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Introduction to Liquid Crystals

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Introduction to Liquid Crystals 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 1 Literature Many excellent books/reviews have been published covering various aspects of liquid crystals. Among them: 1. The bible on liqud crystals: P. G. de Gennes and J. Prost “The Physics of Liquid Crystals”, Ref. [1]. 2. Excellent review of basic properties (many topics below are taken from this review): M. J. Stephen, J. P. Straley “Physics of liquid crystals”, Ref. [2]. 3. Symmetries, hydrodynamics, theory: P. M. Chaikin and T. C. Lubensky “Principles of Condensed Matter Physics”, Ref. [3]. 4. Defects: O. D. Lavrentovich and M. Kleman, “Defects and Topology of Cholesteric Liquid Crys- tals”, Ref. [4]; Oleg Lavrentovich “Defects in Liquid Crystals: Computer Simulations, Theory and Experiments”, Ref. [5]. 5. Optics: Iam-Choon Khoo, Shin-Tson Wu, “Optics and Nonlinear Optics of Liquid Crystals”, Ref. [6]. 6. Textures: Ingo Dierking “Textures of Liquid Crystals”, Ref. [7]. 7. Simulations: Michael P. Allen and Dominic J. Tildesley “Computer simulation of liquids”, Ref. [8]. 8. Phenomenological theories: Epifanio G. Virga “Variational Theories for Liquid Crystals”, Ref. [9]. Finally, the pdf file of the lecture notes can be downloaded from http://www.mpip-mainz.mpg.de:/~andrienk/lectures/IMPRS/liquid_crystals.pdf. 1 What is a liquid crystal There is always a reason behind everything and this principle does not exclude naming conventions in physics. Hence, to understand what is a “liquid crystal” we need to clarify why would one give such a name to a substance. At first, the notion “liquid crystal” seems to be absurd. It, however, suggests that it is an intermediate state of a matter, in between the liquid and the crystal. It must possess some typical properties of a liquid (e. g. fluidity, inability to support shear, formation and coalescence of droplets) as well as some crystalline properties (anisotropy in optical, electrical, and magnetic properties, periodic arrangement of molecules in one spatial direction, etc.). Certain structural features are often found in molecules forming liquid crystal phases, and they may be summarized as follows: 1. The molecules have anisotropic shape (e. g. are elongated). Liquid crystallinity is more likely to occur if the molecules have flat segments, e. g. benzene rings. 2. A fairly good rigid backbone containing double bonds defines the long axis of the molecule. 3. The existence of strong dipoles and easily polarizable groups in the molecule seems important. 4. The groups attached to the extremities of the molecules are generally of lesser importance. 2 Figure 1: The arrangement of molecules in liquid crystal phases. (a) The nematic phase. The molecules tend to have the same alignment but their positions are not correlated. (b) The cholesteric phase. The molecules tend to have the same alignment which varies regularly through the medium with a periodicity distance p/2. The positions of the molecules are not correlated. (c) smectic A phase. The molecules tend to lie in the planes with no configurational order within the planes and to be oriented perpendicular to the planes. 1.1 Nematics The nematic phase is characterized by long-range orientational order, i. e. the long axes of the molecules tend to align along a preferred direction. The locally preferred direction may vary throughout the medium, although in the unstrained nematic it does not. Much of the interesting phenomenology of liquid crystals involves the geometry and dynamics of the preferred axis, which is defined by a vector n(r) giving its local orientation. This vector is called a director. Since its magnitude has no significance, it is taken to be unity. There is no long-range order in the positions of the centers of mass of the molecules of a nematic, but a certain amount of short-range order may exist as in ordinary liquids. The molecules appear to be able to rotate about their long axes and also there seems to be no preferential arrangement of the two ends of the molecules if they differ. Hence the sign of the director is of no physical significance, n = −n. Optically a nematic behaves as a uniaxial material with a center of symmetry. A simplified picture of the relative arrangement of the molecules in the nematic phase is shown in Fig. (1a). The long planar molecules are symbolized by ellipses. Figure 2: (a) Schlieren texture of a nematic film with surface point defects (boojums). (b) Thin nematic film on isotropic surface: 1-dimensional periodicity. Photos courtesy of Oleg Lavrentovich http://www. lci.kent.edu/ALCOM/oleg.html. (c) Nematic thread-like texture. After these textures the nematic phase was named, as “nematic” comes from the Greek word for “thread”. Photo courtesy of Ingo Dierking. On optical examination of a nematic, one rarely sees the idealized equilibrium configuration. Some very prominent structural perturbation appear as threads from which nematics take their name (Greek 3 “νηµα” means thread). These threads are analogous to dislocations in solids and have been termed disclinations by Frank. Several typical textures of nematics are shown in Fig. (2). The first one is a schlieren texture of a nematic film. This picture was taken under a polarization microscope with polarizer and analyzer crossed. From every point defect emerge four dark brushes. For these directions the director is parallel either to the polarizer or to the analyzer. The colors are newton colors of thin films and depend on the thickness of the sample. Point defects can only exist in pairs. One can see two types of boojums with “opposite sign of topological charge”; one type with yellow and red brushes, the other kind not that colorful. The difference in appearance is due to different core structures for these defects of different “charge”. The second texture is a thin film on isotropic surface. Here the periodic stripe structure is a spectacular consequence of the confined nature of the film. It is a result of the competition between elastic inner forces and surface anchoring forces. The surface anchoring forces want to align the liquid crystals parallel to the bottom surface and perpendicular to the top surface of the film. The elastic forces work against the resulting “vertical” distortions of the director field. When the film is sufficiently thin, the lowest energy state is surprisingly archived by “horizontal” director deformations in the plane of the film. The current picture shows a 1-dimensional periodic pattern. Many compounds are known to form nematic mesophase. A few typical examples are sketched in Fig. (3). From a steric point of view, molecules are rigid rods with the breadth to width ratio from 3:1 to 20:1. Figure 3: Typical compounds forming nematic mesophases: (PAA) p-azoxyanisole. From a rough steric point of view, this is a rigid rod of length ∼ 20A˚ and width ∼ 5A.˚ The nematic state is found at high temperatures (between 1160C and 1350C at atmospheric pressure). (MMBA) N-(p-methoxybenzylidene)- p-butylaniline. The nematic state is found at room temperatures (between 200C to 470C). Lacks chemical stability. (5CB) 4-pentyl-4’-cyanobiphenyl. The nematic state is found at room temperatures (between 240C and 350C). 1.2 Cholesterics The cholesteric phase is like the nematic phase in having long-range orientation order and no long-range order in positions of the centers of mass of molecules. It differs from the nematic phase in that the director varies in direction throughout the medium in a regular way. The configuration is precisely what one would obtain by twisting about the x axis a nematic initially aligned along the y axis. In any plane perpendicular to the twist axis the long axes of the molecules tend to align along a single preferred direction in this 4 plane, but in a series of equidistant parallel planes, the preferred direction rotates through a fixed angle, as illustrated in Fig. (1b). The secondary structure of the cholesteric is characterized by the distance measured along the twist axis over which the director rotates through a full circle. This distance is called the pitch of the cholesteric. The periodicity length of the cholesteric is actually only half this distance since n and −n are indistin- guishable. A nematic liquid crystal is just a cholesteric of infinite pitch, and is not really an independent case. In particular, there is no phase transition between nematic and cholesteric phases in a given material, and nematic liquid crystals doped with enantiomorphic materials become cholesterics of long (but finite) pitch. The molecules forming this phase are always optically active, i. e. they have distinct right- and left-handed forms. Figure 4: (a) Cholesteric fingerprint texture. The line pattern is due to the helical structure of the cholesteric phase, with the helical axis in the plane of the substrate. Photo courtesy of Ingo Dierking. (b) A short-pitch cholesteric liquid crystal in Grandjean or standing helix texture, viewed between crossed polarizers.
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