It has been postulated [38] that whether a collision between two particles results in the formation of an aggregate depends on the size of the particles involved in the collision. In other words, the size of the particles involved in a collision influences whether the particles will stick together or not. Models have been developed to estimate the lower and upper limits of such "collision efficiency." Note that these models are applicable only to the free-molecular regime.

Assuming that a moving particle does not experience friction (due to collisions) from the surrounding gas, the lower limit for the free-molecular regime is established using the Lennard-Jones potential. It is written as

(19–68)

In Equation 19–68, is the Lennard Jones potential well depth and is the dimensionless surface temperature. The latter accounts for the additional energy involved in a collision due to the potential and is given by

(19–69)

The potential well depth is a function of particle size and is computed using the Hamaker () constant as

(19–70)

where , , and are functions of particle size (diameter ) and the minimum separation distance ().

(19–71)

The lower limit given by Equation 19–68 is assumed to be valid for "sufficiently small" particles. On the other hand, for "sufficiently large" particles, it is argued that the particles moving away after a collision experience friction and there is more chance that they get trapped in the potential energy well. The upper limit for the collision efficiency is thus considered to be unity.

As suggested in the literature [17], Ansys Chemkin uses a mixing function to blend the upper and lower limits. This mixing function and the effective collision efficiency are then expressed as

(19–72)

(19–73)

As mentioned before, the upper limit used in Equation 19–73 is taken to be unity.

In Equation 19–72, and are the parameters that control the blending effect of the mixing function. Note that the mixing function approaching 1 means the upper limit of the collision efficiency is the dominant part in the blended effective value whereas its value of 0 means the lower limit is dominant. The latter implies that the value of should be such that for small particles the Tanh term becomes -1. Thus, one may write

(19–74)

In Equation 19–74, is a constant and is the diameter at which the mixing function approaches zero. A small integer value of (around 5) is then sufficient to yield such that the Tanh function approaches -1 as the particle diameter approaches . It should be noted that Equation 19–74 is just one of the many possible choices of setting . Noting that the value of the mixing function is 0.5 for = , represents the threshold on either side of which one of the limits of the collision efficiency starts to dominate. For a given , the larger the value of the quicker is the either limit reached by the mixing function and hence the effective collision efficiency. In Ansys Chemkin, the default value of is chosen to be 7 whereas is the particle diameter corresponding to the smallest section as specified by the user.

For simulations involving sooting flames [17], has been correlated with the maximum flame temperature and the corresponding values are of the order of 10 nm. Ansys Chemkin expects that a value of is given by the user. The default values of the Hamaker constant and the minimum separation distance used by Chemkin are also based on those used in [17]. These default values are only a guideline and may not be applicable to particulate systems other than soot.

The final general expression for the coagulation kernel in the free molecular regime involves multiplication of Equation 19–50 by the collision efficiency as calculated in Equation 19–73 and by a constant factor () accounting for the van der Waals enhancement factor. Thus,

(19–75)