Freedericksz instability for the twisted nematic device: A three-dimensional analysis

Of interest here is the fully three-dimensional analysis of the Freedericksz transition for the twisted nematic device (TND), which is widely used in liquid-crystal display monitors. Using a coupled electromechanical variational formulation, the problem is treated as a bifurcation instability triggered by an externally applied electric field. More specifically, we study a finite thickness liquid-crystal layer, anchored between two infinite parallel plates relatively rotated with respect to each other by a given twist angle and subjected to a uniform electric field perpendicular to these bounding plates. The novelty of the proposed analysis lies in the fully three-dimensional formulation of the TND problem that considers all possible bounded perturbations about the principal solution. By scanning a wide range of the liquid crystal's material parameter space, we establish whether the Freedericksz transition is global, i.e., has an eigenmode depending solely on the layer thickness coordinate, or local (also termed the periodic Freedericksz transition), i.e., has an eigenmode with finite wavelengths in one or both directions parallel to the plate. It is found that global modes are typical for low values, while local modes appear at large values of the twist angle. Moreover, for certain TND's, the increase in twist angle can lower the critical electric field, findings that could be useful in guiding liquid-crystal selection for applications.

Figure 1

A typical twisted nematic device with a ${90}^{\circ }$ twist angle. In (a) the strength of the electric field is below its critical value and the nematic molecules in each $x_3=\text{const}$ plane are parallel to the bounding plates, while in (b) the strength of the electric field is above its critical value and the molecules in each $x_3=\text{const}$ plane rotate out of their plane, tending to align with the externally applied electric field.

Figure 2

Contours of the lowest dimensionless electric field ${\zeta }_m$ at the onset of bifurcation, as a function of the dimensionless eigenmode wave numbers ${\omega }_1$ and ${\omega }_2$, for four different values of the twist angle $\mathrm{\Delta }\varphi$ calculated for a 5CB liquid crystal. Its lowest (critical) value ${\zeta }_c$ occurs at the location indicated by a small (red) dot; notice that for $\mathrm{\Delta }\varphi =0,\pi /2$, the critical electric field corresponds to a long-wavelength eigenmode $({\omega }_{1c},{\omega }_{2c})=(0,0)$, while for $\mathrm{\Delta }\varphi =\pi ,3\pi /2$, the minimum corresponds to a finite-wavelength eigenmode.

Figure 3

Dependence of the dimensionless critical electric field ${\zeta }_c$ (top) and the corresponding dimensionless eigenmode wave numbers ${\omega }_1$ and ${\omega }_2$ (bottom) on the twist angle $\mathrm{\Delta }\varphi$ for two different sets of Frank-Oseen constants. Notice that above a certain value of $\mathrm{\Delta }\varphi$, the critical eigenmode changes from global to local (from infinite to finite wavelengths), as indicated by a dashed (red) vertical line. The electric susceptibility constants (${\chi }_n=17.5$ and $\chi =6.0$) are the same in both cases. Observe the critical electric field ${\zeta }_c\left(\mathrm{\Delta }\varphi \right)$ in the liquid crystal with the lower bend constant $k_3/k_1=0.1$ (right) that drops below its lowest value in the global mode regime, e.g., ${\zeta }_c(3\pi /2)<{\zeta }_c\left(0\right)$.

Figure 4

Gnomonic projection used in plotting the influence of the three Frank-Oseen constants $k_i$ on the critical eigenmode and corresponding electric field; the projection is motivated by the need to provide an easy comparison of the anisotropic cases with the isotropic, one-constant approximation $k_1=k_2=k_3=k$ (midpoint of the triangular domain).

Figure 5

Influence of the dimensionless Frank-Oseen constants $k_1,k_2$, and $k_3$ on the critical eigenmode. Contours of ${\mathrm{\Omega }}_c\equiv {[{\left({\omega }_{1c}\right)}^2+{\left({\omega }_{2c}\right)}^2]}^{1/2}$, corresponding to the critical dimensionless electric field ${\zeta }_c$, are presented in gnomonic projection (see the inset) as functions of $k_i$ for four different values of the twist angle $\mathrm{\Delta }\varphi$. The noncolored (white) area in the above graphs corresponds to global (long-wavelength) critical eigenmodes ${\mathrm{\Omega }}_c=0$. Results for the Frank-Oseen constants of the 5CB liquid crystal are indicated by a small (black) dot.

Figure 6

Influence of the dimensionless Frank-Oseen constants $k_1,k_2$, and $k_3$ on the critical electric field. Contours of the critical dimensionless electric field ${\zeta }_c$ are presented in gnomonic projection (see the inset) as functions of $k_i$ for four different values of the twist angle $\mathrm{\Delta }\varphi$. The unshaded (white) area in the above graphs corresponds to global (long-wavelength) critical eigenmodes ${\mathrm{\Omega }}_c=0$, while the shaded (gray) area corresponds to local (finite-wavelength) critical eigenmodes ${\mathrm{\Omega }}_c>0$, with the boundary of the two regions depicted by a dashed (red) line. Results for the Frank-Oseen constants of the 5CB liquid crystal are indicated by a small (black) dot.

Figure 7

Influence of electric susceptibility constants $\chi$ and ${\chi }_n$ on the critical electric field. Contours of ${\zeta }_c$ are presented as functions of $\chi$ and $\mathrm{\Delta }\chi \equiv {\chi }_n-\chi$ for different values of $\mathrm{\Delta }\varphi$. The unshaded (white) area in the above graphs corresponds to global (long-wavelength) critical eigenmodes ${\mathrm{\Omega }}_c=0$, while the shaded (gray) area corresponds to local (finite-wavelength) critical eigenmodes ${\mathrm{\Omega }}_c>0$ with the boundary of the two regions depicted by a dashed (red) line. In all calculations, the Frank-Oseen constants are those of the 5CB liquid crystal, while the small (black) dots indicate results where the susceptibilities are also those of the 5CB.

Figure 8

Influence of the dimensionless Frank-Oseen constants $k_1,k_2$, and $k_3$ on the dropping of the critical electric field for the supertwisted nematic device (STND) presented in gnomonic projection. Contours of the difference of critical dimensionless electric field ${\zeta }_c(3\pi /2)-{\zeta }_c\left(0\right)$ as functions of the ratios $k_2/k_1$ and $k_3/k_1$. Results for the 5CB liquid crystal are indicated by a small (black) dot.