Liquid crystalline mesophases Short- and long-range ordering Phenomenological descriptions Optical properties Defects Simulation of liquid crystals Applications Literature What are liquid crystals? Introduction to Liquid Crystals Denis Andrienko IMPRS school, Bad Marienberg September 14, 2006 Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
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Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
Optical propertiesDefects
Simulation of liquid crystalsApplications
LiteratureWhat are liquid crystals?
Introduction to Liquid Crystals
Denis AndrienkoIMPRS school, Bad Marienberg
September 14, 2006
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
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 themedium with a periodicity distance p/2.
(c) smectic A phase. The molecules tend to lie in the planes with no configurational order within the planesand to be oriented perpendicular to the planes.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
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 themedium with a periodicity distance p/2.
(c) smectic A phase. The molecules tend to lie in the planes with no configurational order within the planesand to be oriented perpendicular to the planes.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
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 themedium with a periodicity distance p/2.
(c) smectic A phase. The molecules tend to lie in the planes with no configurational order within the planesand to be oriented perpendicular to the planes.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
Figure: (1) Columnar phase formed by the disc-shaped molecules and themost common arrangements of columns in two-dimensional lattices: (a)hexagonal, (b) rectangular, and (c) herringbone. (2,3) MD simulationresults: snapshot of the hexabenzocoronene system with the C12 sidechains.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
Optical propertiesDefects
Simulation of liquid crystalsApplications
Order tensorProperties of the order tensorDirector
Definition of the order tensor
Figure: A unit vector u(i) along the axis of ith molecule describes itsorientation. The director n shows the average alignment.
Sαβ(r) =1
N
∑i
(u(i)
α u(i)β − 1
3δαβ
)Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
Optical propertiesDefects
Simulation of liquid crystalsApplications
Order tensorProperties of the order tensorDirector
Properties of the order tensor
1. Sαβ is a symmetric tensor since u(i)α u
(i)β = u
(i)β u
(i)α and
δαβ = δβα:
Sαβ = Sβα
2. It is traceless
TrSαβ =∑
α=(x ,y ,z)
Sαα
=1
N
∑i
[(u
(i)x )2 + (u
(i)y )2 + (u
(i)z )2 − 1
33
]= 0,
since u is a unit vector.3. Two previous properties (symmetries) reduce the number of
independent components (3 by 3 matrix) from 9 to 5.Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
Optical propertiesDefects
Simulation of liquid crystalsApplications
Order tensorProperties of the order tensorDirector
Properties of the order tensor
1. Sαβ is a symmetric tensor since u(i)α u
(i)β = u
(i)β u
(i)α and
δαβ = δβα:
Sαβ = Sβα
2. It is traceless
TrSαβ =∑
α=(x ,y ,z)
Sαα
=1
N
∑i
[(u
(i)x )2 + (u
(i)y )2 + (u
(i)z )2 − 1
33
]= 0,
since u is a unit vector.3. Two previous properties (symmetries) reduce the number of
independent components (3 by 3 matrix) from 9 to 5.Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
Optical propertiesDefects
Simulation of liquid crystalsApplications
Order tensorProperties of the order tensorDirector
Properties of the order tensor
1. Sαβ is a symmetric tensor since u(i)α u
(i)β = u
(i)β u
(i)α and
δαβ = δβα:
Sαβ = Sβα
2. It is traceless
TrSαβ =∑
α=(x ,y ,z)
Sαα
=1
N
∑i
[(u
(i)x )2 + (u
(i)y )2 + (u
(i)z )2 − 1
33
]= 0,
since u is a unit vector.3. Two previous properties (symmetries) reduce the number of
independent components (3 by 3 matrix) from 9 to 5.Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
Optical propertiesDefects
Simulation of liquid crystalsApplications
Order tensorProperties of the order tensorDirector
Properties of the order tensor
4. In the isotropic phase S isoαβ = 0.
ux = sin θ cosφ, uy = sin θ sinφ, uz = cos θ.
Sαβ =
∫ 2π
0dφ
∫ π
0sin θdθP(θ, φ)
(uαuβ −
1
3δαβ
),
Sxy = Syz = Szx = 0 because of the integration over φ. Forthe Szz component we obtain
Szz = 2
∫ 2π
0dφ
∫ π/2
0sin θdθP(θ, φ)
(cos2 θ − 1
3
)=
4πP iso
∫ 1
0
(cos2 θ − 1
3
)d(cos θ) =
2
3π (x3 − x)
∣∣10
= 0.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
Optical propertiesDefects
Simulation of liquid crystalsApplications
Order tensorProperties of the order tensorDirector
Properties of the order tensor
5. In a perfectly aligned nematic (with the molecules along the zaxis), prolate geometry
Sprolate =
−1/3 0 00 −1/3 00 0 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 weobtain (1).
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
Optical propertiesDefects
Simulation of liquid crystalsApplications
Order tensorProperties of the order tensorDirector
Properties of the order tensor
6. In a perfectly aligned oblate geometry (uz = 0)
Soblate =
1/6 0 00 1/6 00 0 −1/3
.
Try to follow previous arguments and show this!
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
Optical propertiesDefects
Simulation of liquid crystalsApplications
Order tensorProperties of the order tensorDirector
Definition of the director
In general, any symmetric second-order tensor has 3 realeigenvalues 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
(nαnβ −
1
3δαβ
)Vector n is called a director.
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 mesophasesShort- and long-range orderingPhenomenological descriptions
If we consider a nematic liquid crystal in which the order parameteris slowly varying in space, the free energy will also contain termswhich depend on the gradient of the order parameter. These termsmust 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 mesophasesShort- and long-range orderingPhenomenological descriptions
The question we would like to address here is: how much energywill it take to deform the director filed?We will refer to the deformation of relative orientations away fromequilibrium position as curvature strains. The restoring forceswhich arise to oppose these deformations we will call curvaturestresses 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
∂zDenis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
We now postulate that the Gibbs free energy density g of a liquidcrystal, relative to its free energy density in the state of uniformorientation can be expanded in terms of six curvature strains
g =6∑
i=1
kiai +1
2
6∑i ,j=1
kijaiaj
where the ki and kij = kji are the curvature elastic constants andfor convenience in notation we have puta1 = s1, a2 = t2, a3 = b1, a4 = −t1, a5 = s2, a6 = b2.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
For the purpose of qualitative calculations it is sometimes useful toconsider 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
[(∇ · n)2 + (∇× n)2
].
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
Figure: Landau theory: dependence of the Gibbs free energy density onthe 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 mesophasesShort- and long-range orderingPhenomenological descriptions
The magnetic susceptibility of a liquid crystal, owing to theanisotropic 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 thefree energy of
gm = −1
2χ⊥H2 − 1
2χa(n ·H)2.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
Figure: Frederiks transition. The liquid crystal is constrained to beperpendicular to the boundary surfaces and a magnetic field is applied inthe direction shown. (a) Below a certain critical field Hc , the alignmentis 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 mesophasesShort- and long-range orderingPhenomenological descriptions
For fields weaker than Hc only the trivial solution exists, and thereis no distortion on the nematic structure.Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
Optical propertiesDefects
Simulation of liquid crystalsApplications
NematicsColorsCholesterics
Refractive indexes
Susceptibility is a tensor
εij = ε⊥δij + εaninj .
Correspondingly, we can introduce ordinary and extraordinaryrefractive indexes
ne =√ε‖, no =
√ε⊥, ∆n = ne − no .
Typically no ∼ 1.5, ∆n ∼ 0.05− 0.5.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
Optical propertiesDefects
Simulation of liquid crystalsApplications
NematicsColorsCholesterics
Ordinary and extraordinary light waves
Figure: Light travelling through a birefringent medium will take one oftwo paths depending on its polarization.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
Optical propertiesDefects
Simulation of liquid crystalsApplications
NematicsColorsCholesterics
Nematic cell between crossed polarizers
The incoming linearly polarized light
Eincident =
(Ex
Ey
)=
(E0 cosαE0 sinα
)becomes elliptically polarized
Ecell(z) =
(Ex exp(ikez)Ey exp(ikoz)
)Using Jones calculus for optical polarizer we obtain the outputintensity
Iout = |Eout|2 = E 20 sin2(2α) sin2
(∆kL
2
)= I0 sin2(2α) sin2 π∆nL
λ.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
Optical propertiesDefects
Simulation of liquid crystalsApplications
NematicsColorsCholesterics
Colors arising from polarized light studies
Birefringence can lead to multicolored images in the examinationof liquid crystals under polarized white light.
∆n = ∆n(λ)
Different wavelengths will experience different retardation andemerge in a variety of polarization states. The components of thislight passed by the analyzer will then form the complementarycolor to λ.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
Optical propertiesDefects
Simulation of liquid crystalsApplications
NematicsColorsCholesterics
Optical properties of cholesterics
This will be your home work.
I Cholesteric pitch is of the order of the wavelength of visiblelight
I Chiral structure - circularly polarized eigenmodes of Maxwell’sequations
I Pitch depends on temperature (thermometer)
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
Optical propertiesDefects
Simulation of liquid crystalsApplications
NematicsColorsCholesterics
Optical properties of cholesterics
This will be your home work.
I Cholesteric pitch is of the order of the wavelength of visiblelight
I Chiral structure - circularly polarized eigenmodes of Maxwell’sequations
I Pitch depends on temperature (thermometer)
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
Optical propertiesDefects
Simulation of liquid crystalsApplications
NematicsColorsCholesterics
Optical properties of cholesterics
This will be your home work.
I Cholesteric pitch is of the order of the wavelength of visiblelight
I Chiral structure - circularly polarized eigenmodes of Maxwell’sequations
I Pitch depends on temperature (thermometer)
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
Optical propertiesDefects
Simulation of liquid crystalsApplications
Linear defectsLinear defectsNematic colloids
Defects in nematics
Examples of disclinations in a nematic.
Figure: (a) m = +1, (b) the parabolic disclination, m = +1/2, (c) thehyperbolic disclination (topologically equivalent to the parabolic one),m = −1/2.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
Optical propertiesDefects
Simulation of liquid crystalsApplications
Linear defectsLinear defectsNematic colloids
Energy of disclinations
The axial solutions of the Euler-Lagrange equations representingdisclination lines are
φ = mψ + φ0,
where nx = cosφ, ψ is the azimuthal angle, x = r cosψ, m is apositive or negative integer or half-integer. The elastic energy perunit 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, theformation of disclinations with large m is energetically unfavorable.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions
Optical propertiesDefects
Simulation of liquid crystalsApplications
Linear defectsLinear defectsNematic colloids
Nematic-mediated interactions
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 mesophasesShort- and long-range orderingPhenomenological descriptions
Optical propertiesDefects
Simulation of liquid crystalsApplications
Forces, torques and gorquesGay-Berne potentialPhase diagramsNematic colloids
Forces, torques and gorques
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 mesophasesShort- and long-range orderingPhenomenological descriptions
Optical propertiesDefects
Simulation of liquid crystalsApplications
Forces, torques and gorquesGay-Berne potentialPhase diagramsNematic colloids
Gay-Berne potential
The complete Gay-Berne potential can be expressed as follows
Figure: In a typical PDLC sample, there are many droplets with differentconfigurations and orientations. When an electric field is applied,however, the molecules within the droplets align along the field and havecorresponding optical properties.
Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals
Liquid crystalline mesophasesShort- and long-range orderingPhenomenological descriptions