Self-Assembly of Cubes into 2D Hexagonal and Honeycomb Lattices by Hexapolar Capillary Interactions

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Self-Assembly of Cubes into 2D Hexagonal and Honeycomb Lattices by Hexapolar Capillary Interactions

Giuseppe Soligno, Marjolein Dijkstra, and René van Roij

Phys. Rev. Lett. 116, 258001 – Published 20 June 2016

Abstract

Particles adsorbed at a fluid-fluid interface induce capillary deformations that determine their orientations and generate mutual capillary interactions which drive them to assemble into 2D ordered structures. We numerically calculate, by energy minimization, the capillary deformations induced by adsorbed cubes for various Young’s contact angles. First, we show that capillarity is crucial not only for quantitative, but also for qualitative predictions of equilibrium configurations of a single cube. For a Young’s contact angle close to 90°, we show that a single-adsorbed cube generates a hexapolar interface deformation with three rises and three depressions. Thanks to the threefold symmetry of this hexapole, strongly directional capillary interactions drive the cubes to self-assemble into hexagonal or graphenelike honeycomb lattices. By a simple free-energy model, we predict a density-temperature phase diagram in which both the honeycomb and hexagonal lattice phases are present as stable states.

Received 25 April 2016

DOI:https://doi.org/10.1103/PhysRevLett.116.258001

Physics Subject Headings (PhySH)

Self-assembly

Colloids Liquid-liquid interfaces

Polymers & Soft Matter

Authors & Affiliations

Giuseppe Soligno¹, Marjolein Dijkstra², and René van Roij¹

¹Institute for Theoretical Physics, Center for Extreme Matter and Emergent Phenomena, Utrecht University, Princetonplein 5, Utrecht 3584 CC, The Netherlands
²Soft Condensed Matter, Debye Institute for Nanomaterials Science, Department of Physics and Astronomy, Utrecht University, Princetonplein 5, Utrecht 3584 CC, The Netherlands

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Vol. 116, Iss. 25 — 24 June 2016

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Images

Figure 1
(a) Configuration of a cubic particle at a fluid-fluid interface. (b) Adsorption energy $E_{1}$ [Eq. (1)] of a single cube (side $L$ ) at a fluid-fluid interface, in units of $Σ γ$ (see text), minimized over the center of mass height $z_{c}$ and the internal Euler angle $ψ$ [see (a)], as a function of the polar angle $φ$ , for Young’s contact angle $\cos θ = 0$ and $\cos θ = 0.3$ . The blue and red lines include and neglect capillarity, respectively. The labels ${100}$ , ${110}$ , and ${111}$ indicate the cube’s orientation in each minimum of the energy. The insets show, for the equilibrium configurations, a 3D view of the interface shape (blue grid) close to the particle (black grid), as calculated by our method. (c) 3D illustration of the ${100}$ , ${110}$ , and ${111}$ orientations of a cube, where the red grid represents a plane parallel to the flat interface. (d) Contour plots of the deformed-interface height profile for the global minimum-energy configuration of the cube. For $\cos θ = 0$ , a hexapolar deformation emerges, while for $\cos θ = 0.3$ , the interface is essentially undeformed.
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Figure 2
Results for ${111}$ -oriented adsorbed cubes with side $L$ and Young’s contact angle $θ = 90 °$ . (a) Interaction energy per particle ${\tilde{E}}_{2}$ [Eq. (2)] of two cubes, in units of $Σ γ$ (see text), as a function of their center-of-mass distance $D$ , for five relative orientations of their hexapole deformations, sketched in the insets with blue spots for depressions and red spots for rises. The main graph shows the two attractive configurations, where (violet curve) a red-blue dipole approaches another red-blue dipole, and (green curve) a red-blue-red tripole approaches another red-blue-red tripole. The inset shows the two repulsive counterparts and an almost “neutral” dipole-tripole pair. The violet and green vertical dotted lines represent the cube contact distance for the dipole-dipole and tripole-tripole attachments, respectively. As $\cos θ = 0$ , the system is invariant under exchange of red and blue. (b), (c) Sketch of the hexagonal (b) and honeycomb (c) lattices, formed by dipole-dipole and tripole-tripole attached cubes, respectively. Particle-particle distances are only schematic. In the hexagonal lattice, all cubes have the same azimuthal orientation, whereas in the honeycomb, each neighbor is rotated by $π$ . (d) Interaction energy per particle ${\tilde{E}}_{\infty}$ [Eq. (2)], as a function of $D$ , for a periodic ( $N \to \infty$ ) hexagonal (violet curve) and honeycomb (green curve) lattice, formed by dipole-dipole and tripole-tripole interacting cubes, respectively. The two insets illustrate the lattice unit cells.
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Figure 3
Temperature-density phase diagram for the adsorbed cubes. In colors, we show the honeycomb-lattice ( $h$ ), hexagonal-lattice ( $x$ ), and disordered-fluid ( $f$ ) phases. The gray area indicates phase coexistence. The normalized density $ϑ^{*}$ is 1 for the $x$ -phase closest packing density.
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