Topological defects in buckled colloidal monolayers

Abstract

When colloidal particles are vertically confined to a gap of between 1.3-1.6 particle diameters, they pack into buckled crystals of particles in either "up" or "down" states. Neighboring particles tend to occupy opposite states, analogous to the behavior of antiferromagnetic spins. The particles sit on a nearly-triangular lattice, and the spins of trios of adjacent particles are geometrically frustrated. Two levels of translational order exist in this system: that of the underlying triangular lattice in the horizontal plane, and that of the emergent frustrated spin lattice in the vertical dimension. We study the topological defects of both levels of translational order, and we find that both types of defects play a role in crystal grain boundary structure and spin domain coarsening. We classify the spin defects and outline the basic rules for their motion, and we observe interactions between dislocations and spin defects. Finally, we map the phase space of spin coarsening in the buckled monolayer, characterizing which types of defects drive the dynamics. Understanding defect formation, motion, and interaction in the buckled monolayer is the first step in predicting the material properties and aging of this geometrically frustrated, self-assembled system.

Figures (8)

• Figure 1: Frustrated spin order in a buckled colloidal monolayer. (a) Simplified side view schematic of sample with gap height $h = \ell D+\Delta z$ in the $z$ direction, where $D$ is the particle diameter, $\ell$ is the dimensionless looseness parameter, and $\Delta z$ is the "free height" between particle centers of opposite spin particles. In the schematic, the centers of same-spin neighbors are separated by distance $f=\ell D$ along the $y$ direction, while the centers of opposite-spin neighbors are spaced by only $u=\sqrt{\ell^2D^2-(h-\ell D)^2}$. (b) A buckled monolayer of colloidal particles. Particles pack into one of many ground state domain configurations, consisting of stripe and zig-zag motifs (highlighted in inset). Ground state domains are composed of rows of alternating spins along a single axis, such as the three rows highlighted with yellow lines.
• Figure 2: Anisotropic compressibility in stripe and zig-zag spin domains. (a) Lattice dislocations cause compression (red shading) and expansion (yellow shading) on the sides of the particle with five nearest neighbors (magenta) and seven nearest neighbors (blue), respectively. The magenta arrow with black outline indicates the Burgers vector of the lattice dislocation shown. (b) The mean free length between a pair of adjacent same-spin particles is $\Delta y_\mathrm{f} = D(\ell-1)$, while the lateral separation between opposite-spin particles is $\Delta y_\mathrm{u} = \sqrt{\ell^2 D^2 - \Delta z^2 } - \sqrt{D ^2 - \Delta z^2}$. This additional free distance gives alternating spin rows more room to be compressed along the lattice direction. Note that the looseness $\ell$ is exaggerated here for visual clarity. (c) A spin domain of stripe motifs (left) has one same-spin lattice axis and two alternating spin lattice axes (dotted lines). A spin domain of zig-zag motifs has two half-frustrated axes, where half of the edges are same-spin and half are opposite-spin, and one alternating spin axis (dotted line). (d) An experimentally observed lattice dislocation glides within a spin domain so that the five (magenta) moves along an alternating-spin segment of a lattice line. The glide region is bounded by two pairs of adjacent up spins at the lower left and upper right, confining lattice dislocation motion to a segment of length $x=8$ lattice constants.
• Figure 3: Spin defects correspond to dislocations in the unfrustrated lattice. (a) Any spin domain consisting of combinations of stripe and zig-zag motifs is made up of lines of alternating spin particles. The unfrustrated lattice contains all particle $x$-$y$ positions (white dots - spin up, grey dots - spin down), connected only by opposite-spin nearest-neighbor edges (black segments). The unfrustrated lattice can be continuously deformed into a square lattice. (b) The presence of a spin defect (in this case a "pitchfork" defect) corresponds to a double dislocation in the unfrustrated lattice, which has the effect of adding two lattice rows, as illustrated by the Burgers circuit (green) and the Burgers vector (purple with white outline). (c) Dipole compression-extension field surrounding a simulated pitchfork spin defect. Plaquettes in the unfrustrated lattice are colored linearly with their area, with smaller area in darker color and larger area in brighter color.
• Figure 4: Experimental images containing the five basic spin defects that can be isolated in a spin domain: pitchfork (a), flower (b), bump (c), diamond (d), and antidiamond (e). Purple arrows (white outlines) indicate the Burgers vector, calculated using a Burgers circuit as shown in Fig. 3. Yellow arrows (black outlines) indicate the direction in which each defect can move due to spin flips along a single alternating spin axis. Note that only the pitchfork has its Burgers vector aligned with its direction of motion, and that the diamond and antidiamond have zero Burgers vector.
• Figure 5: Spin defect motion. (a)-(b) Pitchfork defect glide by two steps ($2\Delta$) via two spin flips (outlined in green). Displacement is along the direction of the Burgers vector (purple arrow, white outline). (c)-(d) Bump and flower defects cannot move independently without emitting additional spin defects. Here a bump and flower defect move together by one step ($\Delta$) via five spin flips (outlined in green). Their displacement is along the direction of the vector sum of their Burgers vectors (dark purple arrow, white outline).