Nanoparticle self-assembly at the interface of liquid crystal droplets

Abstract

Nanoparticles adsorbed at the interface of nematic liquid crystals are known to form ordered structures whose morphology depends on the orientation of the underlying nematic field. The origin of such structures is believed to result from an interplay between the liquid crystal orientation at the particles' surface, the orientation at the liquid crystal's air interface, and the bulk elasticity of the underlying liquid crystal. In this work, we consider nanoparticle assembly at the interface of nematic droplets. We present a systematic study of the free energy of nanoparticle-laden droplets in terms of experiments and a Landau-de Gennes formalism. The results of that study indicate that, even for conditions under which particles interact only weakly at flat interfaces, particles aggregate at the poles of bipolar droplets and assemble into robust, quantized arrangements that can be mapped onto hexagonal lattices. The contributions of elasticity and interfacial energy corresponding to different arrangements are used to explain the resulting morphologies, and the predictions of the model are shown to be consistent with experimental observations. The findings presented here suggest that particle-laden liquid crystal droplets could provide a unique and versatile route toward building blocks for hierarchical materials assembly.

Keywords: defect; interface; liquid crystal; nanoparticle; self-assembly.

Fig. 1.

(A) Free energy of one particle at the droplet surface as a function of particle anchoring strength for two types of configurations: the particle located at the boojum, and the particle placed in the equator. The difference of free energies indicates that there is a strong tendency to adsorb particles at the boojum. The scalar order around the particle’s surface for both configurations is shown. There is a critical anchoring strength when the particle is located at the boojum, after which the free energy and the local scalar order suddenly increase. At that point, the nematic field twists and stands along the planar anchoring. Insets in the plot correspond to the top view of weak and strong anchoring. The director field is illustrated with black lines, and the color represents the z component of director field. (B and C) Polarized-light, bright-field, and fluorescence images of a 5CB droplet with a PS particle localized at the boojum. (C) Polarized-light image shows the twisting of the nematic field near the boojum.

Fig. 2.

(A) Free energy as a function of r and φ for two particles at a planar LC interface. One particle is located at the origin of a polar coordinate system, and the other is driven around the origin. The nematic director is along the x axis. (B and C) Top view and side view of two particles at closest distance, r = 215 nm, and angle φ = 0°, respectively. The anchoring strength is 10⁻⁵ J/m².

Fig. 3.

(A) Free energy for two particles, one of which is located at the boojum along the z axis, and the other is driven along the droplet surface. The polar angle θ controls particle separation. Inset pictures in the plot correspond to the final phase of systems at θ = 13° and θ = 170°. The field lines and color that represents the z component of director field show the position of two boojums. In the first minimum, at θ = 13°, the boojum adsorbs only one particle. At the angle 170° the opposite boojum diffuses to the particle to reduce the elastic free energy. Inset plot shows a local minimum at θ = 90°, which corresponds to the equator of the droplet. (B) Fluorescent micrographs of a 5CB droplet in water with four PS particles. Particles are localized at the positions predicted by simulations. (C) Localization of four particles at the droplet surface corresponding to the minimum free energy, in agreement with the experimental observation of B.

Fig. 4.

Arrangement of three particles at the droplet surface, one of which is located at the boojum; the other two particles are driven along the droplet surface. Two particles are constrained to have the same polar angle from the boojum. The polar angle θ controls the distance of the two particles with the last particle fixed at the boojum, and the difference of corresponding azimuthal angles Δϕ monitors the separation of two particles. (A) The free-energy profile indicates that the boojum strongly adsorbs particles to reduce the free energy. (B) Free energy as a function of Δϕ at θ = 13°, closest distance to the boojum. Two minima are observed at θ = 60 and 180, which correspond to an equilateral triangle and a line arrangement of the three particles, respectively. (C and D) Top view of two stable particle arrangements. The director field is illustrated with black lines, and color represents the z component of the director field. Insets in the figures correspond to fluorescence micrographs of 5CB droplets in water emulsions with three PS particles. (E and F) Field lines within the droplet for equilateral triangle and a line arrangement, respectively.

Fig. 5.

(A) Free-energy profile for four particles as a function of θ and ϕ. Three particles are fixed in a stable equilateral triangle arrangement, and the other one is driven along the droplet surface. (B and C) Top view of particle arrangements at the droplet surface. The director field is illustrated with black lines, and color represents the z component of the director field. (B) Diamond arrangement of four particles with minimum free energy. (C) A metastable arrangement for four particles. (D) Field lines within the droplet for the metastable arrangement. Insets correspond to fluorescent micrographs of 5CB droplets in water emulsions with four PS particles.

Fig. 6.

(A–C) Multiparticle arrangements at the pole of bipolar droplet. Insets in the figures correspond to fluorescence micrographs of 5CB droplets in water emulsions. (D and E) Two arrangements for eight particles: a diamond and a line arrangement. The diamond arrangement has 11 kT lower free energy than the line arrangement.

Fig. 7.

Hexagonal lattices arrangement of multiparticle at the pole of bipolar droplet, which covers the pole.

Fig. 8.

Top view of five particle arrangements at the pole of bipolar droplet. (A) Stable arrangement with minimum free energy. (B–D) Metastable arrangements have free energies that are 66, 142, and 153 kT higher than that of the stable arrangement, respectively. The director field is illustrated with black lines, and color represents the z component of director field. Insets in the figures correspond to the fluorescent micrographs of 5CB droplets in water emulsions. (E) Field lines within the droplet for arrangement in A. (F and G) Field lines within the droplet for pentameric ring pattern. (H) Field lines within the droplet for arrangement in D.