It is possible to apply collisional forces that are deterministic or random. A deterministic force based on a friction model can be applied using the Friction Force node. Random collisional forces can be applied using the Collisions node.

The Friction Force feature adds the following contribution to Ft:

where v and u (SI unit: m/s) are the particle velocity and background fluid velocity. The collision frequency ν (SI unit: 1/s) is either specified directly, or via a collision cross section σ (SI unit: m2) and background number density Nd (SI unit: 1/m3):

The Collisions node uses a stochastic approach to modeling particle collisions with the atoms or molecules in a background gas. Individual collisions between the model particles and the atoms or molecules in the background gas are detected by sampling random numbers from a distribution and comparing them to the collision probability, which usually varies over time. During a collision, the velocity of the model particle changes discontinuously. Depending on the type of reaction, secondary particles may be emitted as a result of the collision.

A single Collisions node usually corresponds to a single species in the background gas. Add subnodes to the Collisions node to specify the different reaction types that can occur between the model particles and the background gas, such as elastic collisions and charge exchange reactions.

Consider a model particle interacting with a background gas with an average collision frequency ν(t) (SI unit: 1/s) given by the expression

where the νi(t) are the frequencies associated with individual reaction types between the model particle and each species in the background gas. For a time interval Δt (SI unit: s) much smaller than the average time between collisions, the probability PT(t) (dimensionless) that a collision takes place within a time interval of width Δt centered at a time t is

Let Q(t) (dimensionless) be the probability that a collision does not occur during the time interval (0, t). Similarly, Q(t + Δt) is the probability that a collision does not occur during the interval (0,t + Δt). Q(t + Δt) can be expressed the product of the probability that no collision occurs in (0, t) (given by Q(t)), and furthermore that no collision occurs in the interval (t, t + Δt). The latter can be expressed as 1 − PT(t) or 1 − ν(t)Δt. Thus the relationship between Q(t) and Q(t + Δt) can be expressed as

Taking the limit as Δt approaches zero, the left-hand side becomes a time derivative,

Integrating and noting that Q(0) = 1 (that is, given no time, there is no possibility of a collision) yields

The probability that the next collision does occur in the interval (0, t) is thus

The Collisions node supports two algorithms for predicting the collision times. The algorithm is controlled by the Collision detection list in the settings window for the Collisions node. The two options are At time steps taken by solver and Null collision method, cold gas approximation.

The default option, At time steps taken by solver, can only detect and apply collisions at the discrete times used by the time dependent solver. At each one of these solution times, an uncorrelated, uniformly distributed random number U (dimensionless) is sampled from the interval (0, 1) for each particle. The time step is assumed to be significantly smaller than the free time between collisions,

This assumption is important because the collision frequency is assumed constant over each time step taken by the solver. For constant collision frequency, Equation 4-5 reduces to

To detect whether a collision occurs, the collision probability for each particle is compared to the random number U,

Because U is not used for any other purpose, and its actual value within the distribution has not yet been specified, we can take advantage of the equivalent probability distributions of U and 1 − U to simplify this expression slightly:

When this option At time steps taken by solver is used, each model particle is only allowed to undergo at most one collision per time step taken by the solver. If the collision frequency is large enough so that particles are likely to undergo two or more collisions in a single time step taken by the solver, then the second and later collisions within the step are disregarded, potentially skewing the particle statistics.

In practice, the collision frequencies are usually not constant over time. In its most general form (see for example Ref. 3), the collision frequency for the jth reaction can be expressed in terms of the corresponding collision cross section σj (SI unit: m2),

The integral is taken over all velocity space. The probability distribution function f(vg) (SI unit: s3/m3) is usually assumed to be a drifting Maxwell distribution,

Alternatively, the mass mg can be expressed in terms of the molar mass Mg of the background species (SI unit: kg/mol),

where NA = 6.02214076 × 1023 1/mol is the Avogadro constant. Optionally, the molar mass can be assigned unique values or expressions for each reaction type.

For the option At time steps taken by solver, the following simplification is made to Equation 4-7 to remove the integration over velocity space. First, a gas molecule velocity is sampled at random from Equation 4-8, using a built-in randomnormal function that creates uncorrelated Gaussian pseudorandom numbers for each velocity component. Then the velocity distribution function in Equation 4-7 is replaced with a Dirac delta function at the sampled velocity components, yielding the simplified expression

where vg now denotes the specific value of the background gas velocity sampled from the distribution.

The limiting time step size for the option At time steps taken by solver is determined from the following criteria:

The null collision method (Ref. 4) is one of a class of computational methods known as rejection methods. This approach is used by selecting Null collision method, cold gas approximation from the Collision detection list. The basic premise of the null collision method is to avoid inverting Equation 4-5 for an arbitrary functional form of ν(t) by first assigning a large, constant value νm to the collision frequency. A number of collision times are then sampled from a distribution based on this artificially large, constant collision frequency. The sampled collision times are then either used or rejected based on an integral of the real collision frequency ν(t) up to each trial time.

The implementation of the null collision method in the Collisions node uses a simplifying assumption known as the cold gas approximation. In applications with extremely high particle speeds, such as ions or electrons accelerating in strong electric fields, the cold gas approximation can be used to simplify the expression for collision frequency. In the cold gas approximation, the background gas velocity is set to zero, so Equation 4-9 simplifies to

As is the case with the option At time steps taken by solver, the first step to predicting the collision times is to generate a uniformly distributed random number U1 for each particle and compare it to the collision probability over the time interval; we use the subscript 1 here because the null collision method requires multiple uncorrelated random numbers for each model particle. Instead of the simplification made in Equation 4-6, however, the random number is substituted directly into Equation 4-5,

The objective is to solve for t.

The next step is to assign a constant trial frequency νm, such that νm > ν(t) over the interval (0, t). Substituting the corresponding trial time tm for t, and νm for ν(t), into Equation 4-10 and solving for tm then yields

After a trial time for the particle has been obtained, the next step is to check whether the collision detected at the trial time actually occurs. A second uniformly distributed random number U2, not correlated with U1, is also randomly sampled from the interval (0, 1). If the inequality

In the event that a null collision occurs, the particle moves along its trajectory unaffected by the background gas until time tm, when a new trial frequency and trial time are computed, and the process repeats itself.

There is often no analytic expression for the integral in Equation 4-11, since the functional form that relates the collision cross section and collision frequency is not known a priori. However, the integral can be approximated as

If the integral in Equation 4-11 were ever greater than unity, that would imply that the selected value of the trial frequency was too low and the process must be repeated, without advancing the particle in time to tm.

After a real collision is detected, meaning that the trial frequency was sufficiently high but a null collision wasn’t detected, the final step is to determine the type of collision to apply. If multiple types of collision are present, the probability pj of a specific collision type occurring is

The Maximum number of consecutive null collisions is available in the physics interface Advanced Settings section. The default is 100. If any particle undergoes more null collisions than the specified value, without undergoing any real collisions, then the particle cannot collide anymore until the next time step. When this happens, a Warning is produced in the solver sequence. Consider a modest reduction in the maximum time step size if this warning appears.

A single Collisions node corresponds to a single species in the background gas. Add subnodes to the Collisions node to specify the types of reaction that the model particles undergo with the background gas, such as elastic collisions and charge exchange reactions.

The Elastic collision force causes a particle to collide with a background gas molecule in such a way that the total energy of the system is conserved. The post-collision velocity of the particle is defined by the expression:

where is the relative velocity in the center of mass reference frame, mg is the mass of the background gas atoms or molecules. The post-collision relative velocity is where R is a uniformly distributed random unit vector. No secondary particles are produced.

The Excitation collision force causes a particle to collide with a background gas molecule in such a way that the total energy of the system is not conserved. The post-collision velocity of the particle is defined by the expression:

where R is a uniformly distributed random unit vector, and ΔE (SI unit: J) is the kinetic energy lost as a result of the collision. No secondary particles are produced.

The Attachment node causes the model particle to be annihilated, as in the Disappear condition for the Wall node. Optionally, the attached species can be released as a secondary particle that is initially at rest.

The Ionization node causes an electron to collide with a background gas molecule in such a way that the total energy of the system is not conserved. The post-collision velocity of the particle is defined by the expression:

where R is a uniformly distributed random unit vector, and ΔE (SI unit: J) is the kinetic energy lost as a result of the collision. No secondary particles are produced.

The background gas molecule is assumed to be ionized; that is, one electron collides with the background gas molecule and causes a secondary electron to be released. For the results to be physical, this particle should inherit its properties from a Particle Properties node that is appropriate for electrons. This Particle Properties node should therefore have a mass (or rest mass) equal to me_const and a charge number of −1.

The Resonant Charge Exchange and Nonresonant Charge Exchange nodes are used to model collisions in which charge is exchanged between a model particle and a background gas particle. It is possible to compute the post-collision trajectories of either or both products of the charge exchange reaction. The post-collision velocity of the ionized species and neutralized species are defined by the expressions:

and χ (SI unit: rad) is the scattering angle in the center of mass coordinate system. The vectors and are defined by the expressions:

where φ (SI unit: rad) is a uniformly distributed angle between 0 and 2π. The unit vectors and are defined so that they form an orthonormal basis with the normalized pre-collision relative velocity .The post-collision relative velocity magnitude is defined by the expression:

where ΔE (SI unit: J) is the energy loss. During resonant charge exchange collisions the energy loss is 0 and .