## Random, optimal, and thermal packing of aspherical shapes

 Yoav Kallus Packing of Continua WorkshopAspen, June 21, 2017

### Equilibrium and metastability

 Haji-Akbari, Engel, & Glotzer. J Chem Phys (2011).

### Random close packing

 Jiao & Torquato, Phys Rev E (2011) Jaoshvili, Esakia, Porrati, & Chaikin. Phys Rev Lett, (2010)

Isostatic if face-face = 3 contacts, face-edge = 2, vertex-face = 1, edge-edge = 1

### Ellipsoids

 Donev, Stillinger, Chaikin, & Torquato. Phys. Rev. Lett. (2004) Donev, Connelly, Stillinger, & Torquato. Phys. Rev. E (2007)

### Ulam's conjecture

 “Stanislaw Ulam told me in 1972 that he suspected the sphere was the worst case of dense packing of identical convex solids, but that this would be difficult to prove.” Damasceno, Engel, Glotzer (2012)

### Disk is not worst in 2D

 $\phi=0.9069$ $\phi=0.9062$ $\phi=0.9024$

### Optimal packing of nearly-circular shapes

 In general, for centrally-symmetric shapes: if $r_K(\theta) = 1 + \delta r(\theta)$ describes the boundary of the shape (i.e. $K=\{(\lambda\cos\theta,\lambda\sin\theta):\lambda\le 1, 0\le \theta\le 2\pi\}$), then $\displaystyle{\frac{\phi(K) - \phi(C)}{\phi(C)} = 2\overline{\delta r} - 2\min_\omega \sum_{n=0}^{5}\delta r(\tfrac{n\pi}{3}+\omega) + o(\|\delta r\|)}$.

So, for a family $r_K(\theta) = 1 + \alpha \rho(\theta) + o(\alpha)$,
we have $d\phi/d\alpha \ge 0$.

$d\phi/d\alpha = 0$ is possible, e.g., for $\rho(\theta) = \cos(8\theta)$.

### Optimal packing of nearly spherical shape

 In 3D, for centrally-symmetric shapes: $\displaystyle{\Delta\phi/\phi = 3\overline{\delta r} - 3\min_R \sum_{n=0}^{11}\delta r(R\mathbf{x}_i) + o(\|\delta r\|)}$, where $\delta R:S^2\to\mathbb{R}$, $R\in SO(3)$, and $\mathbf{x}_i$ are the 12 f.c.c. contact directions. Lemma: $\min_R \sum_{n=0}^{11}\delta r(R\mathbf{x}_i) = \overline{\delta r}$ iff $\delta r(\mathbf{x}) = a + (\mathbf{b}\cdot\mathbf{x})^2$. So, any centrally symmetric shape, apart from ellipsoids has $\phi(K)-\phi(B) \ge c \|\delta r\|$, for some constant $c$.

### Random close packing (no friction)

Start from a RCP of spheres, then continuously to another shape, keeping particle $i$'s rotation $R_i$ locked:
$\displaystyle{p\Delta V = \sum_i \sum_{j\in \partial i} f_{ij} \delta r(R_i \mathbf{n}_{ij}) + O(V\|\delta r\|^{3/2})}$

Rotation is free for spheres, but at start of deformation, particles under pressure will take rotation that minimizes volume, so

for $r(\mathbf{x}) = 1 + \alpha \rho(\mathbf{x}) + o(\alpha)$
$\displaystyle{\frac{1}{3\phi}\frac{d\phi}{d\alpha} = \overline{\rho} - \frac{1}{\langle|\partial i|\rangle \langle f\rangle} \big\langle \min_R \sum_{j\in \partial i} f_{ij} \rho(R \mathbf{n}_{ij})\big\rangle}$

$d\phi/d\alpha>0$ for all nonconstant $\rho$ (and for all dimensions)

We can numerically calculate $\eta=(\tfrac1{3\phi} \tfrac{d\phi}{d\alpha})/(\overline{|\rho-\overline{\rho}|})$ for given $\rho$

### Some 1-parameter shape families

 $\eta=0.94$ $\eta=0.79$ $\eta=0.86$ $\eta=1.08$ $\eta=1.36$ $\eta=0.77$ $\eta=1.45$ $\eta=1.06$ $\eta=1.31$ $\eta=1.01$ $\eta=1.32$ $\eta=1.20$