Role of Interparticle Forces in the Formation of Random Loose Packing

We present a physical and numerical study of the settling of uniform spheres in liquids and show that interparticle forces play a critical role in forming the so-called random loose packing (RLP). Different packing conditions give different interparticle forces and, hence, different RLP. Two types of interparticle forces are identified: process dependent and process independent. The van der Waals force, as the major cohesive force in the present study, plays a critical role in effecting the process-dependent forces such as drag and lift forces. An equation is formulated to describe the relationship between the macroscopic packing fraction and microscopic interparticle forces in a packing. We argue there is no lowest packing fraction for a mechanically stable RLP; hence, the packing fractions of RLP can range from 0 to 0.64 depending on the cohesive and frictional conditions between particles.

Figure 1

The packing fraction of different sized glass beads as a function of the effective gravitational acceleration. Points are the measured results: ▲, $d=500\mu \mathrm{m}$; □, $d=250\mu \mathrm{m}$; ◆, $d=110\mu \mathrm{m}$; and ×, results of Onoda and Liniger [16]. Lines are the simulated results. The inset shows the variation of viscosity (solid line) and the Hamaker constant (dashed line) with liquid composition.

Figure 2

The packing fraction as a function of: (a) the Hamaker constant when ${\rho }_f=2200\mathrm{kg}/{\mathrm{m}}^3$ (◆) and ${\rho }_f=2400\mathrm{kg}/{\mathrm{m}}^3$ (□); (b) particle size when $A=6.5\times {10}^{-20}\mathrm{J}$ and ${\rho }_f=0$ (▲) or $1000\mathrm{kg}/{\mathrm{m}}^3$ (◆), and $A=0\mathrm{J}$ and ${\rho }_f=1000\mathrm{kg}/{\mathrm{m}}^3$ (□); (c) liquid density when $A=6.5\times {10}^{-20}\mathrm{J}$ and $d=1000$ (◇) or $d=250\mu \mathrm{m}$ (■), and $A=0\mathrm{J}$ and $d=250\mu \mathrm{m}$ (×); (d) liquid viscosity when $A=6.5\times {10}^{-20}\mathrm{J}$ and $d=1000$ (△) or $d=250\mu \mathrm{m}$ (◆), and $A=0\mathrm{J}$ and $d=250\mu \mathrm{m}$ (×). The default values listed in Table  are used for unspecified parameters.

Figure 3

Evolutions of the averaged process-dependent forces $|{\mathbf{F}}_{\mathrm{vdw}}|$, $|{\mathbf{F}}_{\mathrm{drag}}|$, and $|{\mathbf{F}}_{\mathrm{lift}}|$ with time.

Figure 4

The packing fraction as a function of the force ratio between the averaged van der Waals force and effective gravity acting on individual particles in a packing: ◆, ${\chi }^{\prime }$; and □, $\chi$.