# Particle Tracing Module

# Particle Tracing Module

### Analyze the Behavior of Particles with the Particle Tracing Module

#### Extend the Functionality of the COMSOL Environment with Particle Tracing

The Particle Tracing Module extends the functionality of the COMSOL environment for computing the trajectory of particles in a fluid or electromagnetic field, including particle-particle, fluid-particle, and particle-field interactions. You can seamlessly combine any application-specific module with the Particle Tracing Module for computing the fields that drive particle motion. Particles can have mass or be mass-less. The movement is governed by either the Newtonian, Lagrangian, or Hamiltonian formulations from classical mechanics. Boundary conditions can be imposed on the particles on the walls of the geometry to allow particles to freeze, stick, bounce, disappear, or reflect diffusely. User-defined wall conditions may also be specified, where the post collision particle velocity is typically a function of the incoming particle velocity and the wall normal vector. Secondary particles released when an incoming particle strikes a wall can be included. The number of secondary particles and their velocity distribution function can be functions of the primary particle velocity and the wall geometry. Particles can also stick to the wall according to an arbitrary expression or a sticking probability. Additional dependent variables can be added to the model which allows you to compute quantities like particle mass, temperature, or spin.

Particles can be released on boundaries and domains uniformly, according to the underlying mesh, as defined by a grid or according to an arbitrary expression. A wide range of predefined forces is available to describe specifically how the particles interact with the fields. You can then add arbitrary forces as defined by a suitable expression. It is also possible to model the two-way interaction between the particles and the fields (particle-field interaction), as well as the interaction of particles between each other (particle-particle interaction).

#### Animations

*Static mixers, also known as motionless or in-line mixers, are pipes containing stationary blades that mix fluids as they are pumped through. This type of mixing technique is great for laminar flow mixing as the pressure losses it generates in this flow regime are small. The example shown here studies the flow in a static mixer containing twisted blades. The mixing performance is evaluated via the calculation of the suspended particles' trajectory through the mixer. The model uses the Laminar Flow and Particle Tracing for Fluid Flow interfaces.*

#### Additional Images:

*A scanning electron microscope samples images by scanning a target with a high-energy beam of electrons. The subsequent electron interactions produce signals such as secondary and back-scattered electrons that contain information about the sample surface topography. Electromagnetic lenses are used to focus this electron beam down to a spot about 10 nm wide on the sample surface. This model requires both the Particle Tracing Module and the AC/DC Module.*

*The principle component of a quadrupole mass spectrometer is the mass filter that is used to filter ions with different charge-to-mass ratios. The quadrupole mass filter has been well-studied over the years, and the physics and optimal design are well understood. In a real quadrupole mass spectrometer, fringe fields exist at both the entrance and exit of the mass filter. These fringe fields can play an important role in determining the transmission probability of a specific ion through the mass filter. This model computes the ion trajectories in a quadrupole mass spectrometer, including the effects of fringe fields.*

*Trajectories of protons in Earth's magnetic field. Three types of oscillatory motion can be observed on three different time scales. On the shortest time scale, protons follow helical paths around Earth's magnetic field lines. At the intermediate time scale, these helical paths alternate between the magnetic north and south poles as the proton approaches latitudes known as mirror points, which are a function of the equatorial pitch angle. On the longer time scale, these oscillating trajectories precess around Earth from east to west due to asymmetries in the magnetic force.*

#### Powerful Processing Tools

Powerful processing tools allow for sophisticated visualization of the computed particle trajectories. Particle trajectories can be represented by points, comet tails, arrows, lines, tubes, or ribbons. Animations can easily be created and viewed directly in the graphical user interface (GUI) or exported to file. The particle trajectories can be colored with arbitrary expressions that can depend on the particles, the fields, or any combination of the two. In cases where the trajectory of many particles are simulated, it is possible to filter out specific particle trajectories according to a logical expression. The group behavior of the particles can be projected onto a lower dimension and visualized using Poincaré maps or phase portraits. It is also possible to perform operations on the particles to compute and plot the maximum, minimum, average, or integral of some quantity over all the particles. The particle trajectory data itself can be evaluated and written to the Results table or exported to a file. You can conveniently visualize the velocity and energy distributions of the particles, using 1D or 2D histograms.

#### Charged Particles in Electric and Magnetic Fields

Charged particles, such as electrons, individual ions, or small ion clusters, are affected by three primary forces in electric and magnetic fields:

- The electric force, which arises either due to a gradient in the electric potential or due to a time-varying magnetic vector potential. Particles with negative charge move in the opposite direction to the electric field, and particles with positive charge move in the same direction as the electric field. The electric force does work on these particles.
- The magnetic force, which does no work on the charged particles but can significantly alter their trajectory. The magnetic force often results in “banana” orbits for charged particles, causing them to orbit around magnetic field lines with a distance proportional to their mass.
- Collisional forces, which occur when charged particles collide with a background gas. The higher the background pressure, the more important the collisional forces.

If the number density of charged species is less than around 10^{13} 1/m^{3}, the effect of the particles on the fields can be neglected. This allows you to compute the fields independently from the particle trajectories. The fields are then used to compute the electric, magnetic, and collisional forces on the particles. The fact that the particle trajectories can be computed in their own study allows efficient and computationally inexpensive iterative solvers to be used.

#### Solving for Particle Tracing

For each particle, an ordinary differential equation is solved for each component of the position vector. This means that three ordinary differential equations are solved for each particle in 3D, and two in 2D. At each time step, the forces acting on each particle are queried from the computed fields at the current particle position. If particle-particle interaction forces are included in the model, they are added to the total force. The particle position is then updated, and the process repeats until the specified end time for the simulation is reached. Since the Particle Tracing Module uses a very general formulation for computing particle trajectories, the Particle Tracing interfaces can be used to model charged particle motion in electromagnetic fields, large scale planetary and galactic movement, and particle motion in laminar, turbulent, and two-phase fluid systems.

#### Studying Particle Tracking in a Fluid

The motion of microscopic- and macroscopic-sized particles is typically dominated by the drag force acting on particles immersed in a fluid. There are two phases in the system: a discrete phase consisting of bubbles, particles, or droplets, and a continuous phase in which the particles are immersed. In order for the particle tracking approach to be valid, the system should be a dilute or dispersed flow. This means that the volume fraction of the discrete phase should be much smaller than the volume fraction of the continuous phase (generally less than 1%). When the volume fraction of the particles is not small, the fluid system is categorized as a dense flow and you are required to take a different modeling approach. It is important to realize that, with the particle tracking approach, particles do not displace the fluid they occupy.

In a sparse flow, the continuous phase affects the motion of the particles, but not vice versa. This is often referred to as “one-way coupling”. When modeling such a system, it is usually most efficient to solve for the continuous phase first, then to compute the trajectories of the dispersed phases.

In a dilute flow, the continuous phase affects the motion of the particles, and the particle motion in turn disrupts the continuous phase. This is often referred to as “two-way coupling”. In order to model this effect, you must compute the continuous phase and disperse phase simultaneously. Thus, the computational demand is significantly higher when modeling dilute flows than when modeling sparse flows.

# Particle Tracing Module

### Product Features

- Charged Particle Tracing interface to model ion and electron trajectories in electric and magnetic fields including elastic collisions with a background gas
- Particle Tracing for Fluid Flow interface to model the motion of microscopic and macroscopic particles in a fluid
- Mathematical Particle Tracing interface, which offers complete freedom over the equations solved
- Massless, Newtonian, Lagrangian, and Hamiltonian formulations
- Predefined forces to facilitate model set-up
- Electric
- Magnetic
- Collisional
- Lift
- Drag
- Brownian
- Thermophoretic
- Gravity
- Acoustophoretic
- Dielectrophoretic
- Magnetophoretic

- User-defined forces
- Fictitious forces for rotating frames
- Particle-field interactions
- Particle-particle interactions
- Reinitialization of the particle velocity vector based on some logical expression allows for general purpose Monte Carlo modeling
- Particle release mechanisms
- Mesh-based where a specific number of particles are released in each mesh element
- Uniform distribution of particles on a given boundary
- Expression based which allows the density of particles to be greater in specific locations
- Grid
- Thermionic emission of electrons

- Boundary conditions
- Freeze
- Stick
- Bounce
- Disappear
- General reflection
- Diffuse scattering
- Periodic conditions
- Symmetry
- Secondary emission
- Sticking probabilities

- Postprocessing
- Particle trajectory plots (lines, tubes, points and comet tails)
- Color trajectories with arbitrary expressions
- Filter particles to plot
- Animations
- Poincaré sections and maps
- Phase portraits
- Compute maximum, minimum, average and integrals over all particles
- Write particle data to tables
- Export particle data
- 1D and 2D histograms
- Transmission probabilities

- Add auxiliary dependent variables to compute particle mass, spin, etc.
- Fully compatible with moving mesh

### Application Areas

- Mass spectrometry
- Beam physics
- Brownian motion
- Ion optics
- Ion mobility spectrometry

- Fluid flow visualization
- Sprays
- Aerosol dynamics
- Mixers
- Secondary emission

- Separation and filtration
- Ion energy distribution function visualization
- Acoustophoresis
- Classical mechanics

Ion Cyclotron Motion

*This model computes the trajectory of an ion in a uniform magnetic field using the Newtonian, Lagrangian and Hamiltonian formulations available in the Mathematical Particle Tracing interface.*

Red Blood Cell Separation

*Dielectrophoresis (DEP) occurs when a force is exerted on a dielectric particle as it is subjected to a nonuniform electric field. DEP has many applications in the field of biomedical devices used for biosensors, diagnostics, particle manipulation and filtration (sorting), particle assembly, and more.
The DEP force is sensitive to the size, ...*

Einzel Lens

*An Einzel lens is an electrostatic device used for focusing charged particle beams. It may be found in cathode ray tubes, ion beam and electron beam experiments, and ion propulsion systems.
This particular model consists of three axially aligned cylinders. The outer cylinders are grounded, while the cylinder in the middle is held at a fixed ...*

Particle Trajectories in a Laminar Static Mixer

*In static mixers, also called motionless or in-line mixers, a fluid is pumped through a pipe containing stationary blades. This mixing technique is particularly well suited for laminar flow mixing because it generates only small pressure losses in this flow regime. This example studies the flow in a twisted-blade static mixer. It evaluates the ...*

Brownian Motion

*Transport which is purely diffusive in nature can be modeled using a Brownian force. This model shows how to add such a force in the Particle Tracing for Fluid Flow physics interface. Particle diffusion in a fluid is modeled with the diffusion equation and the Particle Tracing for Fluid flow interfaces and the results are compared.*

Ideal Cloak

*This model demonstrates the use of optical tracing for studying optically large gradient-index structures with anisotropic optical properties. Additionally, the model introduces a smoothing technique for handling discontinuities of refractive index on curved surfaces, which are typical in conventional optical devices such as lenses.*

Rotating Galaxy

*This tutorial model shows how to add customized particle-particle interaction forces. In this example the gravitational force between 2500 stars in a galaxy is modeled. The galaxy initially rotates as a rigid body, then begins to change shape due to gravitational forces.*

Molecular Flow Through an RF Coupler

*This model computes the transmission probability through an RF coupler using both the angular coefficient method available in the Free Molecular Flow interface and a Monte Carlo method using the Mathematical Particle Tracing interface. The computed transmission probability determined by the two methods is in excellent agreement with less than a ...*

Electron Beam Diverging Due to Self Potential

*When modeling the propagation of charged particle beams at high currents, the space charge force generated by the beam significantly affects the trajectories of the charged particles. Perturbations to these trajectories, in turn, affect the space charge distribution.
The Charged Particle Tracing interface can use an iterative procedure to ...*

Laminar Static Particle Mixer Designer

*In static mixers, a fluid is pumped through a pipe containing stationary mixing blades. This mixing technique is well suited for laminar flow mixing, because it generates only small pressure losses in this flow regime. When a fluid is pumped through the channel, the alternating directions of the cross-sectional blades mix the fluid as it passes ...*

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