Interactions of carbon nanotubes in a nematic liquid crystal. I. Theory

Elongated and rodlike objects such as carbon nanotubes (CNTs) are studied when immersed in a nematic liquid crystal. Their interaction energy in a uniform nematic field depends on their orientation relative to the director $\mathit{n}$, and its minimum determines if they stabilize parallel or perpendicular to $\mathit{n}$. Using free energy calculations, we deduce the orientation at equilibrium that they choose in a uniform director field $\mathit{n}$ or when they are in contact with a splay-bend disclination line. Naturally, the CNT orientations also depend on the anchoring conditions at their surface. Essentially, three types of anchorings are considered, planar, homeotropic, and Janus anchorings in the cases of weak and strong anchoring strengths. In the presence of a splay-bend disclination line, they are attracted toward it and ultimately, they get out of the colloidal dispersion to stick on it. Their orientation relative to the line is found to be parallel or perpendicular to it, again depending on the anchoring conditions. When a sufficient number of particles are deposited on a disclination line, we finally obtain a micro- or nanonecklace in the shape of a thin thread or of a bottle brush, according to the CNTs being oriented parallel or perpendicular to the disclination line, respectively. The system exhibits a rich versatility even if up to now the weak anchorings appear to be difficult to control. As discussed in the associated experimental paper, these necklaces could be a step toward interesting applications for realizing nanowires self-connected in three dimensions to predesignated electrodes. This method could provide a way to increase the number of transistors that may be connected together on a small volume.

Figure 1

Schematic representation of the $\mathit{n}$ distortion around a CNT aligned parallel to the director $\mathit{n}$ in a uniformly oriented nematic LC. The oval red dots depict the two boojums on each pole of the CNT.

Figure 2

Cross sections of a CNT with homeotropic anchoring conditions, and oriented parallel to $\mathit{n}$. The whole director field is symmetric around the horizontal axis. Being topologically equivalent to a +1 defect, the CNT is accompanied (a) with a −1 point defect at a distance R from the tip, (b) that may open up into a Saturn ring of $-\frac{1}{2}$ disclination line shown as red dots in the cross section, and represented close to the CNT center, this symmetric configuration being probably the stable one.

Figure 3

Director field around a CNT with homeotropic anchoring conditions, introduced in the nematic perpendicularly to $\mathit{n}$. (a) Cross section perpendicular to the CNT, and (b) in the plane of both the CNT axis and of $\mathit{n}$.

Figure 4

Cross sections of a Janus CNT and its director field in the case where (a) the CNT axis is parallel to $\mathit{n}$, and (b) perpendicular to $\mathit{n}$. The yellow (white) parts indicate the areas treated for a homeotropic (planar) anchoring, respectively. The oval dots mark surface disclination lines. The circular dot at a distance R from the CNT tip is a $-\frac{1}{2}$ point defect.

Figure 5

(a) Pure splay-bend disclination line perpendicular to the figure (red dot). The pure splay (α) and pure bend (β) areas belong to the symmetry plane that contains the disclination line. (b) CNT with planar anchoring, located in the area α, and perpendicular to the disclination. (c) CNT perpendicular to the disclination line, positioned close to the area β, but out of the symmetry plane. The boojums on the CNT tips are not shown.

Figure 6

CNT in interaction with a pure splay-bend disclination line perpendicular to the figure. (a) Case of a strong homeotropic anchoring. The CNT (black open circle) is located at a distance $d<L/2$ in the domain α below the disclination line (red dot). The supplementary distortion that is added to the initial distortion of the disclination when introducing a CNT is roughly contained inside the hatched cylinder of diameter $d$ and length L. (b) Case of a weak homeotropic anchoring. The anchoring breaks on the surface of the CNT are marked in red (gray). This allows the disclination line to vanish inside the CNT. (c) Case of a Janus anchoring. The disclination line vanishes inside the CNT as in (b), but without breaking the anchoring. The lateral disclination lines along the CNT surface are not shown.