Uniform distortions and generalized elasticity of liquid crystals

Ordinary nematic liquid crystals are characterized by having a uniform director field as ground state. In such a state, the director is the same everywhere and no distortion is to be seen at all. We give a definition of uniform distortion which makes precise the intuitive notion of seeing everywhere the same director landscape. We characterize all such distortions and prove that they fall into two families, each described by two scalar parameters. Uniform distortions exhaust R. Meyer's heliconical structures, which, as it has recently been recognized, include the ground state of twist-bend nematics. The generalized elasticity of these new phases is treated with a simple free-energy density, which can be minimized by both uniform and nonuniform distortions, the latter injecting a germ of elastic frustration.

Figure 1

The heliconical nematic fields with negative (blue) and positive (brown) eigenvalue ${\lambda }_3$, as delivered by (35) and (41), respectively. Panels (a) and (b) also illustrate the symmetries of the helix axes embodied by (40) and (43). The blue field (for which ${\lambda }_3=-3q$) precesses counterclockwise around the helix axis, whereas the brown field (for which ${\lambda }_3=3q$) precesses clockwise around the helix axis.

Figure 2

The same heliconical director fields as in Fig. 1 but for $b=0$. The two helix axes are perpendicular to one another, as prescribed by (46), and the cone angle takes on the limiting value $\alpha =\pi /2$.

Figure 3

Phase diagram for the minimizers of $F_{\mathrm{TB}}$ in the half-plane $(k_2,k_3)$ with $k_2\geqq 0$ (arbitrary units). The blue (lower) region is delimited by the straight line (63). In this region, $F_{\mathrm{TB}}$ is minimized by the uniform state (64). In the white (middle) region, the state of minimum energy is the nonuniform pure bend (60). In the red (upper) region, the minimum energy is attained at the uniform state (59).

Figure 4

The critical values of ${\mathrm{\Phi }}_{\mathit{A}}$ in (C8) as functions of the parameter $\kappa$. They are given by (C12) and are all positive for $-4\sqrt{2}<\kappa <4\sqrt{2}$. All but $\lambda =1$ correspond to discrete critical points on the unit sphere (C10e). The points marked by (red) circles are bifurcations points, where different eigenvalues meet and their number may change; they are located at $\kappa =0$, $\kappa =\pm 4$, and $\kappa =\pm 8$.