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Introduction to liquid crystals
Denis Andrienko
International Max Planck Research School
Modelling of soft matter
11-15 September 2006, Bad Marienberg
September 14, 2006
Contents
1 What is a liquid crystal 2
1.1 Nematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Cholesterics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Smectics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Columnar phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Lyotropic liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Long- and short-range ordering 7
2.1 Order tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Director . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Phenomenological descriptions 10
3.1 Landau-de Gennes free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Frank-Oseen free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Nematic-isotropic phase transition 14
4.1 Landau theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.2 Maier-Saupe theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3 Onsager theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5 Response to external fields 18
5.1 Frederiks transition in nematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6 Optical properties 20
6.1 Nematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6.2 Cholesterics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
7 Defects 23
8 Computer simulation of liquid crystals 24
9 Applications 27



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