Introduction to liquid crystals

Denis Andrienko

International Max Planck Research SchoolModelling of soft matter

11-15 September 2006, Bad Marienberg

September 14, 2006

Contents

1 What is a liquid crystal 21.1 Nematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Cholesterics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Smectics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Columnar phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Lyotropic liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Long- and short-range ordering 72.1 Order tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Director . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Phenomenological descriptions 103.1 Landau-de Gennes free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Frank-Oseen free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Nematic-isotropic phase transition 144.1 Landau theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Maier-Saupe theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3 Onsager theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Response to external fields 185.1 Frederiks transition in nematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6 Optical properties 206.1 Nematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.2 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. Amongthem:

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 CondensedMatter 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 andExperiments”, 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 inphysics. Hence, to understand what is a “liquid crystal” we need to clarify why would one give such aname to a substance.

At first, the notion “liquid crystal” seems to be absurd. It, however, suggests that it is an intermediatestate of a matter, in between the liquid and the crystal. It must possess some typical properties of aliquid (e. g. fluidity, inability to support shear, formation and coalescence of droplets) as well as somecrystalline properties (anisotropy in optical, electrical, and magnetic properties, periodic arrangement ofmolecules in one spatial direction, etc.).

Certain structural features are often found in molecules forming liquid crystal phases, and they maybe summarized as follows:

1. The molecules have anisotropic shape (e. g. are elongated). Liquid crystallinity is more likely tooccur 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.

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Figure 1: The arrangement of molecules in liquid crystal phases. (a) The nematic phase. The moleculestend to have the same alignment but their positions are not correlated. (b) The cholesteric phase. Themolecules tend to have the same alignment which varies regularly through the medium with a periodicitydistance p/2. The positions of the molecules are not correlated. (c) smectic A phase. The molecules tendto lie in the planes with no configurational order within the planes and to be oriented perpendicular tothe planes.

1.1 Nematics

The nematic phase is characterized by long-range orientational order, i. e. the long axes of the moleculestend 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 crystalsinvolves the geometry and dynamics of the preferred axis, which is defined by a vector n(r) giving itslocal orientation. This vector is called a director. Since its magnitude has no significance, it is taken tobe 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 beable to rotate about their long axes and also there seems to be no preferential arrangement of the twoends 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 ofthe relative arrangement of the molecules in the nematic phase is shown in Fig. (1a). The long planarmolecules are symbolized by ellipses.

Figure 2: (a) Schlieren texture of a nematic film with surface point defects (boojums). (b) Thin nematicfilm 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 nematicphase was named, as “nematic” comes from the Greek word for “thread”. Photo courtesy of IngoDierking.

On optical examination of a nematic, one rarely sees the idealized equilibrium configuration. Somevery prominent structural perturbation appear as threads from which nematics take their name (Greek

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“νηµα” means thread). These threads are analogous to dislocations in solids and have been termeddisclinations by Frank.

Several typical textures of nematics are shown in Fig. (2). The first one is a schlieren texture of anematic 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 tothe polarizer or to the analyzer. The colors are newton colors of thin films and depend on the thicknessof the sample. Point defects can only exist in pairs. One can see two types of boojums with “oppositesign of topological charge”; one type with yellow and red brushes, the other kind not that colorful. Thedifference 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 spectacularconsequence of the confined nature of the film. It is a result of the competition between elastic innerforces and surface anchoring forces. The surface anchoring forces want to align the liquid crystals parallelto the bottom surface and perpendicular to the top surface of the film. The elastic forces work againstthe resulting “vertical” distortions of the director field. When the film is sufficiently thin, the lowestenergy state is surprisingly archived by “horizontal” director deformations in the plane of the film. Thecurrent picture shows a 1-dimensional periodic pattern.

Many compounds are known to form nematic mesophase. A few typical examples are sketched inFig. (3). From a steric point of view, molecules are rigid rods with the breadth to width ratio from 3:1to 20:1.

Figure 3: Typical compounds forming nematic mesophases: (PAA) p-azoxyanisole. From a rough stericpoint of view, this is a rigid rod of length ∼ 20A and width ∼ 5A. The nematic state is found at hightemperatures (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 chemicalstability. (5CB) 4-pentyl-4’-cyanobiphenyl. The nematic state is found at room temperatures (between240C and 350C).

1.2 Cholesterics

The cholesteric phase is like the nematic phase in having long-range orientation order and no long-rangeorder in positions of the centers of mass of molecules. It differs from the nematic phase in that the directorvaries in direction throughout the medium in a regular way. The configuration is precisely what one wouldobtain by twisting about the x axis a nematic initially aligned along the y axis. In any plane perpendicularto the twist axis the long axes of the molecules tend to align along a single preferred direction in this

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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 twistaxis 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- andleft-handed forms.

Figure 4: (a) Cholesteric fingerprint texture. The line pattern is due to the helical structure of thecholesteric 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 crossedpolarizers. The bright colors are due to the difference in rotatory power arising from domains withdifferent cholesteric pitch occuring on rapid cooling close to the smectic A* phase where the pitch stronglydiverges with decreasing temperature. Photo courtesy of Per Rudqvist. (c) Long-range orientation ofcholesteric liquid crystalline DNA mesophases occurs at magnetic field strengths exceeding 2 Tesla. Theimage presented above illustrates this long-range order in DNA solutions approaching 300 milligrams permilliliter. Parallel lines denoting the periodicity of the cholesteric mesophase appear at approximately45-degrees from the axis of the image boundaries.

The pitch of the common cholesterics is of the order of several hundreds nanometers, and thus com-parable with the wavelength of visible light. The spiral arrangement is responsible for the characteristiccolors of cholesterics in reflection (through Bragg reflection by the periodic structure) and their very largerotatory power. The pitch can be quite sensitive to temperature, flow, chemical composition, and appliedmagnetic or electric field. Typical cholesteric textures are shown in Fig. (4).

1.3 Smectics

The important feature of the smectic phase, which distinguishes it from the nematic, is its stratification.The molecules are arranged in layers and exhibit some correlations in their positions in addition to theorientational ordering. A number of different classes of smectics have been recognized. In smectic Aphase the molecules are aligned perpendicular to the layers, with no long-range crystalline order withina layer (see Fig. (1c)). The layers can slide freely over one another. In the smectic C phase the preferredaxis is not perpendicular to the layers, so that the phase has biaxial symmetry. In smectic B phase thereis hexagonal crystalline order within the layers.

In general a smectic, when placed between glass slides, does not assume the simple form of Fig. (1c).The layers, preserving their thickness, become distorted and can slide over one another in order to adjustto the surface conditions. The optical properties (focal conic texture) of the smectic state arise fromthese distortions of the layers. Typical textures formed by smectics are shown in Fig. (5).

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Figure 5: (a,b) Focal-conic fan texture of a chiral smectic A liquid crystal (courtesy of ChandrasekharS., Krishna Prasad and Gita Nair) (c) Focal-conic fan texture of a chiral smectic C liquid crystal. Moreimages are available here: http://www.chem.warwick.ac.uk/jpr/lq/liqcry.html

A number of substances exhibit nematic (or cholesteric) and smectic phases. The general rule appearsto be that the lower temperature phases have a greater degree of crystalline order. Examples are: (i) thenematic phase always occurs at at a higher temperature than the smectic phase; (ii) the smectic phasesoccur in the order A → C → B as the temperature decreases.

1.4 Columnar phases

Figure 6: (a) hexagonal columnar phase Colh (with typical spherulitic texture); (b) Rectangular phaseof a discotic liquid crystal (c) hexagonal columnar liquid-crystalline phase.

Figure 7: Typical discotics: derivative of a hexabenzocoronene and 2,3,6,7,10,11-hexakishexyloxytriphenylene. K(70K) → Colh(100K) → I.

Disk-shaped mesogens can orient themselves in a layer-like fashion known as the discotic nematicphase. If the disks pack into stacks, the phase is called a discotic columnar. The columns themselves

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Figure 8: (1) Columnar phase formed by the disc-shaped molecules and the most common arrangementsof columns in two-dimensional lattices: (a) hexagonal, (b) rectangular, and (c) herringbone. (2,3) MDsimulation results: snapshot of the hexabenzocoronene system with the C12 side chains. Aromatic coresare highlighted. Both top and side views are shown. T = 400 K, P = 0.1 MPa.

may be organized into rectangular or hexagonal arrays, see Fig. (8). Chiral discotic phases, similar tothe chiral nematic phase, are also known.

The columnar phase is a class of liquid-crystalline phases in which molecules assemble into cylindricalstructures to act as mesogens. Originally, these kinds of liquid crystals were called discotic liquid crystalsbecause the columnar structures are composed of flat-shaped discotic molecules stacked one-dimensionally.Since recent findings provide a number of columnar liquid crystals consisting of non-discoid mesogens, itis more common now to classify this state of matter and compounds with these properties as columnarliquid crystals.

Columnar liquid crystals are grouped by their structural order and the ways of packing of the columns.Nematic columnar liquid crystals have no long-range order and are less organized than other columnarliquid crystals. Other columnar phases with long-range order are classified by their two-dimensionallattices: hexagonal, tetragonal, rectangular, and oblique phases.

The discotic nematic phase includes nematic liquid crystals composed of flat-shaped discotic moleculeswithout long-range order. In this phase, molecules do not form specific columnar assemblies but onlyfloat with their short axes in parallel to the director (a unit vector which defines the liquid-crystallinealignment and order).

1.5 Lyotropic liquid crystals

Liquid crystals which are obtained by melting a crystalline solid are called thermotropic. Liquid crys-talline behavior is also found in certain colloidal solutions, such as aqueous solutions of tobacco mosaicvirus and certain polymers. This class of liquid crystals is called lyotropic. For this class concentration(and secondarily temperature) is the important controllable parameter, rather then temperature (andsecondarily pressure) as in the thermotropic phase. Most of the theories outlined below is equally validfor either class, although we will generally have a thermotropic liquid crystal in mind.

2 Long- and short-range ordering

2.1 Order tensor

We will first try to characterize a nematic mesophase. Suppose that the molecules composing a nematicare rigid and rod-like in shape. Then we can introduce a unit vector u(i) along the axis of ith moleculewhich describes its orientation. Since liquid crystals possess a center of symmetry, the average of u(i)

vanishes. It is thus not possible to introduce a vector order parameter for a liquid crystals analogous to

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Figure 9: Orientation of mesogens in a nematic phase. A unit vector u(i) along the axis of ith moleculedescribes its orientation. The director n shows the average alignment.

the magnetization in a ferromagnet, and it is necessary to consider higher harmonics (i. e. tensors). Anatural order parameter to describe the ordering in nematics (or cholesteric) is the second rank tensor

Sαβ(r) =1N

∑i

(u(i)

α u(i)β − 1

3δαβ

)(1)

when the sum is over all the N molecules in a small but macroscopic volume located at the point r;indexes α, β = (x, y, z).

Properties of the order parameter tensor

1. Sαβ is a symmetric tensor

Sαβ = Sβα, (2)

since u(i)α u

(i)β = u

(i)β u

(i)α and δαβ = δβα.

2. It is traceless

TrSαβ =∑

α=(x,y,z)

Sαα =1N

∑i

[(u(i)

x )2 + (u(i)y )2 + (u(i)

z )2 − 133]

= 1− 1 = 0, (3)

since u is a unit vector.

3. Two previous properties (symmetries) reduce the number of independent components (3 by 3 ma-trix) from 9 to 5.

4. In the isotropic phase Sisoαβ = 0.

To prove this, let us transform everything to the spherical coordinate system (in the isotropic phaseall directions are equivalent - spherical coordinates allow simple formulation of this fact)

ux = sin θ cosφ (4)uy = sin θ sinφuz = cos θ

Then

Sαβ =∫ 2π

0

dφ

∫ π

0

sin θdθP (θ, φ)(uαuβ −

13δαβ

), (5)

where P (θ, φ) is the probability to find a molecule with the orientation given by the angles θ, φ.

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It is clear that in the isotropic phase Piso is a constant. Normalization conditions require thatPiso(θ, φ) = 1

4π . Indeed,

2∫ 2π

0

dφ

∫ π/2

0

sin θdθ = 4π. (6)

Note that the integration over θ is from 0 to π/2, but we double the value of the integral tocompensate for the conventional integration from 0 to π in spherical coordinates. This is due thefact the u and −u are indistinguishable (for unpolar molecules), but we nevertheless describe theirorientations using vectors, for which the opposite directions are of course distinguishable.

It is easy to see that in the isotropic phase the cross terms Sxy = Syz = Sxz = 0 because of theintegration over φ (periodic function integrated over its full period). Let us consider one of thediagonal components, for example Szz

Szz = 2∫ 2π

0

dφ

∫ π/2

0

sin θdθP (θ, φ)(

cos2 θ − 13

)= (7)

4πPiso

∫ 1

0

(cos2 θ − 1

3

)d(cos θ) =

23π (x3 − x)

∣∣10

= 0.

In a similar fashion one can show that the other two diagonal components are also equal to zero.

5. In a perfectly aligned nematic (with the molecules along the z axis), prolate geometry

Sprolate =

−1/3 0 00 −1/3 00 0 2/3

. (8)

To prove this it is sufficient to calculate only the Szz component:

Szz = uzuz − 1/3 = 1− 1/3 = 2/3. (9)

Keeping in mind that S is symmetric and traceless we obtain (8).

6. In a perfectly aligned oblate geometry (uz = 0)

Soblate =

1/6 0 00 1/6 00 0 −1/3

. (10)

These properties tell us that Sαβ is a very good candidate for the order parameter of our system.It equals to zero in the isotropic phase, it is sensitive to the direction of the average orientation of themolecules as well as to the molecular distribution of the molecules around the their average orientation.It is indeed often used to describe “difficult” situations, which include presence of defects, melting ofparts of the system, i. e. where the order parameter is a function of spatial coordinates (in the defectcore region, at the nematic-isotropic interface, etc.).

2.2 Director

In the previous section we showed that the order tensor is a suitable candidate for an order parameter of anematic mesophase. It is, however, rather inconvenient to use since it lacks clear physical interpretation.Indeed, in some special cases we can find a more convenient representation, especially in those systemswhere the degree of ordering is not changing and therefore only the average orientation of the moleculesis of interest.

In general, any symmetric second-order tensor has 3 real eigenvalues and three corresponding orthog-onal eigenvectors. (Recall gyration tensor or mass and inertia tensor). Since this tensor is a symmetric

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3 × 3 matrix, a Cartesian coordinate system can be found in which it is diagonal with the diagonalelements 2

3S, − 13S +B, − 1

3S −B (principal moments in case of a gyration tensor). Here the eigenvaluesare ordered such as |S| has the biggest aboslute value. The corresponding eigenvector is called a director,n(r). B is a biaxiality of the molecular distribution.

For a uniaxial nematic phase (molecular distribution function has an axial symmetry) two smallereigenvalues are equal (B = 0) and the order tensor reads

Sαβ = S

(nαnβ −

13δαβ

)(11)

where nα are the components of n in a laboratory (fixed) coordinate system.If we choose n along the z axis of the coordinate system, the three nonzero components of S are

Szz =23S, Sxx = Syy = −1

2S. (12)

The scalar quantity S is a measure of the degree of alignment of molecules. Quantitatively, if P (θ) sin θdθis the fraction of molecules whose axes make angles between θ and θ + dθ with the preferred direction

S =∫ π

0

(1− 3

2sin2 θ

)P (θ)dθ. (13)

In the isotropic phase S = 0, and in the nematic (cholesteric) phase 0 < S < 1. The limit S = 1corresponds to perfect alignment of all the molecules and, of course, can never be realized in practice. Ifwe compare Eq. (11) to two previous cases of ideal prolate and oblate molecular distributions, we willsee that negative values of S correspond to a pancake-like (oblate) molecular distribution, positive valuesdescribe prolate-like distributions, and isotropic liquid has S = 0. In general, it can be shown that thescalar order parameter S changes in the range from −1/2 to 1 (from perfect oblate to perfect prolategeometry via isotropic phase).

The most important classes of liquid crystals belong to the uniaxial case (the smectic C is a typicalcounter example). The order parameter for all such systems can be characterized by a magnitude S anda direction n, where the latter is the principal axis of the order parameter tensor.

The theoretical development depends merely upon the existence of n and is by no means restrictedto the case of rigid symmetric molecules.

3 Phenomenological descriptions

3.1 Landau-de Gennes free energy

Once the appropriate order parameter of the system is identified (here the order tensor Sαβ) we canassume, in a spirit of Landau theories, that the free energy density g(P, T, Sαβ) is an analytic function ofthe order parameter tensor Sαβ . To the extent that Sαβ is a small parameter, we may expand g in powerseries. In particular this expansion is supposed to work well near the transition temperature. Since thefree energy must be invariant under rigid rotations, all terms of the expansion must be scalar functionsof the tensor Sαβ . The most general such expansion has the form

g = gi +12ASαβSαβ −

13BSαβSβγSγα +

14CSαβSαβSγδSγδ (14)

correct to fourth order in Sαβ . Here gi is the free energy density of the isotropic phase. There is just onedistinct invariant of the order 2,3, and 4, as shown: other forms (such as SαβSβγSγδSδα) can be reducedfor uniaxial crystals to one of the forms given when consideration is taken of the fact that the trace Sαα

vanishes.

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The coefficients A, B, and C are in general functions of P and T . Typical to Landau-type theories,this model equation of state predicts a phase transition near the temperature where A vanishes. It isnormally assumed that A has the form

A = A′ (T − T ∗) . (15)

The transition temperature itself will prove to be somewhat above T ∗ (see Sec. 4.1). The coefficients Band C need have no particular properties near T ∗; we will regard them as constants.

If we consider a nematic liquid crystal in which the order parameter is slowly varying in space, thefree energy will also contain terms which depend on the gradient of the order parameter. These termsmust be scalars and consistent with the symmetry of a nematic. If we confine ourselves to terms O(S2),the most general form for the inhomogeneous part of the free energy density is

ge =12L1∂Sij

∂xk

∂Sij

∂xk+

12L2∂Sij

∂xj

∂Sik

∂xk(16)

We will refer to the constants L1 and L2 as elastic constants. In the Oseen-Frank curvature elasticitytheory (see Sec. 3.2) it will be shown that there are three independent constants for nematics in contrastto just two here. Evidently it is necessary to retain higher terms in the expansion in S.

The constants L1 and L2 are related to Frank-Oseen elastic constants by K11 = K33 = 9Q2b(L1 +

L2/2)/2 and K22 = 9Q2bL1/2 and Sb is the bulk nematic order parameter.

Typical values for a nematic compound 5CB [10]: A′ = 0.044 × 106 J/m3K, B = 0.816 × 106 J/m3,C = 0.45 × 106 J/m3, L1 = 6 × 10−12 J/m, L2 = 12 × 10−12 J/m T ∗ = 307K. The nematic-isotropictransition temperature for 5CB is TNI = 308.5 K.

3.2 Frank-Oseen free energy

We have learned that in a given microscopic region of a liquid crystal there is definitive preferred axisalong which the molecules orient themselves (the director). Globally, the orientation of the liquid crystalis then characterized by a vector field, which we denote as a “director filed”. The question we would liketo 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 curvaturestrains. The restoring forces which arise to oppose these deformations we will call curvature stresses ortorques. If these changes in molecular orientation vary slowly in space relative to the molecular distancescale, we will be able to describe the response of the liquid crystal with a version of a continuum elastictheory.

We will assume that the curvature stresses are proportional to the curvature strains when these aresufficiently small; or, equivalently, that the free energy density is a quadratic function of the curvaturestrains in which the analog of elastic moduli appear as coefficients.

We consider a uniaxial liquid crystal and let n(r) be a unit vector giving the direction of the preferredorientation at the point r. As we discussed before, the sign of this vector has no physical significance.Once n is defined at some point we assume that it varies slowly from point to point. At r we introducea local right-handed Cartesian coordinate system x, y, z with z parallel to n. The x and y axis canbe chosen arbitrarily because the liquid crystal is uniaxial. Referred to this coordinate system, the sixcomponents of curvature at this point are defined as

splay s1 =∂nx

∂x, s2 =

∂ny

∂y(17)

twist t1 = −∂ny

∂x, t2 =

∂nx

∂y(18)

bend b1 =∂nx

∂z, b2 =

∂ny

∂z(19)

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Figure 10: The three distinct curvature strains of a liquid crystal: (a) splay, (b) twist, and (c) bend.

These three curvature strains can also be defined by expanding n(r) in a Taylor series in powers ofx, y, z measured from the origin

nx(r) = s1x+ t2y + b1z +O(r2), (20)ny(r) = −t1x+ s2y + b2z +O(r2), (21)nz(r) = 1 +O(r2). (22)

We now postulate that the Gibbs free energy density g of a liquid crystal, relative to its free energydensity in the state of uniform orientation can be expanded in terms of six curvature strains

g =6∑

i=1

kiai +12

6∑i,j=1

kijaiaj (23)

where the ki and kij = kji are the curvature elastic constants and for convenience in notation we haveput a1 = s1, a2 = t2, a3 = b1, a4 = −t1, a5 = s2, a6 = b2.

Because the crystal is uniaxial, a rotation about z will make no change in the physical descriptionof the substance, and consequently the free energy density g should be invariant under such rotations.By consideration of a few special cases (such as rotations of 1

2π and 14π about z), it is readily shown

that there are only two independent moduli ki, and that of the thirty-six kij , only five are independent.The general expression for the free energy density, written in terms of a set of eight independent moduli,becomes

g = k1(s1 + s2) + k2(t1 + t2) +12k11(s1 + s2)2

12

+ k22(t1 + t2)2 (24)

+12k33(b21 + b22) + k12(s1 + s2)(t1 + t2)− (k22 + k24)(s1s2 + t1t2).

The last term can be written

s1s2 + t1t2 =∂nx

∂x

∂ny

∂y− ∂ny

∂x

∂nx

∂y=

∂

∂x

(nx∂ny

∂y

)− ∂

∂y

(nx∂ny

∂x

)(25)

and consequently it will contribute only to to surface energies. Thus we can omit the last term inconsiderations involving the properties of the bulk liquid crystal.

In the presence of further symmetries g can have still simpler forms

1. If the molecules are nonpolar or, if polar, are distributed with equal probability in the two directions,then the choice of the sign of n is arbitrary. We have chosen a right-handed coordinate system inwhich z is positive in the direction of n. A reversal of n which retains the chirality of the coordinatesystem generates the transformation

n → −n, x→ x, y → −y, z → −z. (26)

Invariance of the free energy under this transformation requires

k1 = k12 = 0 (nonpolar). (27)

If k1 6= 0, the equilibrium state has finite splay.

12

2. In the absence of enantiomorphism (chiral molecules which have different mirror images) g shouldbe invariant under reflections under reflections in plane containing the z axis, such as the transfor-mation

x→ x, y → −y, z → z. (28)

This introduces the constraints

k2 = k12 = 0 (mirror symmetry). (29)

If k2 6= 0 then the equilibrium state has finite twist.

In summary, k1 vanishes in the absence of polarity, k2 in the absence of enantiomorphy, and k12

vanishes unless both polarity and enantiomorphy occur together.It is convenient to define

s0 = − k1

k11t0 = − k2

k22(30)

and to add constants to g

g = g +12k11s

20 +

12k22t

20 (31)

=12k11(s1 + s2 − s0)2

12

+ k22(t1 + t2 − t0)2

+12k33(b21 + b22) + k12(s1 + s2)(t1 + t2)− (k22 + k24)(s1 + s2)(t1 + t2).

so that now evident that s0 and t0 are the splay and twist of the state which minimizes the free energy -the equilibrium state. Thus the cholesteric phase is characterized by t0 6= 0.

The free energy density can be written in a vector notation. We note that

s1 + s2 =∂nx

∂x+∂ny

∂y= ∇ · n, (32)

−(t1 + t2) =∂ny

∂x− ∂nx

∂y= n · (∇× n), (33)

b21 + b22 =(∂nx

∂z

)2

+(∂ny

∂z

)2

= (n ·∇n)2 (34)

which may be substituted back to the expression for the free energy to give

g =12k11(∇ · n− s0)2 +

12k22(n · curln + t0)2 +

12k33(n× curln)2 − k12(∇ · n)(∇ · curln) (35)

This is the famous Frank-Oseen elastic free energy density for nematics and cholesterics.Briefly about the meaning of different symbols:

div n = ∇ · n =∂

∂xnx +

∂

∂yny +

∂

∂znz

curln = ∇× n =

∣∣∣∣∣∣i j k∂∂x

∂∂y

∂∂z

nx ny nz

∣∣∣∣∣∣ = i

(∂

∂ynz −

∂

∂zny

)+ j

(∂

∂znx −

∂

∂xnz

)+ k

(∂

∂xny −

∂

∂ynx

)

For the purpose of qualitative calculations it is sometimes useful to consider a nonpolar, nonena-tiomorphic liquid crystal whose bend, splay, and twist constants are equal (one-constant approximation).The free energy density for this theoretician’s substance is

g =12k[(∇ · n)2 + (∇× n)2

]. (36)

13

Equations for the director can be obtained by minimization of the free energy∫

Vgdr with an additional

constraint n2 = 1 and have the following form

− k11∇ (∇ · n) + k22 A[∇× n] + [∇× (An)]+ k33 [[∇× n]×B] + [∇× [B × n]]+ µn = 0,

where A = n · [∇ × n], B = [n× [∇× n]], µ is an unknown Lagrange multiplier which must be foundfrom the condition n2 = 1.

4 Nematic-isotropic phase transition

In this section we will apply existing theories to describe the isotropic to nematic phase transition. Weconsider thermotropic liquid crystals, where temperature is the parameter driving the transition.

4.1 Landau theory

In Sec. 2.2 we derived that in a uniaxial liquid crystal the order parameter Sαβ takes the form

Sαβ = S

(nαnβ −

13δαβ

)(37)

Substituting this into Eq. (14) we obtain

g = gi +13AS2 − 2

27BS3 +

19CS4. (38)

The equilibrium value of S is that which gives the minimum value for the free energy. The dependenceof gi on S for several choices of T is shown in Fig. (11). There is a discontinuous phase transition at atemperature Tc slightly above T ∗. The source of this first-order phase transition lies in the presence ofthe odd-order powers of S in the expansion; the existence of these terms is in turn due to the fact thatthe sign of S has physical meaning.

Figure 11: Landau theory: dependence of the Gibbs free energy density on the order parameter. The caseof the three special temperatures, T ∗∗, Tc, and T ∗ are shown. For illustration we use A = B = C = 1.

The value of S which minimizes Eq. (38) can be found algebraically. It will be a root of the derivativewhich is

AS − 13BS2 +

23CS3 = 0. (39)

14

The solutions of this equations are

S = 0 isotropic phase (40)S = (B/4C)[1 + (1− 24β)1/2] nematic phase, (41)

where β = AC/B2. A third solution, corresponding to a maximum of the free energy, has been suppressed.The transition temperature Tc will be such that the free energies of isotropic and nematic phases are equal.From Eqs. (38) and (41) this point can be determined to be

β =127

; Tc = T ∗ +127

B2

A′C. (42)

Above Tc the isotropic phase is stable; below Tc the nematic is stable.The value of the order parameter at the phase transition is

Sc = B/3C. (43)

The temperature T ∗ corresponds to the limit of metastability of the isotropic phase. It should bepossible, in principle, to supercool the isotropic liquid to this temperature. At T ∗, where the coefficientA in the free energy changes sign, the isotropic phase becomes unstable (because S = 0 is not a localminimum of the free energy).

Likewise, the nematic phase becomes unstable when β > 1/24. This determines a temperatureT ∗∗ = T ∗ + B2/(24A′C) which is the limit of the nematic phase on heating. The temperatures T ∗

and T ∗∗ also have the significance that they are apparent “critical points” for the isotropic and orderedphases, respectively. Thus susceptibilities and correlation lengths, which increase as the transition pointis approached, will appear to headed for a divergence at a temperature slightly beyond the transitiontemperature Tc.

4.2 Maier-Saupe theory

Maier and Saupe in a series of paper (1958-1960) have given a microscopic model for the phase transitionin a nematic liquid crystal.

The Maier-Saupe theory is based on three assumptions:

1. An attractive, orientation-dependent van der Waals interaction between the molecules.

2. The configuration of the centers of mass is not affected by the orientation-dependent interaction.

3. The mean field approximation.

We consider rod-like, nonpolar molecules interacting via van der Waals interactions. We restrict ourselvesto the orientation-dependent part and assume that a particular molecule feels the averaged interactionof the surrounding molecules. We also assume that molecules are arranged in a spherically symmetricway (positional correlations are isotropic) and that the distribution of orientations of each molecule isdescribed by the average order parameter Sαβ .

This assumptions allow to derive the effective potential for each molecule with orientation u

V (u, S) = −32ASαβ(uαuβ −

13δαβ) (44)

The probability distribution for the orientation of a molecule in the presence of this external field is

f(u) =1Z

exp[−V (u, S)

kT

](45)

where Z =∫du exp(−V (u, S)/kT ) is the normalization factor, du indicates an integration over all

orientations of u.

15

The theory is made self-consistent by requiring that the average value of uαuβ − 13δαβ be equal to

Sαβ .

Sαβ =∫du

(uαuβ −

13δαβ

)f(u). (46)

In the case of a uniaxial liquid crystal this equation can be simplified. Taking the preferred direction(director) to be the z axis. and defining θ to be the angle between u and z, the zz component of Eq. (46)can be written

S =2πZ

∫ π

0

(32

cos2 θ − 12

)exp

[−V (θ, S)

kT

]sin θdθ, (47)

where, according to Eq. (44)

V (θ, S) = −AS(

32

cos2 θ − 12

)(48)

and Z can now be written

Z = 2π∫ π

0

exp[−V (θ, S)

kT

]sin θdθ. (49)

It is readily seen that one solution of Eqs. (47) and (49) is S = 0, which corresponds to the isotropicphase. Nontrivial solutions also exist. After integration by parts Eq. (47) can be written in a parametricform

S =34

[exp(x2)xD(x)

− 1x2

]− 1

2(50)

kT

A=

32S

x2, (51)

where D(x) =∫ x

0exp(y2)dy is Dawson’s integral.

For each value of x Eq. (51) determines a corresponding value of S and T . The resulting relationshipbetween S and T is shown in Fig. (12).

Figure 12: Maier-Saupe theory: dependence of free energy F and order parameter S on temperature.

The stable mesophase is given by the solution which minimizes the free energy per particle, whichmay be calculated as

F (T, S) = −12A0S

2 + kT lnC/4π (52)

The Maier-Saupe theory predicts a first-order phase transition at a temperature Tc defined by

kBTc = 0.22A0, (53)

and the value of the order parameter at the transition is

Sc = 0.43. (54)

16

4.3 Onsager theory

Onsager theory [11] is a form of molecular field theory, with the difference that the configurational averageenergy is replaced by the orientation-dependent configurational entropy. Onsager theory is a forerunnerof modern density-functional theory [12].

Consider an ensemble of elongated particles interacting pairwise through some potential V(1,2) whichdepends on both positions and orientations of the molecules: 1 = (r1,Ω1). Onsager has shown how theMayer cluster theory may be used to give an expansion for the equation of state of this system [11].Onsager’s expression for the Helmholtz free energy per particle is expressed in terms of the single-particledensity, ρ(1), which gives the number of molecules per unit solid angle and per unit volume

βF [ρ] =∫ρ(1)

ln ρ(1)Λ3 − 1− βµ+ βU(1)

d(1)− 1

2

∫f(1,2)ρ(1)ρ(2)d(1)d(2). (55)

Here β = 1/kBT , Λ is the de Broglie wavelength, µ is the chemical potential, U(1) is the external potentialenergy,

f(1,2) = exp [−V(1,2)/kBT ]− 1 (56)

is the Mayer f -function.The equilibrium single-particle density that minimizes the free energy (55) is a solution of the following

Euler–Lagrange equation

ln ρ(1)Λ3 − βµ+ βU(1)−∫f(1,2)ρ(2)d(2) = 0, (57)

which can be obtained from the variation of the functional (55).In principle, the equation (57) can be solved numerically for any form of the second virial coefficient

which can be expanded in Legendre polynomials. However, sometimes it is more convenient to minimizethe functional (55) instead of solving the integral equation (57).

The solutions are qualitatively the same as the solutions of the Maier-Saupe equation. The differenceis that the Maier-Saupe theory is commonly applied to liquids, which are only slightly compressible,whereas the Onsager expansion is applied to dilute suspensions of particles for which the change in freeenergy with density is relatively small. The Onsager theory predicts large changes in density at thetransition, as is observed for such systems.

Onsager theory can be explained using simple arguments of a volume taken by a molecule (excludedvolume). In the nematic state, the excluded volume decreases, and this gives rise to the translationalentropy of the system [13]

∆Str = −kB ln (1− Vexcl/V ) ∼ kBρL2D, (58)

where ρ = N/V is the density of rods with length L and diameter D. However, there is also a decreasein orientational entropy in the nematic state, SN = kB lnΩN , in comparison to the isotropic state,SI = kB lnΩI ,

∆Sor = kB lnΩN/ΩI ∼ kB, (59)

where ΩI,N gives the number of orientational states in the isotropic or nematic mesophase.At the nematic-isotropic transition, these two contributions compensate each other, ∆Sor +∆Str = 0

and the critical density is ρc ∼ 1/(L2D), or a critical volume fraction is given by φc = Vrods/V ∼(NLD2)/(NL2D) = D/L. Therefore, increasing length-to-breadth ratio we favor nematic formation.

In conclusion, we point out that the Onsager theory does not describe the bulk equation of stateperfectly. The reason for this is truncation of the virial expansion of the free energy at the leading,pairwise term. However, systematic improvements are possible which improve the agreement with thebulk equation of state [14, 15, 16, 17].

17

5 Response to external fields

The director field is easily distorted and can be aligned by magnetic and electric fields, and by surfaceswhich have been properly prepared.

The magnetic susceptibility of a liquid crystal, owing to the anisotropic form of the molecules com-posing it, is also anisotropic. In the uniaxial case it is a second-rank tensor with two components χ‖ andχ⊥, which are the susceptibilities per unit volume along and perpendicular to the axis. The susceptibilitytensor thus takes the form

χij = χ⊥δij + χaninj , (60)

where χa = χ‖ − χ⊥ is the anisotropy and is generally positive. It is thus possible to exert torques onthe liquid crystal by applying a field. The presence of a magnetic field H leads to an extra term in thefree energy of

gm = −12χ⊥H

2 − 12χa(n ·H)2. (61)

The first term usually will be omitted as it is independent of the orientation of the director. The lastterm gives rise to a torque on the liquid crystal - if χa is positive the molecules will align parallel to thefield.

The dielectric susceptibility of a liquid crystal is also anisotropic and has the same form as the magneticsusceptibility. Thus, in principle, we can achieve the same effect with an electric field as with a magneticfield. In an electric field E there will be an additional free energy

ge = − 18πε⊥E

2 − 18πεa(n ·E)2. (62)

in practice the alignment of a liquid crystal by an electric field is complicated by the presence of conductingimpurities which make it necessary to use alternating electric fields.

5.1 Frederiks transition in nematics

Consider a nematic liquid crystal between two glass slides. The interaction between the nematic andthe glass is such that the director is constrained to lie perpendicular to the glass at the boundaries.When a magnetic field, applied perpendicular to the director, exceeds a certain critical value, the opticalproperties of the system change abruptly. The reason is that both the magnetic field and the boundariesexert torques on the molecules and when the field exceeds Hc it becomes energetically favorable for themolecules in the bulk of the sample to turn in the direction of the field. This effect first observed byFrederiks and Zolina and can be used to measure some of the elastic constants.

Figure 13: Frederiks transition. The liquid crystal is constrained to be perpendicular to the boundarysurfaces and a magnetic field is applied in the direction shown. (a) Below a certain critical field Hc, thealignment is not affected. (b) slightly above Hc, deviation of the alignment sets in. (c) field is increasedfurther, the deviation increases.

18

Let the z axis be perpendicular to the glass surfaces and the field H lie along the x direction (seeFig. (13)). The director will have the form

nx = sin θ(z), ny = 0, nz = cos θ(z) (63)

so that θ is the angle between the director and the z axis.First we need to calculate the divergence and curl of our vector

∇ · n = − sin θ∂θ

∂z,

∇× n = j cos θ∂θ

∂z,

n · (∇× n) = 0,

n× curln = (knx − inz) cos θ∂θ

∂z.

The elastic energy per unit area now takes the form

g =12

∫ d/2

−d/2

dz

[(k11 sin2 θ + k33 cos2 θ

)(∂θ∂z

)2

− χaH2 sin2 θ

], (64)

where d is the thickness of the sample.In the undistorted structure (θ = 0) the field does not exert a torque on the molecules - they are in

metastable equilibrium. We will suppose one constant approximation, k11 = k33 = k.We define the length ξ =

√k/χaH2 and Eq. (64) becomes

g =12k

ξ2

∫ d/2

−d/2

dz

[ξ2(∂θ

∂z

)2

− sin2 θ

]. (65)

Now we have a variational problem - we have to minimize the free energy satisfying the boundaryconditions

θ|z=−d/2,d/2 = 0. (66)

Variation of the free energy leads to the differential (Euler-Lagrange) equation

ξ2∂2θ

∂2z+ sin θ cos θ = 0. (67)

There is a trivial solution θ = 0 which satisfies the boundary conditions. If the maximum distortion θm

is smallθ = θm cos

z

ξ(68)

is a good approximate solution. The boundary conditions require that d = ξπ, or, equivalently,

Hc =

√k33

χa

π

d(69)

which holds even if k11 6= k33. Thus a measurement of Hc is a measurement of k33, provided that χa

is known. For fields weaker than Hc only the trivial solution exists, and there is no distortion on thenematic structure.

The general solution to Eq. (67) is readily found. The first integral is

ξ2(∂θ

∂z

)2

+ sin2 θ = sin2 θm. (70)

19

where the constant of integration has been identified by observing that dθdz = 0, where θ takes on itsmaximum value. This maximum presumably lies halfway between the glass surfaces at z = 0. Theequation can be further integrated to give

12d− z = ξ

∫ θ

0

dθ′(sin2 θm − sin2 θ′

)1/2= ξ csc θmF (csc θm, θ), (71)

where F is the incomplete elliptic integral of the first kind, csc θ = 1/ sin θ, and we have used the boundarycondition θ = 0 at z = 1

2d. The maximum distortion is found by putting z = 0, θ = θm,

12d = ξ csc θmF (csc θm, θm) = ξK(sin θm), (72)

where K is the complete elliptic integral of the first kind.For fields just above Hc

θm ∼ 2[H

Hc− 1]1/2

. (73)

The dependence of θm on H is shown in in Fig. (14). In fact, ξ can be directly interpreted as the distancewhich a disturbance can propagate into the liquid crystal in the presence of an ordering field. The lengthξ is called the magnetic coherence length and arises in many problems involving the distortion producedby a magnetic field.

Figure 14: Dependence of θm on H, as given by Eq. (72).

6 Optical properties

6.1 Nematics

It has already been mentioned in Section 5 that the dielectric susceptibility of a nematic is anisotropic andin a uniaxial state is a second-rank tensor with two components ε‖ and ε⊥, which are the susceptibilitiesper unit volume along and perpendicular to the axis.

εij = ε⊥δij + εaninj . (74)

Correspondingly, we can introduce ordinary and extraordinary refractive indexes

ne = √ε‖, no =

√ε⊥, ∆n = ne − no. (75)

For typical nematic liquid crystals, no is approximately 1.5 and the maximum difference, ∆n, may rangebetween 0.05 and 0.5.

20

Thus, when light enters a birefringent material, such as a nematic liquid crystal sample, the processis modelled in terms of the light being broken up into the fast (called the ordinary ray) and slow (calledthe extraordinary ray) components. Because the two components travel at different velocities, the wavesget out of phase. When the rays are recombined as they exit the birefringent material, the polarizationstate has changed because of this phase difference.

Figure 15: Light traveling through a birefringent medium will take one of two paths depending on itspolarization.

The length of the sample is another important parameter because the phase shift accumulates as longas the light propagates in the birefringent material. Any polarization state can be produced with theright combination of the birefringence and length parameters.

Consider the case where a liquid crystal sample is placed between crossed polarizers whose transmissionaxes are aligned at some angle α between the fast and slow direction of the material.

Because of the birefringent nature of the sample, the incoming linearly polarized light

Eincident =(Ex

Ey

)=(E0 cosαE0 sinα

)(76)

becomes elliptically polarized

Esample(z) =(Ex exp(ikez)Ey exp(ikoz)

)(77)

Using Jones calculus for optical polarizer

A =(

sin2 α − cosα sinα− cosα sinα cos2 α

)(78)

We obtain the light behind the second polarizer (analyzer)

Eout = AEsample(z = L) = E0 sin(2α) sin(∆kL/2) exp[i(ko + ke)L/2](

sinαcosα

)(79)

and the output intensity becomes

Iout = |Eout|2 = E20 sin2(2α) sin2

(∆kL

2

)= I0 sin2(2α) sin2 π∆nL

λ. (80)

Therefore, when this ray reaches the second polarizer, there is now a component that can pass through,and the region appears bright. For monochromatic light (single frequency), the magnitude of the phasedifference is determined by the length and the birefringence of the material. If the sample is very thin,the ordinary and extraordinary components do not get very far out of phase. Likewise, if the sampleis thick, the phase difference can be large. If the phase difference equals 2π, the wave returns to itsoriginal polarization state and is blocked by the second polarizer. The size of the phase shift determinesthe intensity of the transmitted light.

21

If the transmission axis of the first polarizer is parallel to either the ordinary or extraordinary direc-tions, the light is not broken up into components, and no change in the polarization state occurs. In thiscase, there is not a transmitted component and the region appears dark.

In a typical liquid crystal, the birefringence and length are not constant over the entire sample.This means that some areas appear light and others appear dark, as shown in the microscope pictureof a nematic liquid crystal, Fig. (2a), taken between crossed polarizers. The light and dark areas thatdenote regions of differing director orientation, birefringence, and length. The dark regions that representalignment parallel or perpendicular to the director are called brushes.

Colors arising from polarized light studies

Birefringence can lead to multicolored images in the examination of liquid crystals under polarized whitelight.

Up to this point, we have dealt only with monochromatic light is considering the optical properties ofmaterials. In understanding the origin of the colors which are observed in the studies of liquid crystalsplaced between crossed linear polarizers, it will be helpful to return to the examples of retarding plates.They are designed for a specific wavelength and thus will produce the desired results for a relativelynarrow band of wavelengths around that particular value. If, for example, a full-wave plate designed forwavelength is λ is placed between crossed polarizers at some arbitrary orientation and the combinationilluminated by white light, the wavelength λ will not be affected by the retarder and so will be extinguished(absorbed) by the analyzer. However, all other wavelengths will experience some retardation and emergefrom the full-wave plate in a variety of polarization states. The components of this light passed by theanalyzer will then form the complementary color to λ.

Color patterns observed in the polarizing microscope are very useful in the study of liquid crystals inmany situations, including the identification of textures, of liquid crystal phases and the observations ofphase changes.

Frederiks transition

The distortion of the director by the filed when Frederiks transition (see Section 5.1) occur can be detectedoptically because there is a change in the refractive index of the material. The refractive index locally is(the light is polarized along z axis

n =neno(

n2e sin2 θ + no cos2 θ

)1/2(81)

The average change in the refractive inde is

δ =1d

∫ d/2

−d/2

dz(ne − no). (82)

Using Eq. (70), this can be put in the form

δ

ne= 1− 2ξ

d

∫ θm

0

dθ

(1 + ν sin2 θ)1/2(sin2 θm − sin2 θ)1/2= 1− 2ξ

d

1(1 + ν sin2 θm)1/2

K

(1 + ν

csc2 θm + ν

).

(83)where ν = (n2

e − n2o)/n

2o. For small deformations the change in the refractive index is

δ = neνH −Hc

Hc(84)

and for large fields approaches ne − no.

6.2 Cholesterics

This is your homework.

22

Figure 16: Examples of axial disclinations in a nematic: (a) m = +1, (b) the parabolic disclination,m = +1/2, (c) the hyperbolic disclination (topologically equivalent to the parabolic one), m = −1/2.

7 Defects

Topological defects appear in physics as a consequence of broken continuous symmetry [3]. They existalmost in every branch of physics: biological systems [18], superfluid helium [19], ferromagnets [20],crystalline solids [21, 22], liquid crystals [23, 24, 25, 26, 27], quantum Hall fluids [28], and even opticalfields [29] playing an important role in such phenomena as response to external stresses, the nature andtype of phase transitions. They even arise in certain cosmological models [30].

Liquid crystals are ideal materials for studying topological defects. Distortions yielding defects areeasily produced through control of boundary conditions, surface geometries, and external fields. Theresulting defects are easily imaged optically. The simplest, nematic liquid crystalline phase owes its nameto the typical threadlike defect which can be seen under a microscope in a nematic or cholesteric phase[31].

First explanations were given by Friedel [32] who suggested that these threads are lines on which thedirector changes its direction discontinuously. In analogy with dislocations in crystals, Frank proposed tocall them disclinations [33]. To classify topological defects the homotopy theory can be employed to studythe order parameter space [34]. For the case of nematics, there are two kinds of stable topological defectsin three dimensions: point defects, called hedgehogs and line defects, called disclinations. Hedgehogs arecharacterized by an integer topological charge q specifying the number of times the unit sphere is wrappedby the director on any surface enclosing the defect core. An analytical expression for q is

q =18π

∫dSiεijkn · (∂jn× ∂kn) , (85)

where ∂α denotes differentiation with respect to xα, εijk is the Levi-Civita symbol, and the integral isover any surface enclosing the defect core. For an order parameter with O3, or vector symmetry, theorder-parameter space is S2, and hedgehogs can have positive or negative charges. Nematic inversionsymmetry makes positive and negative charges equivalent, and we may, as a result, take all charges tobe positive.

The axial solutions of the Euler-Lagrange equations representing disclination lines are

φ = mψ + φ0, (86)

where nx = cosφ, ψ is the azimuthal angle, x = r cosψ, m is a positive or negative integer or half-integer[2]. Examples of disclinations for several m are given in Fig. (16). The elastic energy per unit lengthassociated with a disclination is πKm2 ln(R/r0), where R is the size of the sample and r0 is a lower cutoffradius (the core size) [2]. Since the elastic energy increases as m2, the formation of disclinations withlarge Frank indices m is energetically unfavorable.

As has already been mentioned, within the continuum Frank theory, disclinations are singular lineswhere the gradient in the director becomes infinite; this signals a breakdown in the Frank theory. The

23

region near the singularity where the Frank theory fails is called the disclination core. The phenomeno-logical elastic theory predicts that a uniaxial nematic either melts or exhibits a complex biaxial structurein the core region [35].

Therefore, the core of the defect cannot be represented by the director field only, because of thepossible biaxiality and variation of the order parameter. For this reason, a more general theory based onthe alignment tensor (1) should be applied to provide the correct description of the core region [36, 37, 38].

8 Computer simulation of liquid crystals

Computer simulation is a well established method to study liquid crystalline mesophase. It has alreadybeen applied to study various bulk properties of liquid crystals, such as bulk elastic constants [39, 40, 41,42, 43, 44, 45], viscosities [46, 47], helical twisting power [48, 49], parameters of the isotropic–nematicinterface [50]. Moreover, recent works have also proved that computer simulation can be effectivelyapplied to study confined geometries [51, 52]. Another objective accessible for computer simulation isthe study of topological defects in liquid crystals. Indeed, computer simulation provide us with the exactstructure of the disclination core [53, 54] which is not extractable from phenomenological theories.

Molecular dynamics of non-spherical particles

Molecular dynamics is a numerical technique which allows to solve the classical equations of motion fora system of molecules interacting via a given potential vij .

As has already been mentioned in sec. 1.1, liquid crystal molecules have elongated shape. There-fore, simulation techniques should take into account orientational degrees of freedom, in addition to thetranslational degrees. As an example, we consider equations of motion for linear molecules. For a linearmolecule one of the principal moments of inertia vanishes, and the other two inertia components areequal.

There are different ways of discretising of equations of motion in order to advance from one set of thecoordinates to the next: a quaternion algorithm [55], rotation matrix algorithm, involving equations forEuler angles [8], a Gear predictor-corrector method [56]. They vary in convenience of implementationand accuracy. The approach we describe here uses the leap-frog method [57].

For a linear molecule, the angular velocity and the torque are both perpendicular to the molecularaxis at all times. If ei is a unit vector along the molecular axis, then the torque on a molecule can bewritten as

τ i = ei × gi, (87)

where gi = −∇eivij is a ‘gorque’ to be determined from the intermolecular forces. The vector gi can

always be replaced by its component perpendicular to the molecular axis, without affecting eqn (87)

τ i = ei × g⊥i , (88)

with g⊥i = gi − (gi · ei)ei.The equations for rotational motion can be written as two first-order differential equations [8]

ei = ui,

ui = g⊥i /Ii + λei, (89)

where Ii is the moment of inertia, λ is a Lagrange multiplier, which constrains the bond length (moleculelength) to be a constant of a motion.

The equations of rotational motion (89) and Newton’s equation of motion

miri = fi, (90)

describe completely the dynamics of motion of a linear molecule.

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To solve these equations numerically, we have to adopt a proper discretisation of them. There arecertain criteria that this discretisation has to fulfill. The original equations of motion are time reversible.Therefore, discretised system has to be time reversible and conserve energy. The most widely usedalgorithms are leap-frog and velocity-Verlet algorithms [8].

To solve equation (90) for motion of the center of mass of the molecule using the leap-frog algorithm,we advance the velocities by half a step, using current accelerations a(t) and the previous velocitiesv(t− 1

2∆t). Then the velocities and a new set of forces can be calculated:

r (t+ ∆t) = r(t) + ∆tv(t+ 12∆t), (91)

v(t+ 12∆t) = v(t− 1

2∆t) + ∆ta(t). (92)

To evaluate the kinetic energy at time t the current velocities for time t can be calculated using

v(t) = 12

[v(t+ 1

2∆t)

+ v(t− 1

2∆t)]. (93)

To solve the equations of rotational motion (89), one first obtains a Lagrange multiplier λ by consid-ering the advancement of coordinates over half a time step

ui = ui(t− 12∆t) + 1

2∆t[g⊥i (t)/I + λ(t)ei(t)

]. (94)

Taking the scalar product of both sides with the vector ei(t) gives

λ(t)∆t = −2ui(t− 12∆t) · ei(t). (95)

Using eqn (95) one can calculate ui(t) from eqn (89) and then advance a full step in the integrationalgorithm

ui(t+ 12∆t) = ui(t− 1

2∆t) + ∆tui(t). (96)

The step is completed usingei(t+ ∆t) = ei(t) + ∆tui(t+ 1

2∆t). (97)

Forces and torques

To complete the problem, we have to derive forces and torques (or ‘gorques’) from the pairwise inter-particle potential vij(rij , ei, ej). Let the centers of the molecules be separated by a vector rij , andmolecular axes be along unit vectors ei and ej . Then the force on molecule i due to j is

fij = −∇rijvij . (98)

Using the chain rule, one obtains

fij = −(∂vij

∂rij

)∇rijrij −

∑α=i,j

(∂vij

∂ (nij · eα)

)∇rij (nij · eα) , (99)

where nij = rij/rij .Taking into account that

∇rij(nij · eα) = − (nij · eα)

rij

r2ij+

eα

rij, (100)

where α = i, j we obtain

fij = −(∂vij

∂rij

)nij −

∑α=i,j

(∂vij

∂ (nij · eα)

)(eα

rij− rij

(nij · eα)r2ij

). (101)

25

To evaluate the torque on molecule i due to molecule j

τ ij = −ei ×∇eivij

we again apply the chain rule, and obtain

τ ij = −ei ×[rij

rij

(∂vij

∂ (nij · ei)

)+ ej

(∂vij

∂ (ei · ej)

)]. (102)

One can see that fij = −fji but τ ij 6= −τ ji.

Gay-Berne intermolecular potential

The most widely used pair potential for liquid crystal simulations is the Gay-Berne potential. It isbased on the Lennard-Jones potential extended to an anisotropic pair interaction [58, 59]. The completeGay-Berne potential can be expressed as follows

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

), (103)

where ρ = [rij − σ (ei, ej ,nij) + σs]/σs, ri defines the position of the i-th molecule, ei is a unit vectoralong its anisotropy axis, nij = rij/rij , rij = |rij |.

The molecular shape parameter σ and the energy parameter ε depend on the molecular unit vectorsei, ej as well as on the separation vector nij between the pair of molecules. This dependence is definedby the following set of equations

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

, (104)

ε (ei, ej ,nij) = εs [ε′ (ei, ej ,nij)]µ × [ε′′ (ei, ej)]

ν, (105)

where

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

], (106)

ε′′ (ei, ej) =[1− χ2 (ei · ej)

2]−1/2

. (107)

Here χ and χ′ denote the anisotropy of the molecular shape and of the potential energy, respectively,

χ =κ2 − 1κ2 + 1

, χ′ =κ′1/µ − 1κ′1/µ + 1

. (108)

The most important parameters of the potential are the anisotropy parameters κ and κ′. The κ = σe/σs

is roughly the molecular elongation and the κ′ = εs/εe is the well-depth ratio for the side-by-side andend-to-end configurations.

As was demonstrated by simulations [60], the Gay–Berne potential describes quite well thermotropicliquid crystals. The exponents µ and ν define different variants of the potential. The most studied andwidely used parameterizations are given in Table 1.

The Monte Carlo technique

To sample configurational space using the Monte Carlo technique, one begins with some (usually randomlyselected) configuration of molecular positions and orientations and then randomly moves and rotatesmolecules (avoiding molecular overlaps) to generate trial configuration. When the trial configuration isgenerated, the difference in potential energy ∆U = Utrial−Uold of the trial and old systems is calculated

26

Table 1: Parameterizations of the Gay–Berne potentialk k′ µ ν

3 5 2 1 De Miguel and Rull parameters [61].Isotropic, nematic, smectic B, vapor, and solid phases [62].

3 5 1 3 Berardi and Zannoni parameters. Wider nematic phase [63].3 5 1 2 Luckhurst et al [64].3 - 0 0 Soft repulsive potential. Allows bigger timestep.

4.4 20 1 1 Bates and Luckhurst parameters [65].

from the sum of the pair potentials. If ∆U < 0, the trial configuration is accepted and the processis repeated, choosing another molecule to move and rotate. If ∆U > 0 then the Boltzmann weightκ = exp(−∆U/kBT ) is compared to a generated random number, α ∈ [0..1]. If α < κ, the move isaccepted; otherwise it is rejected. Therefore, if ∆U is small, the Boltzmann factor is close to 1 andthe acceptance of the trial configuration is highly probable. As ∆U increases, the trial moves becomeaccepted with lower and lower probability. This means that big fluctuations in the energy are possible,but suppressed, leading to efficient phase space sampling. It is usual practice that about half of trialmoves should be accepted [8].

Note that forces and torques are never calculated, only potential energies. Neither real time nor realparticle trajectories exist during MC phase sampling.

The MC technique of a canonical (NV T ) ensemble can be adapted to the constant pressure (NPT )ensemble. The volume changes should be implemented, for example by scaling of all the molecularpositions. The probability function also changes, according to the ensemble, κ = exp[−∆U + P∆V −NkBT ln(1−∆V/V )/kBT ]. Volume changes are attempted far less often than molecular moves, to insurethey are slow compared to molecular motion.

The Metropolis description [66], described above, generates configurations which are weighted to favorthermally populated states of the system. However, a number of important properties, for example freeenergies, requires substantial sampling over higher-energy configurations. Then non-Boltzmann samplingcan be used, with the old and trial states additionally weighted by a suitable function. This weight-ing function should have the form which encourages the system to explore regions of phase space notfrequently sampled by the Metropolis method. The weighted averages than should be corrected, givingdesired ensemble averages. A particular example of non-Boltzmann sampling is the umbrella-samplingtechnique [8].

9 Applications

Liquid crystal technology has had a major effect many areas of science and engineering, as well as devicetechnology. Applications for this special kind of material are still being discovered and continue to provideeffective solutions to many different problems.

Liquid Crystal Displays

The most common application of liquid crystal technology is liquid crystal displays (LCDs.) This fieldhas grown into a multi-billion dollar industry, and many significant scientific and engineering discoverieshave been made. The principle of LCD operation is shown in Fig. (17).

Liquid Crystal Thermometers

As demonstrated earlier, chiral nematic (cholesteric) liquid crystals reflect light with a wavelength equalto the pitch. Because the pitch is dependent upon temperature, the color reflected also is dependent

27

Figure 17: Active-matrix liquid crystal display.

Figure 18: Temperature sensitive cholesteric liquid crystalline film. See the following link for more details:http://www.prospectonellc.com/lcr.htm. “Reversible Temperature Indicating paints, slurries, labels,Liquid Crystal Thermometers”.

upon temperature. Liquid crystals make it possible to accurately gauge temperature just by looking atthe color of the thermometer, see Fig. (18). By mixing different compounds, a device for practically anytemperature range can be built.

Optical Imaging

An application of liquid crystals that is only now being explored is optical imaging and recording. In thistechnology, a liquid crystal cell is placed between two layers of photoconductor. Light is applied to thephotoconductor, which increases the material’s conductivity. This causes an electric field to develop inthe liquid crystal corresponding to the intensity of the light. The electric pattern can be transmitted byan electrode, which enables the image to be recorded.

Polymer dispersed liquid crystals

PDLCs consist of liquid crystal droplets that are dispersed in a solid polymer matrix (see Fig. (19)). Theresulting material is a sort of “swiss cheese” polymer with liquid crystal droplets filling in the holes. Thesetiny droplets (a few microns across for practical applications) are responsible for the unique behavior ofthe material. By changing the orientation of the liquid crystal molecules with an electric field, it ispossible to vary the intensity of transmitted light.

Other Liquid Crystal Applications

Liquid crystals have a multitude of other uses. They are used for nondestructive mechanical testing ofmaterials under stress. This technique is also used for the visualization of RF (radio frequency) wavesin waveguides. They are used in medical applications where, for example, transient pressure transmitted

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Figure 19: In a typical PDLC sample, there are many droplets with different configurations and orien-tations. When an electric field is applied, however, the molecules within the droplets align along thefield and have corresponding optical properties. In the following diagram, the director orientation isrepresented by the black lines on the droplet.

by a walking foot on the ground is measured. Low molar mass (LMM) liquid crystals have applicationsincluding erasable optical disks, full color “electronic slides” for computer-aided drawing (CAD), andlight modulators for color electronic imaging.

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