### Geometrical trapping threshold

Figure 1: This figure illustrates the critical ratio *Φ _{a}*. 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 *Φ _{a}* of the ratio

*D/d*(Oger, 1987). The ratio

*Φ*corresponds to the case where a particle of diameter

_{a}*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 *Φ _{b}*, 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.

*Φ _{b}* is then given by:

If we consider trapping due only to geometrical reasons:

- if
*D/d>Φ*, particles can pass through all pores of the porous medium,_{a} - if
*D/d<Φ*, some particles can be trapped in the porous medium._{a}

### More about flow of particles through porous media:

- F. Lominé & L. Oger,
*Transit time during the interparticle percolation process*, Physical Review E, 82, 4, 041301, 2010 - F. Lominé and L. Oger,
*Dispersion of particles by spontaneous interparticle percolation through unconsolidated porous media*, Physical Review E, 79, 5, 051307, 2009 - Franck Lominé,
*Ecoulements de particules dans un milieu poreux*, PhD thesis, Unniversity of Rennes 1, 2007 - F. Lominé and L. Oger,
*Transport of small particles through a 3D packing of spheres: experimental and numerical approaches*, Journal of Statistical Mechanics: Theory and Experiment, P07019, 2006 - L. Oger & F. Lominé,
*Particle percolation through a porous medium*, Proceeding of Powders & Grains 2005, Edition Balkema, 1, 57-61