Laboratory of Civil and Mechanical Engineering
Figure 1: This figure illustrates the critical ratio Φa. Above this ratio, particles are likely to remain trapped in the packing.
In disordered sphere packings, pores can have several sizes. If we consider a compact packing of spheres with diameter D and if small particles of diameter d are allowed to transit through the packing, it is possible to define a critical value Φa of the ratio D/d (Oger, 1987). The ratio Φa corresponds to the case where a particle of diameter d can pass through the hole formed by three spheres in contact, where each one is in contact with two others as illustrated in figure 1.
It is also possible to define another critical value of ratio D/d. This ratio, which will be denoted Φb, corresponds to the case where small particle of diameter d can just be inserted into smallest cavity of the packing of larger spheres. Smallest cavities that can exist in a packing of spheres of diameter D, are the ones formed by four spheres in a tetrahedral arrangement (see figure 2).
Figure 2: Illustration of a tetrahedral cavity. It is the smallest cavity than we can observe in a monosize packing of spheres. The small sphere can just be inserted in this cavity. The diameter ratio between a large and small sphere is roughly 4.45.
Φb is then given by:
If we consider trapping due only to geometrical reasons: