Elastic continuum analysis of the liquid crystal polarization grating

We apply elastic continuum theory to model critical parameters influencing the free-energy equilibrium configuration and the dynamic performance of a continuous and in-plane liquid crystal profile acting as a polarization grating. We present analytical expressions for the threshold voltage, critical thickness, and the dynamic switching times under strong anchoring conditions, negligible flow, and arbitrary splay, twist, and bend constants. We also study the influence of weak anchoring, and derive expressions describing a dramatic reduction of the critical thickness and voltage threshold, even for modest grating periods and surface anchoring strengths. Good correlation exists with previously reported experimental data, except in the dynamic response; we therefore show that flow effects (backflow and kickback) likely play an essential role in the fall times, presumably due to the prominent splay-bend deformation of the zero-field configuration. We consider the impact of surface pretilt, and validate our entire analysis with numerical simulations. The approximation technique we employ is likely broadly useful for many problems which include nano- or micropatterned surfaces.

Figure 1

(a) LCPG geometry. (b) Distorted LC configuration when $V>V_{th}$. (c) Definition of nematic director $\mathbf{n}$.

Figure 2

(a) Calculated maximum tilt angle ${\theta }_m$ for nematic MLC-6080 (and $\alpha =0.4$), showing the critical thickness $d_C$. (b) Total energy $W_{TOT}$ for both potential solutions. (c) Comparison of $d_C$ for various LC materials (Merck [20, 22]) analytically calculated using Eq. (10) as a function of $\alpha$ and numerically calculated with LC3D (bars). The solution $\alpha =0.4$ gives the best match.

Figure 3

Threshold voltage $V_{th}$ for the material MLC-6080 analytically calculated (curves) using Eq. (13) and numerically calculated (circles) using LC3D.

Figure 4

No-flow rise $\left(t_{on}\right)$ and fall $\left(t_{off}\right)$ times for MLC-6080 analytically calculated (curves) using Eq. (18) and numerically calculated (circles) using LC3D ($d=2\mathrm{\mu }\mathrm{m}$, $\Lambda =11\mathrm{\mu }\mathrm{m}$).

Figure 5

(a) LCPG nematic director with a nonzero pretilt ${\theta }_0=5°$ and $d∕\Lambda =0.5$, predicted by LC3D. (b) Maximum tilt angle ${\theta }_m$ for MLC-6080 analytically calculated using Eq. (19) (curves) and numerically calculated using LC3D (circles).

Figure 6

(a) Weak-anchoring critical thickness $d_C^{\prime }$ and (b) voltage threshold $V_{th}^{\prime }$ dependence on polar anchoring strength $W_p$ predicted by Eqs. (24, 25), for various periods $\Lambda$. The plots are normalized to unprimed quantities $d_C$ and $V_{th}$ (the strong anchoring thresholds). The solid circles in (a) show the point on each curve where $d_C=2\mathrm{\mu }\mathrm{m}$, the thickness used for the calculation in (b).