liquid crystal theory and modeling · liquid crystal theory and modeling n. j. mottrama* and c. j....

Liquid Crystal Theory and Modeling

N. J. Mottrama* and C. J. P. NewtonbaDepartment of Mathematics and Statistics, University of Strathclyde, Glasgow, UKbPeartree Cottage, Little London, Longhope, Gloucestershire, UK

Abstract

In this chapter, we explain the rationale behind the theoretical modeling of liquid crystals and explain theimportant steps to construct a realistic and accurate model for a particular physical system. We thensummarize two commonly used theories of nematics: one based on using the director as a dependentvariable and one based on using the tensor order parameter. Using an example problem, the p-cell,we show the advantages and disadvantages of these two theoretical approaches, demonstrating theimportance of carefully considering the choice of model before embarking on simulations.

Introduction

The theory and modeling of liquid crystalline materials has been an area of research for over 100 years,when the first fundamental theories of the state were proposed. Various theories have fallen by thewayside, such as the “swarm” theory of Bose (1908), and others have grown in popularity, such as theOseen–Frank (Oseen 1933; Frank 1958) theory for nematics (see Kelker (1973) and Sluckin et al. (2004)for a survey of the history of the area). However, the topic remains one of intense activity and debate.The use of mathematical models and underlying theory can be hugely rewarding, leading to insights intothe fundamental behavior of a system and providing the opportunity to optimize the performance of liquidcrystalline devices, but it is not without risks. While one model of a liquid crystalline system may beuseful for a particular application, it may also lead to erroneous conclusions in a different setting.Care must be taken when developing a model and selecting a theoretical basis for the model. The misuseof mathematical models can have considerable and unpredictable consequences with far-reaching impact.

This chapter introduces the motivation and fundamentals of the use of theory and models in liquidcrystal systems and briefly considers two specific theories which have previously been used successfully.An example of a liquid crystal device (the p-cell) is examined using these two theories in order todemonstrate when, in one instance, simple models can fail dramatically and, for another instance, anunnecessarily complicated model leads to a waste of computational resources.

Fundamentals of Modeling

MotivationThe motivations for developing mathematical theories or using models to describe liquid crystals andliquid crystal devices are varied. On the most fundamental level, there is the desire to understand thebehavior of the liquid crystal phase, how the fluid orders orientationally and positionally, and whichmolecular interactions exist and give rise to larger-scale phenomena. In this case, it is natural to build atheoretical framework for liquid crystals from other, well-understood, physical effects or the inherent

*Email: [email protected]

Handbook of Visual Display TechnologyDOI 10.1007/978-3-642-35947-7_87-2# Springer-Verlag Berlin Heidelberg 2014

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symmetries contained within the system and, if necessary, to introduce additional elements in an attemptto understand other phenomena. This process of developing fundamental theories has value in itself but isusually of particular benefit when used to explain experimental evidence or to predict behavior. The cycleof theoretical development, experimental comparison, and theory refinement is often used and rarelyreaches a final conclusion. However, insight into the fundamental physics and the performance of physicaldevices is often gained along the way.

Length and TimescalesAsmentioned above, the selection of a theory or model to use in any given instance is of prime importanceand is not always a simple undertaking. One approach to selecting the most appropriate theory is toconsider the important length and timescales involved in the situation to be modeled. The range ofrelevant scales in the area of liquid crystal physics is large. The orientational and positional self-organization of molecules to form a liquid crystal phase is influenced by the atomic and molecularstructure which exists at the sub-nanometer length scale and the picosecond timescale. However, largeliquid crystal devices can have dimensions of meters (although the relevant length scale is probably eitherthe liquid crystal layer thickness, typically of the order of micrometers, or the interpixel distance, typicallyhundreds of micrometers) and timescales of milliseconds. The range of length and timescales thereforecovers at least six orders of magnitude and a useable theory that accurately models such disparate scalesdoes not exist. However, in many cases, the problem at hand does not require such an all-encompassingtheory, and we may concentrate on a smaller section of the complete behavior, assuming that the influenceof phenomena at other length and timescales is either negligible or can be averaged. For instance, in thecontinuum modeling described below, we assume that molecular properties can be averaged to createmacroscopic quantities such as the director n, the average orientation of a group of molecules at a point inspace and at a specific time.

Continuum ModelingWith the necessary restriction of length and timescales mentioned above in mind, the main focus of thischapter will be on continuum theories of liquid crystalline materials. While models based on atomistic andmolecular considerations have considerable value in understanding the small-scale behavior of theseclasses of molecules, and their interactions with other effects such as surfaces, the computational resourcenecessary to enable even the smallest liquid crystal device to be simulated is beyond present capabilities.For the reader interested in such models, a review and discussion can be found in the following referenceand the references therein (Care and Cleaver 2005).

For most scientists and engineers who wish to simulate the operation of a liquid crystal device, it istherefore to a continuum theory that they look. However, there are many types of continuum theory, and itcan be difficult to distinguish between them and to decide on the most appropriate theory, given aparticular experiment or device.

Here we provide a brief summary of two of the most commonly used theories for nematic liquid crystalsand provide references to other, more specialized, theories. We will first briefly summarize the commonlyused Ericksen–Leslie theory, sometimes called the Ericksen–Leslie–Parodi theory, which can beextremely useful when modeling most “standard” liquid crystal devices. This theory is now over40 years old, and there have been many reviews and research papers written on this subject (see Stewart(2004) for a complete description and many examples of the use of Ericksen–Leslie theory). For thisreason, we will only summarize the main points of the model, including some additional physicalphenomena (flexoelectricity, ionic motion) which may be useful for more accurate modeling of liquidcrystal devices, and point the reader to references where this theory is extensively discussed. We will alsoconcentrate on a macroscopic theory, the so-called Landau–de Gennes or Q-tensor theory, which allows


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the modeling of regions of low order, such as defects and surface regions. Because of the inclusion oforder as a dependent variable, such theories are sometimes called mesoscopic theories, since they containinformation about the molecular ordering which is more commonly addressed using smaller length-scaleapproaches. In this chapter, we will review the main aspects of the simplest form of this type of continuummodel, discussing various additional physical phenomena through additional references to other work inthe area.

Dependent Variables, Independent Variables, and ParametersAssuming that the length and timescales of the system being considered mean that it is appropriate to use acontinuum theory, one of the key considerations when embarking on the simulation of a device is theclassification of relevant dependent and independent variables and the system parameters. The classifi-cation used in this chapter is as follows:

• Parameters are constants of the systemwhich will not vary in space or time and are (at least potentially)experimentally measurable quantities. It is possible to use derived parameters which are themselvesfunctions of other parameters, but these quantities will also be constants of the system.

• Independent variables are quantities which vary in the system (i.e., spatial coordinates or time) and donot depend on the parameters in the system. By changing an independent variable (i.e., considering adifferent time or location during the simulation), the observation of the dependent variables willchange.

• Dependent variables are quantities that describe the state of the system and will depend on one or moreof the independent variables and may also depend on the system parameters. Dependent variables maybe predetermined (i.e., the electric potential at an electrode may be assumed to be a specific function oftime) or to be determined through the simulation of the system. Each of the dependent variables whichare not predetermined will have associated with it an equation which governs how it depends on theindependent variables, the predetermined dependent variables, and parameters.

An example system would be a nematic liquid crystalline material in a simple Freedericksz cell(Stewart 2004; de Gennes and Prost 1993), where the order parameter S is assumed to be constant. Forthis case, we might choose the director n, fluid velocity v, fluid pressure p, and the electric field E as thedependent variables. These variables would depend on the independent variables which might be theposition in the cell x = (x, y, z) and time t. The dependent variables would also depend on the materialand cell parameters such as the elastic constants, fluid viscosities, density, and the relative permittivities ofthe liquid crystalline material; the dimensions of the liquid crystal layer; details of the anchoring at thesubstrates; the constant applied voltages at the substrate electrodes; etc.

With these definitions, we then use various physically derived equations, for instance, the balanceequations for mass, linear, and angular momentum or Maxwell’s equations, in order to produce governingequations for the spatial and temporal form of the dependent variables. In order to determine theseunknown dependent variables, it is necessary to have the same number of governing equations. Forinstance, for the example given above, the dependent variables are the director n (a vector in ℝ3 so thereare three unknowns), the velocity v (again three unknowns), the pressure p (a scalar, so just one unknown),and the electric field E (three unknowns), so we have 11 unknowns and require 11 governing equations.As we will see later, the appropriate equations are the three components of the balance of linearmomentum (which govern the fluid velocities), the three components of the balance of angular momen-tum (governing the director), the scalar mass balance equation (sometimes called the continuity equation,which effectively governs the pressure), and the three components of the relevant Maxwell’s equation(Gauss’s law) for the electric field. In fact, because the director is a unit vector, one more equation is


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needed, n2 = 1, but this extra equation is balanced by an unknown Lagrange multiplier. The number ofunknowns (dependent variables and the Lagrange multiplier) is therefore 12, and the number of governingequations is 12.

Because the governing equations are often in the form of differential equations, additional unknowns,namely, the integration constants, are often introduced once these equations are solved. Conditions on thedependent variables must therefore be specified at the boundaries of the region of interest and possibly anyinternal boundaries where material properties change abruptly. For dynamic problems, there should alsobe an initial condition specified, which prescribes the initial form of the dependent variables at the start ofthe simulation. Any specific modeling problem is specified by the appropriate set of governing equations,which govern the spatial and temporal behavior of the dependent variables, together with the correctboundary and initial conditions. This set of equations must then be solved (exactly or approximately,analytically, or numerically).

Nematic Liquid Crystal Theories

Although there are many existing and developmental electro-optic devices which contain smectic liquidcrystals, we shall here concentrate on theories for nematic liquid crystals. For the reader who is interestedin smectic devices there, we refer them to the discussions in the following references (Stewart 2004; deGennes and Prost 1993).

We have split this section into two parts: the first concerns a very commonly used director-basedapproach, and the second additionally considers the ordering of molecules as a dependent variable. Weillustrate these two approaches using the liquid crystal p-cell.

The p-cell is a liquid crystal cell where a sample of nematic liquid crystal is confined between two flatsurfaces which are treated so that the molecules close to the surfaces have a preferred direction. Thispreferential direction manifests itself in the form of a “pretilt” of the director, a fixed direction which thedirector must take at the surface. In the p-cell, the surfaces are arranged in the so-called parallel alignmentwhere the pretilt angle (from the horizontal) takes values of opposite sign at the two surfaces (see Fig. 1).This type of alignment often gives rise to two stable states: the splay state and the bend state (see Fig. 1).Given a liquid crystal with a positive dielectric anisotropy, the cell can be switched from the splay to thebend state by the application of an electric field across the cell (Barberi et al. 2004a). At first sight, wemight assume that this device could be modeled using a theory which models the director without the need

θ = θp

θ = −θp

E

Z

a b

Fig. 1 The stable states of the p-cell: (a) the splay state and (b) the bend state


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to consider the possibility of small-scale phenomena such as defects. However, we shall see later that theobvious length scale (the cell thickness, of the order of microns) is not the only one in the system. As theelectric field strength increases, we will observe a second smaller length scale, an internal reorientationlayer which, eventually, necessitates the use of a more complex theory. In the sections below, we willtherefore consider this device using two different theories, showing the advantages and disadvantages ofeach.

Director-Based ModelFor most devices based on the electro-optic effects of nematics, the most important macroscopic variableof concern to the scientist or engineer is the director n(x, t), the average orientation of the liquid crystallinemolecules, which may depend on the spatial and temporal variables, position x = (x, y, z) and t. Bydefinition, the director is a unit vector and we must ensure that any theory we develop satisfies thisconstraint. Other important dependent variables may be the electric fieldE(x, t), the magnetic fieldH(x, t),the fluid velocity v(x, t) and pressure p(x, t), and the concentration of positive or negative ion species nh (x,t), ne (x, t).

In this section, we summarize the standard governing equations for these variables which we hope mayact as a relatively complete reference for those attempting to model nematic liquid crystal devices wherethere are no regions of reduced order (i.e., no regions of very high distortion such as close to defects).A number of examples of the use of these equations may be found in Stewart (2004) and de Gennes(de Gennes and Prost 1993).

Governing EquationsThe governing equations for the fluid velocity and pressure are given by the balances of mass and linearmomentum:

∇ � v ¼ 0, rdvdt

¼ rFþ ∇ � t (1)

where v is the fluid velocity; ∇ = (@/@x,@/@y,@/@z) is shorthand for the gradient vector, the derivatives ineach direction; r is the (constant) fluid density; F is the external body force per unit mass; and t isthe stress tensor. In the last term of the second equation of (1), the term ∇ � t is a vector with elements(t11,1 + t12,2 + t13,3, t21,1 + t22,2 + t23,3, + t23,3, t31,1 + t32,2 + t33,3) where tij,j= @tij/@xj, the derivative of the ijthelement of the tensor twith respect to the jth independent spatial variable xj, and x= (x, y, z)= (x1, x2, x3).

In this form, Eq. 1 is the same as the Navier–Stokes equations for an incompressible fluid. However, itis in the form of the stress tensor that liquid crystals differ from Newtonian fluids, as we will see later.

The governing equations for the director are the constraint that n is a unit vector and the balance ofangular momentum which can be written as

n � n ¼ 1, Gþ gþ ∇ � s ¼ 0; (2)

where the function s is defined through the couple stress tensor l= n� s, g is the intrinsic body force, andG is related to the external body couple through K = n � G.

In order to complete these equations, we need to “close” the system by specifying various quantitiesmentioned in Eqs. 1 and 2, which will be functions of the dependent variables andmaterial parameters: thestress tensor t, the couple stress s, the intrinsic body force g, and the external forces and couples F andK, ifpresent.


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The equations for the stress, couple stress, and intrinsic body force, the “constitutive equations,” can beshown (Leslie 1968) to be as follows:

tij ¼ �pdij � @w

@np, jnp, j þ ~t ij, sij ¼ @w

@ni, jþ ~sij; (3)

gi ¼ lni � @w

@niþ ~gi, w ¼ w ni, ni, j

� �; (4)

where p is the pressure arising from the fact that the fluid is assumed to be incompressible, l is a Lagrangemultiplier deriving from the n � n= 1 condition,~t, ~s, and ~g (which is actually related to the dynamic part ofthe stress tensor~t througheijknj~gk ¼ eijk~tkj, where the summation of repeated indices has been used and eijkis the alternator, see Stewart (2004)) are dynamic contributions, and w is the elastic energy density. Theselast four quantities will be specific to the type of liquid crystal we are considering and must obey certainconstraints which are derived from the symmetry of the phase and from basic thermodynamic consider-ations. Further details can be found in Stewart (2004) and are summarized below.

For the elastic energy density w of a nematic liquid crystal, we assume that the director is “headless” sothat n and�n are equivalent; therefore, replacing nwith�n inwmust leave the elastic energy unchanged;any translation of the system leaves the energy unchanged; any rigid rotation of the system leaves theenergy unchanged; and there are only small distortions present so that we may neglect terms smaller than

@n=@xð Þ2. These assumptions lead to the general form of the elastic energy density

w ¼ 1

2K1 ∇ � nð Þ2 þ 1

2K2 n � ∇� n� qð Þ2 þ 1

2K3 n� ∇� nð Þ2

þ 1

2K2 þ K4ð Þ∇ � n � ∇ð Þn� ∇ � nð Þnð Þ:

(5)

It is relatively straightforward (de Gennes and Prost 1993) to see that the first distortion term ∇ � n isassociated with splaying of the director configuration, the second term n � ∇ � n is associated withtwisting of the configuration, the third term n � ∇ � n is associated with bending of the directorconfiguration, and the fourth term is associated with a distortion in two directions (a saddle-splay term).

In Eq. 5 the Ki are elastic constants, specific to each liquid crystal material, are temperature dependent,and are usually called the Frank elastic constants. The constant q is the natural twist of the material whichis nonzero only in chiral nematics. It can also be shown that (K2 + K4) = 0 in a chiral nematic (Stewart2004). It is common (but rarely accurate) to use the “one constant approximation” which asserts thatK1 = K2 = K3 = K and K4 = 0 (some authors write K2 + K4 = 0 instead) which can greatly simplify thegoverning equations and may allow further mathematical analysis than would normally be possible.However, if the governing equations are to be solved numerically, there is little to be gained by using suchan approximation and the specific values of each constant can be used. For the liquid crystal material 5CB(at 26 �C), the first three elastic constants take the values K1 = 6.2 � 10�12 N, K2 = 3.9 � 10�12 N, andK3 = 8.2 � 10�12 N (Dunmur et al. 2001).

The dynamic contributions ~t , ~s, and ~g are also assumed to vanish in a rigid rotation so that they arefunctions of velocity gradients and director rotation through the rate of strain tensorD, the vorticity tensorW, and a vector N, where

Dij ¼ 1

2

@vi@xj

þ @vj@xi

� �,Wij ¼ 1

2

@vi@xj

� @vj@xi

� �Ni ¼ dni

dt�Wiknk; (6)


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The dynamic contributions must also be invariant to translations, rotations, and the transformationn ! �n and can therefore be shown to be

~t ij ¼ a1nknpDkpninj þ a2Ninj þ a3Njni þ a4Dij þ a5Diknknj þ a6Djknkni; (7)

~gi ¼ a2 � a3ð ÞNi þ a5 � a6ð ÞDiknk; (8)

~sij ¼ 0; (9)

where the ai are the Leslie viscosities and can be related to the (experimentally more useful) Miesowiczviscosities (Miesowicz 1935, 1936).

The equations above therefore specify the Ericksen–Leslie equations for a nematic liquid crystal.Parodi later added a restriction to the Leslie coefficients (ai) through an Onsager relation, which showedthat a2 + a3 = a6 � a5 (Parodi 1970).

With this assumption, there are therefore five independent viscosities and, augmented with theParodi relationship, the governing equations are often called the Ericksen–Leslie–Parodi equations.In 5CB (at 26 �C), the five viscosity parameters take the values a1 = �0.0060 Pa s, a2 = �0.0812 Pa s,a3 = �0.0036 Pa s, a4 = 0.0652 Pa s, and a5 = 0.0640 Pa s (Dunmur et al. 2001).

All that remains is to consider possible external forces and couples that occur within liquid crystalsamples. The most common body force (per unit mass) F, in Eq. 1, will be a gravity force F= (0, 0,�g),although, because of the small-scale nature of most liquid crystalline flows, this term will usually benegligible.

With regard to the body torque, K, the usual terms will be those derived from electromagnetic forces.We will treat applied magnetic and electric fields separately in the following discussion. In both cases,however, the application of a field can be either seen as an externally applied torque or, more usually, it isseen as a change in the free energy density. The most usual way to include such fields is therefore toaugment the elastic energy with an additional magnetic or electric term. This is how we will proceedbelow. We first consider the simpler magnetic field case, followed by the electric field case.

A liquid crystal molecule placed in a magnetic field H will distort that field, inducing a magnetizationM and increasing the magnetic energy. It will distort the magnetic field by different amounts depending ofthe orientation of the molecule, and, for a general director, the magnetization can be written asM ¼ wm⊥Hþ wma (n � H)n. The magnetic susceptibilities wm | | and wm⊥ are usually negative while thediamagnetic anisotropy wma = wm | | � wm⊥ positive. The magnetic induction is

B ¼ m0 HþMð Þ ¼ m0 1þ wm⊥ð ÞHþ wma n �Hð Þnð Þ;

where m0 is the permeability of free space. The magnetic energy density is

wm ¼ �ðB � dH ¼ � 1

2m0m⊥H

2 � 1

2m0ma n �Hð Þ2; (10)

where we have defined m⊥= 1 + wm⊥, m| |= 1 + wm,| | and ma= m| |� m⊥= wma. This energy density shouldbe added to the elastic energy density w in Eq. 5 and, when integrated over the region, leads to the totalfree energy. To be accurate, for a specific director configuration, the magnetic fieldH should be calculatedfrom Maxwell’s equations for the field and the magnetic induction B, i.e., ∇ � H = 0 and∇ �B= 0. However, since ma is usually very small (approximately 10�6, unitless if SI units are considered)compared with m⊥, then we are usually safe to assume that B is directly proportional to H. In that case,


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Maxwell’s equations become ∇�H= 0 and ∇ �H= 0, which are solved by settingH to be constant andprescribed by the external source of the magnetic field. It is therefore usual to assume that the magneticfield is constant, although the magnetic energy density in Eq. 10 will still depend on the directororientation.

A similar approach can be used for the electric field situation. The polarization P (the equivalent to themagnetizationM) is made up of an induced (dielectric) component and other components, for instance, ifa flexoelectric polarization is present. The dielectric polarization is related to the applied electric fieldthrough the susceptibility tensor, so that the total polarization can be written as the sum of the dielectricpolarization and other components Ps; thus, P ¼ we⊥Eþ wea n � Eð Þnþ Ps The electric displacementD (equivalent to the magnetic induction B) is then D = e0(E + P) so that

D ¼ e0 1þ we⊥ð ÞEþwea n � Eð ÞEð Þ þ e0Ps ¼ e0 e⊥Eþ ea n � Eð ÞEð Þ þ e0Ps; (11)

where e⊥ ¼ 1þ we⊥, ejj ¼ 1þ wejj and ea ¼ wejj � wejj � we⊥ are the dielectric anisotropy (sometimesdenoted by De).

The most commonly used form of the additional polarization term in nematics is the flexoelectricpolarization, e0Ps ¼ e11 ∇ � nð Þnþ e33n� ∇� n . There are unfortunately two conflicting conventionsfor the sign of the e33 term, and here we have opted for the original form suggested by Meyer (1969).The factor of e0 in the definition of the flexoelectric polarization acts only as a scaling factor and is veryoften incorporated into the definitions of e11 and e33. It is more usual therefore not to see the factor of e0 inthe final term in Eq. 11. Using this latter definition of the flexoelectric polarization, the electrostatic energydensity is (similar to the magnetic case) then

we ¼ �ðD � dE ¼ 1

2e0e⊥E2 � 1

2e0ea n � Eð Þ2 � e0Ps � E: (12)

However, because the electric susceptibilities are much larger than the magnetic susceptibilities, theappropriate Maxwell’s equations must be solved in order to obtain the (nonconstant) form of the electricfield E throughout the liquid crystal sample. Maxwell’s equations for the electric field and the electricdisplacement are ∇�E= 0 and∇ �D= 0. The first of these equations implies that the electric field may bewritten in terms of an electric potential U, such that E = �∇U, and the second equation may be solved,together with the equations for the director, fluid velocity, and pressure, to provide the solution for theelectric potential U(x, t).

Boundary ConditionsIn most liquid crystal devices, the glass or plastic substrates within which the liquid crystal material issandwiched are treated to provide some form of alignment of the director. The main forms of alignmentare:

• Strong (or infinite) anchoring: the alignment layer at the substrate is strong enough to fix the liquidcrystal molecules there. The director is therefore fixed to be a specific value n = ns at the substrate.

• Weak anchoring: the alignment method defines a fixed direction, a preferred direction, ns. The directorat the surface may take a different value from ns, but this will cost energy.

Mathematically, strong anchoring gives a fixed, “Dirichlet” boundary condition, n = ns. For weakanchoring, we specify a surface energy, such as the commonly used Rapini–Papoular energy(Stewart 2004; Rapini and Papoular 1969) ws = Ws (1 � (n � ns)2), which is added to the bulk energy.


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A solution for n should then minimize the total energy. The inclusions of surface energy terms mean thatthe boundary conditions change from Dirichlet conditions to Neumann conditions, where a balance oftorques at the surface leads to the gradient of the free energy, normal to the substrate, being specified at theboundary (see Stewart (2004) for more details).

Boundary conditions for the fluid velocity and pressure are dependent on the problem considered.However, in most liquid crystal display situations, the appropriate boundary conditions will be that thefluid is stationary at a solid boundary, i.e., v = 0 at the substrate, and that pressure at either end of thedevice is equal, i.e., p = pa at x = � Lx and y = � Ly, where Lx, Ly is the xy extent of the device underconsideration. The usual boundary conditions for the electric potential are that, at electrodes, the potentialtakes a constant or time-varying value, i.e., U = 0 at one electrode and U = V(t) at the other electrode.

Additional Physical PhenomenaAs well as the relatively standard consideration of the director, flow velocity, electric field, etc., it issometimes appropriate to consider additional dependent variables in order to accurately model liquidcrystal devices. One of the most common examples of this is the inclusion of mobile ionic species in thesystem. It is often the case that the complete removal of ionic contaminants is nearly impossible and, inany case, the repeated application of electric fields can cause the introduction of ions (from surface layersand/or the liquid crystal molecules decomposing). Such ionic species often shield the applied voltages andcan influence the behavior of the director and thus the electro-optic response of the device. As an exampleof the type of governing equations that result, we here consider two ionic species which may move withinthe fluid due to the application of an electric field as well as through convection of the ions with the fluidflow and natural diffusion.

The governing equations for the number densities of the two species ne and nh (negative and positiveions) are

@ne@t

þ v � ∇ð Þne ¼ �me∇ � ne∇U � De

me∇ne

� �þ Re; (13)

@nh@t

þ v � ∇ð Þnh ¼ mh∇ � nh∇U þ Dh

mh∇nh

� �þ Rh; (14)

where me and mh are the negative and positive ion mobilities and De and Dh are the negative and positiveion diffusivities. The quantities Re and Rh model any possible effects associated with the production ordestruction of ions within the device, which may be functions of parameters such as the applied voltage oreven of the ionic densities.

For such a system, it is usual to consider the boundaries to be insulated, i.e., that no current can flow intoor out of the cell. Mathematically, these conditions can be expressed in the following forms:

0 ¼ ne∇U � De

me∇ne, 0 ¼ nh∇U þ Dh

mh∇nh (15)

The only change to the previously stated governing equations is that Maxwell’s equation must be alteredto be

∇ � D ¼ qhnh � qene; (16)


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where qe and qh are the charges on the negative and positive ions, respectively.This set of equations, together with appropriate initial condition for ne and nh at t = 0, should then be

solved at the same time as the governing equations for the director, flow velocity, electric fieldpotential, etc.

Example: The p-CellWe illustrate the use of these equations using a simple model of the p-cell (Bos and Koehler-Beran 1984)(see Fig. 1). To allow easy comparisons with the Q-tensor model we describe later, we will ignore floweffects and ionic contaminants and assume strong anchoring at the boundaries. The only equation thatremains is the angular momentum balance Eq. 2, and substituting from Eqs. 3 and 4 and Eqs. 7, 8, and 9gives a set of equations which govern the components of the director.

For static solutions @n=@t ¼ 0, and these equations are equivalent to the Euler–Lagrange equations forthe minimization of the total energy in the system subject to the constraint that n is a unit vector. However,in this example, we consider the dynamic equations.

If we assume that the director always lies in a single plane (the xz-plane in our example) and thatthe director does not vary in the x and y directions, then the director can be written as n= (cos y(z, t), 0, siny(z, t)) where y, the tilt angle, is the angle that the director makes with the positive x-axis and is a functionof z and t only. For this example, we use the “one constant approximation” and we neglect any additionalpolarization terms, such as Ps. As mentioned above, we write the electric field as the gradient of theelectric potential, E=�∇U, and then, after some relatively simple but long-winded manipulations of theequations (see, e.g., (Stewart 2004)), we obtain governing equations for the y(z, t) and U(z, t):

g1@y@t

¼ Kyzz � e0ea sin y cos yU2z : (17)

�e0 e⊥ þ ea sin2 y� �

Uzz � e0ea sin 2yyzUz ¼ 0; (18)

where the subscript z denotes differentiation with respect to z and g1= a3� a2 is the rotational viscosity ofthe director. Equations 17 and 18 must be solved subject to appropriate boundary conditions. y at theboundaries is given by the pretilts at the surfaces (see below), and for the electric potential, we setU= 0 atthe bottom surface and U = V, the applied voltage, at the top.

In the p-cell, we have equal and opposite pretilts at the boundaries, and this allows us to set theboundary conditions for the problem. However, we do need to be careful when assigning values at theboundary to the director angle y. To see this, we consider static solutions @y/@t= 0, in the zero field case,when V= 0. For this situation, Eq. 18, with the boundary conditions, leads to U(z)� 0. Equation 17 thenleads to yzz = 0, and solutions are therefore linear in z. For the splay state, Fig. 1a, the situation isstraightforward; the director tilt changes linearly, from � yp to yp; and the tilt in the center of the cell iszero. To set the boundary conditions for the bend state, Fig. 1b, we first note that n and�n are equivalentand so y and y� p are equivalent. We can therefore consider the tilt varying linearly from�yp to yp � p.The tilt in the center of the cell is then �p/2.

There is a third possible state: the twist state. This state is an important one since the bend state will,once the electric field is removed, normally relax into the twist state since it is of a lower energy. However,since we will be concerned with the transition between states when the field is applied, we have restrictedthe director to the xz-plane and will not consider the relaxation to the twist state.

Figure 2 shows the tilt profiles obtained when we apply a number of different voltages to the splay andthe bend states. For this situation, where the liquid crystal has a positive dielectric anisotropy, theelectrostatic energy is a minimum when the director is aligned with the field (i.e., when y = p/2, or


of 26

equivalently y = �p/2). Aligning the director with the field increases the distortion, and the associatedelastic energy, and the final configuration results from a balance between the electric field and elasticeffects, subject of course to the boundary conditions.

Figure 2 shows the solutions to Eqs. 17 and 18 for voltages V= 0, 2, 5, 30. (Given our comments aboveabout the twist state, the solution for the bend state at 0 V is only the transient solution, before relaxation tothe twist state.) The parameters used for this simulation were typical for a nematic liquid crystal: g1= 0.02Pas, K = 14.7 � 10�12 N, e0 = 8.854 � 10�12 F m�1, ea ¼ 14, e⊥ ¼ 5:1; yp = p/30 rads. The extent ofthe liquid crystal region (the cell gap) was taken to be 2 � 10�6 m.

In Fig. 2a we see that as the voltage increases, for the splay state, the director becomes increasinglyvertical (y= p/2 or�p/2) in the majority of the cell, with regions of distortion at the boundaries and in thecenter of the cell as well. For the splay state, the tilt at the center of the cell remains at zero. We see that theincreasing electric field is producing a region of relatively small length scale. For the bend state, in Fig. 2b,the bulk of the cell, including the center of the cell, can align vertically, with distortion regions only at theboundaries. For both states, when the voltage is removed, the director configuration reverts back to theinitial state, splay, or bend, respectively. For these range of voltages, switching between the two statesdoes not occur. In fact, as will be discussed later, the fact that this model does not lead to switchingbetween the two states is a consequence, and a problem, with the chosen model. We will see later that amore realistic model does in fact allow switching between states.

2

−2.5

−2

2

1.5

1

0.5

1.51

z, distance through cell (x10−6m)

0.5

0

0

−0.5

−1

−1.5

−1.5

−2

θ, d

irect

or a

ngle

(ra

dian

s)

0

21.51


0.50

−0.5

−1

−3

−3.5

θ, d

irect

or a

ngle

(ra

dian

s)

a

b0V2V5V30V

Fig. 2 Plots of the director angle for the p-cell in the (a) splay state and (b) the bend state


of 26

Q-Tensor Models

Uniaxial OrderIn addition to the director n, the average molecular orientation direction, we could also compute thestandard deviation of the distribution or orientations of molecules. This would provide additionalinformation about the state of matter at that point in space. However, rather than the standard deviationof this distribution, the usual measure of this amount of order is the scalar order parameter, usuallydenoted by S, which is a weighted average of the molecular orientation angles ym between the longmolecular axes and the director

S ¼ 1

2< 3 cos2ym � 1 > ; (19)

where < > denotes the thermal or statistical average, i.e.,

S ¼ 1

2

ðℬ

3 cos2 ym � 1� �

f ymð ÞdV ; (20)

where f(ym) is the probability distribution function of the molecular angle ym. When the material iscrystalline, all molecules align exactly with the director and so ym = 0 for all molecules, which meansthat < cos2 ym > = 1 and S = 1. When all molecules lie in the plane perpendicular to the director, butrandomly oriented in that plane, the average still leads to the same director but ym = p/2 for all moleculesso that< cos2 ym>= 0 and S=�1/2. In the isotropic liquid phase, the molecules are randomly orientedand so f(ym) is constant and equal to 1/(2p) (this can be derived from the property of probabilitydistributions that

Rf(ym)dA= 1). Therefore, performing the integration in Eq. 20 in spherical coordinates,

where ym is the angle between the molecule and the director and fm is the azimuthal angle, i.e., the anglebetween a fixed direction in the plane perpendicular to the director and the projection of the director ontothat plane, we obtain (using the substitution X = cos(ym))

ðdℬþ

cos2 ymf ymð ÞdA ¼ 1

2p

ð2p0

ðp=20

cos2 ym sin ym dym dfm ¼ 1

3(21)

and so < cos2 ym > = 1/3 and from Eq. 20, S = 0.Although it is possible to achieve a molecular configuration for which S is negative, i.e.,

�1/2 < S < 0, it is more usual that, in the equilibrium liquid crystal state, the scalar order parameter ispositive, 0 < S < 1. As the temperature of the material changes, the scalar order parameter will changefrom S = 0 in the isotropic state, at high temperature, to S > 0 in the liquid crystalline state, at lowtemperature. A typical scalar order parameter in the middle of the phase region, for this type of liquidcrystal, might be S = 0.6.

It is possible to now construct a theory, similar to the Ericksen–Leslie theory discussed in the previoussection, which treats S(x, t) as well as n(x, t) as dependent variables. Ericksen developed just such a theory(Ericksen 1991), and it is possible to consider regions of low order, and to model defects, with this type ofmodel. However, a single order parameter theory does not describe the most general form of a nematicliquid crystal consisting of rodlike, uniaxial molecules. For more general cases, we must considerbiaxiality.


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Biaxial Order

The fundamental principle of any biaxial system is that there is no axis of complete rotational symmetry(i.e., no axis about which a rotation of any angle leaves the system unchanged), unlike a uniaxial systemwhich has an axes of rotational symmetry (such as the director n in uniaxial liquid crystals). However,there can be defined a set of perpendicular axes (only two need to be defined as the third is then specifiedas perpendicular to the other two, n � m) for each of which there is a reflection symmetry. In liquidcrystals, the two axes, or directors, n and m are therefore defined and the symmetries are the reflectionsn ! �n and m ! �m. The most general nematic state for uniaxial molecules is then the biaxial statedescribed by two vector-valued variables and two scalar-valued variables, the directors n(x, t) andm(x, t)and the scalar order parameters S1(x, t) and S2(x, t), the measure of order with respect to the two directors,all of which may depend on the spatial coordinates x and time. We assume, without loss of generality, thatthe directors are of unit length, |n| = 1 and |m| = 1. These directors can be represented in terms of thestandard Euler rotation angles. Since the director n is of unit length, it can be written as

n ¼ cos y cosf, cos y sinf, sin yð Þ: (22)

The director m is perpendicular to n and therefore has one remaining degree of freedom, the angle c, sothat

m ¼ sinf cosc� cosf sinc sin y,� sinf sinc sin y� cosf cosc, sinc cos yð Þ: (23)

The angle c is the angle from m to the direction (sin f, � cos f, 0) in the xy-plane, which is alsoperpendicular to n. A theory can now be constructed using the five dependent variables, y(x, t), f (x, t),c(x, t), S1(x, t), and S2(x, t). However, there may be problems with a theory such as this based on Eulerangles. When the zenithal angle y equals p/2, the azimuthal angle f is undefined and, as with all suchangle variables, there may be a problem with multivaluedness since f = 0 is equivalent to f = 2p.Care must therefore be taken when solving the governing differential equations if the dependent variablesy, f, and c are to be used. This problem of degeneracy of rotation angles can however be avoided if thedirectors are not described in terms of these angles, but left in terms of the six components of the directorsn= (nx, ny, nz) andm= (mx, my, mz). However, in this case, three Lagrangemultipliers (similar to l used inthe Ericksen–Leslie theory) must be used to ensure that the constraints that n, m are unit vectors andperpendicular. This approach can be simplified if an alternative approach is used, using the so-called ordertensor for the system.

The Tensor Order Parameter QWe now define an alternative approach which removes the problems of the angle representation andthe complexities of the Lagrange multipliers described above. Instead of defining the nematic state interms of the five separate variables mentioned above, we construct a 3 � 3 matrix which contains all theinformation about the nematic state, i.e., the information contained in the five variables. The problemsof solving the Euler angle governing equations will be removed with this approach. Consider the3 � 3 matrix (or tensor), M = S1 (n � n) + S2 (m � m), where for any vector h, the ijth element ofthe product h � h is hi hj, the ith element of h multiplied by the jth element of h. This matrix will besymmetric, since ni nj = nj ni and mi mj = mj mi, and the trace of M will be S1 + S2 since |n| = 1 and|m| = 1. The tensor M contains the same information as the five separate variables. However, if weconstruct a theory using M, there will be no problems with the degeneracies of Euler angles. In fact the


of 26

tensorM is the second moment tensor for the molecular distribution and therefore is directly related to theorientational distribution function (Virga 1994).

As will be indicated later, it is actually more useful to use the tensor:

Q ¼ S1 n� nð Þ þ S2 m�mð Þ � 1

3S1 þ S2ð ÞI; (24)

where I is the identity matrix so that the trace of Q is zero. The tensor order parameter Q is therefore asymmetric traceless matrix and can be written as

Q ¼q1 q2 q3q2 q4 q5q3 q5 � q1 �q4

0@

1A; (25)

and from the definitions of n andm and Eqs. 24 and 25, we see that the elements qi can be written in termsof the variables y, f, c, S1, and S2; thus:

q1 ¼ S1 cos2y cos2fþ S2 sinf cosc� cosf sinc sin yð Þ2 � 1

3S1 þ S2ð Þ; (26)

q2 ¼ S1 cos2y sinf cosf� S2 cosf coscþ sinf sinc sin yð Þsinf cosc� cosf sinc sin yð Þ; (27)

q3 ¼ S1 sin y cos y cosfþ S2 sinc cos y sinf cosc� cosf sinc sin yð Þ; (28)

q4 ¼ S1 cos2y sin2fþ S2 cosf coscþ sinf sinc sin yð Þ2 � 1

3S1 þ S2ð Þ; (29)

q5 ¼ S1 cos y sin y sinf� S2 sinc cos y cosf coscþ sinf sinc sin yð Þ: (30)

The eigenvalues of the matrix Q, described by Eqs. 26, 27, 28, 29, and 30, are

l1 ¼ 2S1 � S2ð Þ=3, l2 ¼ � S1 þ S2ð Þ=3, l3 ¼ 2S2 � S1ð Þ=3: (31)

Uniaxial states exist when two of these eigenvalues are the same, i.e., when l1= l2 so that S1= 0 or whenl2= l3 so S2= 0 or when l1= l3 so S1= S2. When all the eigenvalues are the same, we have an isotropicsystem and we have S1= 0 and S2= 0 so thatQ= 0. We shall see later that the use ofQ rather thanM asour dependent tensor variable was in fact dictated by the property that the isotropic state is described byQ = 0.

There will be no problems when solving the governing equations for the five dependent variables q1, q2,q3, q4, and q5 because the Euler angles only appear in sin and cos functions and the problem with themultivaluedness of the angles is removed.

Phenomenological TheoryAs was seen for the director-based theory, it is necessary to construct certain quantities such as the total

free energy F ¼ð ð

wdV , the integral of the free energy density over the volume of the region under

consideration. This energy may include terms such as the elastic energy of any distortion to the structure


of 26

of the material; thermotropic energy which dictates the preferred phase of the material; electric and/ormagnetic energy from an externally applied electric or magnetic field and, in polar materials, the internalself-interaction energy of the polar molecules; and surface energy terms representing the interactionenergy between the bounding surface and the liquid crystal molecules at the surface. The total energy istherefore

F ¼ F distortion þ F thermotropic þ F electromagnetic þ F surface

¼ðVwd þ wt þ weð Þduþ

ðSwsð Þds; (32)

where the energy densities, wd, etc., depend on the dependent variables, the tensor order parameterelements.

In static situations, the governing equations may be derived from the minimization of this energy, usingthe calculus of variations, leading to sets of differential equations in the bulk of the material and at thesurface, for each of the dependent variables. The solution of these bulk equations subject to the surfaceboundary conditions gives the equilibrium configuration of the dependent variables through the sample.A version of such a theory which considers fluid flow as well as variations in the order tensor is beyondthe scope of this summary article and we refer the interested reader to the following reference(Sonnet et al. 2004).

The free energy density is assumed to depend on the tensor order parameter Q and all first-orderdifferentials ofQ. It is assumed that distortions ofQ are small and therefore higher-order differentials andhigh powers of first-order differentials will be negligible. The bulk free energy density,wb=wd +wt + we,is therefore the integral of a function dependent on the elements of Q and all derivatives of the elements,whereas the surface free energy density is assumed to be a function of the elements of Q only:

F ¼ðVwb qi,∇qið Þduþ

ðSws qið Þds: (33)

We write down the different terms that might contribute to the free energy density and assign modelparameters to them. Knowing what values to use for these parameters when we come to solve the resultingequations can be difficult. However, we can often relate these model parameters to measured quantities byassuming that we have a uniaxial state and writing down the energy terms on that basis. We then relate theenergy terms that we obtain to those in the corresponding director model assigning the parametersappropriately.

Dynamic EquationsWhen the dynamic evolution of the Q-tensor is required, a simple dissipation principle can be used toshow that the governing equations will be

g@D@ _qi

¼ ∇:Gi � f i; (34)

whereD is the dissipation functionD ¼ 12 tr @Q=@tð Þ2�

, _qi ¼ @qi=@t, and the viscosity g is related to the

standard nematic viscosity g1 (which is equal to a3� a2 in Leslie viscosities) by g= g1/(2Sexp2), and Sexp is

the uniaxial order parameter of the liquid crystal when the experimental measurement of g1 was taken. Theother quantities in Eq. 34 are gradients of the free energy, given byGj

i= @wb/@qi,j and fi= @wb/@qi. In fact


of 26

a similar dissipation principle approach can be used to derive the Ericksen–Leslie equations (see Stewart2004).

Boundary ConditionsSimilar to the director-based theory, there are two types of boundary conditions which will be considered:strong (infinite) or weak anchoring.

• For strong, or infinite, anchoring, we will assume the Dirichlet conditions. That is, the order tensor willbe fixed at a specified value at the domain boundary that has been dictated by some substrate alignmenttechnique. In this case, the boundary condition isQ=Qs, whereQs is the prescribed order tensor at theboundary. In this case, there is no surface energy and ws = 0 in Eq. 32.

• For a weak-anchoring effect, there exists a surface energy which is added to the free energy and must beminimized at the same time as the bulk free energy. This minimization leads to the condition that, on theboundary, the liquid crystal variables qi satisfy

Xj¼1, 2, 3

nj@wb

@qi, j

!¼ @ws

@qi; (35)

or equivalently n � Gi = Gi where Gi = @ws/@qi and n is the normal at the substrate pointing into theliquid crystal regions. Usually the surface is planar or circular so that the surface normal is relativelysimple, e.g., n = (0, 0, 1), if the surface is a plane parallel to the xy-plane. However, in general thesubstrate normal can be position dependent n(x).

It is now necessary to specify each of the components of the free energy. The following sectionsdescribe the thermotropic, elastic, electrostatic, and surface energies.

Landau–de GennesThe thermotropic energy, wt, is a potential function which dictates which state the liquid crystal wouldprefer to be in, i.e., a uniaxial state, a biaxial state, or the isotropic state. At high temperature, this potentialshould have a minimum energy in the isotropic state, i.e., Q = 0, whereas at low temperatures, thereshould be minima at three uniaxial states, i.e., the states where any two of the eigenvalues ofQ are equal.We will not consider a situation where a biaxial state is a local minimum. The simplest form of such afunction is a Taylor expansion about Q = 0:

wt ¼ a tr Q2� �þ 2b

3tr Q3� �þ c

2tr Q2� �� 2

; (36)

which is a quartic function of the qis. This energy is sometimes written as

wt ¼ a tr Q2� �þ 2b

3tr Q3� �þ c tr Q4

� �: (37)

Equations 36 and 37 are equivalent only if the factor of two in the c term is included.The coefficients a, b, and cwill in general be temperature dependent although it is usual to approximate

this dependency by assuming that b and c are independent of temperature while a = a(T � T) = aDTwhere a > 0 and T is the fixed temperature at which the isotropic state becomes unstable.


of 26

By writing this energy density in terms of the order parameters and directors mentioned above, it isrelatively straightforward to show that this term has stationary points when

S1 ¼ 0; (38)

S1 ¼ 1

4c�bþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � 24ac

p� ; (39)

S1 ¼ 1

4c�b�


p� : (40)

By calculating d2wt/dS12 and comparing the energies of each solution, we find that

• S1 = 0, the isotropic state, is globally stable for a > b2

27c , metastable for 0 < a < b2

27c, and unstable fora < 0.

• S1 ¼ 14c �b�


p� , the nematic state, is globally stable for a < b2

27c , metastable for

b2

27c < a < b2

24c, and not defined for a > b2

24c.

• S1 ¼ 14c �bþ


p� is metastable (but has negative value) for a < 0, unstable for

b2

27c < a < b2

24c, and not defined for a > b2

24c.

The equilibrium nematic scalar order parameter Eq. 39 will be denoted by Seq.

There are clearly three important values of a: a ¼ b2

24c, the high temperature where the nematic statedisappears; a ¼ b2

27c, the temperature at which the energy of the isotropic and nematic states are exactlyequal; and a = 0, the low temperature where the isotropic state loses stability. If we use the notationa = a(T � T), then the critical temperatures are

Tþ ¼ b2

24acþ T;

TNI ¼ b2

27acþ T

Therefore, for this thermotropic energy and depending on the values of the coefficients a, b, and c, in anintermediate temperature region, the minimum at the isotropic state and the minima at the uniaxialnematic states may all be locally stable. However, at sufficiently low temperatures, the isotropic statemust lose stability leaving only the uniaxial nematic states stable, and at sufficiently high temperatures,the uniaxial states must lose stability leaving only the isotropic state stable.

This expression for the thermotropic energy, Eq. 36, is essentially a Taylor series of the true thermo-tropic energy, close to the point Q= 0. Therefore, we must remember that the Landau–de Gennes theoryis only really valid close to the nematic–isotropic transition temperature, TNI, where Q 0. It is for thisreason that we may assume that higher-order powers of Q may be neglected in Eq. 36. The first fivepowers ofQ are included in Eq. 36 (the constant term is neglected as it will not enter into a minimizationof the energy and the linear term is taken to have zero coefficient since tr(Q) = 0 so that there is aminimum at Q = 0) since we assume there are at most four minima in the potential function wt.

It would be energetically favorable for the system to lie in one of the minima of wt. When a liquidcrystal material is forced to contain some form of distortion (e.g., surface or electric field effects), there are


of 26

two mechanisms for undertaking such a distortion. For example, imagine a region of liquid crystalconstrained to lie between two solid surfaces. Through some surface treatment, one of the surfaces mayforce the liquid crystal in contact with that surface to lie in a fixed uniaxial state, whereas the other surfacemay force the liquid crystal in contact with it to lie in a different uniaxial state. Firstly, for the system tomove from one minima of wt to another, the eigenvectors of Q may change. This is equivalent to themolecular frame of reference changing. Alternatively, the eigenvalues of Q may change but the eigen-vectors remain fixed. There may be a combination of changes to the eigenvector and eigenvalue, andexactly how the system distorts will depend on the competition between the thermotropic energy wt andthe elastic energy wd.

ElasticityThe distortional or elastic energy density, wd, of a liquid crystal is derived from the energy induced bydistorting the Q-tensor in space. It is, generally, energetically favorable for Q to be constant throughoutthe material, and any gradients inQwould lead to an increase in distortional energy. wd therefore dependson the spatial derivatives ofQ. Given a fixed distortion in space ofQ, the distortional energy must remainunchanged if we were to translate or rotate the material. Such restrictions (or symmetries) mean that not allcombinations of Q derivatives are allowed. In fact the elastic energy may be simplified to

wd ¼X

i¼1, 2, 3j¼1, 2, 3k¼1, 2, 3

L12

@Qij

@xk

� �2

þ L22

@Qij

@xj

@Qik

@xkþ L3

2

@Qik

@xj

@Qij

@xk

" #

þX

i¼1, 2, 3j¼1, 2, 3k¼1, 2, 3l¼1, 2, 3

L42elikQlj

@Qij

@xkþ L6

2Qlk

@Qij

@xl

@Qij

@xk

� �: (41)

The coordinates (x1, x2, x3)= (x, y, z) are the usual Cartesian coordinate system, andQij is the ijth elementof Q. The first four terms are quadratic in the scalar order parameters S1 and S2, whereas the last term iscubic in the scalar order parameters.

The elastic parameters Li are related to the Frank elastic constants Ki by

L1 ¼ 1

6S2expK3 � K1 þ 3K2ð Þ, L2 ¼ 1

S2expK1 � K2 � K4ð Þ; (42)

L3 ¼ 1

S2expK4ð Þ, L4 ¼ 2

S2expq0K2, L6 ¼ 1

2S3expK3 � K1ð Þ; (43)

where Sexp is the uniaxial order parameter of the liquid crystal when the experimental measurement of the

Li was taken and may not be equal to the current order parameter Seq = �bþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � 24ac

p� = 4cð Þ

mentioned in chapter “Landau–de Gennes.” The parameter q0 is the chirality of the liquid crystal, and if anachiral liquid crystal is under consideration, then L4 = 0.

It has been shown that there are seven elastic terms of cubic order, but we will only include one, the L6term, in Eq. 41, in order to ensure we canmodel a nematic state with nonequal elastic constantsK1,K2, and


of 26

K3. Without the L6 term, we have four elastic parameters L1,. . ., L4 in the Q-tensor elastic energy, but wehave five independent parameters in the Frank approach, K1, K2, K3, K4, and q0. Including the L6 termremoves the degeneracy in the mapping from the Q-tensor to the Frank energy approaches.

ElectrostaticsThe liquid crystal will interact with an externally applied electric field or indeed self-induce an internalelectric field due to the dielectric and spontaneous polarization effects. As in the director-based theory,the electrostatic energy density is calculated from the integration of the displacement D = e0 « E + e0 Ps,which relates the displacement field to the electric field E, the dielectric tensor «, and the polarization Ps.The electrostatic energy density is then

we ¼ �ð

e0«Eþ «0Psð Þ � dE ¼ � 1

2e0 «Eð Þ � E� e0Ps � E: (44)

In a nematic liquid crystal, the dielectric tensor in terms of the Q-tensor is usually approximated toe ¼ De Qþ eIwhere De ¼ ek � e⊥

� �=Sexp is the scaled dielectric anisotropy and e ¼ ek þ 2e⊥

� �=3 is

an average permittivity. Within an isotropic material, such as an alignment layer of photoresist material,the dielectric tensor is diagonal and isotropic, e ¼ eII.

In the present, nematic, situation, a spontaneous polarization vector is assumed to derive only from aflexoelectric type of polarization, i.e., a polarization caused by a distortion of the molecular arrangement.This may be due to a shape asymmetry in the molecules or due to a distortion of a pair-wise coupling ofmolecules. For either of these mechanisms, the polarization may be written, to leading order, in terms ofthe Q-tensor as Ps ¼ e∇ �Q where the ith component of ∇ ∙ Q is, as mentioned before, understood to beX

j¼1, 2, 3@Qij=@xj.

If we consider the flexoelectric polarization of a uniaxial state, where Q ¼ S n� nð Þ � I=3ð Þ, we find

Ps ¼ e S ∇ � nð Þnþ S ∇� nð Þ � nþ n � ∇Sð Þn� 1

3∇S

� �: (45)

Comparing this to the standard expressions for flexoelectric polarization and order electricity,

Pf ¼ e11 ∇ � nð Þnþ e33 ∇� nð Þ � n; (46)

Po ¼ r1 n � ∇Sð Þnþ r2∇S; (47)

and we see that Eq. 45 is equivalent to Eqs. 46 and 47 when we sete11 ¼ e33 ¼ Se and r1 ¼ e, r2 ¼ �e=3.Therefore, without taking higher-order terms in Eq. 45, this expression assumes that the coefficients offlexoelectric and order electric terms are equal. Within a region where S is constant, only the flexoelectricpolarization Eq. 46 would be present.

In order to distinguish between the flexoelectric parameters e11 and e33, higher-order terms are neededin the flexoelectric polarization term in Eq. 45. For instance, if we include second-order terms, the ithcomponent of the polarization vector is


of 26

Psð Þi ¼ p1X

j¼1, 2, 3

@Qij

@xjþ p2

Xj¼1, 2, 3k¼1, 2, 3

Qij

@Qjk

@xkþ p3

@

@xi

Xj¼1, 2, 3k¼1, 2, 3

QjkQjk

0BBBB@

1CCCCAþ p4

Xj¼1, 2, 3k¼1, 2, 3

Qjk

@Qji

@xk(48)

For a uniaxial material, we can substitute, expand, and collect terms as before. This gives

e11 ¼ Sp1 þS2

32p2 � p4ð Þ, e33 ¼ Sp1 þ

S2

32p4 � p2ð Þ,

r1 ¼ p1 þS

3p2 þ p4ð Þ, r2 ¼ 1

3�p1 þ

S

3p2 þ p4ð Þ þ 4Sp3

� �:

Therefore, including second-order terms does allow us to have different values for e1 and e3 if p2 6¼ p4.However, deciding on values for p1 . . . p4 is nontrivial. Barbero et al. (1986) have stated that p1, p2, and p4can be obtained by measuring the flexoelectric parameters as a function of temperature and assume asimilar magnitude for p3, but there seems to be no such measurements in the literature.

At this point, the electric field within the whole cell is unknown but, as in the director-based model,may be found using Maxwell’s equations, one of which is satisfied if we again use the electric potentialE = �∇U. The governing equation for the electric potential U is then

0 ¼ ∇:D ¼ ∇: �e0e∇U þ Psð Þ; (49)

and the free energy density

we ¼ � 1

2e0e∇U :∇U þ Ps:∇U ; (50)

will enter the total free energy density to be minimized in order to obtain the Euler–Lagrange equations forthe qi. Equation 49 is in fact equivalent to the Euler–Lagrange equation derived from minimizing the freeenergy in Eq. 50 with respect to U.

By making the electric potential U continuous through the cell, we ensure the standard conditions forelectrostatics that, at material boundaries, the component of the displacement field normal to the boundaryis continuous and the component of the electric field parallel to the boundary is continuous.

The external boundary conditions are usually set at the electrodes where, for example, one electrode isset to be earthed and so U = 0 and the other electrode is set to be a fixed voltage U = V.

Surface Energy: Alignment DirectionWhen the liquid crystal molecules are close to a solid surface, they will feel a molecular interaction force.Whatever this force is, we would like to model this liquid crystal–surface interaction in a macroscopicframework, i.e., how does the surface interact with the macroscopic variables (the elements of the tensororder parameter Q). If the surface is treated in some way, usually by coating the surface with a chemicaland possibly rubbing the surface in a fixed direction so as to create a preferred surface direction, then wewill assume that at that surface, the directors n and m would prefer to lie in a certain direction. We mayalso assume that the solid surface may affect the amount of order (i.e., the variables S1 and S2) at thatsurface. Associated with this preference for a certain orientation and order will be a surface energy,F surface, which will have a minimum at the preferred state. This surface energy will be a function of the


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value of the dependent variables at the surface. Thus, if S is the surface in contact with the liquid crystal,F surface = F surface (Q|s). If the variables are forced to move out of the minima, usually by the bulk of theliquid crystal being in some alternate state, then the surface energy will increase. There will be acompetition between surface energy and bulk energy. In equilibrium, a balance will be reached suchthat the total energy in Eq. 32 is minimized.

One simple form of the free energy density, Fs, has a single minimum at the point where the dependentvariables take the value dictated by the surface treatment:

ws ¼ W

2tr QjS �QSð Þ2 (51)

whereW is the anchoring energy andQs is the value of the tensor order parameter preferred by the surface.When the system is in equilibrium, the calculus of variations gives the set of boundary conditions inEq. 35 which are essentially a torque balance at the surface of the distortion torque and the torque due tothe surface energy function. If we were to compare this energy to a Rapini–Papoular-type anchoringenergy where, say, a preference for the director to lie in the x direction is given by the energy densityFs ¼ Ws

2 sin2y, then the relationship between the Rapini–Papoular anchoring strengthWs and theQ-tensoranchoring strength in Eq. 51 will be W = Ws/2Ss

2, where Ss is the preferred surface order parameter.An example of the type of anchoring in Eq. 51 is if we have a surface which prefers the nematic to be

uniaxial with the director in the z direction and scalar order parameter of 0.6.We then take S2= 0, y= p/2,and S1 = 0.6 in Eqs. 26, 27, 28, 29, and 30 so that the preferred Q-tensor is

Qs ¼�0:2 0 00 �0:2 00 0 0:4

0@

1A; (52)

or in terms of the qi values, q1 = �0.2, q2 = 0.0, q3 = 0.0, q4 = �0.2, and q5 = 0.0.Another important example of weak anchoring is homeotropic anchoring on a nonplanar surface. Such

anchoring prefers the director to lie perpendicular to the surface at all points. If we denote the unit normalto the surface as n = (nx,ny,nz), then this will be the preferred director. If we assume that the surface willprefer a uniaxial state (as suggested by the local symmetry of homeotropic anchoring), then we may useEq. 24 with n = n, S1 = Ss, and S2 = 0 to obtain the preferred Q-tensor

Qs ¼ Ss

n2x �1

2nxny nxnz

nxny ny2 � 1

3nynz

nxnz nynz2

3� n2x � n2y

0BBBBB@

1CCCCCA : (53)

Strong AnchoringThe case of strong anchoring mentioned in chapter “Boundary Conditions” is equivalent to the limitWs!1. In this case, the Dirichlet conditionQ=Qs is applied on the boundary instead of the weak-anchoringcondition for which a surface energy is minimized.


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Surface Energy: Planar DegenerateAnother common liquid crystal substrate treatment is the planar degenerate surface. In this situation, thepreferred orientation for the directors is to lie parallel to the substrate. There is no preference as to whichdirection on the plane of the surface to lie but simply that the director lies on the surface.

The most general surface energy density in this case is

ws ¼ c1v �Q � vþ c2 v �Q � vð Þ2 þ c3v �Q2 � vþ as tr Q2

� �þ 2bs3

tr Q3� �þ cs

2tr Q2� �� 2

:(54)

In this energy density, the first three terms give the most general energy density, up to quadratic order,which specifies that the eigenvectors ofQ lie parallel to the surface with normal n. The last three terms inEq. 54 are added to specify preferred eigenvalues of Q.

If we assume the liquid crystal has taken a uniaxial state at the surface, S1 = S and S2 = 0, then theplanar degenerate surface energy density is

ws ¼ S

33c2S sin

4yþ 3c1 þ c3 � 2c2ð ÞSð Þ sin2y� �þ f Sð Þ; (55)

which has a minimum at y = 0 when S(3c1 + (c3 � 2c2)S) > 0. The effective anchoring strength, whencompared to a Rapini–Papoular energy, may be thought of asws ¼ 2

3 S 3c1 þ c3 � 2c2ð ÞSð Þ. It is clear thatthere is no dependence on the azimuthal angle f as such a degenerate anchoring condition would suggest.

The p-CellWe again illustrate the use of these equations using the p-cell device (Fig. 1). We use the Q-tensorequivalent of “one constant approximation,” and, as before, we assume that the additional polarizationterm, Ps, is zero. We write the electric field in terms of the electric potential,E=�∇U, and solve for the qiwith i = 1. . . 5 and the electric potential, U.

The problem is again inherently one dimensional, and so the dependent variables are functions of z andt only. Unlike in the director case, we do not constrain the solution as we want to allow the order to changeand biaxial solutions to emerge when necessary. In this model, we do not constrain the director to lie in thexz-plane, which we did in the director model. The only constraints on the behavior of the liquid crystalenter through the governing equations, and since we assume that, in the absence of evidence to thecontrary, the Q-tensor at the boundaries is fixed to be in a uniaxial state with the equilibrium value of theorder parameter.

The equations for the five elements of the Q-tensor and the electric potential are then

g@q1@t

¼ L1q1, z �De

6U2

z � 2aq1 �2

3b q21 þ q22 þ q23 � 2q24 � 2q1q4 � 2q25� �� 2cq1tr Q2

� �;

g@q2@t

¼ L1q2, z � L1q2, z � 2aq2 � 2b q1q2 þ q2q4 þ q3q5ð Þ � 2cq2tr Q2� �

;

g@q3@t

¼ L1q3, z � 2aq3 � 2b q2q5 � q3q4ð Þ � 2cq3tr Q2� �

;


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g@q4@t

¼ L1q4, z �De

6U2

z � 2aq4 �2

3b �2q21 þ q22 � 2q23 þ q24 � 2q1q4 þ q25� �� 2cq4tr Q2

� �;

g@q5@t

¼ L1q5, z � 2aq5 � 2b q2q3 � q1q5ð Þ � 2cq5tr Q2� �

;

with

tr Q2� � ¼ 2 q21 þ q22 þ q23 þ q24 þ q25 þ q1q4

� �;

and the electrostatic displacement equation.We now carry out the same simulations as before, with the same parameter values as the director-based

approach, as well as the additional parameters needed for theQ-tensor model: Sexp = 0.624, Seq = 0.624,a= 0.975� 105 N/Km2, b=�36� 105 N/m2, c= 43.875� 105 N/m2, dT=�4.0K. The results of thesesimulations are shown in Fig. 3, where we have determined the in-plane director angle y from theQ-tensorby calculating the major eigenvalue as discussed above. For the bend state (Fig. 3b), the situation is verymuch the same as before, with a uniform bulk region (aligned with the electric field, y = p/2) and highdistortion regions close to the boundaries. (Note that, as in the director model, the solution for the bendstate at 0 V is an unstable equilibrium solution which would, if allowed, relax to the twist state.) However,the case for the splay state (Fig. 3a) is different. For the lower values of the applied voltage, we see similarbehavior to the director-based model, with a distortion region in the center of the cell. However, at highvoltages (i.e., 30 V), the system has switched from the splay state to the bend state.

By considering the major order parameter S1 for the bend state, Fig. 4, we see that the region of highdistortion at the center of the cell has reduced the order parameter. This eventually leads to an effective“melting” of the liquid crystal (in fact the liquid crystal enters a transient biaxial state), and the liquidcrystal order reforms to take the lower energy bend state. Although this change in order is a high energyevent, it is favorable in order to achieve the low-order bend state.

RemarksIn this section, we have laid out the ingredients for a theory which allows the modeling of regions ofchanging order (i.e., near to defects (Schophol and Sluckin 1987)), including dynamic effects through adissipation functional which considers the rate of change of theQ-tensor. However, a fuller description ofthe dynamics would include dissipation through flow, i.e., the equivalent to the Ericksen–Leslie equationsfor the director-based approach. However, describing such a theory in a short chapter is impossible, andgiven the, as yet, limited use of such a theory to model liquid crystal devices, we have opted to describeonly this simpler no-flow model. We do however refer the reader to (Sonnet et al. 2004) which contains adetailed description of just such a theory, which also contains a summary of other similar theories whichhave started to be used.

Summary

We have tried in this chapter to explain and summarize two commonly used theories of nematics, onebased on using the director as a dependent variable and one based on using the tensor order parameter.Both have their advantages and disadvantages.

The director-based approach is relatively simple and has been used successfully in many differentsituations. All material parameters in this theory have been measured for a few materials, although a


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2

1.5

1

0.5

0

21.510.50

−0.5

−1

−1.5

−2

−2.5

−3

21.51



0.50

−2.5

−2

−1.5

0

−0.5

−1

−3

−3.5

a

b

0V2V5V29V30V

θ, d

irect

or a

ngle

(ra

dian

s)θ,

dire

ctor

ang

le (

radi

ans)

Fig. 3 Plots of the director angle from the Q-tensor simulation, for the p-cell in the (a) splay state and (b) bend state

0.66

0.64

0.62

0.6

0.58

0.56

0.54S, o

rder

par

amet

er

0.52

0.5

0 0.5 1 1.5 20.48


0V2V5V29V30V

Fig. 4 The order parameter S1 as a function of distance through the cell, for various voltages. At V = 29 V, a significantreduction in order is caused by the high distortion in the center of the cell


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complete characterization of all the necessary parameters is time-consuming and rarely undertaken. Forinstance, many of the viscosities remain unknown for most liquid crystal materials. The main problemwith a director-only approach is the inability of such a theory to accurately handle defects, where thereexists a singularity in the director, and the nematic order is reduced from its bulk value, or other moregeneral regions of reduced order such as the example described above. Indeed we have seen that in thiscase, the director-based model fails completely to model the switch between the splay and bend statewhich occurs (Barberi et al. 2004b; Ramage and Newton 2008).

It is in such a situation that a Q-tensor approach is valuable. With the ability to model biaxialconfigurations of molecules, and reductions in order parameter, this theory is able to describe the coresof defects and other instances of changes in order, i.e., near to rough surfaces. However, when using theQ-tensor approach, it is unusual to be able to make any analytic headway, and for most realistic situations,the governing equations must be solved numerically. These numerical computations are often extremelyexpensive (in terms of computational time and memory) because of the large discrepancies in time andlength scales that exist in problems which contain defects. The ratio of defect core size to the devicedimensions is a few orders of magnitude, and it is often necessary to implement sophisticated numericalmethods where time and space adaption are utilized.

As mentioned in the introduction, the crucial step in modeling liquid crystal devices is often the choiceof the appropriate dependent variables and then the choice of theory. If a nematic liquid crystal device isthought to contain no defects, or regions of varying order parameter, the Ericksen–Leslie theory will beappropriate. When defects are present, an order parameter should be included as a dependent variable, anda theory such as the Q-tensor theory described above will be more appropriate.

Further Reading

Barberi R, Ciuchi F, Lombardo G, Bartolino R, Durand GE (2004a) Time resolved experimental analysisof the electric field induced biaxial order reconstruction in nematics. Phys Rev Lett 93:art.137801

Barberi R, Ciuchi F, Durand GE, Iovane M, Sikharulidze D, Sonnet AM, Virga EG (2004b) Electric fieldinduced order reconstruction in a nematic cell. Euro Phys J E 13:61

Barbero G, Dozov I, Palierne JF, Durand GE (1986) Order electricity and surface orientation in nematicliquid-crystals. Phys Rev Lett 56:2056

Bos PJ, Koehler-Beran KR (1984) The pi-cell – a fast liquid-crystal optical-switching device. Mol CrystLiq Cryst 113:329

Bose E (1908) Zur Theorie der anisotropen Flussigkeiten. Phys Z 9:708Care CM, Cleaver DJ (2005) Computer simulation of liquid crystals. Rep Prog Phys 68:2665de Gennes PG, Prost J (1993) The physics of liquid crystals, vol 2. OUP Clarendon Press, OxfordDunmur D, Fukuda A, Luckhurst G (2001) Physical properties of liquid crystals: nematics. Institution of

Engineering and Technology, StevenageEricksen J (1991) Liquid crystals with variable degree of orientation. Arch Ration Mech Anal 113:97Frank FC (1958) On the theory of liquid crystals. Discuss Faraday Soc 25:19Kelker H (1973) History of liquid crystals. Mol Cryst Liq Cryst 21:1Leslie FM (1968) Some constitutive equations for liquid crystals. Arch Ration Mech Anal 28:205Meyer R (1969) Piezoelectric effects in liquid crystals. Phys Rev Lett 22:918Miesowicz M (1935) Influence of a magnetic field on the viscosity of para-azoxyanisol. Nature 136:261Miesowicz M (1936) Der einfluss des magnetischen feldes auf die viskositat der flussigkeiten in der

nematischen phase. Bull Acad Pol A 28:228Oseen CW (1933) The theory of liquid crystals. Trans Faraday Soc 29:883


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Parodi O (1970) Stress tensor for a nematic liquid crystal. J Phys (Paris) 31:581Ramage A, Newton CJP (2008) Adaptive grid methods for Q-tensor theory of liquid crystals: a

one-dimensional feasibility study. Mol Cryst Liq Cryst 480:160Rapini A, Papoular M (1969) Distortion d’une lamelle nématique sous champ magnétique conditions

d’ancrage aux parois. J Phys (Paris) Colloq 30:C4Schophol N, Sluckin TJ (1987) Defect core structure in nematic liquid-crystals. Phys Rev Lett 59:2582Sluckin TJ, Dunmur DA, Stegemeyer H (2004) Crystals that flow: classic papers from the history of liquid

crystals. Taylor & Francis, LondonSonnet AM, Maffettone PL, Virga EG (2004) Continuum theory for nematic liquid crystals with tensorial

order. J Non-Newtonian Fluid Mech 119:51Stewart IW (2004) The static and dynamic continuum theory of liquid crystals: a mathematical introduc-

tion. Taylor & Francis, LondonVirga EG (1994) Variational theories for liquid crystals. Chapman & Hall, London


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liquid crystal theory and modeling · liquid crystal theory and modeling n. j. mottrama* and c. j....

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