Steady state solutions in a model of a cholesteric liquid crystal sample

Abstract

Motivated by recent mathematical studies of Fréedericksz transitions in twist cells and helix unwinding in cholesteric liquid crystal cells [(da Costa et al. in Eur J Appl Math 20:269–287, 2009), (da Costa et al. in Eur J Appl Math 28:243–260, 2017), (McKay in J Eng Math 87:19–28, 2014), (Millar and McKay in Mol Cryst Liq Cryst 435:277/[937]–286/[946], 2005)], we consider a model for the director configuration obtained within the framework of the Frank-Oseen theory and consisting of a nonlinear ordinary differential equation in a bounded interval with non-homogeneous mixed boundary conditions (Dirichlet at one end of the interval, Neumann at the other). We study the structure of the solution set using the depth of the sample as a bifurcation parameter. Employing phase space analysis techniques, time maps, and asymptotic methods to estimate integrals, together with appropriate numerical evidence, we obtain the corresponding novel bifurcation diagram and discuss its implications for liquid crystal display technology. Numerical simulations of the corresponding dynamic problem also provide suggestive evidence about stability of some solution branches, pointing to a promising avenue of further analytical, numerical, and experimental studies.

Appendix

In this Appendix we make some remarks about Conjecture 1.

We recall that the determination of the sign of $\frac{\partial ^2 T_B}{\partial z^2}(z,\phi )$ directly using (33) seems to be a difficult problem due to the balance between the (positive) integral terms and the nonintegral term that has different signs in different regions of the rectangle $(0,2)\times (0,\frac{\pi }{2})$.

Fig. 14

Illustration (in black) of the set $\Omega$ of points $(z,\phi )$ where $3-k(z,\phi )h(z,\phi ,0) > 0$

Fig. 15

Graph of $(z,\phi )\mapsto \Phi (z,\phi )$

Thus, we tried several, less direct, approaches to the convexity of $T_B(z,\phi ).$ One such approach was the standard method presented in Smoller [22, Chap. 13§D]: to establish the convexity of $T_B(z,\phi )$ it is sufficient to prove that

$$\begin{aligned} \Phi (z,\phi ):= 4\frac{\partial ^2 T_B}{\partial z^2} + 2k(z,\phi )\frac{\partial T_B}{\partial z} >0, \end{aligned}$$

(47)

if $(z,\phi )\in (0,2)\times (0,\frac{\pi }{2})$ is a stationary point of $z\mapsto T_B(z,\phi )$, for all functions $k(z,\phi )$, because at stationary points the value of the first derivative $\frac{\partial T_B}{\partial z}$ is zero, by definition. Thus, if we find a function $k(z,\phi )$ such that (47) holds for all points $(z,\phi )$, we conclude that, for each fixed $\phi$, $\Phi (\cdot ,\phi )$ is convex at each of its stationary points, and thus there can exist only one stationary point.

Now, by (27), (33), and (34), choosing

$$\begin{aligned} k(z,\phi )&:= - z^{-1}(2-z)^{-1}(1-\cos 2\phi )^{-1}g(z,\phi ) \nonumber \\&= (1-\cos 2\phi )^{-1} - 2(z-1)z^{-1}(2-z)^{-1}, \end{aligned}$$

(48)

we get

$$\begin{aligned} \Phi (z,\phi ) = \Bigl (\int _0^\phi + \int _0^{{\bar{x}}^{\star }(z)}\Bigr )h^{-5/2}(3-kh)\,dx, \end{aligned}$$

(49)

where

$$\begin{aligned} h = h(z,\phi ,x):= z-\cos 2\phi +\cos 2x. \end{aligned}$$

(50)

From the definition of k it follows that $k <0$ whenever $g>0$ and, from (49), (50), and $h(z,\phi , x) < h(z,\phi ,0),$ we easily conclude that $\Phi (z,\phi ) > 0$ in $\Omega := \left\{ (z,\phi )\in {\mathbb {R}}^2 \,|\, 3-k(z,\phi )h(z,\phi ,0) > 0\right\} .$ This set $\Omega$ is illustrated in Fig. 14.

Outside $\Omega$ the sign of $\Phi$ is much harder to establish since the two integrals can have opposite signs and be divergent on the boundaries of the domain of $\Phi$. However, the positivity of $\Phi (z,\phi )$ everywhere in the rectangle $(0,2)\times (0, \frac{\pi }{2})$ is easy to get numerically, as illustrated in Fig. 15 obtained using the software Mathematica^©.

Alternative approaches to prove the existence of a single minimum of the graph of $z\mapsto T_B(z,\phi )$, based on attempting to define different type of time maps via changes of variables [14, 21] or other analytic approaches [14] were fruitless. Proving Conjecture 1 remains a technical challenge.