M9: Multiscale modelling in liquid crystal science
| Researcher: | Dr Chong Luo |
| Team Leader(s): | Dr Apala Majumdar & Dr Radek Erban |
| Collaborators: | N/A |
Project completed September 30, 2012
Background
Liquid crystal-based technologies are presently at the forefront of global display applications. As we enter the era of portable electronic devices, cost-effective displays are increasingly lucrative for industry and general applications. In contrast to conventional liquid crystal displays, bistable display pixels can support stable optically contrasting states (bright and dark) so that power is needed only to switch between the competing stable states, but not to maintain them individually. This could significantly increase the battery life of mobile devices and is particularly attractive for devices that retain a static image for a long period of time.
A planar bistable device comprises a periodic array of micron-sized square and rectangular liquid crystal wells. These wells are bistable or multistable with at least two different types of stable states: the so-called ‘diagonal’ states and ‘rotated’ states.
This project investigated both the static and dynamic properties of a bistable device, in both macroscopic and microscopic frameworks. Researchers at the Oxford Centre for Collaborative Applied Mathematics (OCCAM) modelled the planar bistable device within the macroscopic Landau–de Gennes framework and also developed two new microscopic molecular models.
Techniques and Challenges
Macroscopic model: The researchers modelled the planar bistable device within the macroscopic Landau–de Gennes framework. The presence of boundary defects makes the macroscopic modelling challenging. The techniques used to formulate the model within the macroscopic framework rely heavily on the theory of partial differential equations (PDEs), finite-difference and finite-element methods. The mathematical formulation, in the static case, is a system of two coupled nonlinear elliptic PDEs, analogous to the Ginzburg–Landau theory for superconductors.
Microscopic models: Existing microscopic models such as the Lebwohl–Lasher lattice-based model, the Gay–Berne model and the Hard-Gaussian-Overlap (HGO) model, reproduce the experimentally observed bistability. However, they are undesirable because the total energy in the system diverges as the number of molecules increases. To resolve this issue, two new models were formulated. The first one is a lattice-based intermolecular interaction potential, which can be viewed as a discretised ‘microscopic’ version of the macroscopic Landau–de Gennes model. The second model is an off-lattice molecular model, which can be viewed as a variant of the Gay–Berne or the HGO model.
Results
Macroscopic model: The model uses the concept of an optimal boundary condition that mimics the experimentally imposed tangent boundary condition. The optimal boundary conditions have been used to define an appropriate surface anchoring energy in terms of an anchoring coefficient, W. The resulting numerical problem is well-posed, yields physically realistic equilibria for all values of W, and makes physically realistic predictions for the order parameters in all relevant cases. It is capable of reproducing the experimentally-observed multistability in the absence of an electric field. Six different classes of equilibrium profiles were found and classified as either diagonal or rotated solutions.
Additionally, bifurcation diagrams for the multiplicity of solutions as a function of the anchoring strength were found. Diagonal solutions exist for all values of the anchoring strength W > 0, while rotated solutions only exist for W > Wc > 0, where Wc is a critical anchoring strength that has been computed as a function of the material parameters.
The researchers have proposed a dynamic model for the switching mechanisms based on only dielectric effects. For sufficiently strong external electric fields, diagonal-to-rotated and rotated-to-diagonal switching was demonstrated by allowing for variable anchoring strength across the domain boundary.
Microscopic models: The microscopic models were analysed using upscaling techniques and Monte Carlo algorithms, and both the diagonal and rotated solutions were observed. The new models do not suffer from the energy blow-up problem present in existing models.
The Future
This project has set up a sound theoretical and computational foundation for future work on bistable liquid crystal devices.
Related Publications
[11/55] Luo C., Majumdar A., Erban R.: The dynamics of bistable liquid crystal wells