The minimum energy state of an aggregate may be considered as the completely coalesced spherical particle. The rate of approach to this state is then considered to be dictated by how much surplus surface area the aggregate has compared to the corresponding spherical particle. Thus,

(19–125)

Equation 19–125 can be derived from the simple two-sphere neck growth model [147]. In the above equation, a is the surface area of the aggregate, a _s is the corresponding surface area of the completely coalesced spherical particle, and τ _f is the characteristic fusion (sintering) time.

The characteristic fusion time can be formulated in terms of physical properties of the aggregate bulk material, such as surface tension and viscosity.

The fusion time scale is the time required for complete coalescence of two individual particles after being brought into contact. For example, implies an instantaneous coalescence of two colliding particles. The fusion time can be defined as[151] :

(19–126)

where η and σ are, respectively, the viscosity and the surface tension of the bulk species in liquid phase and is the "would-be" particle diameter if coagulation takes place. Surface tension σ is a function of temperature and is given as [13]:

(19–127)

where m varies from 0.25 to 0.31. For 0.4 = T r = 0.7, d σ /d T is almost constant, and the surface tension can be expressed as a linear function of temperature[13] :

(19–128)

where the value of is on the order of 10 and on the order of -10-1 for .

The temperature dependence of liquid viscosity η can be approximated by the Andrade equation[13] :

(19–129)

where the value of is generally on the order of -10 and the value of is between 2000 and 4000.

By combining Equations Equation 19–128 and Equation 19–129 into Equation 19–126 , the particle fusion time scale takes the form

(19–130)

While the characteristic fusion time is given by Equation 19–126 , the collision time scale is already computed by the coagulation model of the Particle Tracking feature and is given as

(19–131)

In Equation 19–131 , T is the temperature of the aggregate and d _p is the diameter of the primary particle. Note that the above expression tacitly assumes that all the primary particles in an aggregate are spheres although this is not strictly true once the process of fusion starts. The value of the power n typically is 4.

If the characteristic fusion time is shorter than the time required for collisions, then particle coalescence dominates and most of the particles are spheres. At the other extreme, where collisions occur faster than coalescence, a collection of attached primary particles (aggregates) is formed. As expressed by Equation 19–131 , the fusion time depends on particle temperature and diameter of the primary particles. At high temperatures, the aggregates with smaller primary particles are more likely to coalesce completely.