Stripes, Zigzags, and Slow Dynamics in Buckled Hard Spheres

We study the analogy between buckled colloidal monolayers and the triangular-lattice Ising antiferromagnet. We calculate free-volume-induced Ising interactions, show how lattice deformations favor zigzag stripes that partially remove the Ising model ground-state degeneracy, and identify the martensitic mechanism prohibiting perfect stripes. Slowly inflating the spheres yields jamming as well as logarithmically slow relaxation reminiscent of the glassy dynamics observed experimentally.

Figure 1

Free-volume model. (a) Top view. Free volume originates from motion along axes to each of the neighbors. (b) Side view. Up particle surrounded by down particles has more free volume (lower energy) than a down particle. (c) Surrounding neighbors touch wall and are separated by $2L$; central particle is confined to the vertical plane. Large circles are volumes excluded by neighbors, horizontal bands are volumes excluded by the walls, remaining white region is the central particle’s free volume, divided at the middle of the cell height (dotted line) into $V_{+}$ and $V_{-}$.

Figure 2

Average number of frustrated bonds per particle vs sphere diameter and cell height. Free-volume model (lines) agrees with Monte Carlo simulations (symbols).

Figure 3

(a) Sphere diameter and (b) average number of frustrated bonds per particle following swelling to different sphere diameters. Normalized cell height $h/L=1.3$. Inset: same for Ising model following quench to $T=0$.

Figure 4

Final configurations following swelling to $d/L=1.01$ at $h/L=1.3$. (a) Deformable box. (b and c) Rigid box. The system has $N=1600$ (a and b), 6400 (c) spheres. Spheres touching top (bottom) wall are dark (bright). Simulation boxes are deformed parallelograms with periodic boundary conditions. For ease of presentation we copy the simulated region and plot a rectangular region of the periodic system.

Figure 5

Tiling with (a) equilateral triangles for the Ising ground state and (b) isosceles triangles for close-packed buckled hard spheres. The large angle $\beta$ is blackened.