Nanoparticle localization within chiral liquid crystal defect lines and nanoparticle interactions

Self-assembly of colloidal particles into predefined structures is a promising way to design inexpensive manmade materials with advanced macroscopic properties. Doping of nematic liquid crystals (LCs) with nanoparticles has a series of advantages in addressing these grand scientific and engineering challenges. It also provides a very rich soft matter platform for the discovery of unique condensed matter phases. The LC host naturally allows the realization of diverse anisotropic interparticle interactions, enriched by the spontaneous alignment of anisotropic particles due to the boundary conditions of the LC director. Here we demonstrate theoretically and experimentally that the ability of LC media to host topological defect lines can be used as a tool to probe the behavior of individual nanoparticles as well as effective interactions between them. LC defect lines irreversibly trap nanoparticles enabling controlled particle movement along the defect line with the use of a laser tweezer. Minimization of Landau–de Gennes free energy reveals a sensitivity of the ensuing effective nanoparticle interaction to the shape of the particle, surface anchoring strength, and temperature, which determine not only the strength of the interaction but also its repulsive or attractive character. Theoretical results are supported qualitatively by experimental observations. This work may pave the way toward designing controlled linear assemblies as well as one-dimensional crystals of nanoparticles such as gold nanorods or quantum dots with tunable interparticle spacing.

Figure 1

(a) Polarized optical micrographs of a topological line defect decorating a cholesteric finger. (b) Director configuration in the vertical cross section of the cholesteric finger with two twist line defects, highlighted by red. The depicted structure is invariant along the normal to the cross section. The two right panels illustrate schematically the configurations of the director field (blue rods) around the defect lines at the top and bottom of the cross section. (c), (d) Director configurations (shown by black rods) around a twist disclination line as obtained by numerical minimization of the Landau–de Gennes free energy. The red tube in (c) encloses the region of a reduced nematic scalar order parameter $Q\le 0.25.$ The configuration (c) is taken in the $x=0$ plane, where the center of the defect core is located. The cross-sectional configuration depicted in (d) is taken in the $y=0$ plane which is perpendicular to the line defect. This configuration is translationally invariant along the defect line. Black bars at the top of (c), (d) schematically illustrate the confining surfaces which impose rigid homeotropic boundary conditions on the nematic director. The green line in (d) indicates the location of the twist wall.

Figure 2

Elastic entrapment of colloidal particles within the cores of twist disclination lines. Numerically calculated Landau–de Gennes free energy as a function of the separation $d_{cl}$ between the colloidal particle and the core of the disclination line obtained at two values of the radius $R$ of the colloidal particle: (a) $R\approx 1.7\xi$; (b) $R\approx 3.4\xi$, where $\xi$ is the nematic coherence length defined in Sec. 2. Insets show configurations of the colloidal particle (green spheres) at the disclination lines (red tubes enclosing the regions of the reduced nematic scalar order parameter $Q\le 0.25$). The configurations correspond to data points indicated by black arrows. The results are obtained at the values of the model parameters specified in Sec. 2.

Figure 3

Effective interactions between pairs of colloidal spheres and cubes entrapped within the core of the twist disclination lines. Landau–de Gennes free energy $F_{\mathrm{LdG}}$ [see Eq. (1) in Sec. 2] as a function of the center-to-center distance $d_{\mathrm{cc}}$ between two spherical (a) and cubic (b) particles. Different symbols and fitting curves correspond to different values of temperature quantified by parameter $A$; see Eq. (1) in Sec. 2. Symbols in (a), (b) represent numerically calculated values; solid curves are guides to the eye. In both cases a finite homeotropic anchoring is assumed at the surfaces of the particles, characterized by the surface extrapolation length $K/W\approx {10}^{-3}\mu \mathrm{m}.$ The radius of the spheres, and the side length of the cubes $R\approx 3.4\xi .$ Insets in (a), (b) show superimposed $F_{\mathrm{LdG}}$ and Coulomb repulsive potential assuming that each of the colloids carries an electrostatic charge of $10e$ (a), and $20e$ (b), where $e$ is the elementary charge. (c)–(f) Cross-sectional representation of the nematic director (black rods) around the colloidal particles (depicted by blue surfaces) at two selected separations between the colloids and at low (c), (d), and high (e), (f) values of temperature. Red tubes enclose the regions of the reduced nematic scalar order parameter $Q\le 0.25$ at $A=0.01$, and $Q\le 0.2$ at $A=0.06$. (g) Transmission electron microscope (TEM) micrographs of nanocubes (size ≈20 nm) deposited onto a copper grid. (h), (i) Polarized optical micrographs of a cholesteric finger with line defect and with nanocubes trapped in the singular defect core without (h) and with (i) a retardation plate. (j) Luminescence based confocal microscopy images of the nanocubes in the line defect.

Figure 4

Numerically calculated surface anchoring free energy $F_{\mathrm{Anch}}$ [see Eq. (2) in Sec. 2] as a function of the center-to-center distance $d_{\mathrm{cc}}$ between two spherical (a) and cubic (b) colloidal particles. Different symbols correspond to different values of temperature as given by parameter $A$; see Eq. (1) in Sec. 2. The particle homeotropic anchoring is characterized by the surface extrapolation length $K/W\approx {10}^{-3}\mu \mathrm{m}.$

Figure 5

Numerically calculated Landau–de Gennes free energy $F_{\mathrm{LdG}}$ (b), (d), and surface anchoring free energy $F_{\mathrm{Anch}}$ (a), (c) as functions of the center-to-center distance $d_{\mathrm{cc}}$ between two spherical (a), (b) and cubic (c), (d) colloidal particles. Different symbols correspond to different values of temperature as given by parameter $A$; see Eq. (1) in Sec. 2. The particle homeotropic anchoring is characterized by the surface extrapolation length $K/W\approx 2\times {10}^{-2}\mu \mathrm{m}.$ The radius of the spheres, and the side length of the cubes $R\approx 3.4\xi .$ (e)–(h) Cross-sectional representation of the nematic director (black rods) around the colloidal particles (blue surfaces) at two selected separations between the colloids and at low (e), (g), and high (f), (h) values of temperature. Red tubes enclose the regions of the reduced nematic scalar order parameter, $Q\le 0.25$ at $A=0.01$, and $Q\le 0.2$ at $A=0.06.$

Figure 6

Effective interactions between pairs of spherocylinders entrapped within the core of the twist disclination lines. Landau–de Gennes free energy $F_{\mathrm{LdG}}$ [see Eq. (1) in Sec. 2] as a function of the center-to-center distance $d_{\mathrm{cc}}$ between two spherocyliners with the aspect ratio $\kappa =1.3$ (a) and $\kappa =1.65$ (b), the radius of the spherical caps, and the central cylindrical parts $R\approx 3.4\xi .$ Different symbols and fitting curves correspond to different values of temperature as given by parameter $A$; see Eq. (1) in Sec. 2. Symbols in (a), (b) represent numerically calculated values; solid curves are guides to the eye. In both cases a finite homeotropic anchoring is assumed at the surfaces of the particles, characterized by the surface extrapolation length $K/W\approx {10}^{-3}\mu \mathrm{m}.$ Insets in (a), (b) show superimposed $F_{LdG}$ and Coulomb repulsive potential assuming that each of the particles carries an electrostatic charge of $10e$ (a), and $20e$ (b), where $e$ is the elementary charge. (c)–(f) Cross-sectional representation of the nematic director (black rods) around the colloidal particles (blue surfaces) at two selected separations between the colloids and at low (c), (d), and high (e), (f) values of temperature. Red tubes enclose the regions of the reduced nematic scalar order parameter $Q\le 0.25$ at $A=0.01$, and $Q\le 0.2$ at $A=0.06.$ (g) Schematics of the GNR showing the silica shell and DMOAP surfactant monolayer; real GNRs have the size $\approx 40\times 60\mathrm{nm}$. (h) Variation of center-to-center spacing with time when two GNRs were moved close to each other with the help of an optical tweezer and then released, demonstrating repulsion between the nanoparticles at $T=27{}^{\circ }\mathrm{C}$ and $45{}^{\circ }\mathrm{C}$. Inset shows the dark-field optical micrograph of two GNRs separated by ∼1 µm inside the line defect when they are moved close to each other with the help of an optical tweezer. (i) Interaction potential between two GNRs at $T=27{}^{\circ }\mathrm{C}$ and $45{}^{\circ }\mathrm{C}$ calculated based on the curves in (h). Inset shows the dark-field optical micrographs of two GNRs trapped inside a line defect after releasing the particles from the optical traps. (j) Dark-field optical micrograph, showing the assemblies of individual GNRs inside line defects. (k), (l) Polarized optical micrographs of the assemblies of individual GNRs inside the line defect.

Figure 7

Effective interactions between pairs of spherocylinders entrapped within the core of the twist disclination lines. Landau–de Gennes free energy $F_{\mathrm{LdG}}$ as a function of the center-to-center distance $d_{\mathrm{cc}}$ between two spherocyliners with the aspect ratio $\kappa =2$ (a) and $\kappa =3$ (b), the radius of the spherical caps, and the central cylindrical parts $R\approx 3.4\xi .$ Different symbols and fitting curves correspond to different values of temperature as given by parameter $A$; see Eq. (1) in Sec. 2. Symbols in (a), (b) represent numerically calculated values; solid curves are guides to the eye. In both cases a finite homeotropic anchoring is assumed at the surfaces of the particles, characterized by the surface extrapolation length $K/W\approx {10}^{-3}\mu \mathrm{m}$. Insets in (a), (b) show superimposed $F_{\mathrm{LdG}}$ and Coulomb repulsive potential assuming that each of the particles carries an electrostatic charge of $20e$, where $e$ is the elementary charge. (c) TEM micrograph of upconversion nanorods $(20\mathrm{nm}\times 1.6\mu \mathrm{m})$ deposited on a copper grid. (d)–(f) Polarized optical micrographs of upconversion nanorods trapped inside line defects without (d), (f), and with (e) a retardation plate. (g) Luminescence based confocal microscopy image of upconversion nanorods trapped inside the line defect.

Figure 8

Effective interactions between pairs of colloidal platelets entrapped within the core of the twist disclination lines. Landau–de Gennes free energy $F_{\mathrm{LdG}}$ as a function of the center-to-center distance $d_{\mathrm{cc}}$ between two platelets with the aspect ratio $\kappa =5$ and face-to-face (a), and edge-to-edge (b) orientation. The side length of the square $R\approx 3.4\xi$. Surface anchoring free energy $F_{\mathrm{Anch}}$ obtained for face-to-face (c), and edge-to-edge (d) platelet alignment. Symbols in (a), (b) represent numerically calculated values; solid curves are guides to the eye. A finite homeotropic anchoring is assumed at the surfaces of the particles, characterized by the surface extrapolation length $K/W\approx {10}^{-3}\mu \mathrm{m}$. (e)–(h) Cross-sectional representation of the nematic director (black rods) around the platelets (blue surfaces) at two selected separations between the colloids. Red tubes enclose the regions of the reduced nematic scalar order parameter $Q\le 0.25$. (i) Schematic of CdSe nanoplate with dimensions marked in the illustration. (j) Confocal fluorescence microscopy image of CdSe nanoplates inside the line defect. (k), (l) polarized optical micrographs of CdSe nanoplatelets inside the line defects without (k) and with (l) retardation plate.

Figure 9

Effective interactions between pairs of colloidal platelets entrapped within the core of the twist disclination lines and aligned edge to edge and with the aspect ratio $\kappa =5$ and the side length of the square $R\approx 3.4\xi$. Landau–de Gennes free energy $F_{\mathrm{LdG}}$ (b) and the surface anchoring free energy $F_{\mathrm{Anch}}$ (a) as functions of the center-to-center distance $d_{\mathrm{cc}}$ between platelets. Symbols in (b) represent numerically calculated values; solid curves are guides to the eye. A finite homeotropic anchoring is assumed at the surfaces of the particles, characterized by the surface extrapolation length $K/W\approx 2\times {10}^{-2}\mu \mathrm{m}$. (e)–(h) Cross-sectional representation of the nematic director (black rods) around the platelets (blue surfaces) at two selected separations between the colloids. Red tubes enclose the regions of the reduced nematic scalar order parameter $Q\le 0.25.$