UNIVERSITY OF LJUBLJANA FACULTY OF MATHEMATICS AND PHYSICS DEPARTMENT OF PHYSICS Jadranska 19, 1000 Ljubljana

SEMINAR

Light scattering on liquid crystals

Matej Pregelj

Mentors: prof. dr. Martin Čopič dr. Mojca Vilfan

Lj., 13.4.2005

Abstract I am going to describe the behavior of light when passing through a liquid crystal (LC). Understanding this phenomenon is impossible without knowing liquid crystal’s properties. Therefore I will explain basic optical properties of liquid crystals, such as the index of refraction and introduce the Frank elastic constants. I will describe the fluctuations of the director in the nematic liquid crystal, and relate them to the scattering of light. At the end an example of light scattering experiment will be given.

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Content

1 Introduction 3

2 Liquid crystalline phase 3

2.1 Molecular properties 4

2.2 The refractive index 4

2.3 The elastic constants 5

3 Thermal fluctuations in nematic liquid crystal 6

4 Light scattering on the fluctuations in nematic liquid crystal 9

5 Light scattering experiment 11

6 Conclusion 12

7 Bibliography 13

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1. Introduction Liquid crystals are unique as they exhibit properties of both solid crystal and isotropic liquid [1]. A liquid crystal can flow like an ordinary fluid, while other properties, such as birefringence, remind us of a crystalline phase. These mixed properties make liquid crystals so interesting to observe and study.

Figure 1: Liquid crystal as seen through the polarizing microscope[8]

In the liquid crystalline phase the elongated LC molecules tend to be aligned along the preferred direction, otherwise they behave just like an ordinary liquid. Thermal fluctuations that occur in the order of LC molecules influence the dielectric properties of LC and consequently strongly scatter incident light. That makes light scattering one of the most used methods for observing and identifying these fluctuations, which is the key to comprehending the properties of LC, elastic constants, viscosity, fluctuations relaxation times and also the surface interaction of LC and the aligning substrate.

2. Liquid crystalline phase Liquid crystalline phase is a phase between the solid crystalline state and isotropic liquid state of matter. What enables the formation of such phase is the shape of the liquid crystal molecules, which are elongated and therefore strongly anisotropic. In contrast to the isotropic liquid, that has no ordering on macroscopic scale, liquid crystalline phase exhibits orientational and in some cases partial positional order. In this respect two main groups of liquid crystalline phases are distinguished. The closest to the isotropic liquid is the nematic phase. In this phase the molecules are randomly distributed and show no positional order, while the long range orientational order of long molecular axis is present. Some types of LC have between the nematic and solid state one or more different smectic phases showing liquid structure with both orientational and partial positional order. Orientational order of molecules in a nematic phase is far from perfect. In a uniaxial nematic phase, the direction along which the elongate molecules tend to be aligned, is called director n with length |n| = 1. It should be stated that n is not a proper vector due to the molecular symmetry, n = – n, one can not distinguish the ends of the molecule. In an actual sample the director orientation may vary in space and time as it is induced by the boundary conditions and possibly by the external fields.

10µm

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Molecular orientation deviation from the local director n is described with scalar order parameter S. It measures the average deviation of the molecules’ long axis away from the director. It is defined as:

S = 1/2 <(3 cos 2 θ - 1)>. (1)

The angle θ is the angle between the instantaneous direction of the molecule’s long axis and the director. The expression is averaged over all the molecules in the ensemble. In an ideal nematic with all the molecules aligned S = 1, and in isotropic liquid S = 0. But in the real liquid crystal perfect orientational order can not be achieved, therefore always holds S < 1. Because the transition from isotropic liquid to nematic phase is a first order transition, the order parameter abruptly leaps from S = 0 to S ≈ 0,4 at transition temperature TN. It then increases with decreasing temperature to S ≈ 0,6 at phase transition in to solid, that is again of first order, where S again rapidly changes to 1 [1]. In smectic phases more transitions can be noted.

(a) (b) (c)

Figure 2: The three phases of a condensed matter (a) crystal, (b) liquid crystal and

(c) isotropic liquid

2.1 Molecular properties Liquid crystals are made of organic molecules (Fig. 3). A typical LC molecule is strongly anisotropic and highly polarized, due to benzene rings, which are in a majority of LC molecules. They form a rigid central part to which flexible chains are attached at the ends. One can compare this shape to a 2 – 3 nm long cigar (some have a shape of a disk), with a length to width ratio of approximately five.

Figure 3: Liquid crystal molecule 5CB. Most widely used LC with the liquid crystalline phase at room temperature. It is used in many displays and other devices.

2.2 The refractive index Optical axis in the uniaxial nematic is given by the orientation of the director n, therefore it has two principal refractive indices, no and ne. The ordinary refractive index no is observed when the polarization is perpendicular to the optical axis and extraordinary index ne holds for a parallel polarization.

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In the case that the angle between the director and the electric wave vector of the propagating light α is arbitrary the refractive index for extraordinary beam is:

22sincos

)(1

+

=

eoe nnnαα

α, (2)

Using subscripts || and _|_ for the directions parallel and perpendicular to the director, it can be written: no = n_|_and ne = n||. The level of birefringence is than given by:

∆n = ne – no = n|| – n_|_. (3)

In practice, we usually find n|| > n_|_ and ∆n varies from zero to about 0,4, which makes liquid crystals strongly birefringent. Since ∆n is proportional to the scalar parameter S, it depends on the temperature. The birefringence is most apparent when observing the colors of liquid crystal under a polarizing microscope (Fig. 1).

2.3 The Frank elastic constants In a nematic liquid crystal exist no permanent forces opposing the displacements between points like in solids, but there are restoring torqueses, which directly oppose orientational distortions. If different orientations of n are induced at different part of liquid crystal, deformation in the director field will occur. These modifications of n take place over the macroscopic distances (few microns), and are easily observed optically. Distortions cost energy and this is the basis of the continuum theory of the nematic liquid crystal, which is analogue to classical elastic theory of a solid. In the continuum theory detailed structure on the molecular scale is not important, therefore the defects in the order parameter S are neglected. The liquid crystal is described with the spatially dependent orientation of the director n(r) and the temperature dependent order parameter S(T). With these assumptions, in a uniaxial liquid crystal, at each point r the direction of preferred orientation is given by n(r), whereas sign of n has no physical significance. We assume that n(r) varies slowly and is defined at other points by continuity. At each point r we introduce a local Cartesian coordinate system with z axis along the director n. Assuming that free-energy density f is a quadratic function of the linear components of the curvature strains, it is relevant to consider also higher order terms. However they can be omitted by taking into account the symmetries of liquid crystal. As the free energy is given by

F = ∫f dV, (4)

terms of f, which have the form of ∇ u where u(r) is an arbitrary vector field, can be transformed into surface integrals (Gauss’ theorem) and therefore omitted considering the properties of a bulk nematic liquid crystal. Also the terms which contribute only to surface energy can be omitted. In the end we get the fundamental equation of continuum theory for nematics’ free-energy [2]:

[ ] [ ]( ) rdnnKnnKnKF vvvvvv∫ ×∇×+×∇⋅+⋅∇= 2

32

22

121 )()()( (5)

The constants K1, K2 and K3 are called the Frank elastic constants. They correspond to the three fundamental deformations of director, shown in Fig. 4. The first term describes the splay deformations, where 0≠⋅∇ nv . The second term is associated with the twist distortions, and the third corresponds to the bend deformations. The experimentally observed stability of a uniform director pattern demands that K1, K2 and K3 are all positive, what corresponds to the

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idea that the presence of deformation increases free energy. The Frank elastic constants depend quadraticaly on the order parameter S [3,4] and consequently increase with decreasing temperature.

Figure 4: a) splay deformations, b) twist deformations, c) bend transformations

Achieving the equilibrium configuration of a namatic system is performed as minimization of free-energy density with respect to the variations in the director n(r,t). The Euler-Lagrange equations can be written as:

( ) ii

xn

ji

hnrFxn

F

j

j−==

∂

∂∂∂

−∂∂

∂∂ )(vλ (6)

whit ni being the i-th component of the director n, xi being the i-th Cartesian coordinate and λ the Lagrange multiplier. Here was introduced the molecular field h, which’s direction is in equilibrium parallel to the director n.

3. Thermal fluctuations in a nematic liquid crystal So far the molecular dynamics and time dependence of the director field has not been considered. Motions of the LC molecules range over a wide frequency range. There is very fast (10-9 to 10-11s) rotational and transitional motion of a single molecule. There are density fluctuations of the sample due to the molecular translational mobility which are also present in isotropic liquid, but they weakly scatter light. Specific for the liquid crystal are spatially correlated collective orientational fluctuations, which appear at lower frequencies, are specific for liquid crystalline phase and are therefore of great importance. They include the director fluctuations, describing local fluctuations in the orientation of the director, and the order parameter fluctuations, describing the fluctuations in magnitude of S. In a nematic liquid crystal, the director fluctuations are more important, as they are easier to excite then the fluctuations in order parameter. The latter should be taken in to account only close to the nematic to isotropic phase transition, therefore, far from transition, S can be assumed a constant. In the following part the thermal fluctuations of the director in the bulk nematic LC are going to be further discussed.

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Due to thermal excitations, the actual director in the bulk nematic deviates from the equilibrium orientation. That is why it can be written as a sum of the equilibrium director n0 and the spatially and time dependent term:

n(r, t) = n0(r) + δn(r, t). (7)

The δn(r, t) represents the deviation of the director from equilibrium value and is expected to be small |δn| << |n| = 1. To simplify things, let us assume that the average director orientation in a nematic sample is in z direction, n0 = (0, 0, 1). The fluctuating part has then in the first approximation only two non-zero components, δn = (nx, ny, 0), for the variations nz parallel to the equilibrium director orientation are of higher order and can be neglected. The fluctuational part of Frank free energy is [1]:

rdz

nz

nK

xn

yn

Ky

nxn

KF yxyxyx v∫

∂

∂+

∂∂

+

∂

∂−

∂∂

+

∂

∂+

∂∂

=22

3

2

2

2

121 (8)

The most convenient way to analyze the fluctuating quantities nx(r) and ny(r) is to expand them in terms of planar waves with wave vector q

∑=q

rqixx etqntrn

v

vvvv ),(),( , (9)

and similarly for ny(r). The sum is over all possible wave vectors, limited with the size of the sample on one side and with the requirements of the continuum limit on the other. Hence one finds:

∑=

∂∂

qxx

x qtqnx

nv

v 222

),( , ∑=

∂∂

qyx

x qtqnyn

v

v 222

),( , ∑=

∂∂

qzx

x qtqnz

nv

v 222

),( , (10)

and likewise for ny.

Figure 5: The new coordinate system: e1 and e2 are introduced, and the fluctuations are decupled into two eigenmodes n1 and n2

For each q it is suitable to rotate the coordinate system x, y, z around the z axis, so that the new y’ axis coincides with unit vector e2, which is perpendicular to z axis and to wave vector q. The x’ axis than coincides with e1, which is perpendicular to e2 and to z axis. In this new system we have qy’ = 0, and fluctuation eigenmodes nα(q) along eα (α = 1,2) (Fig. 5). The fluctuation eigenmode n1(q) describes a combination of splay (Fig. 6a) and bend (Fig. 6c) fluctuations and the eigenmode n2(q) a combination of twist (Fig. 6b) and bend (Fig. 6c).The free energy can be than written as:

[ ]∑ +++=q

qKqKtqnqKqKtqnVFv

vv )(),()(),( 2||3

2_|_2

22

2||3

2_|_1

212

1 (11)

where q_|_ = qx’ and q|| = qz.

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Figure 6: Two fluctuation eigenmodes are a combination of splay and bend fluctuations and a combination of twist and bend fluctuations. For simplicity, the fluctuations are plotted

separately: a) splay fluctuations, where the wave vector q is parallel to the fluctuating part δn and thus perpendicular to n0; b) twist fluctuations, where the wave vector, the undistorted

director, and the fluctuating part are perpendicular to each other; c) bend fluctuations, where the wave vector is parallel to the undistorted director n0

It should be noted that at this point free energy is still time dependent. Once the thermal average of the eigenmodes’ amplitudes |nα(q)| is assumed, this dependence fall off. To calculate the amplitude of the eigenmodes one can use the equipartition theorem, which states that for a classical system in the thermal equilibrium, the average energy per degree of freedom is equal to ½ kBT, where kB is the Boltzmann constant. We can assume that every eigenmode excitation on its own costs ½ kBT. As a result the thermal average of |nα(q)|2 is:

( )2||3

2_|_

2 1)(qKqK

TkV

qn B

+=

ααv , α = 1,2. (12)

This is the central equation of fluctuation theory for nematics [2]. The corresponding relaxation times τα(q), depending on the wave vector q and polarization mode α, can be obtained by balancing the viscous and the thermodynamic torque of the molecular filed [1]:

nhtn vvv

λη −=∂∂ , (13)

where η being the effective rotational viscosity and λ the Lagrange multiplicator. Taking the into the account free-energy (Eq. 5) and calculating the molecular field (Eq. 6) the equation of motion for the director filed can be written (to simplify: K1 = K2 = K3 = K):

( )nnnKnKtn vvvvv

22 ∇⋅−∇=∂∂η , (14)

Inserting Eq. 7, linearizing for small fluctuations, and solving the dynamic equation for incompressible nematics, the relaxation rate for given eigenmode can be obtained [2]:

)()(1 2

||32

_|_

qqKqK

q vvα

α

α ητ+

= , α = 1,2, (15)

where ηα are effective viscosities. With the use of the equipartition theorem an expression for the variance of thermal fluctuations of n was accomplished.

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4. Light scattering on the fluctuations in a nematic liquid crystal Let us now clarify how the fluctuations of n influence the optical dielectric tensor ε and consequently scatter incident light. The electric field of the incident light beam can be written:

tirkiin eEitrE ω−=

vvvv0

0),( , tirki

s eREtAftrE ω−= ''0 )/)((),'(

vvvv

(16a)

(16b)

where E0 is the amplitude, and i a unit vector perpendicular to the ingoing wave vector k0, defining the direction of polarization. The electric field Ein(r,t) induces in the sample the polarization P(r,t) = ε0[I – ε(r,t)]Ein(r,t). Our interest is to know the fluctuating part of scattering field Es(r’,t) in some distant point r’ = r + R, with R being large compared with the sample size r. The polarization of the outgoing wave, with wave vector k’ || R, is given by unit vector f and relevant part of the dielectric optical tensor is then:

)( 31−∆+=⋅⋅= fiif nnif εεεε

vv, (17)

where inni

vv ⋅= and fnn f

vv ⋅= are the components of n along the two polarization directions. The fluctuating part of the optical dielectric tensor according to the Eq. 7 equals:

[ ])()( |||| nifnfiifvvvv

δδεδε ⋅+⋅∆= , (18)

where inivv ⋅= 0|| and fnf

vv ⋅= 0|| . If δn is split upon the eigenmodes n1, n2:

δn(q,t) = e1n1(q,t) + e2 n2(q,t), (19)

the fluctuating part of the optical dielectric tensor can be written as:

)(),(),( ||||2,1

ααα

αεδε fifitqntqif +∆= ∑=

vv , α = 1,2, (20)

where αα eii vv⋅= is the component of i along eα, and likewise for the f and q = k’ – k0. We have shown that the orientation fluctuations couple via the optical dielectric tensor to the incident electric field and get a fluctuating polarization P(r) = f ε0[1 - δεif(r,t)] Ein(r,t). We can assume that every fluctuation represents and electric dipole P(r) oscillating at a particular angular frequency. Summing contributions over all sample volume and restricting ourselves to large R >> r, the equation for a differential cross-section per unit solid angle of the outgoing beam can be written, using Eq.12 and Eq.16b, as [2]:

( )∑

∑

=

=

+

+

∆=

+

∆==

Ω

2,12||3

2_|_

2||||

2

2

2

2||||

2,1

22

2

22

)(4

)()(4

)(

α α

αα

ααα

α

πεω

πεωσ

qKqKfifi

TVkc

fifiqnc

VqAdd

B

vv

. (21)

The polarization factor in the Eq. 21 shows that maximum intensity of scattered light is achieved with crossed polarization of incident and outgoing light beams, what is contrary to the situation in isotropic liquid, where scattered light has the same polarization as the incident.

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Figure 7: Scattering geometry, where optical axis is z axis, i and f are polarizations of incoming and outgoing beams, e1 and e2 LC eigenvectors, k0 is incoming and k’ outgoing

wave vector, φ scattering angle and q_|_ = k’ - k0

Let us turn on example shown in Fig. 7 where the incident and the scattered beam are both perpendicular to z, the optical axis of the nematic. The incident beam is linearly polarized in the scattering plane, while the scattered beam is polarized along z axis. Denoting the scattering angle by ϕ , we have

)2/sin(21 ϕkq ≈⊥ , 0|| =q ,

)2/cos(1 ϕ=i , )2/sin(2 ϕ=i ,

021|| === ffi , 1|| =f

(22)

Inserting substitutions into Eq. 21 gives:

212

22

2

21

2

42

42

/)2/(cot)2/(sin)2/(cos16

KKqKqK

Tkcd

dB +∝

+

∆=

Ω ⊥⊥

ϕϕϕπωεσ

. (23)

This relation can also be obtained experimentally and used for obtaining K1/K2 [2] (Fig. 7). Similarly we can obtain the ratio K3/K2 by studying the dependence of cross-section on the angle between q and the optical axis.

Figure 7: Determining the K1 /K2 ratio by extrapolating the straight line [2], where αα eii vv⋅= (α = 1,2) and q_|_ is part of the wave vector, perpendicular to z axis

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5. Light scattering experiment

Figure 8: Experimental setup: incoming laser is polarized P, focused with lens L1 and directed onto sample S. The scattered light passes analyzer A and is directed onto detector D

through a focusing lens L2 and the optical fiber. The correlator (ALV-5000) calculates autocorrelation function g(2) [5]

The amplitude of the scattered light is time dependant due to the fluctuations in the optical dielectric tensor (Eq. 21), which are related to the orientational fluctuations of the director. The most reliable way to study the dynamics in the liquid crystal, and to obtain information on the fluctuation spectrum of the sample, is by measuring the autocorrelation function of the scattered light. The normalized intensity autocorrelation function g(2)(q,t) is defined as [6]

2)2(

)',(

)',()',(),(

tqI

ttqItqItqg r

rrr +

= , (24)

where I(q,t’) is the intensity of the scattered light, proportional to |Es(q,t’) + Estat(q,t’)|2, where Estat denotes the electric field of the statically scattered light. In general two different regimes of measurement can be considered: homodyne and heterodyne. In the former case, the intensity of scattered light by fluctuations is much larger than the scattered light which is independent of time. In the latter case, the intensity contribution of dynamically scattered light is small |Es(q,t’)|2 << |Estat(q,t’)|2 ≈ 1, and as the contributions are statistically independent the autocorrelation function is

+ℜ+= ti

stat

ss etqE

ttqEtqEtqg ω

2

*)2(

)',(

)',()',(21),( r

rrr , (25)

whereℜ denotes the real part. The second term in Eq. 25 is proportional to the dynamically scattered light and it is in this case small compared to 1. Experimentally, this happens when the amplitudes of fluctuations are small, or if the amount of scattering media is small (e.g. thin cells). This regime can also be achieved by splitting the light beam into two, directive one beam on the sample and than correlate it with the direct one. Using the Eq. 25 and Eq. 21 the autocorrelation function of the light scattered on fluctuations in a nematic liquid crystal can be calculated:

( ))(/

2,12||3

2_|_

2||||

20

2

2

2)2( )(

4)(21),( qt

Bstat

eqKqK

fifiTVk

RE

cqEtqg

v

rr

ατ

α α

αα

πεω −

=∑ +

+

∆+= . (26)

From Eq. 26 can be seen that in properly chosen scattering geometry only one fluctuation mode is observed:

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)(/2

02

)2(

)(21),( qt

stat

edd

RE

qEtqg

v

rr

ατ

α

σ −

Ω

+= , (27)

where α depends on chosen eigenmode. This means that by measuring the autocorrelation function, the relaxation times of fluctuation eigenmodes and the differential cross-section per unit solid angle of the outgoing beam can be obtained. As a result we can obtain elastic constant ratios, elastic constant to viscosity ratio K/η, and their thermal dependence. It has been proven, that the method is particularly suitable for thin liquid crystal samples up to a few microns. In such samples, this method is for example used to measure the scatted light spectrum, the strength of anchoring energy, which determines how strong the liquid crystal molecules are anchored in the direction imposed by the surface.

6. Conclusion In this seminar I have passed on a brief theoretical explanation, how does the incident light scatter on liquid crystals. In order to achieve that, I had to give explanation of the basic liquid crystal properties. I have also shed light on the light scattering experimental setup and describe some applications of this experiment. Light scattering on liquid crystals is the basic phenomenon which helps us to distinguish the liquid crystals from the isotropic liquid. The light scattering experiment is most commonly used to determine the relaxation times of different fluctuation modes of LC. Using these data, elastic constant ratios, elastic constant to viscosity ratio K/η, and their thermal dependence can be determined.

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7. Bibliography

[1] P.G. de Gennes, ”The Physics of Liquid Crystals”, Clarendon, Oxford, 1974

[2] W. H. de Jeu, “Physical Properties of Liquid Crystalline Materials", Gordon and Breach, New York, 1980

[3] I. Haller and J. D. Litster, “Temperature dependence of normal modes in a

nematic liquid crystal”, Phys. Rev. Lett. vol. 25, pp. 1550 – 1553, 1970

[4] I. Haller and J. D. Litster, “Light scattering of a nematic liquid”, Mol. Liq. Cryst., vol. 12, pp. 277 – 287, 1971

[5] M. Vilfan, “Influence of Surface Properties on the Fluctuation Spectrum in a

Liquid Crystal”, Dissertation, Ljubljana, 2001

[6] B. J. Berne and R. Pecora, »Dynamic light scattering”, Dover, 2000

[7] http://www.eng.ox.ac.uk/lc/research/introf.html#top

[8] http://bly.colorado.edu/lc/flc.html

[9] http://www.kth.se/fakulteter/TFY/kmf/lcd/lcd~1.htm

[10] http://www-sci.sci.kun.nl/rim/lcl/research/chiral/chiral.htm