Attraction Controls the Inversion of Order by Disorder in Buckled Colloidal Monolayers

We show how including attraction in interparticle interactions reverses the effect of fluctuations in ordering of a prototypical artificial frustrated system. Buckled colloidal monolayers exhibit the same ground state as the Ising antiferromagnet on a deformable triangular lattice, but it is unclear if ordering in the two systems is driven by the same geometric mechanism. By a real-space expansion we find that, for buckled colloids, bent stripes constitute the stable phase, whereas in the Ising antiferromagnet straight stripes are favored. For generic pair potentials we show that attraction governs this selection mechanism, in a manner that is linked to local packing considerations. This supports the geometric origin of entropy in jammed sphere packings and provides a tool for designing self-assembled colloidal structures.

Figure 1

(a) Straight and (b) bent stripe configurations. For the Ising model, we consider $n_s$ shells of $n$ neighbors around the central white particle. Shells are denoted by increasingly darker colors for increasing $n_s$. Thicker lines correspond to frustrated bonds connecting particles at the same state. For straight (c) and bent (d) stripes in the buckled monolayer, we consider different sets of $n$ free particles (white) with $n_{fb}$ frustrated bonds, while other particles (purple) are fixed.

Figure 2

Entropy difference per particle between straight and bent stripe configurations vs deformation angle $\alpha$ for increasing numbers $n$ of fluctuating particles. (a) Ising antiferromagnet: Results with different numbers $n_s$ of shells of free spins converge to the phonon solution $\mathrm{\Delta }{s}_p$. Inset: distance between $\mathrm{\Delta }s$ and $\mathrm{\Delta }{s}_p$ at $\alpha =100°$ vs $n_s$. (b) Hard-sphere monolayer for configurations in Figs. 1 and 1. $n_{fb}$ is the number of frustrated bonds. The line for $n=1$ is an analytic expression, lines for $n>1$ are guides to the eye.

Figure 3

Straight (a)–(c) and bent (d)–(f) configurations. (a), (d) yellow (red) spheres are in contact with the bottom (top) plane. Phase-space volume of one hard sphere free to move given by the three-dimensional volume $V_1$, considering curved (b), (e) and flat (c), (f) surfaces. Here, we take a deformation angle of $\alpha =70°$ and a dimensionless sphere shrinking factor $\delta /c=0.08$ to emphasize the difference between curved and flat surfaces.

Figure 4

Entropy difference per particle between straight and bent configurations vs deformation angle $\alpha$. (a) Harmonic potential $U(\zeta )$. (b) Repulsive harmonic potential $U_r(\zeta )$. (c) Asymmetric potential $U_{\mathrm{asy}}(\zeta )$: $\mathrm{\Delta }s$ vs the ratio ${\gamma }_a/{\gamma }_r$ between the exponents of attraction and repulsion for different values of ${\gamma }_r$. The curves are computed for $n=1$, $U_a^0=U_r^0$, and the dashed vertical lines indicate $\kappa (\alpha )$.