Q-Tensor models of defect dynamics in pure and doped liquid crystals

Liquid crystals are best known for their applications in modern flat screens, but other technologically important applications include colour-sensitive thermometers, temperature-sensitive paints, optoelectronic equipment such as shutters and sensors, materials such as ultra-light body armour, as well as very recently in microfluidic devices-on-a-chip. They are also materials of great fundamental interest because they exhibit some liquid-like properties (they flow) and some solid-like properties (their properties depend on direction). Despite intense theoretical work over the last forty years, it is still the case that new applications require a more fundamental mathematical understanding of liquid crystalline models in order that rapid device prediction can be made.

A major headache in liquid crystal theory is the so-called "defect" problem. Liquid crystals usually exhibit a preferred direction, but this preferred direction can change from place to place. The defects are lines or points near which the preferred direction dissolves. An analogy can be seen when one combs one's hair: it is impossible to comb hair uniformly over a sphere; there will always be some holes in the pattern. Cosmologists have also use the liquid crystal defects as models of the early universe. The internal patterns of the defects also provide the dramatic and beautiful characteristic optical signatures for individual types of liquid crystal.

Describing them has caused much difficulty in mathematical theories of liquid crystals. The standard theory (due to the Scottish mathematician Frank Leslie) just omits them, while alternative approaches ("the Q-tensor" theory) often are computationally inefficient in regions in which the preferred direction changes only slowly. In the latter case, even when there is a computational model for the liquid crystal properties, it may be difficult to provide a physical picture. The end result is that properties of new devices cannot be predicted reliably and quickly.

Our project will provide a more sophisticated model of the motion of defected nematic liquid crystals. We shall build on a new mathematical approximation which we have recently developed. This approach, which we have labelled the Defect-free Q-tensor approximation (DFQTA), uses the best features of the Leslie and Q-tensor approaches. In non-defected liquid crystals, this approach has led to dramatic improvements (of the order of a factor of 100) in the computational time required to solve some benchmark problems, without any loss of accuracy. Our new model will treat defect motion by patching together solutions in the defect-free and defected regions. The defect-free region will be treated using DFQTA. The defect itself will be cut out of the problem, and treated separately using a range of approximations made possible by the fact, amongst others, that the regions that contains them are small. The two regions will be matched using asymptotic analysis. Our approach will combine the algorithmic requirements of engineers (who often downplay the necessity for mathematical analysis), with sophisticated mathematical approximation tools to obtain an accurate and computationally efficient model.

One particular engineering application concerns nanoparticles suspended in liquid crystal solutions. Nanoparticles are of micron size or smaller, and even at very low concentrations (much less than 1% by volume), they can change the properties of the solvent significantly. These particles are being used in a new generation of devices, both in liquid crystals and elsewhere. But in liquid crystals the nanoparticles are attracted to defects. Both the motion and the optical signature of the nanoparticles seem to be governed by defects which trap them. Our new theory will enable such systems to be treated successfully on computationally accessible time-scales.

The mathematical model of liquid crystal dynamics that will result from this project (i) will be a breakthrough in understanding and simulating liquid crystal nanomaterials and (ii) will allow us to run cost effective numerical simulations of realistic liquid crystal devices. It will, therefore, be directly relevant to device modelling and to the exponentially growing body of research and development in nanodoped materials.

To give an example, the number of publications on gold nanoparticles in a liquid crystal environment more than doubled in the last three years. The main thrust of this published research concerns experimental investigation of rich and fascinating effects of nanoparticles self-assembly, their functionalisation and optical response. Only a handful of papers deal with mathematical modelling of such systems. The same trend can be observed for other types of nanoparticles, such as seminconductor or ferroelectric. The model proposed here will address these issues head on and use an in-depth mathematical base to build a comprehensive and efficient model of liquid crystal based nanomaterials.

Examples of areas that will benefit directly from this project include:

(1) International research on defects and self-assembly

There is a clear need for more a more efficient framework to study defects in liquid crystals. The Defect Free Q-Tensor Approximation (DFQTA) will make the numerical analysis of liquid crystal configurations in geometries that vary in shape and size, much faster and more efficient. This will not be of interest just to the liquid crystal community, but also to other disciplines, for example optics and soft-matter. For example, it will be possible to determine conditions to arrange nanoparticles in regular arrays over large volumes and how their configuration changes with the orientation of the liquid crystal: this may well be the route to an electrically tunable soft-metamaterial. From a mathematical point of view, it may be possible to export the results of this projects to other fields where defects play a significant role. In particular, it may be possible to extend the techniques to excise "problem'' areas from "nice'' areas also to other fields where the core hypothesis of the DFQTA does not hold.

(2) Liquid crystal based devices

This project will have a direct impact on the study of liquid crystal based devices. Liquid crystal based devices may involve relatively large volumes of liquid crystals, too large for current numerical methods. For example, an experimental study of liquid crystal waveguides carried out with colleagues at the Optoelectronics Research Centre at Southampton showed that a pincement bifurcation can take place for sufficiently large voltages and zig-zag defects appear across the waveguide. The volume of liquid crystal involved in such device is considerable because of the length of the waveguide.

(3) PDRAs and PhD students in our research groups

The PDRAs and PhD students in our groups will benefit from their direct or indirect involvement in this project. The named PDRA funded by this project will develop a set of skills that will put him in good stead for a research position either in academia or industry. Moreover, the PDRA will benefit from the training sessions set up by the University for early career researchers. Funding for this project will give us the resources to develop training on defects that will be available to other members of our groups and, hopefully, through the training networks we belong to, to other PhD students across the UK.