Nematic liquid crystals in a spatially step-wise magnetic field

We study the molecular reorientation induced by a textured external field in a nematic liquid crystal (nLC). In particular, we consider an infinitely wide cell with strong planar anchoring boundary conditions, subjected to a spatially periodic piecewise magnetic field. In the framework of the Frank's continuum theory, we use the perturbation analysis to study in detail the field-induced splay-bend Fréedericksz transition. A numerical approach, based on the finite differences method, is instead employed to solve the fully nonlinear equations. At high field strengths, an analytic approach allows us to draw the bulk profile of the director in terms of elliptic integrals. Finally, through the application of the Bruggeman texture hydrodynamics theory, we qualitatively discuss on the LCs piecewise director configuration under sliding interfaces, which can be adopted to actively regulate friction. Our study opens the pathway for the application of highly controlled nLC texturing for tribotronics.

Figure 1

Schematic representation of the nematic cell.

Figure 2

Qualitative behavior of the roots of the threshold equation $j=0$.

Figure 3

Schematic representation of the post-buckling director alignment. The vertical external field is applied in the shaded region.

Figure 4

${\omega }_{\mathrm{cr}}d/\pi$, solution of Eqs. (18a) (solid lines) and (18b) (dashed lines) as a function of the aspect ratio $d/L$, for $\overline{x}/L=0.1,0.25$, and $0.5$ (from top to bottom).

Figure 5

Numerical solution of Eq. (18a) (with $L/d=10$) compared with the analytical approximated solution Eqs. (20) and (21). The inset represents the magnification of the region close to $\overline{x}/d=10$.

Figure 6

The predictions of Eqs. (23) and (27) compared with the numerical results, at a constant texture ratio $p_h=\overline{x}/L=0.2$. For several aspect ratios, (a) $d/L=2$, (b) $d/L=0.2$, (c) $d/L=0.02$.

Figure 7

The predictions of Eqs. (23) and (27) compared with the numerical results, at a constant texture ratio $p_h=\overline{x}/L=0.5$. For several aspect ratios, (a) $d/L=5$, (b) $d/L=0.5$, (c) $d/L=0.05$.

Figure 8

Profile of $\theta$ at $z=d/2$ for several values of the strength field. Comparison between the numerical solutions (solid lines) with the linear analytical solutions Eq. (22) (dashed lines).

Figure 9

Profile of $\theta$ at $z=d/2$ for several values of the strength field. Comparison between the numerical solutions (solid lines) and the nonlinear solution Eqs. (31) and (32) (dashed lines).

Figure 10

Schematic of a macroscopic contact pair (assumed, for simplicity, infinitely wide in the out-of-plane direction) in a sliding interaction. Both surfaces are macroscopically smooth; however, the top surface is characterized by a microtexturing on the inlet side. We consider two texture cases, namely (i) a microstructural patterning, constituted by an ordered repetition of microgrooves perpendicular to the sliding direction, and (ii) a microstructural (as before) and microviscosity patterning. In the calculations below we have assumed that the textured side is $1/2$ of the total bearing length, whereas the in-plane size of the generic texture defect is $1/2$ of the patterning lattice. In the BTH theory [1] the viscosity is assumed constant across and along the gap under electrodes, an assumption that is not quite in agreement with the theoretical results provided above.

Figure 11

Normalized load $F_{\mathrm{N}}/F_{\mathrm{N},\mathrm{opt}}$ as a function of the normalized separation $h_0/h_{0\mathrm{opt}}$, for an optimized bearing (i) microstructural patterning (black line) and (ii) microstructural and microviscosity patterning (dashed line).

Figure 12

Normalized friction $F_{\mathrm{N}}/F_{\mathrm{N},\mathrm{opt}}$ as a function of the normalized separation $h_0/h_{0\mathrm{opt}}$, for an optimized bearing (i) microstructural patterning (black line) and (ii) microstructural and microviscosity patterning (dashed line).

Figure 13

LCs normalized viscosity (minimizing the sliding friction) as a function of the normalized load, corresponding to the bearing load curve of Fig. 11 (dashed line) and to the friction curve of Fig. 13 (dashed line).