Self-Assembly of Particles for Densest Packing by Mechanical Vibration

It is shown that by properly controlling vibrational and charging conditions, the transition from disordered to ordered, densest packing of particles can be obtained consistently. The method applies to both spherical and nonspherical particles. For spheres, face centered cubic packing with different orientations can be achieved by monitoring the vibration amplitude and frequency, and the structure of the bottom layer, in particular. The resultant force structures are ordered but do not necessarily correspond to the packing structures fully. The implications of the findings are also discussed.

Figure 1

$\rho$ as a function of $d/D$ ratio under different conditions: (a), method I; and (b), method II. The results are fitted by a linear equation $\rho =a\times (d/D)+{\rho }_{\max }$, and parameters $a$ and ${\rho }_{\max }$, together with the varying range of ${\rho }_{\max }$ when the level of confidence is 95%, are listed. The right photos (top view) were taken when $d/D=0.036$.

Figure 2

Evolution of $\rho$ when $A=0.1d$, $\omega =200\mathrm{rad}/\mathrm{s}$, and $\mathrm{Np}=98\mathrm{\text{particles}}/\mathrm{\text{batch}}$ (method II). For each operational cycle, $A_1$ represents the addition of spheres and start of vibration, and $A_2$ the end of vibration. Note that the addition of spheres stops at 25 s, although the whole packing process ends at $t=59\mathrm{s}$ to produce the final stable packing. The right figure is a snapshot showing the formation of the packing.

Figure 3

Particle density as a function of vibration amplitude under 1D (◆, ◇), 2D (■, □), and 3D (▴, △) conditions: (a), method I when $\omega =100\mathrm{rad}/\mathrm{s}$; (b), method II when $\omega =200\mathrm{rad}/\mathrm{s}$, $\mathrm{Np}=101\mathrm{\text{particles}}/\mathrm{\text{batch}}$.

Figure 4

(a) Influence of Np on $\rho$ when $A=0.1d$, $\omega =200\mathrm{rad}/\mathrm{s}$. The insets show the representative structure for: left, $\{100\}$; and right, $\{111\}$ oriented fcc, obtained when Np is small enough with different A and $\omega$. (b) and (c) show, respectively, the force structure of the $\{100\}$ and $\{111\}$ oriented fcc, where each line represents a normal contact force between spheres whose centers are shown by small spheres, with its thickness proportional to its magnitude.