Introduction to Liquid Crystals Denis Andrienko IMPRS school, Bad - - PowerPoint PPT Presentation
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Literature Optical properties What are liquid crystals? Defects Simulation of liquid crystals Applications Introduction to Liquid Crystals Denis
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Literature What are liquid crystals?
Denis Andrienko IMPRS school, Bad Marienberg September 14, 2006
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Literature What are liquid crystals? Literature What are liquid crystals? Liquid crystalline mesophases Nematics Cholesterics Smectics Molecular arrangement Columnar phases Short- and long-range ordering Order tensor Properties of the order tensor Director Phenomenological descriptions Landau-de Gennes theory Frank-Oseen free energy One elastic constant approximation Nematic-isotropic transition Response to external fields Frederiks transition Optical properties Nematics Colors Cholesterics Defects Linear defects Interaction of defects Nematic colloids Simulation of liquid crystals Forces, torques and gorques Gay-Berne potential Phase diagrams Nematic colloids Applications Liquid Crystal Displays Liquid Crystal Thermometers Polymer dispersed liquid crystals Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Literature What are liquid crystals?
Many excellent books/reviews have been published covering various aspects of liquid crystals. Among them: 1. The bible on liquid crystals: P. G. de Gennes and J. Prost “The Physics of Liquid Crystals”. 2. Excellent review of basic properties (many topics below are taken from this review): M. J. Stephen, J. P. Straley “Physics of liquid crystals”. 3. Symmetries, hydrodynamics, theory: P. M. Chaikin and T. C. Lubensky “Principles of Condensed Matter Physics”. 4. Defects: Oleg Lavrentovich “Defects in Liquid Crystals: Computer Simulations, Theory and Experiments”. 5. Optics: Iam-Choon Khoo, Shin-Tson Wu, “Optics and Nonlinear Optics of Liquid Crystals”. 6. Textures: Ingo Dierking “Textures of Liquid Crystals”. 7. Simulations: Michael P. Allen and Dominic J. Tildesley “Computer simulation of liquids”. 8. Phenomenological theories: Epifanio G. Virga “Variational Theories for Liquid Crystals”. Finally, the pdf file of the lecture notes can be downloaded from http://www.mpip-mainz.mpg.de:/∼andrienk/lectures/IMPRS/liquid crystals.pdf. Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Literature What are liquid crystals?
The name suggests that it is a state of a matter in between the liquid and the crystal.
◮ Liquid
◮ Solid
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Literature What are liquid crystals?
The name suggests that it is a state of a matter in between the liquid and the crystal.
◮ Liquid
◮ Solid
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Nematics Cholesterics Smectics Molecular arrangement Columnar phases
Figure: (a) Schlieren texture. (b) Thin nematic film on isotropic surface. (c) Nematic thread-like texture.
“Nematic” comes from the Greek word for “thread”.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Nematics Cholesterics Smectics Molecular arrangement Columnar phases
From a rough steric point of view, this is a rigid rod of length ∼ 20˚ A and width ∼ 5˚ A.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Nematics Cholesterics Smectics Molecular arrangement Columnar phases
Figure: (a) Fingerprint texture. (b) Grandjean or standing helix texture (c) DNA mesophases
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Nematics Cholesterics Smectics Molecular arrangement Columnar phases
Figure: (a,b) Focal-conic fan texture of a chiral smectic A liquid crystal (c) Focal-conic fan texture of a chiral smectic C liquid crystal.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Nematics Cholesterics Smectics Molecular arrangement Columnar phases
Figure: 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. (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. Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Nematics Cholesterics Smectics Molecular arrangement Columnar phases
Figure: 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. (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. Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Nematics Cholesterics Smectics Molecular arrangement Columnar phases
Figure: 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. (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. Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Nematics Cholesterics Smectics Molecular arrangement Columnar phases
Figure: (a) hexagonal columnar phase Colh (with typical spherulitic texture); (b) Rectangular phase of a discotic liquid crystal (c) hexagonal columnar liquid-crystalline phase.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Nematics Cholesterics Smectics Molecular arrangement Columnar phases
Figure: Typical discotics: derivative of a hexabenzocoronene and 2,3,6,7,10,11-hexakishexyloxytriphenylene. K(70K) → Colh(100K) → I.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Nematics Cholesterics Smectics Molecular arrangement Columnar phases
Figure: (1) Columnar phase formed by the disc-shaped molecules and the most common arrangements of columns in two-dimensional lattices: (a) hexagonal, (b) rectangular, and (c) herringbone. (2,3) MD simulation results: snapshot of the hexabenzocoronene system with the C12 side chains.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Order tensor Properties of the order tensor Director
Figure: A unit vector u(i) along the axis of ith molecule describes its
Sαβ(r) = 1 N
α u(i) β − 1
3δαβ
Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Order tensor Properties of the order tensor Director
α u(i) β = u(i) β u(i) α and
δαβ = δβα: Sαβ = Sβα
TrSαβ =
Sαα = 1 N
x )2 + (u(i) y )2 + (u(i) z )2 − 1
33
since u is a unit vector.
independent components (3 by 3 matrix) from 9 to 5.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Order tensor Properties of the order tensor Director
α u(i) β = u(i) β u(i) α and
δαβ = δβα: Sαβ = Sβα
TrSαβ =
Sαα = 1 N
x )2 + (u(i) y )2 + (u(i) z )2 − 1
33
since u is a unit vector.
independent components (3 by 3 matrix) from 9 to 5.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Order tensor Properties of the order tensor Director
α u(i) β = u(i) β u(i) α and
δαβ = δβα: Sαβ = Sβα
TrSαβ =
Sαα = 1 N
x )2 + (u(i) y )2 + (u(i) z )2 − 1
33
since u is a unit vector.
independent components (3 by 3 matrix) from 9 to 5.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Order tensor Properties of the order tensor Director
αβ = 0.
ux = sin θ cos φ, uy = sin θ sin φ, uz = cos θ. Sαβ = 2π dφ π sin θdθP(θ, φ)
3δαβ
Sxy = Syz = Szx = 0 because of the integration over φ. For the Szz component we obtain Szz = 2 2π dφ π/2 sin θdθP(θ, φ)
3
4πPiso 1
3
3π (x3 − x)
0 = 0.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Order tensor Properties of the order tensor Director
axis), prolate geometry Sprolate = −1/3 −1/3 2/3 . To prove this it is sufficient to calculate only the Szz component: Szz = uzuz − 1/3 = 1 − 1/3 = 2/3. Keeping in mind that S is symmetric and traceless we
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Order tensor Properties of the order tensor Director
Soblate = 1/6 1/6 −1/3 . Try to follow previous arguments and show this!
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Order tensor Properties of the order tensor Director
In general, any symmetric second-order tensor has 3 real eigenvalues and three corresponding orthogonal eigenvectors. (Recall gyration tensor or mass and inertia tensor). For a uniaxial nematic phase two smaller eigenvalues are equal Sαβ = S
3δαβ
In the isotropic phase S = 0, in the nematic phase 0 < S < 1. S = 1 corresponds to perfect alignment of all the molecules.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Landau-de Gennes theory Frank-Oseen free energy One elastic constant approximation Nematic-isotropic transition Response to external fields Frederiks transition
We would like to describe:
◮ isotropic to nematic transition ◮ inhomogeneous systems ◮ influence of external factors (boundaries, fields)
To do this, we need to write down a free energy of our system. There are of course several ways (levels) of doing it.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Landau-de Gennes theory Frank-Oseen free energy One elastic constant approximation Nematic-isotropic transition Response to external fields Frederiks transition
To the extent that Sαβ is a small parameter, we may expand the free energy density g(P, T, Sαβ) in power series g = gi + 1 2ASαβSαβ − 1 3BSαβSβγSγα + 1 4CSαβSαβSγδSγδ This model equation of state predicts a phase transition near the temperature where A vanishes A = A′ (T − T ∗) .
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Landau-de Gennes theory Frank-Oseen free energy One elastic constant approximation Nematic-isotropic transition Response to external fields Frederiks transition
If we consider a nematic liquid crystal in which the order parameter is slowly varying in space, the free energy will also contain terms which depend on the gradient of the order parameter. These terms must be scalars and consistent with the symmetry of a nematic ge = 1 2L1 ∂Sij ∂xk ∂Sij ∂xk + 1 2L2 ∂Sij ∂xj ∂Sik ∂xk We will refer to the constants L1 and L2 as elastic constants.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Landau-de Gennes theory Frank-Oseen free energy One elastic constant approximation Nematic-isotropic transition Response to external fields Frederiks transition
The question we would like to address here is: how much energy will it take to deform the director filed? We will refer to the deformation of relative orientations away from equilibrium position as curvature strains. The restoring forces which arise to oppose these deformations we will call curvature stresses or torques. The six components of curvature are defined as splay s1 = ∂nx ∂x , s2 = ∂ny ∂y twist t1 = −∂ny ∂x , t2 = ∂nx ∂y bend b1 = ∂nx ∂z , b2 = ∂ny ∂z
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Landau-de Gennes theory Frank-Oseen free energy One elastic constant approximation Nematic-isotropic transition Response to external fields Frederiks transition
These three curvature strains can also be defined by expanding n(r) in a Taylor series in powers of x, y, z measured from the origin nx(r) = s1x + t2y + b1z + O(r2), ny(r) = −t1x + s2y + b2z + O(r2), nz(r) = 1 + O(r2).
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Landau-de Gennes theory Frank-Oseen free energy One elastic constant approximation Nematic-isotropic transition Response to external fields Frederiks transition
We now postulate that the Gibbs free energy density g of a liquid crystal, relative to its free energy density in the state of uniform
g =
6
kiai + 1 2
6
kijaiaj where the ki and kij = kji are the curvature elastic constants and for convenience in notation we have put a1 = s1, a2 = t2, a3 = b1, a4 = −t1, a5 = s2, a6 = b2.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Landau-de Gennes theory Frank-Oseen free energy One elastic constant approximation Nematic-isotropic transition Response to external fields Frederiks transition
There are several symmetries which will reduce the number of the elastic constants in this expansion
free energy). Out of the thirty-six kij, only five are independent.
n → −n, x → x, y → −y, z → −z. k1 = k12 = 0 (nonpolar).
x → x, y → −y, z → z. k2 = k12 = 0 (mirror symmetry).
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Landau-de Gennes theory Frank-Oseen free energy One elastic constant approximation Nematic-isotropic transition Response to external fields Frederiks transition
There are several symmetries which will reduce the number of the elastic constants in this expansion
free energy). Out of the thirty-six kij, only five are independent.
n → −n, x → x, y → −y, z → −z. k1 = k12 = 0 (nonpolar).
x → x, y → −y, z → z. k2 = k12 = 0 (mirror symmetry).
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Landau-de Gennes theory Frank-Oseen free energy One elastic constant approximation Nematic-isotropic transition Response to external fields Frederiks transition
There are several symmetries which will reduce the number of the elastic constants in this expansion
free energy). Out of the thirty-six kij, only five are independent.
n → −n, x → x, y → −y, z → −z. k1 = k12 = 0 (nonpolar).
x → x, y → −y, z → z. k2 = k12 = 0 (mirror symmetry).
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Landau-de Gennes theory Frank-Oseen free energy One elastic constant approximation Nematic-isotropic transition Response to external fields Frederiks transition
g = 1 2k11(∇ · n)2 + 1 2k22(n · curl n + t0)2 + 1 2k33(n × curl n)2 This is the famous Frank-Oseen elastic free energy density for nematics and cholesterics.
Figure: The three distinct curvature strains of a liquid crystal: (a) splay, (b) twist, and (c) bend.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Landau-de Gennes theory Frank-Oseen free energy One elastic constant approximation Nematic-isotropic transition Response to external fields Frederiks transition
For the purpose of qualitative calculations it is sometimes useful to consider a nonpolar, nonenatiomorphic liquid crystal whose bend, splay, and twist constants are equal (one-constant approximation) k11 = k22 = k33 = k. The free energy density for this theoretician’s substance is g = 1 2k
.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Landau-de Gennes theory Frank-Oseen free energy One elastic constant approximation Nematic-isotropic transition Response to external fields Frederiks transition
Substituting Sαβ = S
3δαβ
g = gi + 1 3AS2 − 2 27BS3 + 1 9CS4. The equilibrium value of S is that which gives the minimum value for the free energy.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Landau-de Gennes theory Frank-Oseen free energy One elastic constant approximation Nematic-isotropic transition Response to external fields Frederiks transition
Figure: Landau theory: dependence of the Gibbs free energy density on the order parameter. The case of the three special temperatures, T ∗∗, Tc, and T ∗ are shown. For illustration we use A = B = C = 1.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Landau-de Gennes theory Frank-Oseen free energy One elastic constant approximation Nematic-isotropic transition Response to external fields Frederiks transition
The minima of the free energy are S = isotropic phase (1) S = (B/4C)[1 + (1 − 24β)1/2] nematic phase, where β = AC/B2. The transition temperature Tc will be such that the free energies
βc = 1 27; Tc = T ∗ + 1 27 B2 A′C . Above Tc the isotropic phase is stable; below Tc the nematic is stable.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Landau-de Gennes theory Frank-Oseen free energy One elastic constant approximation Nematic-isotropic transition Response to external fields Frederiks transition
The magnetic susceptibility of a liquid crystal, owing to the anisotropic form of the molecules composing it, is also anisotropic. The susceptibility tensor takes the form χij = χ⊥δij + χaninj, where χa = χ − χ⊥ is the anisotropy and is generally positive. The presence of a magnetic field H leads to an extra term in the free energy of gm = −1 2χ⊥H2 − 1 2χa(n · H)2.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Landau-de Gennes theory Frank-Oseen free energy One elastic constant approximation Nematic-isotropic transition Response to external fields Frederiks transition
Figure: Frederiks transition. The liquid crystal is constrained to be perpendicular to the boundary surfaces and a magnetic field is applied in the direction shown. (a) Below a certain critical field Hc, the alignment is not affected. (b) slightly above Hc, deviation of the alignment sets in. (c) field is increased further, the deviation increases.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Landau-de Gennes theory Frank-Oseen free energy One elastic constant approximation Nematic-isotropic transition Response to external fields Frederiks transition
Let θ be the angle between the director and the z axis nx = sin θ(z), ny = 0, nz = cos θ(z) The elastic energy per unit area takes the form g = 1 2 d/2
−d/2
dz
∂θ ∂z 2 − χaH2 sin2 θ
where d is the thickness of the sample.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Landau-de Gennes theory Frank-Oseen free energy One elastic constant approximation Nematic-isotropic transition Response to external fields Frederiks transition
Variation of the free energy leads to the differential equation ξ2 ∂2θ ∂2z + sin θ cos θ = 0. Here we defined the correlation length ξ =
The first integral is (free energy does not have explicit dependence
ξ2 ∂θ ∂z 2 + sin2 θ = sin2 θm.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Landau-de Gennes theory Frank-Oseen free energy One elastic constant approximation Nematic-isotropic transition Response to external fields Frederiks transition
θ = θm cos z ξ
Hc =
χa π d
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Landau-de Gennes theory Frank-Oseen free energy One elastic constant approximation Nematic-isotropic transition Response to external fields Frederiks transition
θ = θm cos z ξ
Hc =
χa π d
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Landau-de Gennes theory Frank-Oseen free energy One elastic constant approximation Nematic-isotropic transition Response to external fields Frederiks transition
θ = θm cos z ξ
Hc =
χa π d
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Landau-de Gennes theory Frank-Oseen free energy One elastic constant approximation Nematic-isotropic transition Response to external fields Frederiks transition
Figure: Dependence of θm on H.
For fields weaker than Hc only the trivial solution exists, and there is no distortion on the nematic structure.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Nematics Colors Cholesterics
Susceptibility is a tensor ǫij = ǫ⊥δij + ǫaninj. Correspondingly, we can introduce ordinary and extraordinary refractive indexes ne = ǫ, no = √ǫ⊥, ∆n = ne − no. Typically no ∼ 1.5, ∆n ∼ 0.05 − 0.5.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Nematics Colors Cholesterics
Figure: Light travelling through a birefringent medium will take one of two paths depending on its polarization.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Nematics Colors Cholesterics
The incoming linearly polarized light Eincident = Ex Ey
E0 cos α E0 sin α
Ecell(z) = Ex exp(ikez) Ey exp(ikoz)
intensity Iout = |Eout|2 = E 2
0 sin2(2α) sin2
∆kL 2
λ .
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Nematics Colors Cholesterics
Birefringence can lead to multicolored images in the examination
∆n = ∆n(λ) Different wavelengths will experience different retardation and emerge in a variety of polarization states. The components of this light passed by the analyzer will then form the complementary color to λ.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Nematics Colors Cholesterics
This will be your home work.
◮ Cholesteric pitch is of the order of the wavelength of visible
light
◮ Chiral structure - circularly polarized eigenmodes of Maxwell’s
equations
◮ Pitch depends on temperature (thermometer)
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Nematics Colors Cholesterics
This will be your home work.
◮ Cholesteric pitch is of the order of the wavelength of visible
light
◮ Chiral structure - circularly polarized eigenmodes of Maxwell’s
equations
◮ Pitch depends on temperature (thermometer)
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Nematics Colors Cholesterics
This will be your home work.
◮ Cholesteric pitch is of the order of the wavelength of visible
light
◮ Chiral structure - circularly polarized eigenmodes of Maxwell’s
equations
◮ Pitch depends on temperature (thermometer)
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Linear defects Linear defects Nematic colloids
Examples of disclinations in a nematic.
Figure: (a) m = +1, (b) the parabolic disclination, m = +1/2, (c) the hyperbolic disclination (topologically equivalent to the parabolic one), m = −1/2.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Linear defects Linear defects Nematic colloids
The axial solutions of the Euler-Lagrange equations representing disclination lines are φ = mψ + φ0, where nx = cos φ, ψ is the azimuthal angle, x = r cos ψ, m is a positive or negative integer or half-integer. The elastic energy per unit length associated with a disclination is πKm2 ln(R/r0), where R is the size of the sample and r0 is a lower cutoff radius (the core size). Since the elastic energy increases as m2, the formation of disclinations with large m is energetically unfavorable.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Linear defects Linear defects Nematic colloids
Figure: Topological defects induced by a colloidal particle.
Interaction of colloidal particles is anisotropic: dipole-dipole, quadruple-quadruple like in the first order.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Forces, torques and gorques Gay-Berne potential Phase diagrams Nematic colloids
The equations for rotational motion (Ii is the moment of inertia) ˙ ei = ui, ˙ ui = g⊥
i /Ii + λei,
and Newton’s equation of motion mi¨ ri = fi describe completely the dynamics of motion of a linear molecule. gi = −∇eiVij (2) is a “gorque”.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Forces, torques and gorques Gay-Berne potential Phase diagrams Nematic colloids
The complete Gay-Berne potential can be expressed as follows v (ei, ej, rij) = 4ε (ei, ej, nij)
, (3) where ρ = [rij − σ (ei, ej, nij) + σs]/σs, nij = rij/rij, rij = |rij|.
σ ` ei , ej , nij ´ = σs ( 1 − χ 2 " ` ei · nij + ej · nij ´2 1 + χei · ej + ` ei · nij − ej · nij ´2 1 − χei · ej #)−1/2 , ε ` ei , ej , nij ´ = εs h ε′ ` ei , ej , nij ´iµ × h ε′′ ` ei , ej ´iν , ε′ ` ei , ej , nij ´ = 1 − χ′ 2 " ` ei · nij + ej · nij ´2 1 + χ′ei · ej + ` ei · nij − ej · nij ´2 1 − χ′ei · ej # , ε′′ ` ei , ej ´ = h 1 − χ2 ` ei · ej ´2i−1/2 . Here χ and χ′ denote the anisotropy of the molecular shape and of the potential energy, respectively, χ = κ2 − 1 κ2 + 1 , χ′ = κ′1/µ − 1 κ′1/µ + 1 . Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Forces, torques and gorques Gay-Berne potential Phase diagrams Nematic colloids
Figure: Phase diagrams of the Gay-Berne fluid.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Forces, torques and gorques Gay-Berne potential Phase diagrams Nematic colloids
Figure: Computer simulation of a Saturn ring and satellite defects using Gay-Berne potential.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Liquid Crystal Displays Liquid Crystal Thermometers Polymer dispersed liquid crystals
Figure: Active-matrix liquid crystal display.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Liquid Crystal Displays Liquid Crystal Thermometers Polymer dispersed liquid crystals
Figure: Temperature sensitive cholesteric liquid crystalline film
http://www.prospectonellc.com/lcr.htm Reversible Temperature Indicating paints, slurries, labels, Liquid Crystal Thermometers
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Liquid Crystal Displays Liquid Crystal Thermometers Polymer dispersed liquid crystals
Figure: In a typical PDLC sample, there are many droplets with different configurations and orientations. When an electric field is applied, however, the molecules within the droplets align along the field and have corresponding optical properties.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Liquid Crystal Displays Liquid Crystal Thermometers Polymer dispersed liquid crystals
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals