Field-triggered vertical positional transition of a microparticle suspended in a nematic liquid crystal cell

In this paper, based on the numerical calculation of total energy utilizing the Green's function method, we investigate how a field-triggered vertical positional transition of a microparticle suspended in a nematic liquid crystal cell is influenced by the direction of the applied field, surface anchoring feature, and nematic's dielectric properties. The new equilibrium position of the translational movement is decided via a competition between the buoyant force and the effective force built on the microparticle by the elastic energy gradient along the vertical direction. The threshold value of external field depends on thickness $L$ and Frank elastic constant $K$ and slightly on the microparticle size and density, in a Fréedericksz-like manner, but by a factor. For a nematic liquid crystal cell with planar surface alignment, a bistable equilibrium structure for the transition is found when the direction of the applied electric field is (a) perpendicular to the two plates of the cell with positive molecular dielectric anisotropy or (b) parallel to the two plates and the anchoring direction of the cell with negative molecular dielectric anisotropy. When the electric field applied is parallel to both plates and perpendicular to the anchoring direction, the microparticle suspended in the nematic liquid crystal tends to be trapped in the midplane, regardless of the sign of the molecular dielectric anisotropy. Such a phenomenon also occurs for negative molecular dielectric anisotropy if the external field is applied perpendicular to the two plates. Explicit formulas proposed for the critical electric field agree extremely well with the numerical calculation.

Figure 1

Sketch of a microparticle suspended in a nematic cell with (a) homeotropic anchoring and (b) planar anchoring in the presence of an external field. The five cases of external fields applied are shown by red arrows in the two coordinate frames.

Figure 2

Dependence of $E_c$ and $\sqrt{K}/L$ for different densities of microparticle (0.99, 1.0, and 1.03 $\mathrm{g}{\mathrm{cm}}^{-3}$).

Figure 3

Total energy profile as a function of the suspended microparticle position for an NLC cell with planar anchoring in the presence of different electric fields perpendicular to the two plates.

Figure 4

Equilibrium position $x_0$ in response to electric field for different (a) cell thicknesses (8, 9, 10, and 11 $\mu \mathrm{m}$) with Frank elastic constant $K=7\mathrm{pN}$ and the radius of microparticle $r=2.2\mu \mathrm{m}$, and (b) Frank elastic constants (8, 9, 10, and 11 $\mathrm{pN}$) with cell thickness $L=10\mu \mathrm{m}$ and radius of microparticle $r=2.2\mu \mathrm{m}$.

Figure 5

Equilibrium position $x_0$ for different (a) radii (2.2, 2.35, 2.5, and 3.0 $\mu \mathrm{m}$) and (b) densities (0.99, 1.0, and 1.04 $\mathrm{g}{\mathrm{cm}}^{-3}$) of a microparticle with $K=7$ pN and $L=7\mu \mathrm{m}$, showing the same critical value $E_c$ of electric field triggering positional transition. The dependence of $E_c$ on $\sqrt{K}/L$ for different (c) radii (2.2, 2.35, 2.5, and 3.0 $\mu \mathrm{m}$) and (d) densities (0.99, 1.0, and 1.04 $\mathrm{g}{\mathrm{cm}}^{-3}$) of the microparticle, obeying strictly a master curve which can be given by the theoretical prediction Eq. (6).

Figure 6

Total energy profile as a function of the suspended microparticle position for an NLC cell with planar anchoring in the presence of four chosen electric fields parallel to the two plates and the anchoring direction as well.

Figure 7

Equilibrium position $x_0$ in response to electric field for different (a) cell thicknesses (8, 10, 12, and 15 $\mu \mathrm{m}$), where the Frank elastic constant and the radius of microparticle are set as $K=7$ pN and $r=2.5\mu \mathrm{m}$, and (b) Frank elastic constants (8, 10, 12, and 15 pN), where the cell thickness and the radius of microparticle are set as $L=10\mu \mathrm{m}$ and $r=2.5\mu \mathrm{m}$.

Figure 8

(a) Equilibrium position $x_0$ for different radii of microparticle with $K=7$ pN and $L=8\mu \mathrm{m}$, showing the same critical value $E_c$ of electric field triggering positional transition. (b) $K=8$ pN and $L=10\mu \mathrm{m}$. The dependence of $E_c$ and $\sqrt{K}/L$ for different (c) radii (2.2, 2.35, 2.5, and 3.0 $\mu \mathrm{m}$) and (d) densities (0.99, 1.0, and 1.03 $\mathrm{g}{\mathrm{cm}}^{-3}$) of microparticle, obeying strictly a master curve given by theoretical prediction Eq. (7).

Figure 9

Total energy profile as a function of the suspended microparticle position for different external electric fields.

Figure 10

Equilibrium position $x_0$ in response to electric field for different (a) cell thicknesses (7, 8, 9, and $10\mu \mathrm{m}$), where the Frank elastic constant and the radius of microparticle are set as $K=7$ pN and $r=2.5\mu \mathrm{m}$, and (b) Frank elastic constants (8, 9, 10, and 11 pN), where the cell thickness and the radius of microparticle are set as $L=10\mu \mathrm{m}$ and $r=2.5\mu \mathrm{m}$.

Figure 11

Equilibrium position $x_0$ for different (a) radii (2.2, 2.35, 2.5, and 3.0 $\mu \mathrm{m}$); (b) densities (0.99, 1.0, and 1.02 $\mathrm{g}{\mathrm{cm}}^{-3}$) of microparticle with $K=8$ pN and $L=10\mu \mathrm{m}$, showing the same critical value $E_c$ of electric field triggering positional transition. The dependence of $E_c$ and $\sqrt{K}/L$ for different (c) radii (2.2, 2.35, 2.5, and 3.0 $\mu \mathrm{m}$) and (d) densities (0.99, 1.0, and 1.02 $\mathrm{g}{\mathrm{cm}}^{-3}$) of microparticle, obeying strictly a master curve given by theoretical prediction Eq. (8).