Capillarity-induced ordering of spherical colloids on an interface with anisotropic curvature

Abstract

Objects floating at a liquid interface, such as breakfast cereals floating in a bowl of milk or bubbles at the surface of a soft drink, clump together as a result of capillary attraction. This attraction arises from deformation of the liquid interface due to gravitational forces; these deformations cause excess surface area that can be reduced if the particles move closer together. For micrometer-sized colloids, however, the gravitational force is too small to produce significant interfacial deformations, so capillary forces between spherical colloids at a flat interface are negligible. Here, we show that this is different when the confining liquid interface has a finite curvature that is also anisotropic. In that case, the condition of constant contact angle along the three-phase contact line can only be satisfied when the interface is deformed. We present experiments and numerical calculations that demonstrate how this leads to quadrupolar capillary interactions between the particles, giving rise to organization into regular square lattices. We demonstrate that the strength of the governing anisotropic interactions can be rescaled with the deviatoric curvature alone, irrespective of the exact shape of the liquid interface. Our results suggest that anisotropic interactions can easily be induced between isotropic colloids through tailoring of the interfacial curvature.

Keywords: Young-Laplace equation; colloidal interactions; pickering; self-assembly.

Fig. 1.

Particle organizations on oil/water interfaces of different shape. Maximum intensity projections of confocal z-stacks, showing fluorescently labeled particles on (A) a flat interface, (B) a spherical interface, (C) a dumbbell-shaped droplet, (D) a droplet pinned to a square patch (only one corner is shown), (E) a toroid-shaped droplet, and (F) a prolate ellipsoid. Inset in F shows square lattice organization. Green lines in C–F indicate the directions of principal curvature. Probability distributions of the angle ϕ between bonds and principal directions for each picture are shown in Fig. S6; values of order parameter 〈cos(4ϕ)〉: A, −0.031; B, −0.013; C, 0.44; D, 0.47; E, 0.40; and F, 0.37. (Scale bar, 10 μm in all images.)

Fig. 2.

Analysis of particle organizations at interfaces with different deviatoric curvature. (A) Radial distribution function g(r) for particles on a dumbbell-shaped interface and for particles on a flat interface. On the anisotropic interface, g(r) shows the characteristic peaks for a square lattice organization, at 1, √2, 2, and √5 times the lattice spacing (1.9 μm in this case). (B) Probability distribution of angles ϕ between interparticle bonds and the local principal curvature axes on a dumbbell-shaped droplet. (C) Order parameter 〈cos(4ϕ)〉 as a function of the deviatoric curvature D for droplets of different shape. Average values and SDs are shown, obtained for ∼4,000 particles on three different droplets for shapes with D > 0 and for ∼2,000 particles on one droplet for D = 0. (D) Interparticle potential for two different deviatoric curvatures (D₁ = 0.016 μm⁻¹ and D₂ = 0.007 μm⁻¹) as obtained from measurement of the interparticle distance distribution (Fig. S8).

Fig. 3.

Calculated deformations and capillary interactions for colloidal particles on a saddle-shaped interface. (A) Calculated deformation field around a spherical particle with radius a = 1 μm on a catenoid interface with D = 0.02 μm⁻¹; blue regions indicate a depression of the interface and red regions a rise of the interface. (B) Calculated interaction energy between two particles (a = 1 μm) on catenoid interfaces of different curvature as a function of separation distance, for two relative orientations with respect to the principal curvature axis of the interface. Interfacial tension γ = 30 mN/m, and contact angle θ = 90° in these calculations. (C) Superimposed interaction curves for particles approaching along principal axis for different a and D, showing −U/[(γa²)(Da)²] as a function of (r/a). The black points indicate the analytical prediction for the far field (24).

Fig. 4.

Transition from square to hexagonal packing at high particle densities. (A) Part of a dumbbell-shaped droplet covered with colloidal particles at relatively low density (0.16 μm⁻²). The particles organize in a square pattern: 〈|ψ₄|〉 = 0.81, 〈|ψ₆|〉 = 0.2. (B) At high density (0.5 μm⁻²) the particles are organized in a hexagonal lattice (Inset): 〈|ψ₄|〉 = 0.22, 〈|ψ₆|〉 = 0.74. (C) Bond order parameters for four- and sixfold symmetry as a function of particle density for interfaces with D = 0.006 μm⁻¹. The red and blue vertical dashed lines indicate maximum densities for a particle separation of 1.74 μm in a square and a hexagonal lattice, respectively.