Colloidal ionic complexes on periodic substrates: Ground-state configurations and pattern switching

We theoretically and numerically studied ordering of “colloidal ionic clusters” on periodic substrate potentials such as those generated by optical trapping. Each cluster consists of three charged spherical colloids: two negatively and one positively charged. The substrate is a square or rectangular array of traps, each confining one such cluster. By varying the lattice constant from large to small, the observed clusters are first rodlike and form ferro- and antiferrolike phases, then they bend into a bananalike shape, and finally they condense into a percolated structure. Remarkably, in a broad parameter range between single-cluster and percolated structures, we have found stable supercomplexes composed of six colloids forming grapelike or rocketlike structures. We investigated the possibility of macroscopic pattern switching by applying external electrical fields.

Figure 1

Sketch of two clusters—each made up of three colloids—in two different potential traps, with definition of the relevant quantities. The center of a blue area denotes the confining potential minimum and $l$ is therefore the intertrap distance. The same color code will be used throughout the rest of the paper: black (white) dots for positively (negatively) charged colloids.

Figure 2

Summary of the most frequent composite clusters encountered in our study (from left to right): straight trimer, banana $\mathcal{B}$, grape $\mathcal{G}$, rocket $\mathcal{R}$.

Figure 3

Sketch of the different ordering patterns that will be reported (from left to right): antiferro $\mathcal{A}$, chain $\mathcal{C}$, ferro $\mathcal{F}$, tilted ferro ${\mathcal{F}}^{*}$, and ladder $\mathcal{L}$. Dashed lines represent a square (or rectangular) lattice of traps.

Figure 4

(a) Ground-state phase diagram at a relatively strong substrate potential $(A=5)$ for colloidal trimers on a square lattice. Typical snapshot configurations are calculated using $(l/d,\kappa l)=(7,2)$ for the $\mathcal{B}$ phase, $(15,4)$ for the $\mathcal{A}$ structure, and $(5,2)$ for the $\mathcal{C}$ phase. The lines between the different phases are shown as a guide to the eye. (b) Values of the three quantities characterizing the cluster shape [red dashed (upper), $r$; black solid, $\vartheta$; and green dot-dashed (lower), $\phi$] as a function of the ratio $l/d$ at constant $\kappa l=7$. (c) Values of the same three quantities as a function of $\kappa l$ at constant $l/d=8$.

Figure 5

Same as Fig. 4 but for a relatively weak substrate potential $(A=2)$. Typical snapshot configurations are calculated using $(l/d,\kappa l)=(7,8)$ for $\mathcal{B}$, $(7,4)$ for $\mathcal{R}$, $(5,2)$ for $\mathcal{C}$, $(20,8)$ for $\mathcal{G}$, and $(15,2)$ for $\mathcal{A}$ phase.

Figure 6

Definition of the notations used to compute triplet-triplet interaction in the (aligned) straight case.

Figure 7

Coulombic self-energy of six colloids (two positively and four negatively charged) forming various complexes.

Figure 8

Ground-state phase diagram for colloidal trimers on a rectangular lattice with $\alpha =0.8$ and (a) $A=5$ or (b) $A=2$. Typical snapshot configurations in (a) are calculated using $(l/d,\kappa l)=(10,8)$ for $\mathcal{B}$, $(7,4)$ for $\mathcal{R}$, $(7,2)$ for $\mathcal{G}$, $(15,6)$ for $\mathcal{F}$*, and $(20,2)$ for $\mathcal{F}$ phase; in (b) $(l/d,\kappa l)=(7,8)$ for $\mathcal{R}$, $(5,6)$ for $\mathcal{L}$, $(7,2)$ for 2-partite $\mathcal{G}$, and $(10,2)$ for 4-partite $\mathcal{G}$ phase.

Figure 9

Comparison of structures calculated within an isotropic parabolic potential (a) and realistic cosine potential $V_T^{\text{real}}$ (b)–(d) with three neighboring magnitudes $A_r$ chosen to roughly correspond to the parabolic potential at the trap midpoint $x/l=0.5$ (left panel). In (a) the parameter values are, from left to right, $(l/d,\kappa l,A,\alpha )=(10,4,5,0.8)$ ($\mathcal{F}$* phase), $(7,2,2,0.8)$ ($\mathcal{G}$ phase), $(15,4,5,1)$ ($\mathcal{A}$ phase), and $(7,4,2,1)$ ($\mathcal{R}$ phase). In (b)–(d) we use the same parameters as in (a) except $V_T^{\text{real}}$ is used instead of parabolic trapping with $A_r=20,15,10$ for $A=5$ and $A_r=8,6,4$ for $A=2$.

Figure 10

Single-trap calculations for parabolic (para) and realistic (real) trap potentials using $A=A_r=50$ and $\alpha =1$ (left panel) and $\alpha =0.8$ (right panel). Shaded areas denote regions in parameter space where the straight trimer has lower energy then banana $\mathcal{B}$ and grape $\mathcal{G}$ clusters. The symbols are comparing full energy-minimization calculations for both trap potentials considering all neighboring traps: black disks denote exact consistency while the other symbols indicate that different phases are found with the “realistic” and parabolic confinement potentials.

Figure 11

Application of the external electric field $\mathbf{E}$ on colloidal cluster shapes in the parabolic pinning regime for three field strengths $E_0$ and directions depicted at the bottom of the figure. The other parameters are $(l/d,\kappa l,A,\alpha )=(10,4,5,0.8)$ in the first row, $(7,4,2,1)$ in the second row, and $(7,2,2,0.8)$ in the third row. The asterisk indicates that a cluster (when viewed periodically on a lattice) forms a percolated structure along the field direction.

Figure 12

Influence of an external electric field $\mathbf{E}$ on the $\mathcal{A}$ phase within the parabolic trap approximation for $(l/d,\kappa l,A,\alpha )=(15,4,5,1)$. The field directions and magnitudes are shown with arrows in the inset of each plot; arrow lengths correspond to magnitudes $E_0=0.5,2,5$ of Eq. (8). For $E_0=2$ and $\beta =0,\pi /4$ (the middle column), we observe a pattern switching to the $\mathcal{F}$* phase (actually, its close approximation, since the trimers here are slightly bent with ${\phi }_{1,2}\approx 0.91\pi$).

Figure 13

Local energy minima configurations represented by trimer angles ${\vartheta }_i$ (solid line) and ${\phi }_i$ (dashed) as a function of external electric field $E_0$ using the cosine confinement potential Eq. (7). The index $i=1,\dots ,4$ represents the four different values for a 4-partite lattice assumption. The upper row is calculated from initial ($E_0=0$) $\mathcal{A}$ phase with $(l/d,\kappa l,A_r/(2\pi {}^2),\alpha )=(15,4,1.5,1)$ for two field directions (a) $\beta =0$ and (b) $\beta =\pi /4$. In the lower row, the initial ($E_0=0$) state is the $\mathcal{F}$ phase with $(l/d,\kappa l,A_r/(2\pi {}^2),\alpha )=(10,4,0.75,0.8)$ and field directions are again (c) $\beta =0$ and (d) $\beta =\pi /4$. In all the plots, we used a field resolution $\delta E_0=0.01$. A few representative snapshots are added for illustration (see the supplemental material [29] for the animated cartoons).