Prediction and Observation of Crystal Structures of Oppositely Charged Colloids

We studied crystal structures in mixtures of large and small oppositely charged spherical colloids with size ratio 0.31 using Monte Carlo simulations and confocal microscopy. We developed an interactive method based on simulated annealing to predict new binary crystal structures with stoichiometries from 1 to 8. Employing these structures in Madelung energy calculations using a screened Coulomb potential, we constructed a ground-state phase diagram, which shows a remarkably rich variety of crystals. Our phase diagram displays colloidal analogs of simple-salt structures and of the doped fullerene ${\mathrm{C}}_{60}$ structures, but also novel structures that do not have an atomic or molecular analog. We found three of the predicted structures experimentally, which provides confidence that our method yields reliable results.

Figure 1

The ground-state phase diagram of a mixture of large and small oppositely charged colloids with size ratio 0.31: (a) and (b), as a function of the Debye screening parameter $\kappa$ and the charge ratio $Q$, and (c), as a function of composition $x$ and $Q$ for $\kappa a_L=2.5$. In (a), the structures coexist with a gas of pure small colloids, and in (b) with one of large colloids. In (c), “${\mathrm{gas}}_L$” and “${\mathrm{gas}}_S$” refer to gas of large and small colloids, respectively. Note that the path $\kappa a_L=2.5$ in (a) and (b) corresponds to the coexisting crystals at $x=1^{-}$ and $x=0^{+}$ in (c), respectively.

Figure 2

Theoretically predicted stable binary crystal structures: (a) ${\mathrm{ReO}}_3$, (b) ${\mathrm{Li}}_3\mathrm{N}$, (c) ${\mathrm{SiF}}_4$, (d) $L{S}_6^{\mathrm{fcc}}$, (e) $L{S}_6^{\mathrm{hcp}}$, (f) wurtzite, (g) ${\mathrm{CaF}}_2$, (h) ${\mathrm{Cr}}_3\mathrm{Si}$, (i) $L{S}_6^{\mathrm{hp}}$, (j) $A_4{\mathrm{C}}_{60}$, and (k) $L{S}_4^{\mathrm{bct}}$.

Figure 3

Experimental observations and the corresponding theoretical predictions of binary structures. Confocal images of small positive light (green) and large negative dark (red) PMMA particles: (b) superposition of several layers of $L{S}_8^{\mathrm{hcp}}$; (d) (100) plane of $L{S}_8^{\mathrm{fcc}}$; (e) (110) plane of $L{S}_8^{\mathrm{fcc}}$; (g) (100) plane of $A_6{\mathrm{C}}_{60}^{\mathrm{bcc}}$; (h) (110) plane of $A_6{\mathrm{C}}_{60}^{\mathrm{bcc}}$. All scale bars are $3\mu \mathrm{m}$. The insets in (d), (e), (g), and (h) show the corresponding plane in the theoretical predictions. Unit cells of (a) $L{S}_8^{\mathrm{hcp}}$, (c) $L{S}_8^{\mathrm{fcc}}$, and (f) $A_6{\mathrm{C}}_{60}^{\mathrm{bcc}}$ structures.