MSE11: Bistable liquid crystal displays: modelling, simulation and applications

Background

In recent years, researchers have developed different types of bistable liquid crystal displays in response to a growing demand for energy-efficient portable electronic devices.

In this project, we focus on the mathematical modelling of one such popular bistable device on the market - the Zenithally Bistable Nematic Device (ZBND). The ZBND geometry typically comprises a liquid crystal layer sandwiched between two surfaces – one planar and the other featured by a sinusoidal-like grating.  Both surfaces are treated to induce homeotropic alignment, i.e. the molecules align with the surface normal. This combination of a micropatterned surface and homeotropic boundary conditions yields at least two stable states of comparable energy. In Figure 1 b we see the 'defect state' with two degree-1/2 singularities of opposite signs attached to the device boundary.  In Figure 1 c we see the 'defect-free state' in which molecules are largely vertically aligned throughout the geometry.

Techniques and Challenges

We are considering a two-dimensional periodic domain with a sinusoidal grating on the bottom. We are working within the Landau-de Gennes framework using symmetric traceless Q-tensors to model the liquid crystal configuration.

The stable states correspond to local minimisers of the corresponding free energy functional . This free energy consists of an elastic and a bulk energy, e.g. [11/55]. Boundary values can be imposed strongly as Dirichlet conditions or weakly by adding a pinning potential with support on the boundary. This free energy is equivalent to a standard Ginzburg-Landau functional where the temperature-dependence is explicitly incorporated into a penalty factor, e.g. [11/55].

Simulating switching processes between the two stable states involves adding an electrostatic energy to the functional and deducing the appropriate notion of evolution dynamics. The use of finite elements for computations is not straightforward since the mesh size must be sufficiently small to properly resolve the defects. Adaptive methods yield neat results but convergence as the mesh size tends to zero must be established.

Results and the Future

In the absence of external fields, our model contains four parameters: two are of geometric nature (the amplitude of the grating and the height of the cell) the others are penalising factors (one mainly temperature dependent in front of the bulk energy and the other material dependent in front of the pinning potential). A first aim is to understand the energy landscape for varying parameters and the two co-existing stable states. We will compute explicit energy bounds to quantify the bistability regimes. This will be followed by a detailed stability analysis of the two competing stable states and numerical computation of bifurcation diagrams as a function of the anchoring strength.  In the second part of the project, we will undertake detailed studies of the different switching mechanisms in the ZBND device and obtain quantitative estimates for critical fields and switching times. It is expected that our work can give novel insight into the design of the `optimal’ ZBND geometry and three-dimensional analogues of this two-dimensional prototype device.

References

[11/55] Luo C., Majumdar A., Erban R.: The dynamics of bistable liquid crystal wells, Accepted for publication in Physical Review E, 2012