Chap. 5. Liquid Crystals 5.1. Introduction - Liquid crystals: a new state of matter (4th) isotropic liquid <-> meso (or intermediate) phase <-> crystalline solid liquid crystal phases (fluidity + elasticity) - Thermotropic (temperature) vs lyotropic (concentration: surfactants and lipids) *About 1% of all organic molecules melt from the solid crystal phase to a thermotropic liquid crystal (no solvent is needed). *Lyotropic liquid crystal phases form in solution. - A degree of order intermediate between the molecular disorder of a liquid and the regular structure of a crystal. -> anisotropic molecular shape (rod-like or disk-like) needed: orientational order
- Thermotropic (temperature) vs lyotropic (concentration: surfactants and lipids)
*About 1% of all organic molecules melt from the solid crystal phase to a thermotropic liquid crystal (no solvent is needed). *Lyotropic liquid crystal phases form in solution.
- A degree of order intermediate between the molecular disorder of a liquid and the regular structure of a crystal.
-> anisotropic molecular shape (rod-like or disk-like) needed: orientational order
5.2. Types of Liquid Crystals I
5.2.2. Thermotropic Liquid Crystals
- An understanding of the correlation between molecular structure and physical properties of thermotropic mesogens is important to optimize the material parameters
*Calamitic (or rod-like) mesogens: a rigid aromatic (or phenyl groups) core + terminal units such as alkyl chains (steric interactions) and cyano groups (polar interactions).
1) Nematic and chiral nematic phases
- no long-range translational order but orientational order (anisotropic liquid)
*chiral molecules: helical twist of the average orientation of the molecules (director)
*helical pitch: 100 nm to infinity -> selective reflection of light: color change with pitch
(2) Smectic phases
- Weak layering of molecules: density modulation in 1-dim (liquid-like in 2-dim layers)
Smectic A (or A1) and smectic C (tilted smectic A)
- Density modulation in smectic A1
cos with (d = layer spacing)
- Long-range order suppressed by logarithmic growth of thermal fluctuations:
Landau-Peierls instability (1-dim)
- Several molecular arrangements: layer spacing (d) vs molecular length (l)
A1: normal, ≈ ,
Ad: interdigitated, antiparallel dimers of polar molecules,
A2: antiparallel arrangement of polar molecules,
- Other smectic phases with higher degree of order *true long-range order in 3-dim -> crystalline phases (not liquid crystalline phases)
- Chiral smectic C phase (Sm C*): improper ferroelectric -> ferroelectric when unwound
(3) Smectic D, blue phases (I, II, II), twist-grain-boundary (TGB) phases, etc.
(4) Columnar phases
- Molecular structures:
- Molecular arrangements
5.3. Characteristics of Liquid Crystal Phases
5.3.1. Degree of Order
- The positional order is described by the radial distribution function : the probability that a molecule is located in the range dr at a distance r.
- In crystals, peaks of g(r) from the periodic spacing of molecules in the lattice.
*thermal fluctuations broaden the peaks
- In LCs, quasi-long range positional order (1-D in Sm A), ∼
- In liquids, only short range order, ∼ exp with the positional correlation
- In a Sm A, thermal fluctuations of the mean displacement [layer position u(r)]: Using the theorem of the equipartition energy,
ln
with C = the relevant elastic constant L = the size of the system a = the molecular dimension
-> logarithmic divergence as →∞ (typical for quasi-long range order).
- In a crystal, , constant indep. of the size of the system
(true long range order)
5.3.2. Anisotropy of Properties
(1) under electrical and magnetic fields,
- Anisotropic dielectric constant and diamagnetic susceptibility
⊥ and ⊥
*the dielectric free energy: ⋅ ,
parallel to E ( ) or perpendicular to E ( )
(2) optical anisotropy (birefringence)
⊥
(3) Surface alignment: homogeneous or planar (parallel to substrate) homeotropic or vertical (perpendicular to substrate)
(4) Three elastic constants in a nematic LC: (splay, twist, bend)
≈∼ in a mean-field theory
(5) Alignment by flow: three geometries of shear flow
(Miesowicz viscosities)
5.4. Identification of Liquid Crystal Phases
5.4.1. Textures
- Schlieren textures in a nematic- Fan-like textures of focal-conics in a smectic A
5.5.1. Definition of the Orientational Order Parameter
- Euler angle with respect to the director along the z-axis.
anisotropic intermolecular potential, orientational distribution function
→ exp
with exp
cos
- Orientational order parameter: ⟨ cos ⟩ or cos cos *In a tensor form,
⋅ ⋅
5.5.2. Mean-Field Theories for I-N and N-Sm A Transitions
- In the Onsager theory for the I-N transitions, only steric repulsions are considered using the virial expansion (truncated at the second-order: two-body excluded volume). - In the Maier-Saupe theory (anisotropic attractions) and its variants, long-range attractive and short-range repulsive contributions are taken into account.
cos : put into and find self-consistently by iteration.
- For the N-Sm A transition, McMillan introduced the coupling of the orientational order to the translational order for the density modulation normal to the layers:
⟨cos× cos ⟩ with d = the period of density modulation
*The dependence of the anisotropic intermolecular potential on the intermolecular separation in a Gaussian form: exp
5.6. Elastic Properties
- The tensor order parameter for the director field:
*In the continuum theory, varies slowly and smoothly with spatial position , so that details at a molecular scale can be ignored.
- The elastic distortion energy per unit volume in a nematic LC:
∇⋅ ⋅∇× × ∇×
- In the smectic A phase,
with B = the compression
5.7. Phase Transitions in LCs
5.7.1. N-Sm A Transitions: Pretransitional fluctuations:
- Landau-de Gennes model: layer periodicity and the layer fluctuations:
Using the order parameter with , a phase factor,
the free energy near the N-Sm A transition is given by
⋅⋅ gradient terms
5.7.2. Sm A-Sm C and Sm A-Sm B Transitions
- For Sm A-Sm C transitions, the order parameter - For Sm A-Sm B transitions, the order parameter
(six-fold symmetry)
5.8. Applications of Liquid Crystals
5.8.1. Field-Induced Distortions of Nematic Liquid Crystals
- (positive dielectric anisotropy)
or
-> Electric field-induced (controlled) optical modulation between two polarizers.
5.8.2. Twisted-Nematic (TN) and Super TN Displays
- The amount of the twist angle form one substrate to the other.
