Anisotropic nanoparticles immersed in a nematic liquid crystal: Defect structures and potentials of mean force

We report results for the potential of mean force (PMF) and the defect structures that arise when spherocylindrical nanoparticles are immersed in a nematic liquid crystal. Using a dynamic field theory for the tensor order parameter $\mathbf{Q}$ of the liquid crystal, we analyzed configurations, including one, two, and three elongated particles, with strong homeotropic anchoring at their surfaces. For systems with one nanoparticle, the most stable configuration is achieved when the spherocylinder is placed with its long axis perpendicular to the far-field director, for which the defect structure consists of an elongated Saturn ring. For systems with two or three nanoparticles with their long axes placed perpendicular to the far-field director, at small separations the defect structures consist of incomplete Saturn rings fused with new disclination rings orthogonal to the original ones, in analogy to results previously observed for spherical nanoparticles. The shape of these orthogonal rings depends on the nanoparticles’ configuration, i.e., triangular, linear, or parallel with respect to their long axis. A comparison of the PMFs indicates that the latter configuration is the most stable. The stability of the different arrays depends on whether orthogonal disclination rings form or not, their size, and the curvature effects in the interparticle regions. Our results suggest that the one-elastic-constant approximation is valid for the considered systems; similar results were obtained when a three-constant expression is used to represent the elastic free energy. The attractive interactions between the elongated particles were compared to those observed for spheres of similar diameters. Similar interparticle energies were observed for linear arrays; in contrast, parallel and triangular arrays of spherocylinders yielded interactions that were up to 3.4 times stronger than those observed for spherical particles.

Figure 1

Scheme of the model system for one anisotropic nanoparticle.

Figure 2

(Color online) 3D visualizations of the defect structures (represented as the contour $S=0.25$, top), and contour maps of the scalar order parameter $S$ superimposed with the director field in the $x\text{−}z$ plane (bottom), for (a) $\theta =0°$, (b) $\theta =60°$, and (c) $\theta =90°$.

Figure 3

Potential of mean force (PMF) for one nanoparticle immersed in a nematic liquid crystal, as a function of the angle $\theta$ with respect to the $x$ axis. (a) Total PMF; (b) Landau–de Gennes (LdG, squares), and elastic (circles) contributions to the PMF.

Figure 4

Potential of mean force (PMF) for two spherocylindrical nanoparticles immersed in a nematic liquid crystal, as a function of the minimum surface-to-surface interparticle distance $d$. (a) Total PMF; (b) Landau–de Gennes contribution; and (c) elastic contribution. Squares: linear array, diamonds: parallel $x\text{−}y$ array, circles: parallel $x\text{−}z$ array. Visualizations of the configurations corresponding to the points colored in grey and black are presented in Figs. 5, 6, respectively.

Figure 5

(Color online) 3D visualizations of the defect structures (represented as the contour $S=0.25$, top), and contour maps of the scalar order parameter $S$ superimposed with the director field in different planes (bottom), for two elongated nanoparticles in (a) linear array, $d=10.9\mathrm{nm}$, (b) parallel $x\text{−}y$ array, $d=15.6\mathrm{nm}$, and (c) parallel $x\text{−}z$ array, $d=3.6\mathrm{nm}$. These visualizations correspond to the maxima in the Landau–de Gennes contribution to the PMF (the points colored in grey in Fig. 4).

Figure 6

(Color online) 3D visualizations of the defect structures (represented as the contour $S=0.25$, top), and contour maps of the scalar order parameter $S$ superimposed with the director field in different planes (bottom), for two elongated nanoparticles in (a) linear array, $d=2.9\mathrm{nm}$, (b) parallel $x\text{−}y$ array, $d=3.6\mathrm{nm}$, and (c) parallel $x\text{−}z$ array, $d=3.6\mathrm{nm}$. These visualizations correspond to points close to the minima in the elastic contribution to the PMF (the points colored in black in Fig. 4). The visualizations for the parallel $x\text{−}z$ array are equivalent to those presented in Fig. 5c, since for this array the minimum and the maximum in the elastic and Landau–de Gennes contributions to the PMF occur at the same distance $d$.

Figure 7

(Color online) Potential of mean force (PMF) for two spherocylindrical nanoparticles immersed in a nematic liquid crystal, as a function of the minimum surface-to-surface interparticle distance $d$. (a) Total PMF; (b) Landau–de Gennes contribution; and (c) elastic contribution. Diamonds, squares, and circles represent values for the parallel $x\text{−}y$, linear, and parallel $x\text{−}z$ arrays, respectively. Open symbols: one-elastic-constant approximation; filled symbols: three-elastic-constant expression of Edwards et al. [38, 39, 40].

Figure 8

(Color online) Elastic part of the potential of mean force (PMF) for two spherocylindrical nanoparticles immersed in a nematic liquid crystal, as a function of the minimum surface-to-surface interparticle distance $d$. (a) Parallel $x\text{−}y$ array; (b) linear array; and (c) parallel $x\text{−}z$ array. White triangles: one-elastic-constant approximation; black triangles: total elastic PMF from the three-elastic-constant expression of Edwards et al. [38, 39, 40]. Diamonds, squares, and circles represent the partial contributions from the $L_1$, $L_2$, and $L_3$ terms.

Figure 9

Potential of mean force (PMF) for three spherocylindrical nanoparticles immersed in a nematic liquid crystal, as a function of the minimum surface-to-surface interparticle distance $d$. (a) Total PMF; (b) Landau–de Gennes contribution; and (c) elastic contribution. Squares, linear array; diamond, parallel $x\text{−}y$ array; circles, parallel $x\text{−}z$ array; triangles, triangular array. Visualizations of the configurations corresponding to the points colored in grey and black are presented in Figs. 10, 11, respectively.

Figure 10

(Color online) 3D visualization of the defect structure (represented as the contour $S=0.25$, top), and contour map of the scalar order parameter $S$ superimposed with the director field in the $x\text{−}y$ plane (bottom), for three elongated nanoparticles in a triangular array, $d=11.6\mathrm{nm}$. These visualizations correspond to the maxima in the Landau–de Gennes contribution to the PMF (the point colored in grey in Fig. 7).

Figure 11

(Color online) 3D visualizations of the defect structures (represented as the contour $S=0.25$), for three elongated nanoparticles in (a) linear array, $d=2.4\mathrm{nm}$, (b) parallel $x\text{−}y$ array, $d=3.1\mathrm{nm}$, (c) parallel $x\text{−}z$ array, $d=3.1\mathrm{nm}$, and (d) triangular array, $d=2.1\mathrm{nm}$. A contour map of the scalar order parameter $S$ superimposed with the director field in the $x\text{−}y$ plane is also included for the triangular array. These visualizations corresponds to points close to the minima in the PMF curves (the points colored in black in Fig. 9).