Colloidal interactions in a homeotropic nematic cell with different elastic constants

We propose a theoretical description of the interaction mediated by a nematic-liquid-crystal host with different Frank elastic constants. A general expression for the energy of such an interaction between colloidal particles of arbitrary size and shape suspended in a homeotropic cell is obtained. In the cells of large thickness, the presented potential converges to that found previously for small particles in the nematic bulk. In general, our results confirm the validity of the one-constant approximation for weakly elastically anisotropic nematic liquid crystals. For nematics with a high splay-to-bend ratio we predict a larger range of the interaction. Using the dependence of this range on the elastic constants, we show that there exists a qualitative similarity between the interactions in a nematic and in a smectic-$A$ phase. It manifests itself, in particular, in a decrease of the angle between a chain of quadrupole particles and the uniform far-field director across a nematic–smectic-$A$ phase transition. We also demonstrate that the anisotropy of the elastic constants can lead to the formation of thermodynamically stable linear superstructures of asymmetric particles (elastic monopoles) with large, compared to usual dipole chains, interparticle distances.

Figure 1

Coordinate system used in this study. For a given wave vector $\mathbf{q}$ the basis $({\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3)$ is rotated with respect to the fixed basis $({\mathbf{e}}_x,{\mathbf{e}}_y,{\mathbf{e}}_z)$ by the angle ${\varphi }_{\mathbf{q}}$ around the axis ${\mathbf{e}}_z\parallel {\mathbf{e}}_3$, which is normal to the plane of the drawing, in such a way that ${\mathbf{e}}_1\parallel {\mathbf{q}}_{\perp }$.

Figure 2

Interaction between two monopoles $q_x$ and $q_x^{\prime }$ of opposite sign. The particles are placed in the middle of the cell $\left(z=z^{\prime }=L/2\right)$. (a) Map of the interaction. Arrow lines indicate the local direction of the force $\mathbf{F}=-\mathbf{\nabla }{U}_{\text{qq}}$. The shaded region is a repulsion zone $\frac{\partial U_{\text{qq}}}{\partial \rho }<0$. The stars indicate minima of the energy. Here ${\kappa }_1/{\kappa }_3=3.5$ and ${\kappa }_2/{\kappa }_3=0.35$ [25]. (b) Energy of the interaction along the $y$ axis $\left(\phi =\pi /2\right)$. Here $q_x=q_x^{\prime }=d,d=3\mu \mathrm{m},L=12\mu \mathrm{m},K_3=0.88\mathrm{pN}$, and $K=1.4\mathrm{pN}$, the average elastic constant of C1Pbis10BB [25].

Figure 3

Dimensionless energy of the dipole-dipole interaction $V_{\text{pp}}=U_{\text{pp}}{L}^3{\kappa }_1^2/16\pi Kp{p}^{\prime }{\kappa }_3$ as a function of the particle separation $\rho$. At short distances $\rho \lesssim L\sqrt{{\kappa }_1/{\kappa }_3}$ the energy $U_{\text{pp}}\propto 1/{\rho }^3$. When $\rho \gtrsim L\sqrt{{\kappa }_1/{\kappa }_3}$ the interaction decays exponentially. Note that, although the curve ${\kappa }_1>{\kappa }_3$ lies above the others, the corresponding energy of the interaction is not necessarily higher as it depends on the absolute values of the elastic constants.

Figure 4

Dependence of the angle ${\theta }_{\text{min}}^{\text{th}}$ that minimizes the energy of the quadrupole-quadrupole interaction (25) on the ratio $K_3/K_1$. In the limit of vanishing $K_3/K_1$ the angle ${\theta }_{\text{min}}^{\text{th}}$ approaches (but does not reach) ${90}^{\circ }$. Similarly, if $K_3/K_1\to \infty$, then ${\theta }_{\text{min}}^{\text{th}}$ approaches (but does not reach) $0^{\circ }$.

Figure 5

Quadrupole interaction between two beads with tangential anchoring in the cell filled with 5CB. Here $Q=Q^{\prime }=-0.4a^3$ and $\tilde{Q}={\tilde{Q}}^{\prime }=-0.31a^3$, with $a=D/2=2.2\mu \mathrm{m},L=8\mu \mathrm{m},K_1=6.2\mathrm{pN},K_2=3.7\mathrm{pN},K_3=8.3\mathrm{pN}$ [38], and $K=6.1\mathrm{pN}$. The points depict experimental data from [22].