated particle motion should become more accurate when the time intervals (t₁, t₂ – t₁, t₃ – t₂, etc.) are reduced in size.

The trade-off is that if the time steps are smaller, you need to take more time steps to reach the same output time. Ultimately, this may lead to a noticeable increase in wall-clock time, which is how long the user must wait for the simulation to complete. Engineers working with numerical simulation must always seek a reasonable balance between solution accuracy and wall-clock time.

The Particle Tracing for Fluid Flow interface, available with the Particle Tracing Module, an add-on to COMSOL Multiphysics®, simulates the motion of individual particles in the surrounding fluid by solving Newton’s second law numerically. On a fundamental level, this interface solves Eq. 1 while allowing you to add a wide variety of different forces to the right-hand side. It also includes a variety of options for setting the initial particle position and velocity, as well as the detection and handling of particle collisions with surfaces in the surrounding geometry.

Dealing with Small Particles and Long Time Scales

In many practical applications, the range of desired solution times for a particle tracing model is much greater than the Lagrangian time scale τ_p. For example, suppose that we want to track the motion of some 20-μm silica glass particles in water over a total simulation time of 1 second. As we saw in the previous table, the Lagrangian response time for such small particles in water is about 5 × 10^-5 seconds, so the total simulation time is about 2000 τ_p. If we wanted to track even smaller particles over a span of minutes or hours, it is easy to envision scenarios where our total simulation time could be millions of times larger than τ_p.

The following screenshot shows a log of the time steps taken by the time-dependent solver while tracking these 20-μm particles. The range of output times in the Step 1: Time Dependent node has been set to range(0,0.1,1), meaning that it will only store output at multiples of 0.1 s. However, this does not preclude the solver from taking smaller time steps if necessary to get an accurate solution. As shown here, the solver begins by taking time steps on the order of 1 ms or smaller, then gradually takes larger steps as the particle approaches its terminal velocity.

In COMSOL Multiphysics, the particle tracing physics interfaces generally use a Strict time stepping algorithm that requires at least some of the steps taken by the solver to coincide with the output times, such as step 24 below. This is not a general requirement of all physics; for some physics interfaces, the output times can be obtained by interpolation between the nearest steps taken by the solver.

Toward the end of the study, the time steps can be quite large because the particle is hardly accelerating at all. Ultimately, the solver takes 24 time steps to reach the first output time at 0.1 s, but only needs 12 more time steps to reach the final time at 1 s.

The equation of motion of a particle undergoing gravitational settling is an example of a stiff ordinary differential equation or stiff ODE. The default time stepping method used in most particle tracing models, called Generalized alpha, is a second-order implicit time stepping scheme that is rather good at handling stiff problems. If additional stability is required, there is a numerical damping term that can be adjusted in the Time-Dependent Solver settings, called the Amplification for high frequency. For this reason, the time steps are allowed to become larger as the particle velocity approaches the terminal velocity. (In contrast, the explicit Runge–Kutta method RK34 takes 7425 steps to solve the same problem!)

However, if particles were entering the simulation domain at several different release times, or if the background fluid velocity was spatially nonuniform (so that the particles could still accelerate later in the study), it might be necessary for the solver to continue taking such small time steps up until the final time. If we attempt to trace very small particles over a long simulation time, eventually these studies will require a substantial amount of wall-clock time to complete as the solver might take hundreds of thousands, or even millions of steps.

A closely related phenomenon that can be confusing to new COMSOL® software users involves releasing particles into the simulation domain using the Inlet boundary condition. Suppose these particles are assigned an initial velocity pointing into the simulation domain. Note from the previous screenshots that the initial time step size (for a total simulation time of 1 second) was 1 millisecond. If the initial time step size is still much greater than τ_p, the drag force might overcompensate, causing the particle velocity to briefly change direction and point back toward the Inlet boundary. If this happens, the particles might incorrectly detect a collision with the Inlet boundary, causing them to get stuck there.

Dealing with Numerical Stiffness in Particle Tracing Models

There are two main ways to deal with numerically stiff models of particle motion in a fluid — models in which the interval between the output times is several orders of magnitude greater than τ_p.

The first is what we call the “brute force” method: Simply tell the solver to take smaller time steps. If you don’t want to produce an overwhelming amount of output, potentially creating massive file sizes, you can instead leave the output times alone but specify a smaller step size or maximum step size in the settings for the time-dependent solver further down in the solver sequence.

The other approach, possible as of COMSOL Multiphysics® version 5.6, is to drop the inertial term from Eq. 4. First, we rewrite Eq. 4 as a pair of coupled first-order equations,

Now, instead of fully resolving the particle motion during the first few multiples of τ_p, we just assume that the drag force is always in dynamic equilibrium with all other applied forces,

(6)

In other words, we just assume the particle instantly reaches its terminal velocity. This is a fair approximation if the time required to approach the terminal velocity is many orders of magnitude smaller than the total simulation time. Eq. 6 can be solved for v,

or, more generally,

(7)

where F_other is the sum of all applied forces other than drag.

Then, all we have to do is integrate this expression for v over time to obtain the particle position q.

To choose which equation system the Particle Tracing for Fluid Flow interface will solve, locate the Particle Release and Propagation section. From the Formulation list, you can choose one of the following options:

• Newtonian: Solves Eq. 1
• Newtonian, first order: Separates Eq. 1 into a pair of coupled first-order equations for q and v, then solves them
• Newtonian, ignore inertial terms (available as of version 5.6): A simplified formulation that defines the velocity using Eq. 7, then solves for q
• Massless: An even more simplified formulation where you specify v directly in order to solve for q

Note that the number of available built-in forces is slightly greater for the Newtonian and Newtonian, first order formulations than for the Newtonian, ignore inertial terms formulation. Forces that explicitly depend on particle velocity or the relative positions of other particles have been excluded.

Available forces with the Newtonian formulation.

Available forces with the Newtonian, ignore inertial terms formulation.

The following Application Library examples use the Newtonian, ignore inertial terms formulation to trace very small particles for a long solution time:

• Particle Trajectories in a Laminar Static Mixer
• Dielectrophoretic Separation of Platelets from Red Blood Cells

The following examples use the Newtonian formulation because the particles are large enough for inertia to have a significant effect on particle motion:

• Particle Tracing in a Micromixer
• Pipe Erosion Due to Contaminant Particles

Concluding Thoughts on Particle Tracing in Fluids

When simulating the motion of small particles in a fluid using the Particle Tracing for Fluid Flow interface, you should usually begin by estimating the Lagrangian time scale τ_p associated with the particles,

and comparing this time scale to the range of solution times you want to model.

If you have a distribution of different particle sizes, make this estimate based on the smallest particle size, since the smallest inertial particles in the model are the ones that determine how numerically stiff the equations of motion are.

If you want to predict the particle motion over a range of times much larger than the velocity response time (let’s just say, by a factor of several thousands or more), then you should consider whether inertia actually plays a significant role in the particle motion. If not, you can select the option Newtonian, ignore inertial terms (available as of version 5.6) from the Formulation list.

If you still want to consider inertia, you could use the Newtonian or Newtonian, first order formulation. However, note that the equation system being solved is numerically stiff, and you may need to manually reduce the size of the time steps taken by the solver to prevent nonphysical oscillations in the particle position and velocity.