OVERVIEW: Materials science and engineering has witnessed unprecedented growth in recent years with the advent of biological materials, colloids, metamaterials, topological materials etc. Soft matter is at the forefront of contemporary materials research. Soft matter is a phrase used for materials that are intermediate between conventional solids and liquids in terms of their physical properties. Soft matter is fairly ubiquitous in our everyday lives and can be found in display devices, pharmaceuticals, agrochemicals, cosmetics, fibres, and are even found in the human body as biological soft matter e.g. the cell cytoskeleton. Collectively, soft matter products contribute over 200 billion GBP to the world economy and relevant industries employ over 900,000 people.
Nematic liquid crystals are paradigm examples of soft matter. They are complex liquids with a degree of long-range orientational order i.e. they have certain distinguished directions and consequently, direction-dependent physical, optical, mechanical and rheological properties. Indeed, it is the directional nematic response to external light and electric fields that makes them the working material of choice for the multi-billion dollar liquid crystal display industry. Nematic liquid crystals today have applications far beyond the display industry in the health industry, microfluidics and nano-technologies. The mathematics and theory of nematic liquid crystals is deep and challenging. In fact, Pierre de Gennes was awarded the Nobel Prize for Physics in 1991, partially for his theories for nematic liquid crystals and these theories have a profound role in science and technology too.
In this mini-course, we review the mathematical theories theories for nematic liquid crystals and how they can be applied to display devices, as practical case studies. We will review the celebrated Landau-de Gennes and Oseen-Frank theories for nematic liquid crystals, the key concepts of birefringence, response to external fields, anchoring and topological defects. We will then apply these concepts to the Post Aligned Bistable Nematic Device designed by Hewlett Packard, the Zenithally Bistable Nematic Device and the Planar Bistable Nematic Device, with a discussion of the static and time-dependent properties. The course has a two fold purpose. Firstly, the course will serve as an introductory course to the multi-faceted field of nematic liquid crystals and their applications in an interdisciplinary setting. Secondly, the course will describe certain generic concepts for soft matter, the rich underpinning mathematical frameworks and how mathematics can transform materials research into a predictive science.
Day 1: Lecture 1: Introduction to the theory and physics of nematic liquid crystals - history, different phases and introduction to concept of partial order.
Lecture 2: The Oseen Frank theory for nematic liquid crystals - definition of director and the Oseen-Frank free energy.
Day 2: Lecture 3: Birefringence and Dielectric Anisotropy; the classical Freedericksz Transition for Nematic Liquid Crystals.
Lecture 4: The Landau-de Gennes theory for nematic liquid crystals - the Q-tensor order parameter, the free energy and the equilibrium equations.
Day 3: Lecture 5: Case Study I - Mathematical Modelling of the Post Aligned Bistable Nematic Device and Topological Mechanisms for Multistability.
Day 4: Lecture 6: Case Study II - Mathematical Modelling of the Planar Bistable Nematic Device – static equilibria
Lecture 7: Case Study II – Switching Mechanisms for the Planar Bistable Nematic Device
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A.Majumdar, C.J.P.Newton, J.M.Robbins & M.Zyskin. Topology and Bistability in Liquid Crystal Devices. Physical Review E, 75, 2007.
G.Canevari, A.Majumdar & A.Spicer. Order reconstruction for nematics on squares and hexagons: a Landau-de Gennes study. SIAM Journal of Applied Mathematics, 77, 2017.
Y.Wang, G.Canevari and A.Majumdar. Order reconstruction for nematics on squares with isotropic inclusions: a Landau-de Gennes study. SIAM Journal of Applied Mathematics, 79, 2019.
C.Luo, A.Majumdar & R.Erban. Multistability in planar liquid crystal wells. Physical Review E, 85, 2012.