Geometric Frustration and Interparticle Gap Size Distributions in Ordered Hexagonal Polydisperse Disk Packs

This work analyzes the distribution and size of interparticle gaps arising in an ensemble of hexagonal unit structures in the xy plane when packing disks with a Gaussian distribution of radii with mean (r) and standard deviation $\Delta r$. During the course of this investigation an equivalency is established between gaps arising in hexagonal unit structure packs and nine-ball billiard rack patterns. An analytic expression is derived for the probability distribution and location of interparticle gaps of magnitude $\Gamma$. Due to the number of variables and large number of possible arrangements, a Monte Carlo simulation has been conducted to complement and probe the analytic form for three very different systems: i) billiard balls with Billiard Congress of America (BCA) specifications, ii) US pennies with specifications of the US Mint, and iii) a hypothetical system with $r = 1.0 m$ and $\Delta r = 1x10^{-10}$ m corresponding to the scale of one atomic radius. In each case, probability density distributions of gap sizes have been calculated for those $\Delta r$ above, and also for 2$\Delta r$ and 0.5$\Delta r$, respectively. A general result is presented for the probability of a nonzero normalized ($\frac{\Gamma}{\Delta r}$) gap size arising, $P(\Gamma \geq \alpha \Delta r)= 1-0.124\alpha$, where $\alpha$ is a constant $\leq 5.0$. This curious result reflects the phenomenon of geometric frustration; the inability of the system to simultaneously satisfy all geometric constraints required by a perfect-rack sans interparticle gaps.