University of LjubljanaFaculty for mathematics and physics

Department of physics

Seminar - 4. letnik

Confined liquid crystals

Author: Luka Jeromel

Mentor: dr. Gregor Skačej

Ljubljana, march 2011

Abstract

In modern technology liquid crystal with high surface-to-volume ratio have an importantrole. Knowing the molecular organization in such liquid crystals is very important for under-standing some macroscopic properties. In this seminar we treat different confinement geome-tries. Then we introduce continuum theory and free energy of liquid crystal and calculate itfor some interesting applicable cases. At the end we look how this theory explains some knownliquid crystal phenomena and outline physics behind operating of LCD displays.

Contents1 Introduction 1

2 Liquid crystals 22.1 Orientational and positional order . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Classification of liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2.1 Nematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2.2 Cholesteric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2.3 Smectics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.3 Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Continuum theory 43.1 Free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Boundary effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 Liquid crystal between two flat plates . . . . . . . . . . . . . . . . . . . . . . . . . 63.4 Effects of magnetic and electric field . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.4.1 Wall–field competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Application of Continuum theory with confined liquid crystals 104.1 The Freedericksz effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.1.1 Twist case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.1.2 Perpendicular case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.1.3 Parallel case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.1.4 Application of the Freedericksz effect . . . . . . . . . . . . . . . . . . . . . 12

4.2 The twisted nematic device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Conclusion 14

1 IntroductionThe term liquid crystal (LC) signifies a state of aggregation that is intermediate between thecrystalline solid and amorphous liquid. Several thousand of organic compounds are known now toform liquid crystal. In last years the subject of liquid crystals has grown enormously to become afascinating interdisciplinary field of study. On the technological side, the advances have also beenspectacular: liquid crystal displays, thermometers, optical imaging and many other applications.For many of these utilities we have to confine liquid crystals to use them. These means weput sample between borders of some kind and some exact geometry. To understand how differenttechnologies work and to invent new applications it is important to understand theory that describesconfined liquid crystals.

1

2 Liquid crystals

2.1 Orientational and positional order

The difference between crystals and liquids, the two most common condensed matter phases, isthat the molecules in crystal are ordered whereas in a liquid they are not. The order in a crystalis usually both positional and orientational, in that the molecules are constrained both to occupyspecific sites in lattice and to point their molecular axes in specific direction. The molecules inliquids, on the other hand, diffuse randomly throughout the sample container. Interestingly enough,many phases with more order than liquids and less than crystals also exist. These phases are calledliquid crystals.

The molecules in all liquid crystals diffuse about much like the molecules in liquid, but as theydo so they maintain some degree of orientational order and sometimes some positional order also.There is small tendency for the molecules to point more in one direction than others or to spendmore time in various positions than others. The fact that most of the order of a crystal when ittransforms to LC is reveled by the value of latent heat. The corresponding value is around 250J/g, which is typical of crystal to liquid transition. Latent heat for transformation from LC toliquid is much smaller, around 5 J/g. Ref. [1].

2.2 Classification of liquid crystals

2.2.1 Nematic

(a) Nematic phase. (b) Smectic A phase. (c) Smectic C phase. (d) Cholesteric phase

Figure 1: Molecular arrangement in liquid crystal phases.(a) Nematic phase. The molecules tend to havethe same alignment but their positions are not correlated. (b) Smectic A phase. The molecules tend to liein planes, diffusing around within the plane. It is like 2-D liquid. (c) Smectic C phase. The preferred axisis not perpendicular to the layers. (d) Director twists.

A simplified picture of molecular arrangement in a nematic liquid crystal is shown in Fig. 1a.The nematic phase is characterized by long-range orientational order. Long axes of molecules tendto align along a preferred direction to which we assign a vector called director. The locally preferreddirection may differ throughout the medium, although in unstrained (equilibrium) nematic it doesnot. To specify the orientational order in such a liquid crystal phase, an order parameter is defined.

2

The simplest option consistent with n = −n symmetry is to find average of the second Legendrepolynomial,

S = 〈P2(cos θ)〉 =⟨3

2 cos2 θ − 12

⟩(1)

where the brackets denote average over many molecules at the same time or over time for asingle molecule, and θ is the angle between the director and the molecular axis. If the moleculesare perfectly aligned (θ = 0)then S = 1. If there is no orientational order (equal probability forany θ) then S = 0.

2.2.2 Cholesteric

This phase is like nematic. It differs in that the director varies throughout the medium in regularway. The configuration is precisely what one would get by twisting about x axis a nematic initiallyaligned along y axis. Director rotates as one progress along twist axis. Schematically this is shownon picture Fig. 1d. This secondary structure of cholesteric by the distance measured along thetwist axis over which the director rotates through a full circle. We call this distance a pitch. Anematic liquid crystal is just cholesteric with infinite pitch.

2.2.3 Smectics

The important feature of the smectic phase is its stratification absent in the nematic phase.The molecules are arranged in layers and exhibit some positional correlations in addition to theorientational ordering. A number of different smectic classes have been recognized Ref. [2]. In thesmectic A phase, Fig. 1b, the molecules are perpendicular to the layers, with no long-range orderwithin a layer. In the smectic C phase, Fig. 1c, the prefered axis is not perpendicular to the layers,so this phase has biaxial symmetry.

2.3 Confinement

For all practical applications of LC s it is necessary to impose geometrical confinement. Theconfining geometries can be of various kind.

a) The simplest one but not least useful is to put a liquid crystal between two flat plates. If wehave conflicting preferred directions of director wall-imposed then the director field adoptsto minimize free energy. That orientation has applicable properties which are well used intechnologies like LCD displays or to determine liquid crystal properties. We are going toexamine this properties in detail a bit later.

b) Another interesting enclosure of liquid crystals are droplets. Where sample is closed insidesphere made of some suitable substance (polymer). Liquid crystal again orders inside thedroplet according to boundary conditions, and these droplets can then be manipulated withexternal fields. Example of that is PDLC, which is used for privacy windows.

c) Liquid crystals can also be confined inside cylindrical structures like micro pores or morecomplex porous materials.

3

Figure 2: Examples of liquid crystals confined to spherical droplets and cylinders. Ref. [3].

3 Continuum theory

3.1 Free energy

In a given microscopic region of a liquid crystal there is a definite preferred molecular axis. Even inequilibrium the direction of this axis can vary from place to place, and it can be forced to vary bythe action of external forces or boundary conditions. We describe this variation by vector functionn(r), where |n| = 1. We will refer to the deformation of relative orientations away from theequilibrium position as curvature strains. Restoring forces which arise to oppose deformations arethen called curvature stresses or torques.

Let us consider a uniaxial liquid crystal. We assume that n varies slowly from point to point,and thus is defined by continuity at other points in the region. At r we introduce local right-handedCartesian coordinate system x, y, z with n parallel to z. We define curvature strains.

Splay: ∂nx∂x

= a1,∂ny∂y

= a2 (2)

Twist: − ∂ny∂x

= a3,∂ny∂y

= a4 (3)

Bend: ∂nx∂z

= a5,∂ny∂z

= a6. (4)

these quantities measure director deforamtion. The curvature strains represent three differentdistortion types of the director. Called splay, twist, see Fig. 3.

4

Figure 3: The three principal types of deformation in nematic liquid crystal. Ref. [3].

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

f =6∑i=1

kiai + 12

6∑i,j=1

kijaiaj (5)

Fortunately, there are restrictions on the free energy that reduce the number of independentconstants ki, kij . First there can be no term for which n and −n give different energy values.Second, there can be no linear terms. These terms change if the coordinate system is rotatedabout the director, if the director is reversed, or if the coordinate system is inverted. Finally, termsthat can been written as divergence and integrated over the volume of the sample can be changedto a surface integral. These contributions to the free energy must be taken into account whenconsidering surface effects, but in most cases can be ignored in a discussion of the volume freeenergy per unit volume.

Finding all of the terms that satisfy these criteria is a painstaking operation and there is noneed to repeat it here. Derivation can be found in [4]. Thankfully allowed terms can be groupedtogether. The free energy per unite volume of nematic liquid crystal can be written as follows,

f = 12k11(∇ · n)2 + 1

2k22(n · (∇× n))2 + 12k33(n× (∇× n))2, (6)

k11, k22 and k33 are called elastic constants (or Frank [5] constants). These three constantsdescribe how ’stiff’ the liquid crystal is to distortions of the director for each of the characteristicdeformation modes shown in Fig. 3. They are temperature dependent, kii ∝ S2(T ), where S isorder parameter. Typical values of these constants are about 10−11 J/m, with K33 being two orthree times larger than K11 and K22.

The equilibrium state of the liquid crystal is obtained by minimizing the total free energy

F =∫Vf

(ni,

∂ni∂xj

)dr (7)

5

with appropriate boundary conditions and subject to the condition n = 1. Allowing ni to varyin Eq. 7, we obtain well known Euler-Lagrange equations

− ∂

∂xj

∂f

∂ ∂ni∂xj

+ ∂f

∂ni= γni (8)

where γ is Lagrange multiplier due to constrain n2 = 1.

3.2 Boundary effects

A strongly confined liquid crystal is characterized by a large surface to volume ratio A/V , whichmeans that surface influences behavior of liquid crystals. In most practical conditions the surfaceforces are strong enough to impose a well-defined director orientation at the surface; this is whatwe call strong anchoring. In this case it is sufficient to minimize only the bulk energy.

Boundaries can be prepared so that molecules close to them have a desired direction calledeasy axis. For example consider glass plates. They are usually coated with thin layer of polymer.The usual procedure for obtaining requested anchoring is to rub glass plates with dry lens tissue orcloth along a fixed direction. It is more energy consuming for molecules to lie across, Fig. 4a, therubbed direction than for them to lie parallel, Fig. 4b. This energy difference is quite appreciable.If the glass is rubbed equally in both directions there would be a tendency for the molecules tostand upright.

(a) (b)

Figure 4: By rubbing we crate grooves. It is more energy consuming for molecules to lie across the rubbeddirection, picture 4a, than for them to lie parallel, picture 4b.

Perpendicular or homeotropic alignment may also be achieved by coating the surface withsurfactants in which case the detailed intermolecular forces play a significant part.

3.3 Liquid crystal between two flat plates

To see how theory from Sect. 3.1 works let us consider an example. Begin with the most simplegeometry possible, Fig. 5. Imagine that liquid crystal is sandwiched between two flat glass plates,separated by a distance d. The z axis is perpendicular to glass surface. Let us call the anglebetween the director and x-axis θ(z); then n = (cos θ, cos θ, 0). In general, the energy associatedwith distortion involves splay, twist and bend. Calculated terms are

∇ · n = 0, n · (∇× n) = − cos2 θdθ

dz− sin2 θ

dθ

dz, n× (∇× n) = 0. (9)

6

Figure 5: Twisted nematic cell. On the left wall there is easy axis along direction θ0, on the right wall easyaxis is along θd. Ref. [6]

Using this results in Eq. (5) one has

f = 12k22

(dθ

dz

)2(10)

Now we follow the procedure presented at the end of Sec. 3.1, and the appropriate Eulerequation is

d

dz

∂f

∂ dθdz− ∂f

∂θ= 0.

Substituting the expression for f into this equation yields

k22d2θ

dz2 = 0, and, θ = Az +B (11)

We see that θ is linear function of z with y-interception B = θ(0) and slope dθ/dz = A =[θ(d)− θ(0)]/d. This solution is good only for strong anchoring and in the bulk area. This meansthat θ is fixed at z = 0 and z = d because the surface forces are strong enough to impose awell-defined direction to the director n. In real situations it can happen that this is not trueone deals with weak anchoring on one or on both walls. In this case Eq. (10) no longer holds.Weak anchoring can be modeled by an appropriate phenomenological surface term, which increasesenergy if director is not aligned with easy axis, e.g.

fsurf = W

2 sin2(θ(0)− θ0).

where W is surface term strength and θ0 is easy axis. Suppose that we have weak anchoringonly at z = 0, in this case we have

FS = FA + Fsurf =∫ d

0

12k22

(dθ

dz

)2dz + W

2 sin2(θ(0)− θ0) (12)

7

Figure 6: Depiction of θ(z) minimizing free energy Eq. (12). In region of molecular thickness a near thesurface area the twist depends on detailed molecular properties. b is the extrapolation length. Ref. [6].

For that kind of energy a possible solution for θ(z) is shown on Fig. 6. To estimating theanchoring strength we define, Fig. 6, extrapolation length b. We will not derive θ(z) and b atsurface from microscopic calculations but we can obtain estimate of b by assuming equation 6holds down to the surface and at surface we add appropriate term

fS = 12k22d

(θ(d)− θ(0)

d

)2+ W

2 θ2(0), sin(θ) ≈ θ because θ � 1 (13)

Minimizing this with respect to θ(0) we find

θ(0) = k22W

∂θ

∂z

Comparing equation with the definition of b we see that, in our approximation

b = k22W

This length is a good measure of how strong anchoring is in comparison with elastic constantk22. This is the fundamental formula for boundary effects. We already know that we have twopossibilities: strong anchoring, if b is comparable to the molecular dimensions a, this means thatratio FA/Fsurf = b/d ∼ a/d. For weak anchoring, extrapolation length may become much largerthan the molecular dimensions a and angle θ(0) is large.

3.4 Effects of magnetic and electric field

Field causes slight charge separation on the non-polar molecules. Weak dipoles are created. Strongmolecular shape anisotropy of LC molecules results in diamagnetic susceptibility anisotropy. In auniaxial state it is a second rank tensor with two components χ|| and χ⊥. The susceptibility tensorthus takes the form

χij = χδij + χaninj , where χa = χ|| − χ⊥

8

The presence of magnetic field H leads to an extra term in the free energy of

fm = −12χH

2 − 12χa(n ·H)2. (14)

The first term is usually omitted as it is independent of the orientation of the director. Thelast term gives rise to a torque acting on a LC molecule–if χa > 0 molecules will align parallel tothe field.

When an electric field E is present there is analogous anisotropy and there will be an additionalfree energy

fe = − 18πεE

2 − 18πεa(n ·E)2. (15)

Figure 7: The twisted nematic cell. Arrow shows the direction of field applied. On nematic is confinedbetween two walls. On figure we have field H perpendicular to anchoring on first side. Ref. [6]

3.4.1 Wall–field competition

Let us consider the competing effects of a wall and of a magnetic field on the alignment of nematicsample. We have one wall at the origin z = 0 and the nematic lying in the region z > 0, Fig. 7.There is one easy direction, x-axis, for the molecules and strong anchoring prevails. A magneticfield is applied perpendicular to x-axis and parallel to the plates (y-axis). For material with positivemagnetic anisotropy is positive director tends to align along the magnetic field H. Let us denotethe angle between the director and x-axis with θ(z). If we consider dimensions of the flat piecesof glass to be much larger than the separation d, than θ should not be function of x and y.

The density of free energy is the same as in (10) with additional magnetic term. For freeenergy we have

FA =∫ d

0f =

∫ d

0

(12k22

(dθ

dz

)2− 1

2χaH2 sin2 θ

)dz. (16)

Substituting the expression for f into equation (9) yields

9

ξ2(d2θ

dz2

)+ sin θ cos θ = 0, ξ2 = k22

H2χa(17)

We defined length ξ which is called magnetic coherence length and is introduced to measurethe effect of the external field, Ref. [7]. The differential equation Eq. (17) must be multiplied bydθ/dζ, making each side a perfect differential and thus easily integrated. We know that far fromthe wall (z →∞) θ is π/2. Thus integration constant is zero

ξdθ

dz= ± cos θ (18)

Both choices of sign are permissible, corresponding to a right-handed or left-handed transitionregion. Choosing for instance the + sign we have

tan(π/2− θ

2

)= exp(−z/ξ).

This equation shows that the thickness of the transition layer is essentially ξ(H), as definedin (17). Taking k22 = 10−6, χa = 10−10 and H = 8 × 105 A/m, we have ξ ∼ 3 µm . In largeenough samples (d� ξ), most of the material will be aligned in the field direction.

4 Application of Continuum theory with confined liquid crystals

4.1 The Freedericksz effect

Consider a nematic liquid crystal contained between two flat glass slides. The interaction betweenthe nematic and the glass is assumed to be such that the director is constrained to lie eitherperpendicular or parallel to the glass. Assume we have strong anchoring as well. When a magneticfield is applied perpendicular to director and exceeds a certain critical value, the optical propertiesof the system change abruptly. The magnetic field and the boundaries both exert torques on themolecules and when the field exceeds Hc it becomes energetically favorable for the molecules inthe bulk of the sample to align along the magnetic field. The geometry has to be chosen so thatthe orienting effect of the field conflicts with the orientations imposed by the surfaces with whichthe liquid crystal is in contact.

4.1.1 Twist case

The initial undisturbed orientation of the director is throughout parallel to the glass plates. Thegeometry and the coordinate system is same as in Sect. 3.4.1. Only difference is that thera aretwo walls like in Fig. 5. Let us find solution for this case of Eq. (17). Unfortunately, the solution ofthis equation can only be found in terms of elliptic integrals how this is done is shown in Ref. [1].For us only important thing is value of critical field

Hc = π

d

(k22χa

) 12. (19)

10

Figure 8: Freedericks transition, twist case. Below a certain critical field Hc, the LC alignment is notaffected by the field. Ref. [4]

ForH > Hc the deformation at any arbitrary point can be computed and is shown schematicallyin Fig. 8. There are two more important cases considered in next sections.

4.1.2 Perpendicular case

Figure 9: Bend when applied field is under and over the critical field Hc Ref. [4]

In this case the director is constrained to be perpendicular to the glass surfaces. The geometryis shown on picture 10. Now we have n = (sin θ, 0, cos θ) and H = (H, 0, 0). Critical field in thiscase is

Hc = π

d

(k33χa

) 12. (20)

The field of director is shown in figure 9.

4.1.3 Parallel case

Here director without magnetic field is aligned as in twist case n = (cos θ, cos θ, 0), but applied Hhas direction of z axis. Critical field in this case is

Hc = π

d

(k11χa

) 12. (21)

11

Figure 10: Splay when applied field is under and over the critical field Hc

4.1.4 Application of the Freedericksz effect

As we have seen in previous sections the Freedericksz critical field depends only on a single elasticconstant in different geometries. So this effect is ideal to measure the magnitude of these constants.All we have to do is that we put properly prepared liquid crystal between two glass plates andmeasure the critical field.

The experiment for determination of k11 and k33 consist of measuring the variation of bire-fringence for light incident normal to the film. With linearly polarized light incident and a suitableanalyzer (e.g., a combination of two linear polarizers), the transmitted intensity shows a suddenchange when the field attains the threshold value. As the field gradually increased further, theintensity exhibits oscillations, Fig. 11a because of the change in phase retardation.

(a) (b)

Figure 11: (a) Raw record traces of interference oscillations due to change in the sample birefringence withdeformation for hexyloxyazoxybenzene at various temperatures. Polarizer and crossed analyzer are inclinedat 45o to the principal axes specimen. The sudden onset of oscillations occurs at the threshold field. (b)Optical transmission T between parallel polarizers of twisted nematic film as function of voltage.

12

It is very important that magnetic field is exactly perpendicular to the initial undisturbed ori-entation since in other case distortion does not set in abruptly and the experimental determinationof elastic constants becomes somewhat unreliable. In Fig. 11a there is also shown temperaturedependence of elastic constants. Order parameter can be calculated by this information.

4.2 The twisted nematic device

Another similar configuration of much practical interest is the twisted nematic film. The liquidcrystal is sandwiched between two glass plates with the director aligned parallel to the glassboundaries. A twist is now imposed on the liquid crystal so that the director on top wall isperpendicular to director at bottom wall. Electrical field above critical field is applied along thetwist axis results in a deformation as shown schematically in Fig. 12.

(a) (b)

Figure 12: The twisted nematic cell. Arrow shows the direction of field applied. On (a) field is lower thancritical field H < Hc. On figure (b) field is much stronger than critical field H � Hc.

For critical field strength in this case we have

EC ∝(

2d√εa

)(k11

(π

2

) 12

+ (k33 − 2k22)ϕ20

) 12

where ϕ0 is half of primary twist angle. A noteworthy feature of the twisted nematic is thatthe intensity of the transmitted light (through a pair of polarizers set perpendicular to each other)as a function of field is a bit like Heaviside function, Fig. 11a. It can therefore act as an electricallycontrollable optical shutter. This principle is finding extensive application for making liquid crystaldisplay devices for watches, pocket calculators, automobile dashboards, etc.

Usual configuration is a thin nematic film of positive anisotropy sandwiched between two glassplates, the inside surface of which are coated with transparent conducting material. By surfacetreatment, the director is aligned parallel to both plates except that a 90o twist is imposed.

Linearly polarized light incident normal to the plates, with polarization parallel to the directoron the entrance side of the film, will emerge with its polarization plane rotated for 90o, Fig. 13.Light polarization follows the twist, Ref. [8]. The emergent beam will be transmitted by secondpolarizer set at the crossed position eith respect to the first one. If now an electric field is applied,the molecules in the bulk of the sample tend to align themselves normal to the electrodes. Forvoltages over 2dEc the twist is lost over most of the sample, the polarization is no longer rotateby 90o and the transmitted light is extinguished by the second polarizer. With parallel polarizers

13

Figure 13: On figure is schematically shown how one pixel of LCD display works.

we can get opposite effect namely bright field in the ON state and extinction in the OFF state.Typically the threshold voltage is about 1 V, the switching time (for d = 8 µm) a few tens ofmilliseconds. Disadvantage with TN cell is that viewing angle is limited to about ±45o from thenormal, but advantages far out weight this disadvantage, Ref. [8]. Besides TN cell there are alsoother configurations used for LCDs, like IPS, VA ..., Ref. [8].

5 ConclusionThe liquid crystal science represent a rich field of research incorporating many fundamental physicalaspects as well as wide range of potential applications. For decades liquid crystal research hascentered on the information display industry, particularly the development of the liquid crystaldisplay. As the industry matures, new applications are being investigated for liquid crystal materials.

Throughout this seminar we discussed behavior of liquid crystals enclosed inside the wallsof different geometries. Treating liquid crystals as continuum has big success in explaining manyphenomena. Understanding this is very important for use of liquid crystals in every-day applicationsand for purely scientific work.

References[1] P. J. Collings and M. Hird, Introduction to liquid crystals, Chemmistry and Physics, Taylor

& Francis Ltd (1997)

[2] Sackman, H., and D. Demus, 1969, Fortschr. Chem. Forsch. 12, 349.

[3] S. J. Woltman, G. D. Jay, G. P. Crawford, Liquid Crystals: Frontiers in Biomedical Applica-tions, World Scientific Publishing Co. Pte. Ltd. (2007)

[4] S. Chandrasekhar: Liquid Crystals, second edition, Cambridge: Cambridge Universitiy Press(1992)

[5] F. C. Frank, Disc. Faraday Soc., 25, 945 (1933)

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[6] P. G. de Gennes, The physics of liquid crystals, Oxford: Clarendon Press (1974)

[7] P. G. de Gennes, Mol. Cryst. Liquid Cryst., 12, 193 (1971)

[8] Pochi Yech, Claire Gu, Optics of liquid crystals displays, second edition, John Wiley & Sons(2010)

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