introduction to liquid crystals - max planck societyandrienk/teaching/liquid_crystals/... ·...

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 mesophasesNematicsCholestericsSmecticsMolecular arrangementColumnar phases

Short- and long-range orderingOrder tensorProperties of the order tensorDirector

Phenomenological descriptionsLandau-de Gennes theoryFrank-Oseen free energyOne elastic constant approximationNematic-isotropic transitionResponse to external fields

Frederiks transitionOptical properties

NematicsColorsCholesterics

DefectsLinear defectsInteraction of defectsNematic colloids

Simulation of liquid crystalsForces, torques and gorquesGay-Berne potentialPhase diagramsNematic colloids

ApplicationsLiquid Crystal DisplaysLiquid Crystal ThermometersPolymer dispersed liquid crystals






Recommended books

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 MatterPhysics”.

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.


http://www.mpip-mainz.mpg.de:/~andrienk/lectures/IMPRS/liquid_crystals.pdf





What are Liquid Crystals?

The name suggests that it is a state of a matter in between theliquid and the crystal.

I Liquid

- Fluidity- Inability to support shear- Formation and coalescence of droplets

I Solid

- Anisotropy in optical, electrical, and magnetic properties- Periodic arrangement of molecules in one spatial direction





NematicsCholestericsSmecticsMolecular arrangementColumnar phases

Typical textures

Figure: (a) Schlieren texture. (b) Thin nematic film on isotropic surface.(c) Nematic thread-like texture.

“Nematic” comes from the Greek word for “thread”.






Typical compounds

From a rough steric point of view, this is a rigid rod of length∼ 20A and width ∼ 5A.






Typical textures

Figure: (a) Fingerprint texture. (b) Grandjean or standing helix texture(c) DNA mesophases






Typical textures

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.






Molecular arrangement

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.






Typical textures

Figure: (a) hexagonal columnar phase Colh (with typical spherulitictexture); (b) Rectangular phase of a discotic liquid crystal (c) hexagonalcolumnar liquid-crystalline phase.






Typical structures

Figure: Typical discotics: derivative of a hexabenzocoronene and2,3,6,7,10,11-hexakishexyloxytriphenylene. K(70K) → Colh(100K) → I.






Molecular arrangement

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.





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





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






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.







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).







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!






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.





Landau-de Gennes theoryFrank-Oseen free energyOne elastic constant approximationNematic-isotropic transitionResponse to external fieldsFrederiks transition

Problem

We would like to describe:

I isotropic to nematic transition

I inhomogeneous systems

I 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.






Landau-de Gennes free energy

To the extent that Sαβ is a small parameter, we may expand thefree 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 thetemperature where A vanishes

A = A′ (T − T ∗) .






Elastic part of the free energy

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.






Curvature strains and stresses

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





Curvature strains and stresses

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).






Frank-Oseen free energy

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.






Symmetries

There are several symmetries which will reduce the number of theelastic constants in this expansion

1. Uniaxial crystal (a rotation about the z axis does not changefree energy). Out of the thirty-six kij , only five areindependent.

2. For nonpolar molecules, the choice of the sign of n is arbitrary.

n → −n, x → x , y → −y , z → −z .

k1 = k12 = 0 (nonpolar).

3. In the absence of enantiomorphism (chiral molecules)

x → x , y → −y , z → z .

k2 = k12 = 0 (mirror symmetry).Denis Andrienko IMPRS school, Bad Marienberg Introduction to Liquid Crystals





Frank-Oseen free energy

g =1

2k11(∇ · n)2 +

1

2k22(n · curln + t0)

2 +1

2k33(n× curln)2

This is the famous Frank-Oseen elastic free energy density fornematics and cholesterics.

Figure: The three distinct curvature strains of a liquid crystal: (a) splay,(b) twist, and (c) bend.






One elastic constant approximation

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

].






Landau-de Gennes picture

Substituting

Sαβ = S

(nαnβ −

1

3δαβ

)into Landau-de Gennes free energy we obtain

g = gi +1

3AS2 − 2

27BS3 +

1

9CS4.

The equilibrium value of S is that which gives the minimum valuefor the free energy.






Equilibrium order parameter

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.






Equilibrium order parameter

The minima of the free energy are

S = 0 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 energiesof isotropic and nematic phases are equal

βc =1

27; Tc = T ∗ +

1

27

B2

A′C.

Above Tc the isotropic phase is stable; below Tc the nematic isstable.






Magnetic and dielectric susceptibilities

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.






Geometry

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.






Free energy

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/2dz

[(k11 sin2 θ + k33 cos2 θ

) (∂θ

∂z

)2

− χaH2 sin2 θ

],

where d is the thickness of the sample.






Euler-Lagrange equation and first integral

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 =√

k/χaH2.The first integral is (free energy does not have explicit dependenceon z)

ξ2(∂θ

∂z

)2

+ sin2 θ = sin2 θm.






Solutions

1. Trivial solution θ = 0.

2. If the maximum distortion θm is small

θ = θm cosz

ξ

3. The boundary conditions require that d = ξπ, or, equivalently,

Hc =

√k33

χa

π

d






Second order transition

Figure: Dependence of θm on H.

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





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.






Ordinary and extraordinary light waves

Figure: Light travelling through a birefringent medium will take one oftwo paths depending on its polarization.






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.






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.






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)





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.






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.






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.





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





Gay-Berne potential

The complete Gay-Berne potential can be expressed as follows

v (ei , ej , rij) = 4ε (ei , ej ,nij)(ρ−12 − ρ−6

), (3)

where ρ = [rij − σ (ei , ej ,nij) + σs ]/σs , nij = rij/rij , rij = |rij |.

σèi , ej , nij

´= σs

(1 −

χ

2

" èi · nij + ej · nij

´2

1 + χei · ej

+

èi · nij − ej · nij

´2

1 − χei · ej

#)−1/2

,

εèi , ej , nij

´= εs

hε′ `

ei , ej , nijíµ

×hε′′ `

ei , ejíν

,

ε′ `

ei , ej , nij´

= 1 −χ′

2

" èi · nij + ej · nij

´2

1 + χ′ei · ej

+

èi · nij − ej · nij

´2

1 − χ′ei · ej

#,

ε′′ `

ei , ej´

=h1 − χ

2 èi · 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.






Phase diagrams for the Gay-Berne potential

Figure: Phase diagrams of the Gay-Berne fluid.






Computer simulation of nematic colloids

Figure: Computer simulation of a Saturn ring and satellite defects usingGay-Berne potential.





Liquid Crystal DisplaysLiquid Crystal ThermometersPolymer dispersed liquid crystals

Liquid Crystal Displays

Figure: Active-matrix liquid crystal display.






Liquid Crystal Thermometers

Figure: Temperature sensitive cholesteric liquid crystalline film

http://www.prospectonellc.com/lcr.htmReversible Temperature Indicating paints, slurries, labels, LiquidCrystal Thermometers


http://www.prospectonellc.com/lcr.htm





Polymer dispersed liquid crystals

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.






Thank you for your attention!


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