Three Dimensional Finite Element Modelling of Liquid Crystal Electro-Hydrodynamics

Transcription

1 Three Dimensional Finite Element Modelling of Liquid Crystal Electro-Hydrodynamics by Eero Johannes Willman A thesis submitted for the degree of Doctor of Philosophy of University College London Faculty of Engineering Department of Electronic & Electrical Engineering University College London The United Kingdom

2 I, Eero Johannes Willman, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. I

3 Contents 1 Introduction Motivation Outline of the Work Development of a 3-D Finite Element Computer Model Modelling of Weak Anchoring in the Landau-de Gennes Theory Modelling a Post Aligned Bistable Nematic LC Device Achievements Liquid Crystals Introduction Liquid Crystal Phases Liquid Crystal Order Uniaxial Order Biaxial Order and the Q-Tensor Defects and Disclinations Dielectric Properties and Flexoelectric Polarisation Optical Properties of LCs Jones Calculus Theoretical Framework 21 II

4 3.1 Introduction Mean Field Theories Molecular Simulations Continuum Theory Liquid Crystal Elasticity Thermotropic Energy External Interactions Static Equilibrium Q-Tensor Fields Q-Tensor Hydrodynamics Conservation of Energy Frictional Forces The Power Equations of Motion Choice of the Dissipation Function Explicit Expressions for the LC-Hydrodynamics Discussion and Conclusions Modelling of the Liquid Crystal Solid Surface Interface Introduction Background Classification of Different Anchoring Types Anchoring Mechanisms Experimental Measurement of Anchoring Strengths Review of Currently Used Weak Anchoring Expressions Weak Anchoring in Oseen-Frank Theory Weak Anchoring in the Landau-de Gennes Theory III

5 4.4 The Anchoring Energy Density of an Anisotropic Surface in the Landaude Gennes Theory Determining Values for the Anchoring Energy Coefficients Numerical Results Comparison between the Landau-de Gennes and Oseen-Frank Models Effect of Order Variations on the Effective Anchoring Strength Discussion and Conclusions Finite Elements Implementation Introduction The Finite Element Method Weighted Residuals Method Variational Method Enforcing Constraints and Boundary Conditions Solution Process Shape Functions Analytic and Numerical Integration of Shape Functions General Overview of the Program Electrostatic Potential Q-Tensor Implementation Newton s Method Time Integration Implementation of the Hydrodynamics Enforcement of Incompressibility Mesh Adaptation Introduction IV

6 6.2 Mesh Adaptation Assessment of the Error Adapting the Spatial Discretisation Overview of the Mesh Adaption Algorithm Example Defect Movement in a Confined Nematic Liquid Crystal Droplet Hierarchical p-refinement Discussion Validation and Examples Introduction Three Elastic Constant Formulation Switching Dynamics of a TN-Cell, with Back flow Defect Dynamics Defect Loops in the Zenithally Bistable Device Discussion and Conclusions Modelling of the Post Aligned Bistable Nematic Liquid Crystal Structure Introduction Overwiew of the The PABN Device Modelling the PABN Device The Geometry of the Modelling Window Modelling Results A Topological Study of a Single Corner Modelling the Full Structure The Two Stable States Modelling the Full Structure The Switching Dynamics V

7 8.5 Discussion and Conclusions Summary and Future Work Summary or Achievements Future Work A Values of Material Parameters Used in this Work 149 VI

8 List of Figures 2.1 The molecular configurations of the isotropic, nematic and smectic A and C phases The nematic director n and the scalar order parameter S (a) Isotropic Q-tensor, (b) uniaxial Q-tensor, (c) biaxial Q-tensor, (d) uniaxial Q-tensor, but with negative scalar order parameter Director profiles for defects of whole m = ±1 and m = ± 1 strengths Splay, bend and twist deformations Bulk energy as a function of order parameter for various temperatures Twist angle in a cell of thickness d. Dashed line, strong anchoring. Solid line, weak anchoring Normalised anisotropic parts of the anchoring energy density for a surface with ê = [1, 0, 0], ˆv 1 = [0, 1, 0] and ˆv 2 = [0, 0, 1]. (a) R = 1. (b) R = 3. (c) R = 0. (d) R =. (R = W 2 /W 1 ) Eigenvalues of a Q-tensor that minimises the surface energy density as a function of R, when S e is unity (a) Scalar order parameter S and (b) biaxiality parameter P as functions of the distance from the surface (in µm) and the ratio R between W 2 and W VII

9 4.5 Normalised eigenvalues of Q at the surface as a function of W 2 for R = 1, 3 and, when a is set according to expression 4.23 (no markers) and for the linear case a s = 0 and R = 1 (circles) (a) (c) Tilt and twist angles as a function of V, with a constant R = 1. 3 (d) (f) Tilt and twist angles as a function of R, with a constant applied voltage V = Ratio of the effective azimuthal anchoring strength coefficient and W 1 as a function of R Local coordinates of a tetrahedron Flowchart of the program execution Container with 90 bend for testing the stabilised Stokes flow Flow magnitude (top row) and pressure (bottom row) solutions obtained using three different values the stabilisation parameter ɛ = 10 4, 10 6 and 10 9 ) Element refinement by the red-green method. Bisected edges are drawn in bold. Original nodes are labelled with capital letters whereas new nodes resulting from edge bisection are labelled using lower case letters Example of error introduced by linear interpolation of the components of a Q-tensor field representing a rotation of the director field of a constant order. Black dots represent the original nodes and gray dots the new added node Partial 3-Dimensional view of initial unrefined mesh for LC droplet inside a cube of fixed isotropic dielectric material. Approximately a quarter of the dielectric region (coloured white) and half of the liquid crystal (coloured grey) are shown VIII

10 6.4 (a), (c), (d) 2-Dimensional slices through the centre of a nematic droplet during switching by an external electric field. Director colour indicates scalar order parameter and background electric potential. (b), (d), (e) 3-Dimensional views of corresponding meshes (a) Second, third and fourth order hierarchical shape functions for a one dimensional finite elements implementation. (b) Example of superposition of first and second order hierarchical element shape functions. Linear element (dashed line) is p-refined by the addition of a second order (solid line) shape function (a) 1 2 defect in two dimensions (left) and the one dimensional director profile through the centre (right). (b) Eigenvalues of the Q-tensor in the one dimensional case plotted against the z dimension Comparison between results obtained using hierarchical elements of different order. (a) Total free energy as a function of element size. (b) The effective number of degrees of freedom as a function of element size Magnitudes of higher order hierarchical degrees of freedom as a function of the z-dimension. The number of 1-D elements is 50, resulting in an element size of 2 nm Comparison of tilt angles at z = 0.5µm as a function of time using the Oseen-Frank (dashed line) and the Landau-de Gennes (solid line) theories Switching dynamics of a twisted nematic cell, with and without flow of the LC material The two initial director configurations for the defect annihilation cases (a) and (b), and the corresponding flow solutions (c) and (d) at time = 20 µs IX

11 7.4 Defect positions with respect to time for the two initial configurations, with and without flow. In both cases when flow is ignored, identical results are obtained. The solid line represents the position of the positive defect and the dashed line the position of the negative defect The continuous (a) and discontinuous (b) states found in the two dimensional representation of the ZBD grating structure Three different surface profiles for the ZBD structure, with the height of the slip region set to 0, 0.5 and 1 times the ridge height in (a), (b) and (c) respectively Iso-surfaces of reduced order parameter showing the locations of the defect lines. Circles are drawn to indicate the regions of the ± 1 defect 2 transitions The geometries of the 3-D modelling windows (a) for the full device, and (b) for the isolated corner Director profiles for the horizontal (a) and continuous vertical (b) states on a regular grid along the (x, y) plane through the centre of the isolated corner structure at z = 0.3µm. The discontinuous vertical state is not shown, as it appears nearly identical to the continuous vertical state from this point of view The director field on a regular grid along the diagonal (x = y, z) plane through the separated corner structure. (a) Stable continuous vertical configuration, (b) stable discontinuous vertical configuration Defect line along a post edge during switching. (a) a magnified view of (x, y) plane at z = 0.3 µm cutting through the post. Darker background colour indicates a reduction in the order parameter near the defect core. (b) 3-D view of same post edge with a dark iso-surface for the order parameter showing the extent of the line defect X

12 8.5 Sums of elastic, thermotropic and surface energies for the four director configurations using the modified thermotropic coefficients (black) and for the 5CB material (white). The energies are normalised with respect to the respective horizontal states The director field (x, y) plane at z = 0.3µm for the (a) planar and (b) tilted states. The planar (c) and tilted (d) states in the (x = y, z) plane running diagonally through the modelling window. In (a) and (b), the background color corresponds to the z-component of the director, where positive z direction is out of the page The tilt angles of the stable planar and tilted states along a corner of the modelling window as a function of z Simulation results of planar to tilted to planar switching The sum of the total thermotropic, elastic and surface anchoring energies during the planar-tilted-planar switching sequence XI

13 List of Symbols and Abbreviations δ ij ɛ ijk n S S 0 P Q λ i Ē D ε ε ε ε P e 11 e 33 n n n the Kroenecker Delta the Levi-Civita anti-symmetric tensor liquid crystal director scalar order parameter equilibrium order parameter biaxiality parameter Q-tensor representing the nematic distribution of order eigenvalue of the Q-tensor, corresponding to the eigenvector i electric field electric displacement field liquid crystal relative permittivity tensor relative permittivity parallel to n relative permittivity perpendicular to n dielectric anisotropy, ε = ε ε flexoelectric polarisation vector flexoelectric splay coefficient flexoelectric bend coefficient refractive index parallel to n refractive index perpendicular to n birefringence, n = n n XII

14 f d f th f f f s K 11 K 22 K 33 L 1 L 6 elastic distortion energy density thermotropic energy density electric field induced energy density surface energy density splay elastic coefficient in the Oseen-Frank theory twist elastic coefficient in the Oseen-Frank theory bend elastic coefficient in the Oseen-Frank theory elastic coefficients in the Q-tensor theory T, T c, T temperature, clearing temperature and nematic-isotropic transition temperatures respectively A = a(t T ) temperature dependent thermotropic energy coefficient in the Landau-de gennes theory B, C thermotropic energy coefficients in the Landau-de gennes theory v p D ij W ij γ 1, γ 2, α 1 α 6 flow field hydrostatic pressure symmetric flow gradient tensor antisymmetric flow gradient tensor Ericksen-Leslie viscosities µ 1, µ 2, β 1 β 6 Qian-Sheng viscosities ê ˆv 1, ˆv 2 θ φ a s W i easy axis of anchoring principal axes of weak anisotropic anchoring tilt angle twist angle isotropic anchoring strength coefficient anchoring strength corresponding to ˆv i R anchoring anisotropy ratio R = W 2 /W 1 XIII

15 N i finite element shape function at node i r, s, t local tetrahedral coordinates ˆη q 1 q 5 LC ZBD PABN TN FE surface normal unit vector five independent components of the Q-tensor Liquid Crystal Zenithally bistable Device Post Aligned Bistable Nematic Twisted Nematic Finite Element XIV

16 Acknowledgements I would like to thank the following people who have contributed to this Ph.D. and made the past few years both enjoyable and unforgettable. I am grateful to my supervisors Dr. Aníbal Fernández and Dr. Sally Day who have provided me with guidance and have patiently helped me with all aspects related to this work. I would also like to thank Dr. Richard James, Dr. Mark Gardner and Dr. Jeroen Beeckman with whom I had the pleasure of sharing the office with. The long hours spent in the office never felt like a chore, and I can t imagine the outcome of this work without their expertise and advice. Other people who have been helpful include Mr. David Selviah, who was my M.Phil./Ph.D. transfer thesis examiner and has been a useful resource of constructive critique and many what if questions. Also, a considerable portion of this Ph.D. deals with the modelling of bistable liquid crystal devices. Many informative conversations on this topic have been held with Dr. Christopher Newton from HP Labs and Dr. Cliff Jones from ZBD Displays. The whole Ph.D. experience would not be complete without both past and present friends and flatmates who have contributed indirectly to this work by livening up the non-academic moments. Last but not least, I d like to thank my parents who have always been supportive and made this work financially possible. XV

17 Abstract Liquid crystals (LC) are used in new applications of increasing complexity and smaller dimensions. This includes complicated electrode patterns and devices incorporating three dimensional geometric shapes, e.g. grating surfaces and colloidal dispersions. In these cases, defects in the liquid crystal director field often play an important part in the operation of the device. Modelling of these devices not only allows for a faster and cheaper means of optimising the design, but sometimes also provides information that would be difficult to obtain experimentally. As device dimensions shrink and complex geometries are introduced, one and two dimensional approximations become increasingly inaccurate. For this reason, a three dimensional finite element computer model for calculating the liquid crystal electrohydrodynamics is programmed. The program uses the Q-tensor description allowing for variations in the liquid crystal order and is capable of accurately modelling defects in the director field. The aligning effect solid surfaces has on liquid crystals, known as anchoring, is essential to the operation of nearly all LC devices. A simplifying assumption often made in LC modelling is that of strong anchoring (the LC orientation is fixed at the LC- solid surface interface). However, in small scale structures with high electric fields and curved surfaces this assumption is often not accurate. A general expression that can be used to represent various weak anchoring types in the Landau-de Gennes theory is introduced. It is shown how experimentally measurable values can be assigned to the coefficients of the expression. Using the Q-tensor model incorporating the weak anchoring expression, the operation of the Post Aligned Bistable Nematic (PABN) device is modelled. Two stable states, one of higher and the other of lower director tilt angle, are identified. Then, the switching dynamics between these two states is simulated. XVI

18 Chapter 1 Introduction 1

19 1.1 Motivation Nematic liquid crystals (LC) possess anisotropic properties making them useful in a wide range of electro-optical applications. Traditionally these include for example LC displays and beam steering devices for optical communication. However, nematic LCs also find new applications as solvents for micro emulsions and particle dispersions, in e.g. bio-molecular sensors[1] or in the self-assembly of crystal structures[2]. Traditional applications can be relatively simple; some LC material sandwiched between two glass plates with electrodes. In these cases the orientation of the liquid crystal director varies in a continuous fashion throughout the device. However, the drive for devices with higher resolution and faster switching implies smaller dimensions and more complicated electrode shapes. In addition, applications increasingly incorporate complex three dimensional geometries, as is the case e.g. with some bistable devices and colloids. Frequently this results in discontinuities in the director field orientation, known as defects or disclinations. Computer modelling often allows for faster and cheaper design and optimisation of novel LC devices than manufacturing actual prototype devices. Furthermore, additional information that may be difficult or impossible to gather experimentally can be obtained. In general, modelling of a device involves two steps: First, the orientation of the liquid crystal is found. Then, based on the previously obtained director field the corresponding optical performance of the device can be calculated. Different methods for finding the alignment of the liquid crystal exist. It is possible to consider the interactions between each LC molecule one by one on a molecular or even atomistic scale. However, currently this process is computationally too expensive and time consuming for practical device modelling due to the large number of molecules involved. Instead, continuum elastic theories that describe the LC material in terms of local averages of the molecules can be used. Two continuum theories that have been 2

20 extensively used are the so-called Oseen-Frank theory [3, 4, 5] and the Landau-de Gennes theory [6]. The Oseen-Frank theory represents the local average orientation of the LC molecules with the unit vector n, known as the director. The molecular order is assumed constant and uniaxial, limiting the validity of the theory to relatively large, defect free structures. When defects are present, the Landau-de Gennes theory which allows for biaxiality and variations in the order parameter gives a better description. In this theory, the liquid crystal is represented using the rank two, traceless, symmetric tensor order parameter, the Q-tensor. The Ericksen-Leslie [7, 8] and Qian-Sheng [9] formalisms are extensions to the Oseen-Frank and Landau-de Gennes theories respectively that include the effect of flow of the LC material. 1.2 Outline of the Work The work described in this thesis concentrates on the static and dynamic three dimensional computer modelling of the Q-tensor field in small scale LC devices containing topological defects. Three main topics can be identified: Development of a 3-D Finite Element Computer Model The finite element method has been used to discretise the equations of the Landau-de Gennes theory [6] and its extension, the Qian-Sheng formalism [9], in three dimensions. Previously, the Qian-Sheng formalism has been used in one and two dimensional modelling of LCs (e.g. [10, 11, 12]), but to my knowledge, this is the first three dimensional finite element implementation of the theory. A Brezzi-Pitkäranta stabilisation scheme [13] has been used in the flow solver making it possible to use linear elements for both the flow and pressure solutions without the commonly encountered instability of the pressure solution [14]. A three dimensional mesh adaptation algorithm which performs local h-refinement 3

21 in regions selected using an empirical error indicator has been implemented. This, in conjunction with a stable non-linear Crank-Nicholson time integrator with variable time step makes modelling of three dimensional defect dynamics feasible on a standard PC workstation. The finite element program can be used for the modelling of both the switching dynamics and the static equilibrium states of arbitrarily shaped domains including multiple electrodes and non-liquid crystal regions Modelling of Weak Anchoring in the Landau-de Gennes Theory The operation of LC devices relies on the aligning effect of anchoring the LC to the solid surfaces of the cells. This effect can be achieved by treating the surfaces by a number of means. The physical/chemical processes behind the anchoring are complex and not always well known. Instead, a phenomenological approach describing the observed effect the surfaces have on the LC as an energy density is more useful in device modelling. The assumption of a surface energy density that varies in a W sin 2 Θ fashion as the director at the surface deviates from the preferred easy direction by an angle Θ has become common (known as the Rapini-Papoular assumption [15]). However, usually the anchoring is anisotropic, the polar and azimuthal anchoring strengths being unequal. For this reason, various generalisations that take into account the difference between the two directions have been proposed in the Oseen- Frank theory (e.g. [16, 17, 18]). The Landau-de Gennes theory has been used in the past to explain various aspects of the fundamental physics of the solid surface-lc interface. However, the inclusion of anisotropic weak anchoring characterised by experimentally measurable parameters into a numerical model has so far not received much attention within this framework. Here, a power expansion on the Q-tensor and two mutually orthogonal unit vectors is used as a surface energy density. The expression is shown to simplify in the 4

22 limit of uniaxial constant order parameter to a well known anisotropic anchoring expression in the Oseen-Frank theory. This makes it possible to assign experimentally measurable values with a physical meaning to the coefficients of the tensor order parameter expansion. The two expressions in the Oseen-Frank and Landau-de Gennes are compared using numerical simulations and shown to agree well. The validity of the assumption of constant uniaxial order used in the determination of the coefficients of the expansion is examined by measuring the effective polar and azimuthal anchoring strengths by simulating the torque balance method Modelling a Post Aligned Bistable Nematic LC Device Bistable LC devices have two distinct stable configurations to which the director field may relax, and in which they remain without applied holding voltages. Advantages of bistability include lower power consumption and the possibility of passive addressing of high resolution LC devices. The switching dynamics and the two stable states of the Post Aligned Bistable Nematic (PABN) LC device [19, 20] are modelled using the finite element implementation of the Landau-de Gennes theory. In the past, the Oseen-Frank theory has been used to find the two stable director configurations [21], but the dynamics of the switching has not been reported. The two stable states are found to be separated by a pair of line defects extending along the edges of the post. These defect lines act as energy barriers separating the two stable states. In order to switch between the two topologically distinct states, energy must be provided by externally applied electric fields. 5

23 1.3 Achievements The work described in this thesis has resulted in the following publications, conferences and prizes: Publications R. James, E. Willman, F. A. Fernández and S. E. Day, Finite-Element Modelling of Liquid Crystal Hydrodynamics with a Variable Degree of Order, IEEE Transactions on Electron Devices, 53, no. 7, (2006). E. Willman, F. A. Fernández, R. James and S. E. Day, Computer Modelling of Weak Anisotropic Anchoring of Nematic Liquid Crystals in the Landau-de Gennes theory, IEEE Transactions on Electron Devices, 54, pp , (2007). E. Willman, F. A. Fernández, R. James and S. E. Day, Switching Dynamics of a Post Aligned Bistable Nematic Liquid Crystal Device, IEEE J. Disp. Tech., 4, pp (2008). R. James, E. Willman, F. A. Fernández and S. E Day, Computer Modeling of Liquid Crystal Hydrodynamics, IEEE Transactions on Magnetics, 44, pp , (2008). J. Beekman, F. A. Fernández, R. James, E. Willman and K. Neyts, Finite Element Analysis of Liquid Crystal Optical Waveguides, 12th International Topical Meeting on Optics of Liquid Crystals, Puebla, Mexico. (2007) S. E. Day, E. Willman, R. James and F. A. Fernández, P-67.4: Defect Loops in the Zenithally Bistable Device, Society for Information Display International Symposium Digest of Technical Papers, 39, pp , (2008) 6

24 Conferences 2D and 3D Modelling of Liquid Crystal Hydrodynamics Including Order Parameter Changes, International Workshop on Liquid Crystals for Photonics, April , Ghent (Belgium), Oral Presentation. Three Dimensional Modelling of Nematic Liquid Crystal Devices,Flexoelectricity in Liquid Crystals, September , Oxford, Poster Presentation. Prizes Winner of SHARP-SID Best Student award

25 Chapter 2 Liquid Crystals 8

26 2.1 Introduction Liquid Crystal (LC) is a general term used for a type of mesophase of matter that exists between the solid and liquid phases. LC materials consist typically of organic molecules that are free to move about and flow like a liquid, while retaining a degree of orientational and sometimes positional order [6, 22, 23]. Different LC phases can be classified according to the distribution of molecular order. LC materials exist in different phases depending on the temperature or concentration of a solvent. When the phase depends on the temperature, the LC material is said to be thermotropic, and when it depends on the the concentration of a solvent the LC is said to be lyotropic. Lyotropic LC materials consist of amphiphilic molecules with a hydrophobic tail and a hydrophilic head [22]. When mixed with a polar solvent (e.g. water), the molecules tend to arrange themselves so that the tails group together, while the hydrophilic heads are attracted to the solvent. Soaps are an example of lyotropic liquid crystals. Thermotropic LC materials consist usually of rigid, anisotropically shaped molecules. The molecules are generally shaped either like rods (calamitic) or disks (discotic). Variations in these are possible, e.g wedge shaped or bent-core mesogens have been observed [24]. Currently, most electro optic LC devices make use of calamitic thermotropic materials in the nematic phase. For this reason, throughout the rest of this thesis, it is understood that referring to liquid crystals means thermotropic calamitic LC materials, unless otherwise stated. 9

27 Figure 2.1: The molecular configurations of the isotropic, nematic and smectic A and C phases. 2.2 Liquid Crystal Phases Thermotropic LC materials undergo phase transitions as the temperature is varied. At high temperatures the LC material is in the isotropic phase, where the molecules are randomly distributed. No long range positional or orientational order exists. As the temperature is lowered, at some critical temperature a phase transition occurs. Depending on the exact compound, the LC material becomes either nematic or smectic. In the nematic phase the LC molecules are free to move (no positional order), but an average direction along which the molecules tend to orient their long axes can be observed (long range orientational order exists). This is known as the director and represented by the unit vector n. In the smectic phase both positional and orientational order can be identified: The LC molecules tend to arrange themselves in layers of identical orientation. Depending on the orientation of the molecules within the layers, the smectic phases can further be classified into sub categories A, B, C,... Additionally, cholesteric or chiral variants of the nematic and smectic phases exist. The chiral nematic phase exhibits a continuous twisting of the molecules perpendicular to the long axis of the molecules. In the chiral smectic phases, a finite twist angle 10

28 from one layer to another can be observed. The distance over which the director undergoes a full 360 rotation is known as the chiral pitch length. 2.3 Liquid Crystal Order Typically when below the nematic-isotropic transition temperature, nematic LC materials exist in a uniaxial configuration in the bulk. That is, a single axis of symmetry exists. However, biaxial order, when more than one axis of symmetry exist, may occur e.g. near confining surfaces or in the vicinity of defects Uniaxial Order The uniaxial nematic phase can be characterised by the degree of orientational order, S, and the macroscopic average direction of the constituent molecules, n. The scalar order parameter S can be defined as a measure of the degree of orientational order. In a small volume containing N molecules, with the orientations of their long axes denoted by the unit vectors ū, the scalar order parameter can be defined as the second order Legendre polynomial: S = cos 2 θ 1 = 1 2N N { 3 ( n ūi ) 2 1 }, (2.1) i=1 where θ is the angle between each molecule and the nematic director n (see Fig. 2.2). In the isotropic phase, where no order exists, S = 0. In the nematic phase S is typically within the range from 0.4 to 0.7, depending on the temperature. A negative scalar order parameter is also possible. This corresponds to the molecules lying randomly oriented in a plane perpendicular to n. 11

29 Figure 2.2: The nematic director n and the scalar order parameter S. Many experimentally measurable parameters of a LC material are related to the value of the order parameter, and it can be determined e.g. by means of NMR spectroscopy, Raman scattering, X-ray scattering or birefringence studies [23, 22, 6] Biaxial Order and the Q-Tensor In the case of a biaxial distribution of the LC molecules, more than one order parameter is needed. It is then more convenient to characterise the LC material in terms of a tensor order parameter called the Q-tensor. The Q-tensor is a symmetric traceless rank 2 tensor (a three by three matrix). Q has 9 components, but only five of them are independent. This gives three spatial degrees of freedom and two orientational degrees of freedom. The three eigenvalues λ 1, λ 2 and λ 3 of Q are a measure of the nematic order in the three orthogonal directions defined by the corresponding eigenvectors n, k and l. The Q-tensor can be written in terms of the eigenvalues and eigenvectors as: Q = λ 1 ( n n) + λ 2 ( k k) + λ 3 ( l l). (2.2) However, since only two of the eigenvalues are independent, the definition S = λ 1 12

30 and P = 1 2 (λ 2 λ 3 ) can be made. Then the Q-tensor can be written in terms of the scalar order parameter S, the biaxiality parameter P and the three eigenvectors n, k and l as: Q ij = S 2 (3n in j δ ij ) + P (k i k j l i l j ). (2.3) When the eigenvectors coincide with the x, y and z axes of the frame of reference, the eigenvalues appear along the diagonal of the Q-tensor: Q = λ λ λ 3 = S S 2 + P S 2 P (2.4) A Visual Representation of the Q-Tensor Figure 2.3 is a visual representation of the different distributions of nematic order that can be described using the Q-tensor description. The pictured cuboids or boxes can be imagined to contain rigid rods representing LC molecules, and to be shaken in order to simulate the effect of thermal vibrations. The relative lengths of the sides of the boxes then affect the average orientations of the contained rods and are proportional to the eigenvalues of the Q-tensor describing the corresponding order distribution within the box (with an additional positive factor to avoid negative side lengths): a) λ 1 = λ 2 = λ 3 = 0. The three eigenvalues are equal (and zero due to the tracelessness of the Q-tensor) in the disordered isotropic phase. In this case, the sides of the box are of equal lengths so that the container does not impose a preferred direction on the rods. b) λ 2 = λ 3 = 1 2 λ 1. The dominant eigenvalue is positive while the other two 13

31 (a) (b) (c) (d) Figure 2.3: (a) Isotropic Q-tensor, (b) uniaxial Q-tensor, (c) biaxial Q-tensor, (d) uniaxial Q-tensor, but with negative scalar order parameter. are equal and negative, resulting in the uniaxial configuration S = λ 1 and P = λ 2 λ 3 = 0. The rods are most likely to be oriented in the direction along the longest side λ 1, with smaller but equal probabilities of being oriented in the directions corresponding to λ 2 and λ 3. c) λ 1 λ 2 λ 3. In the biaxial configuration the three eigenvalues are different, so that in this case S = λ 1 and P = λ 3 λ 2 > 0. The lengths of the sides of the box are then related by λ 1 > λ 3 > λ 2. d) λ 1 < 0, λ 2 = λ 3 = 1λ 2 1. The dominant eigenvalue is negative while the two others are positive and equal, resulting in the uniaxial configuration with S = λ 1 < 0 and P = 1(λ 2 3 λ 2 ) = 0. The shape of the box is then in this case a flattened cube with side lengths λ 2 = λ 3 > λ 1. 14

32 2.4 Defects and Disclinations It has been stated in the previous sections that the nematic phase is characterised by an average direction, the director n, along which the constituent molecules orient themselves. The orientation of the director is not fixed and may vary within a sample of the LC material. Mostly the variation is continuous and gradual, but often locations exist where the director orientation changes in a discontinuous fashion and is not well defined. These can be points, lines or surfaces and are commonly known as defects. The discontinuity associated with defect surfaces is not stable and smears into a continuous change of director orientation. However, in the presence of electric or magnetic fields, the continuous distortion may be compressed and contained within a short distance known as the coherence length of the field, resulting in two continuous domains separated by a thin wall. The coherence length depends on the strength of the field and the properties of the LC material. For example, in the case of a twist wall caused by an aligning magnetic field H, the magnetic coherence length ξ M is (see e.g. [6] p. 120 ): ξ M = 1 K 22 H χ, (2.5) where χ and K 22 are the magnetic anisotropy and twist elastic coefficient respectively (see sections 2.5 and 3.4). Line and point defects can be stable, and are classified according to the strength of the defect. The strength, m, of a defect is the number of 2π rotations the director field makes around the defect core. Defects of whole integer strengths are only stable in confined geometries and tend to split into half integer defects [25]. Figure 2.4 shows the director fields around defects of m = ±1 and m = ± 1 strengths. 2 15

33 Figure 2.4: Director profiles for defects of whole m = ±1 and m = ± 1 2 strengths. 2.5 Dielectric Properties and Flexoelectric Polarisation The anisotropy in shape of the LC molecules affects its dielectric permittivity and magnetic susceptibilities. The dielectric permittivity when measured parallel to the long axis of the molecules, ε, is different from that measured perpendicular to the same axis, ε. The dielectric anisotropy, defined as ε = ε ε, may be either positive or negative depending on the specific LC compound. The permittivity may then be expressed as a tensor in terms of the director: ε ij = ε δ ij + εn i n j. (2.6) An approximation to (2.6) written in terms of the Q-tensor is: ( 2 ε ij = ε δ ij + ε Q ij + 1 ) 3S 0 3 δ ij, (2.7) where S 0 is the equilibrium order parameter of the LC material. The magnetic susceptibility tensor χ may be defined in a similar fashion. 16

34 Flexoelectric Polarisation A polarisation associated with deformations in the director field is observed in many LC materials consisting of wedge or bent core molecules carrying a permanent electric dipole moment [6]. It is also present in LC materials consisting of straight molecules, but carrying a quadrupolar moment [26]. The flexoelectric polarisation vector can be written in terms of the director as: P = e 11 ( n n) e 33 (( n) n), (2.8) where e 11 and e 33 are flexoelectric polarisability coefficients corresponding to splay and bend deformations respectively. In the limit of constant uniaxial order parameter, an expression equivalent to (2.8) can be written in terms of a Q-tensor expansion as [27, 28]: P i = ξ 1 Q ij,j + ξ 2 Q ij Q jk,k, (2.9) where ξ 1 = 2 9S 0 (e e 33 ) ξ 2 = 4 (e 9S e 33 ). If only the term linear in Q is taken into account, (2.9) reduces to the special case of (2.8) when 3S 0 2 ξ 1 = e 11 = e 33. Typical values for e 11 and e 33 found experimentally [29] and by theoretical predictions [30] lie in the range 0 to ± Cm. When the order is not considered constant, (2.9) also describes polarisation induced by spatial variations of the order parameter. This effect has been observed for example near the interface of a LC material and a solid surface, where rapid spatial variations in order may occur [31]. 17

35 2.6 Optical Properties of LCs The dielectric anisotropy of LCs discussed previously extends to the optical frequencies, resulting in an anisotropic refractive index. Two indexes of refraction and their differences are defined, n, n and n = n n (Often these are referred to as the ordinary and extraordinary indexes of refraction respectively). The subscripts have the same meaning as described in the case for the dielectric anisotropy. The speed of an electromagnetic wave propagating in an isotropic medium is v = c/n, where c is the speed of light in vacuum and n is the refractive index of the material. The electric field of light propagating through a sample of LC material can be decomposed into two orthogonal components. If the orientation of the director is such that the two components of the electric field experience different values of the refractive index, the two components will propagate at different velocities. This results in a change of the polarisation state of the propagating light. Most LC display devices consist of a layer of LC material whose orientation may be controlled by some configuration of transparent electrodes sandwiched between a pair of polarisers. Incoming unpolarised light, typically from a back light, is polarised by the first polariser. The linearly polarised light then passes through the LC layer which may change the orientation of the polarisation of the light, depending on the orientation of the director. Finally the light is either transmitted or blocked by the second polariser, so that the device may appear bright or dark. The electrodes are used for creating electric fields which align the LC director in such a way that the polarisation of the light is parallel to the last polariser for the bright and perpendicular to it for the dark state. Different approaches for calculating the optical output of an LC device exist. The Jones [32] and Berreman [33] methods are two commonly used methods. These methods are valid when the lateral variation in the director field is small over distances comparable to the wavelength of the propagating light. When lateral variations in 18

36 the director field are rapid, diffractive effects not taken into account by the Jones or the Berreman approaches become significant. In that case a grating method [34, 35] which takes into account lateral variations in the refractive index is more accurate Jones Calculus Jones calculus (or Jones method) [32] is probably the simplest method used in calculating LC optics. Only changes in the light polarisation are considered, but not reflections or diffractive effects. The linearly polarised light (propagating in the z-direction) is described by a Jones vector J = [E x, E y ] T, which represents the polarisation state of the wavefront. The medium through which the light propagates, is considered to consist of k layers. Each layer is described by a 2 2 Jones matrix, M. The combined effect of the layers on the polarisation state of the propagating wavefront is then described by a series of multiplications: J k = M k M k 1... M 1 J 0, (2.10) where J 0 and J k are the incoming and outgoing Jones vectors. Each Jones matrix may represent either an optical element (e.g. a polariser or a retarder) or a slice of the LC material. In general, M may be written as: M = S(φ)NS( φ), (2.11) where S(φ) is a rotation matrix, with φ defined in the (x, y) plane: S = cos(φ) sin(φ) sin(φ) cos(φ), (2.12) 19

37 and N describes the retardation independent of its orientation: N = exp( iψ x) 0 0 exp( iψ y ). (2.13) Ψ x Ψ y = Ψ is the relative phase difference introduced to the polarisation of the electromagnetic field propagating in the z-direction as it passes through the medium. In the case of a slice of LC material with director tilt and twist angles θ and φ respectively, the two phase angles are calculated from the refractive indexes in the x and y directions and the thickness d of the layer as: Ψ x = n x 2π λ d, 2π Ψ y = n d, (2.14) λ where λ is the wavelength of the propagating light. The refractive index in the x- direction depends on θ, and is obtained using [36]: 1 n 2 x = sin2 (θ) n 2 + cos2 (θ) n 2 (2.15) 20

38 Chapter 3 Theoretical Framework 21

39 3.1 Introduction Liquid crystal device modelling is typically a two step process: First, the LC director field orientation within the device is estimated. Then, the corresponding optical performance can be calculated. In this chapter, a theoretical background for the method used throughout the rest of this thesis for calculating LC director fields is introduced. For completeness, this chapter starts with a brief review of some well known theories that can be used to describe LC physics, but are in general not suitable for practical device modelling. In section 3.2, statistical mean field theories explaining LC phase changes are introduced. Then, in section 3.3 methods and applications of molecular simulations are outlined. A good estimate of the orientation of the LC director field over length scales comparable to LC device dimensions can be obtained using arguments based on continuum elasticity. This is the approach taken here, and the majority of this chapter, starting from section 3.4, is devoted to explaining the underlying theory. 3.2 Mean Field Theories Mean field theories attempt to explain what happens to a large number of molecules by making the assumption that on average all the molecular interactions are equal. This means that the macroscopic properties of many molecules can be deduced from the microscopic properties of only a few. Two such theories are the Onsager hard-rod theory [37] and the Maier-Saupe theory [38, 39, 40]. Both of these theories describe the nematic-isotropic phase transition. In the Onsager theory, the constituent molecules are considered to be hard rods, whose lengths are much greater than their widths. The basic assumption is that of a balance of positional and orientational entropy of the rods that cannot interpenetrate 22

40 each other. An interaction potential for a pair of rods is written in terms of the relative positions and orientations of them and the concentration of rods. The solution of the Onsager theory is independent of the temperature and predicts a first order isotropic to nematic phase transition occurring when the concentration of molecules is sufficiently high, it is an early proof that shape anisotropy alone is sufficient to induce nematic order. In the Maier-Saupe theory, an intermolecular attractive contribution due to van der Waals force is additionally taken into account. Furthermore, the probability of finding a molecule being oriented at a given angle from the director can be written as a function involving the temperature of the system, making it possible to predict a first order thermotropic nematic to isotropic transition. 3.3 Molecular Simulations In contrast to mean field theories, molecular theories consider a large number of individual molecules or particles (usually some simplified representation is used). Reviews of the method and its many variations can be found e.g. in [41, 42, 43]. Due to the involved computational cost the number of simulated particles is necessarily limited to far less than what is required in full-scale device modelling. However, molecular simulations have been used to explain links between molecular and observed bulk macroscopic properties of LC materials. For example, the values of elastic constants, viscous parameters and flexoelectric coefficients can be estimated in this way [44, 45, 46, 30]. The core of a molecular simulation is an interaction potential which represents the pairwise potential energies between each of the the constituent molecules. Different assumption on the form of the interaction potential have been made in the past. For example, both hard and soft particle interaction potentials are possible. 23

41 Hard ellipsoids have first been considered in [47, 48], whereas different variants of the soft particle Gay-Berne model [49], which includes both attractive and repulsive forces, have been popular. Additionally, all atomistic interactions are possible but computationally more demanding [30, 50]. Typically a simulation is started from some initial molecular configuration and allowed to evolve to an equilibrium state, after which the sought properties of the system are measured. Two common methods for evolving to equilibrium are the molecular dynamics and Monte Carlo methods. In the molecular dynamics method, the forces acting on each molecule are derived from the interaction potentials. Using this, the accelerations and velocities of each molecule can be calculated, and subsequently the locations are updated. This process is repeated in an iterative fashion, giving the dynamic behaviour of the molecular ensemble. In the Monte Carlo method, the positions and orientations of the molecules are typically updated in a random/pseudo random fashion. However, a decision based on some rule must be made whether an update is accepted or not. For example, only moves which do not result in an increase in the interaction potential energy could be accepted. Due to the nature of the Monte Carlo method, dynamic properties of the LC material, e.g. values of the viscous coefficients, cannot usually be obtained. 3.4 Continuum Theory A phenomenological continuum theory description can be used in modelling the behaviour of the LC material on sufficiently large length and time scales to be useful in device modelling. Instead of considering each of the molecules, the director n or the Q-tensor is used to describe the LC orientation. In this approach, the basic assumption is that a free energy density, f, for the 24

42 sample of LC material can be written as a function of the director or the Q-tensor. The LC material then prefers to exist in a state (director orientation, order parameter distribution) that minimises the total energy within that region. The energy density consist of a number of terms, each accounting for some physical property of the material and its interaction with external effects. The total free energy within a sample Ω with boundaries Γ is given by: F = f d + f th f f dω + f s dγ, (3.1) Ω Γ where f d is the elastic distortion energy density, f th is the thermotropic (or Landau) energy density, f f is an external field induced energy density and f s is a surface energy density appearing at interfaces between the LC material and its surroundings. Each of these terms will be described in more detail in the following sections Liquid Crystal Elasticity A distortion energy density, f d is written as a function of the director and its spatial derivatives. For nematics, this term is minimised when the director field is in an undistorted configuration and for chiral LCs the minimum occurs when a twist deformation with a pitch length p is present. The distortion energy density introduced by Oseen [3], Frank [4] and Zocher [5] identifies three possible distortion types of the bulk nematic director field. These are the so-called splay, twist and bend distortions, depicted in figure (3.1). From Figs. (3.1) a, b and c, showing the vector presentation of the possible distortions in a director field with n = [0, 0, 1], it is easy to verify that the vector expressions satisfying the distortions are given by: 25

43 Splay = n x,x + n y,y = n, Twist = n x,y n y,x = n n, (3.2) Bend = n x,z + n y,z = n n, where a comma in the subscript indicates differentiation with respect to the direction following it. The bulk distortion energy density can then be written as the weighted sum of the terms in (3.2) squared: f d = 1 2 K 11( n) K 22( n n) K 33( n n) 2, (3.3) where K 11, K 22 and K 33 are elastic constants assigned to the three distortion types. (a) (b) (c) (d) (e) (f) Figure 3.1: Splay, bend and twist deformations. A more rigorous way of obtaining the general distortion energy of a nematic LC material is to write the elastic energy density as a power expansion in all the possible 26

44 gradients n i,j of the director field and identifying the terms that satisfy the requirements that the energy is frame invariant, n = 1 and n = n. Following [51], this process is outlined next. In the case when n = [0, 0, 1], all spatial derivatives of n z must vanish in order to satisfy n = 1, i.e. n z,j = 0. The energy density, taking into account up to second order terms in n i,j, can be written as: f d = k i a i K ija i a j, (3.4) where a i contains the nonzero components of n i,j, written as the vector a = [n x,x, n x,y, n x,z, n y,x, n y,y, n y,z ], k and K contain the elastic coefficients of the expression. Since k is a vector of length 6 and K a symmetric matrix of size 6 by 6, the expression contains 21 possible elastic coefficients. However, due to the frame invariance requirement some of these reduce to zero. The non-zero coefficients can be identified by taking into account the uniaxial symmetry of the director field: k i a i + K ij 1 2 a ia j = k i a i K ija ia j. (3.5) In (3.5), the frame invariance is enforced by requiring that the energy of the system is equal in two different frames of reference; a is defined in the same way as a, but in a different frame, and can be obtained by a simple rotation of the x and y axes around the symmetry axis z: n i,j = Rn i,j R T, (3.6) 27

45 where R = , (3.7) is the rotation matrix corresponding to a π 2 rotation around the z-axis, giving a = [ n y,y, n y,x, n y,z, n x,y, n x,x, n x,z ]. Substituting a back into (3.5) and collecting terms in the gradients of n gives k = [k 1, k 2, 0, k 2, k 1, 0] for the linear term, and: K 11 = K 55, K 22 = K 44, K 33 = K 66, K 12 = K 45, K 14 = K 25, (3.8) K 13 = K 16 = K 23 = K 26 = K 34 = K 35 = K 36 = K 46 = K 56 = 0, (3.9) for the terms containing higher order terms of n i,j. After another rotation around the z-axis, by e.g. π 4 given by R = , (3.10) and rearrangement of terms gives the final terms K 14 = K 12 and K 15 = K 11 K 22 K 24, reducing the total K-matrix to: 28

46 K = K 11 K 12 0 K 12 (K 11 K 22 K 24 ) 0 K 12 K 22 0 K 24 K K K 12 K 24 0 K 22 K 12 0 (K 11 K 22 K 24 ) K 12 0 K 12 K K 33. (3.11) With the k and K elastic coefficients identified, equation (3.4) can be expanded and using (3.2) written in vector notation as: f d = 1 2 K 11( n s 0 ) K 22( n n + t 0 ) K 33( n n) 2 K 12 ( n)( n n) (K 22 + K 24 ) ( n n + n n), (3.12) where the terms linear in gradients of the director have been included by making the substitutions s 0 = k 1 /K 11 and t 0 = k 2 /K 22. Finally, taking into account the head-tail symmetry n = n, resulting in k 1 = 0, K 12 = 0, the nematic distortion energy density is written as: f d = 1 2 K 11( n) K 22( n n + 2π p 0 ) K 33( n n) (K 22 + K 24 ) ( n n + n n), (3.13) where K 11, K 22 and K 33 are Frank elastic constants corresponding to the splay, twist and bend LC deformations and p 0 is a chiral pitch length, which is zero for ordinary nematics and non-zero for cholesterics. The last term appears as a surface integral due to the Gauss divergence theorem, but is often ignored in calculations due to enforced boundary conditions. 29

47 Typically values for the elastic constants lie in the range 5-15 pn and usually the relation K 33 > K 11 K 22 holds (see e.g. [6] p ). It is common to make a single elastic coefficient assumption K = K 11 = K 22 = K 33 to simplify calculations. In this case, after some manipulations of (3.13) (see e.g. p. 23 in [51] for details), the elastic energy density reduces to: f d 1 2 K n 2 = 1 2 Kn i,jn i,j. (3.14) The elastic energy density can also be expressed in terms of the Q-tensor and its spatial derivatives as: f d = 1 2 L 1Q ij,k Q ij,k L 2Q ij,j Q ik,k L 3Q ik,j Q ij,k L 4ɛ lik Q lj Q ij,k L 6Q lk Q ij,l Q ij,k, (3.15) where L i are elastic coefficients. The relation between the elastic coefficients in equations (3.13) and (3.15) can be found by replacing the Q-tensor in (3.15) by its uniaxial definition S 0 2 (3n i n j δ ij ) and comparing the expressions. This has been done in [52, 53], resulting in: L 1 = 1 (K 27S K K 22 ) L 2 = 2 (K 9S K 22 K 24 ) L 3 = 2 L 4 = K 9S π p 0 9S 2 0 K 22 L 6 = 2 (K 27S K 11 ) (3.16) 30

48 The single elastic coefficients simplification in terms of Q is f d 1 2 L 1Q ij,k Q ij,k Thermotropic Energy A thermotropic energy density, f th, is used to describe the LC order variations. The bulk, or thermotropic, energy density f th is a power expansion on the tensor order parameter: f b = 1 2 A(T )Tr(Q2 ) B(T )Tr(Q3 ) C(T )Tr(Q2 ) 2 + O(Q 5 ), (3.17) where A, B and C are temperature dependent material parameters. Expression (3.17) describes the first order nematic-isotropic phase transition with respect to temperature T. In practice, B and C are assumed independent of temperature and only the lowest order material parameter A is taken as A(T ) = a(t T ) [6]. The values of a, B and C can be determined e.g. by fitting expression (3.17) with experimentally obtained data of order parameter variation with respect to temperature [22] (p. 250). Substituting Q in terms of the scalar order parameter S and the biaxiality parameter P as defined in (2.3) into (3.17) gives: f th = 3 4 AS BS CS4 + (A BS CS2 )P 2 + CP 4. (3.18) The value of S that minimises (3.18) at any given temperature is known as the equilibrium order parameter S 0, and is given by S 0 = B + B 2 24AC/(6C). The bulk energy, as written here, always favours uniaxiality, i.e. P 0 = 0. Higher order terms would be needed in order to describe a nematic LC with bulk biaxiality P 0 0 [54] or [6] p Such materials have recently been observed experimentally in [24]. 31

49 0 f b T>T c T=T c T=T * T<T * S Figure 3.2: Bulk energy as a function of order parameter for various temperatures The variation of the bulk energy density with respect to the uniaxial order parameter is plotted in figure (3.2) for various temperatures. Two critical temperatures can be identified, the clearing temperature T c and the nematic-isotropic transition temperature T. When the temperature is above T c, the isotropic state (S = 0) is energetically the most favourable. At T = T c, both the nematic and the isotropic states are possible. When T = T, the isotropic state becomes unstable, and at T < T, only the nematic state is stable External Interactions Additional effects, such as the aligning effect of external electric fields or solid surfaces can be included by introducing energy density terms accounting for them. For example, the effect of an external electric field can be expressed by writing the electric field energy, f f in the usual way for a dielectric material: f f = 1 2 D Ē = 1 2 ε 0 εē Ē + P Ē, (3.19) where Ē is the electric field and D is the dielectric displacement and P is a polarisation 32

50 vector as defined in (2.8) and (2.9). The permittivity tensor ε can be defined in terms of the director or the Q-tensor as in (2.6) and (2.7) respectively. A similar expression can be written for magnetic fields, where the magnetic susceptibility tensor χ replaces the dielectric tensor ε. In addition to interactions between external electric or magnetic fields, solid surfaces in contact with the LC material have an aligning effect on the director field. This effect, known as anchoring, can be either strong or weak. In the case of strong anchoring the surface energy density, f s, is assumed infinite and the director or Q- tensor is fixed at the surface. When the anchoring is weak, the surface energy density is some finite function involving the director or the Q-tensor. The effect of solid surfaces on the LC can be complex and the surface energy density is described in more detail in chapter Static Equilibrium Q-Tensor Fields In the continuum elastic theory explained in section 3.4, a free energy density f is written in terms of the Q-tensor and its spatial derivatives f = f(q ij, Q ij,k ). LC configurations resulting in minima in the total energy for the complete region of interest Ω are stable. These are the states to which the LC director field and order parameter distribution relaxes to in the limit of time (here, tens of milliseconds is enough in most cases of interest). The process of finding these states is a task of variational calculus, see e.g. [55, 31]. Stable LC configurations correspond to nulls of the first variation of the total energy with respect to the Q-tensor: δf = Ω [ f δq ij + f ] k δq ij dω = 0. (3.20) Q ij Q ij,k 33

51 Integrating the second term by parts: δf = Ω ( f f k )δq ij dω + Q ij Q ij,k Γ f ˆη k δq ij dγ = 0, (3.21) Q ij,k where ˆη is a unit vector normal to the bounding surface Γ. Since δq ij is an arbitrary variation, in order for (3.21) to be true, the following must be satisfied: f f k Q ij Q ij,k = 0 in Ω. (3.22) f ˆη k Q ij,k = 0 on Γ. (3.23) These are the Euler-Lagrange equations for the problem. Analytic solutions that satisfy the Euler-Lagrange equations are usually only possible in simplified cases, whereas in most cases numerical methods must be used. 3.6 Q-Tensor Hydrodynamics In the previous sections, only the orientation and order distribution of the LC material has been considered. However, since LCs are fluids, they flow and this needs to be taken into account for a more comprehensive description of the material. It is known that director re-orientation induces flow and similarly flow causes director reorientation. An example of this is the observed optical bounce [56] due to backflow in a twisted nematic cell after a holding voltage is removed. Probably the most successful theory describing the liquid crystal hydrodynamics is that by Ericksen and Leslie [7, 8]. This theory, commonly known as the Ericksen-Leslie (EL) theory describes the viscous behaviour of liquid crystals with six phenomenological coefficients (known as Leslie viscosities) α 1 α 6, but taking into account the Parodi relation α 2 + α 3 = α 6 α 5, only five of these are independent [57]. In general, the Leslie coefficients are not directly experimentally measurable, but can be 34

52 obtained from the four shear viscosities η 1, η 2, η 3 and η 12 and rotational viscosity γ 1 [58, 59]. Alternatively these can be estimated by means of molecular simulations or by interpolating from known viscous coefficients for other materials using knowledge of other material properties [60, 61]. The EL theory uses the vector description for the LC orientation, and does not take into account order parameter variations making it unsuitable for describing cases where topological defects are present. Other dynamic descriptions that do take into account variations in the LC order have been proposed in the past in e.g. the Beris- Edwards [62] and the Qian-Sheng formulations [9]. Both of these approaches yield qualitatively similar results [12, 63], but the Qian-Sheng equations reduce in the limit of constant uniaxial order to the EL theory allowing for direct mapping of viscous coefficients between the two theories. Because of this, the Qian-Sheng equations are chosen for this work. The hydrodynamic equations of LC materials can be derived by starting from the conservation of linear and angular momentum as is done with the EL theory and the original derivation of the Qian-Sheng formalism. However, more recently in [64, 65] it is argued that these assumptions are not strictly valid when the LC is described using the Q-tensor with variable order. Instead, a more general approach starting from principles of conservation of energy (but reducing to the same final equations) is proposed. Following the approach presented in [64, 65], the theoretical background of the Qian-Sheng equations is outlined in the following sections Conservation of Energy The basic idea is to balance the rate of change of energy against frictional losses in the form of a Rayleigh dissipation function [66]: δẇ + δr = 0, (3.24) 35

53 where Ẇ is the time rate of change of energy (power) and R is the dissipation accounting for frictional losses. In (3.24), variations with respect to the rate of change of the Q-tensor, Q, and the flow field, v, are taken ensuring minimum restrained dissipation. The total power of the system is the sum of the rate of change of the kinetic, T, and potential, F, energy of the system: Ẇ = T + F, (3.25) The equations of motion need to be frame invariant. This can be achieved by writing the dissipation in terms of the tensors Q, D and N. D and N are the symmetric velocity gradient tensor and the co-rotational time derivative respectively, and are related to the total flow gradient tensor v i,j as follows: v i,j = D ij + W ij, (3.26) where D ij = 1(v 2 i,j +v j,i ) is the symmetric and W ij = 1(v 2 i,j v j,i ) is the anti-symmetric (also known as the vorticity tensor) part of flow gradient tensor. N is a measure of the rotational rate of change of the Q-tensor with respect to the background flow field: N ij = Q ij + Q ik W kj W ik Q kj, (3.27) where Q is the total or material time derivative measuring the rate of change of Q in the flow field v, and is defined in the usual manner as: Q ij = t Q ij + v k Q ij,k. (3.28) 36

54 3.6.2 Frictional Forces The dissipation function R represents the effect of friction and can in the most general form be written as a power expansion of the tensors Q, D and N. Then, the total dissipation within a region Ω is given by: R = R(Q, N, D) dω. (3.29) Ω The variation of the dissipation with respect to Q and v then takes the form: δr = Ω ( ) R Q δ Q ij + R j δv i ij v i,j dω. (3.30) Integrating the second term by parts gives: δr = ( ) R Ω Q δ Q ij j ( R )δv i dω ij v i,j R + ˆη j δv i dγ. (3.31) v i,j Γ The derivatives in the volume integral are then evaluated using the chain rule of differentiation: R Q = R, ij N ij (3.32) and R = R N kl W ab + R D kl. (3.33) v i,j N kl W ab v i,j D kl v i,j Taking into account symmetries of the involved tensors, equation (3.31) simplifies to: δr = Ω [ ( R δ N Q ij + j ij Q ik R R ) ] Q kj δv i dω. (3.34) N kj N ik 37

55 The surface integral in (3.31) reduce to zero, since δv = 0 along Γ when boundary conditions for v are enforced The Power The total power of a sample of LC material equals the sum of the rate of change of the kinetic and potential energies: Ẇ = T + F, (3.35) where the kinetic energy is T = Ω 1 ρv 2 iv i dω and the potential energy F is the free energy of the LC material as defined in equation (3.1). In T, ρ is the the density of the LC material. The rate of change of the kinetic energy, after introducing the hydrostatic pressure p as a Lagrange multiplier to enforce incompressibility, v i,i = 0, of the LC material and integrating by parts is: T = (ρv i v i + j (pδ ij )v i ) dω v i pδ ij ˆη j dγ. (3.36) Ω Γ The time rate of change of potential energy is given by: ( F f = Q ij + f dq ij,k Ω Q ij Q ij,k dt ) dω. (3.37) Using the identity d dt Q ij,k = Q ij,k Q ij,l v l,k, and integrating by parts in (3.37) gives: F = [( ) ( f f k Q ij + k Ω Q ij Q ij,k [ f f + ˆη k Q ij,k ˆη k Q ij,l v l Q ij,k Q ij,k Γ f Q ij,l Q ij,k ) v l ] dω ] dγ. (3.38) 38

56 The first variation of the bulk power is then given by: δẇ = Ω [ ( ( ρ v i + j pδ ij + Q lk,i ( f + k Q ij f dq ij,k ) δ Q ij ] f Q lk,j )) δv i dω (3.39) Equations of Motion After the variations of the dissipation and rate of change of energy have been determined in equations (3.34) and (3.39) respectively, these can be substituted into the balance equation (3.24). The terms corresponding to δ Q and δv can be separated, giving the equations governing the time evolution of the Q-tensor: R = F F + k, (3.40) N ij Q ij Q ij,k and for the flow velocity field: ρ v i = j σ ji, (3.41) where σ is a generalised stress tensor: σ ji = pδ ji F Q kl,i + R R + Q jk R Q ki. (3.42) Q kl,j D ji N ki N jk Equation (3.41) is a generalisation of the Navier-Stokes equation governing the conservation of momentum. In (3.42), the second term containing gradients in the Q-tensor can be identified as the distortion stress tensor σ d : σ d ji = F Q kl,j Q kl,i. (3.43) 39

57 The final three terms in (3.42) correspond to the viscous stress tensor σ v : σji v = R R + Q jk R Q ki. (3.44) D ji N ki N jk The total stress tensor can then be written as: σ ji = pδ ji + σ d ji + σ v ji, (3.45) Choice of the Dissipation Function So far, the exact form of the dissipation function R has been undefined. All possible contributions to R, can be found by writing it as a power expansion in D, N and Q. Not all terms are necessary, and depending on the included terms different formulations (corresponding to special cases) of the LC-hydrodynamics can be obtained, as shown in [65]. It is assumed that the dissipation is quadratic in the velocity v, resulting in linear frictional forces. This limits R to consist of terms that are at most quadratic in D and N. Furthermore, limiting all terms to be at most quadratic in Q would result in a dissipaton containing 15 terms, each term containing a corresponding viscous coefficient. However, the Ericksen-Leslie equations, the most common way of characterising LC flow, contain only five independent viscous coefficients. It is then sufficient, to express R as the expansion: R = ζ 1 N ij N ij + ζ 2 D ij N ij + ζ 3 D ij D ij +ζ 4 D ij D ik Q kj + ζ 5 D ij Q ij D kl Q kl, (3.46) where ζ 1 to ζ 5 are scalar coefficients related to the EL-viscosities. Additional terms may be included in (3.46), but the contribution of these would only appear as adjustments of the values of the final viscous coefficients [65]. The 40

58 relation between the EL-viscosities and the coefficients ζ can be determined by replacing Q by its uniaxial definition Q ij = 1 2 S 0(3n i n j δ ij ) and comparing the resulting terms with the dissipation function in the EL theory [65] Explicit Expressions for the LC-Hydrodynamics After performing the steps outlined above and rearranging terms in the viscous tensor, the equations for the hydrodynamics can be written explicitly. The Qian-Sheng formalism governing the Q-tensor evolution is given by: µ 1 N ij = 1 2 µ 2D ij f f + k Q ij (Q ij,k ), (3.47) and the flow of the LC material is governed by: ρ v i = j σ ji, (3.48) where σ is as defined in (3.45), with the viscous stress tensor written as: σ v ij = β 1 Q ij Q kl D kl + β 4 D ij + β 5 Q ik D kj + β 6 Q jk D ki µ 2N ij µ 1 Q ik N kj + µ 1 Q jk N ki. (3.49) Additionally the incompressibility of the LC material should satisfy: v i,i = 0. (3.50) In expressions (3.47) and (3.49), β 1, β 4, β 5, β 6, µ 1 and µ 2 are viscous coefficients consisting of linear combinations of ζ 1 ζ 5. The values of the coefficients β and µ are 41

59 related to the six viscous coefficients α 1 to α 6 in the EL theory by [9]: µ 1 = 2 (α 9S0 2 3 α 2 ) µ 2 = 2 (α 6 α 5 ) 3S 0 β 1 = 4 α 9S0 2 1 (3.51) β 4 = 1 2 S 0(β 5 + β 6 ) + α 4 β 5 = 2 3S 0 α 5 β 6 = 2 3S 0 α 6 In cases when the effect of flow is not considered, (3.48) can be ignored and the Q-tensor evolution (3.47) simplifies to: µ 1 t Q ij = f f + k (3.52) Q ij Q ij,k 3.7 Discussion and Conclusions The theoretical background for the equations used in this work for describing the physics and modelling the operation of LC devices has been presented. A Landau-de Gennes free energy density taking into account elastic deformations and allowing for order parameter variations and biaxiality induced by externally applied electric fields and/or aligning solid surfaces is used. The elastic energy contribution reduces in the limit of constant uniaxial order to the well known Oseen-Frank elastic description of nematics with three independent elastic coefficients, allowing for realistic treatment of the LC elasticity. Similarly, the thermotropic energy contribution, the essence of the Landau-de Gennes approach, allows for localised order variations making a continuum description of defects possible. In regions of the LC material where order variations are allowed but are not 42

60 significant, the obtained results agree well with the Oseen-Frank theory as shown in section 7.2. The Landau-de Gennes theory is known to agree well with experimental observations of the nematic-isotropic phase transition. However, some criticism to its validity at temperatures far away from the transition temperature has been presented e.g. in [67] where the threshold electric field strength required for a topological transition in a π-cell is over-estimated by about a factor of two in the theoretical predictions as compared to experimental results. As a possible remedy for the discrepancy it is suggested that additional order parameters might be needed to describe more accurately the biaxial phase occurring at the centre of the cell during the switching. However, this is not possible with the current Q-tensor definition due to the number of independent degrees of freedom represented by it (five). Due to this limitation, the theory should not always be relied on producing quantitatively exact predictions of order variations and defects, especially at low temperatures. Nevertheless, the results obtained are useful e.g. in predicting general trends and as a qualitative description of defect structures and dynamics in many LC devices. The theory described in this chapter could be further extended by taking into account finite ion concentrations that may be present in some LC mixtures. This could be accomplished by introducing positive and negative charge densities whose distributions are governed by the drift-diffusion equations, coupled with the Poisson equation for the electric potential. 43

61 Chapter 4 Modelling of the Liquid Crystal Solid Surface Interface 44

62 4.1 Introduction Solid surfaces in contact with a LC break the symmetry of the nematic phase, resulting in a non-arbitrary orientation of the director field. This effect of interfaces imposing an orientation on the director is commonly known as anchoring. The operation of virtually all LC devices relies in some way on anchoring. In traditional display devices the solid surfaces are typically the glass plates between which the LC material is sandwiched. Other possible solid surface-lc interfaces include for example spacers used to keep the cell thickness constant throughout the device or colloidal particles immersed in the LC material for various applications, e.g. [1, 2]. The simplest way of including the effect of anchoring into a continuum model is by fixing the director or the Q-tensor at the interface. This is known as strong anchoring. Alternatively, the aligning effect can be included by introducing a surface anchoring energy density which is minimised when the director n is parallel to the anchoring direction ê (also known as the easy direction). In this case, known as weak anchoring, n or Q may vary at the surface with an associated change in energy. Sometimes weak anchoring gives a more realistic description of the aligning effect than strong anchoring and is an important feature to be included in an LC device model. This is especially true in the case of very small structures where torques on the director due to high electric fields or elastic forces may become comparable to even the high (but in reality finite) anchoring energies. In the Oseen-Frank theory, it has become standard practise to include the effect of weak anchoring by making the well known Rapini-Papoular (RP) assumption [15] or some generalisation of it (see section 4.3.1). In the Landau-de Gennes theory, however, although the fundamental physics of the surface interface has been examined, the anchoring phenomenon has received less attention from the LC device modelling point of view. 45

63 The purpose of this chapter is to study the modelling of weak anchoring in LC devices using the Landau-de Gennes theory. First, in section 4.2 various anchoring types are introduced, physical reasons for the aligning effect of various solid surfaces are given and methods for measuring the anchoring strength are presented. In sections and 4.3.2, surface energy densities in the Oseen-Frank and Landau-de Gennes theories respectively are reviewed. Then, starting from section 4.4 new work is presented. A general power expansion on the Q-tensor and two unit vectors describing the local geometry of the surface in contact with the LC material is proposed to represent the surface energy density. It is shown that in the limit of constant uniaxial order, the proposed expression reduces to a well known anisotropic generalisation of the RP expression by Zhao, Wu and Iwamoto [17, 18], developed in the Oseen-Frank framework. In this limit, experimentally measurable values with a physical meaning in the Oseen-Frank theory can be scaled and assigned to the scalar coefficients of the Q-tensor expansion. The validity of this assumption is examined by comparing results of numerical experiments using both theories. 4.2 Background Classification of Different Anchoring Types Different anchoring types can be classified depending on the orientation of the easy direction with respect to the aligning surface. When the easy direction is in the plane of the surface the anchoring is said to be planar. Planar anchoring can be homogeneous or degenerate. In the case of homogeneous planar anchoring only a single easy direction exists. In the case of degenerate planar anchoring all directions in the plane are equal and the director field may rotate in the plane. It is also possible that the easy direction is not in the plane of the surface. That is, a pre-tilt exists. In the degenerate case, the result is conical degenerate anchoring. When the 46

64 easy direction is perpendicular to the surface, the anchoring is called homeotropic. Finally, it is possible that two or more minima or easy directions exist. In this case, the anchoring is termed bi- or multi-stable The extent of the aligning effect, or strength of the anchoring, varies depending on the properties of the surface and specific LC compound in contact with it [68, 69]. When the director at the surface is rigidly fixed to the easy direction the anchoring is said to be strong, corresponding to an infinite anchoring energy. In the case of weak anchoring the anchoring energy density is some finite function of the director orientation at the surface. It is common to make the RP assumption that the anchoring energy density is of the form W sin 2 Θ, where Θ is the angle between n and ê and W is an experimentally measurable anchoring strength coefficient of dimensions J/m 2. In reality weak anchoring is often anisotropic; more specifically, the anchoring tends to be stronger in the polar (departing from the surface) rather than in the azimuthal direction (on the surface) [70]. For example, in the case of planar homogeneous anchoring, reported polar anchoring strengths typically lie in the range from 10 7 to 10 3 J/m 2, whereas azimuthal anchoring strengths are typically one or two orders of magnitude smaller [71, 72, 73, 74]. For this reason, various generalisations that take into account the difference between polar and azimuthal anchoring strengths have been introduced in the Oseen-Frank theory by several authors [17, 18, 16, 75] Anchoring Mechanisms Although anchoring is essential to the operation of almost all LC devices, the exact mechanisms responsible for it are still not fully understood [76, 77]. A number of complex chemical/physical processes occurring at the surface that are thought to contribute to anchoring are briefly summarised next. One popular explanation for the anchoring phenomenon is small scale grooves on the solid surface. The LC molecules at the interface then tend to orient themselves 47

65 along the grooves in order to minimise the elastic distortion energy, resulting in planar homogeneous anchoring. Based on this argument, Berreman [78] has proposed an expression relating the width and separation of the grooves and the bulk elastic constants to the anchoring strength. For example, rubbing of polyimide or oblique evaporation of inorganic compounds produce grooved or rough surfaces favouring planar alignment [6, 79]. It is also argued that the rubbing process orients the polyimide chains in one direction, along which the LC molecules then align. If the surface is covered with a film of a surfactant consisting of aliphatic chains oriented perpendicular to the surface, the LC molecules at the interface may partially penetrate the chains and adopt their orientation. This method can be used to produce surfaces with homeotropic anchoring [80, 81]. It is also suggested e.g in [82, 83, 31] that surface electric fields due to the presence of ions or the so-called ordoelectric polarisation can have an effect on the strength of the anchoring and the orientation of the easy direction ê. Also, non-structured interfaces (not necessarily with a solid surface) have an aligning effect on the LC (see e.g. [74] and references therein). In this case, changes in the properties of the LC material in a thin region (in the order of nanometres) near the surface are responsible for the alignment. This includes changes in the density of the LC, gradients in the order parameter and monolayers of smectic phases. By geometric arguments, a non-structured or isotropic solid surface should produce planar degenerate anchoring with zero azimuthal anchoring strength. It has been shown long ago that this is not necessarily the case [84]. Two different phenomena have been reported to be responsible for a finite azimuthal anchoring strength in LC cells with untreated surfaces: These are the flow [77, 81] and the memory [85, 86, 87] alignment. The flow alignment occurs when a cell is filled with an LC material in the nematic phase. In this case the alignment tends to be in the filling direction. The flow 48

66 alignment mechanism is also known to affect surfaces which are treated to give a particular orientation of the easy axis, especially when the anchoring is weak [77]. If the cell is filled at an elevated temperature, with the LC in the isotropic phase, the effect of the memory alignment can be observed. After the cell is cooled to temperatures in which the LC is in the nematic phase, the resulting director field will give rise to a Schlieren texture of alternating dark and light regions between crossed polarisers [6]. This means that the director field varies slowly in a random fashion in the plane of the cell. External fields can be used to orient the director field, but the original pattern will re-appear after the removal of the fields indicating a nonzero azimuthal anchoring strength. When the applied field is strong enough the orientation of the easy axis may change (known as surface gliding), so that the anchoring becomes dependent on the past history of the cell. It has been suggested that adsorption of the LC molecules at the surfaces and anisotropic interactions between polymer molecules at the surfaces and the LC material are responsible for the memory effect [86, 87]. However, in [86] these two mechanisms have been eliminated by surface passivation by trimethoxysilane (3- glycidoxypropyl), resulting in truly planar degenerate anchoring. Because of the complex nature of the exact underlying physics and chemistry of the LC solid surface interfaces it is often not feasible to attempt to include all of this in a macroscopic model due to the associated computational cost. Instead, a phenomenological approach describing the effect the surface has on the LC material can be more useful in device modelling Experimental Measurement of Anchoring Strengths It is of great practical importance to be able to measure the strength of anchoring of an aligning surface. This can be achieved in various ways, but in general it involves observing the orientation of n under the action of a distorting torque of a known mag- 49

67 nitude. From this information it is then possible to estimate the anchoring strength by fitting parameters to a model. The torque can be generated either by applying external electric/magnetic fields (field on techniques) [88, 89] or by a distortion in the director field due to the chosen geometry of the test cell used in the measurement (field off techniques). Perhaps the simplest (field off) technique is the torque balance method [90, 91] which relies on calculating the elastic torque energy in a twisted nematic cell of thickness d and with a known total twist angle φ t between the anchoring directions on both surfaces. The distortion in the bulk produces an elastic torque that causes the director at the surfaces to deviate from the easy directions on both surfaces by angles φ which can be found experimentally e.g. by measuring the retardation of polarised light transmitted through the cell. In the case of zero tilt, the twist angle varies linearly through the cell, see figure 4.1. The total energy is then a sum of the bulk distortion energy and the anchoring energies: F tot = F d + 2F s. (4.1) The total bulk twist distortion energy F d in a cell with φ = φ t 2 φ radians of twist is: F d = K 22 2d φ2. (4.2) The surface energy F s at each interface is taken as the RP anchoring energy: F s = W φ sin 2 φ. (4.3) In (4.2) and (4.3) W φ and K 22 are the azimuthal anchoring energy strength and the twist elastic constant respectively. By minimising equation (4.1), with respect to φ 50

68 (and using 2 cos φ sin φ = sin 2 φ), a balance between the two opposing torques and the resulting value of W φ can be found: W φ = K 22φ d sin 2 φ, (4.4) Figure 4.1: Twist angle in a cell of thickness d. Dashed line, strong anchoring. Solid line, weak anchoring. 4.3 Review of Currently Used Weak Anchoring Expressions Weak Anchoring in Oseen-Frank Theory Probably the first and best known expression describing the weak anchoring effect in the Oseen-Frank theory is the Rapini-Papoular (RP) expression [15]. This assumes that the anchoring energy density increases in a sin 2 fashion as the director deviates 51

69 from the easy direction: F RP = W sin 2 (Θ), (4.5) where W is a scalar value known as the anchoring strength, and Θ is the angle of departure of the director n from the easy direction ê. Alternatively, this can be written as: F RP = W ( n ê) 2. (4.6) One weakness of (4.5), is its inability to distinguish between different directions of angular departures from ê. This means that the difference between polar and azimuthal anchoring strengths cannot be taken into account. Furthermore, it has been suggested that higher order terms (e.g. terms in sin 4 Θ) should be taken into account when Θ is large [92]. Despite this, the RP anchoring is a widely used approximation and often used as a reference to which other anchoring representations are compared. Various generalisations to 4.5 exist. One that differentiates between polar and azimuthal anchoring strengths is (e.g. [16]): F RP gen = A 1 sin 2 (θ θ e ) + A 2 sin 2 (φ φ e ), (4.7) where A 1 and A 2 refer to polar and azimuthal anchoring strengths and θ, φ, θ e and φ e to the tilt and azimuthal angles of the director and easy direction, respectively. However, this approach completely decouples the two angles in an unrealistic way giving rise to complications: Firstly, the decoupling of the two angles makes the anchoring energy density discontinuous with respect to θ and φ [18]. Secondly, the azimuthal anchoring energy density should also depend on the tilt angle of the director and this effect is not included. Furthermore, expression (4.7) is periodic with a period 52

70 of π radians, resulting in a bistable anchoring when the tilt angle of the easy direction lies in the range 0 < θ e < π/2. It has later been shown by Zhao, Wu and Iwamoto [17, 18], that a representation of the anisotropic surface energy density without the complications outlined above is: F ZW I = B 1 sin 2 (Θ) cos 2 (Φ Ψ 0 ) +B 2 sin 2 (Θ) sin 2 (Φ Ψ 0 ), (4.8) where (Θ, Φ) are angular deviations of the director from ê in a local coordinate system defined by the orthonormal vector triplet (ˆv 1, ˆv 2, ê) describing the principal axes of anchoring. Equation (4.8) can also be expressed more compactly as [17]: F ZW I = B 1 (ˆv 1 n) 2 + B 2 (ˆv 2 n) 2 (4.9) where B 1 and B 2 are anchoring strength coefficients corresponding to deformations in the (ˆv 1, ê) and (ˆv 2, ê) planes respectively Weak Anchoring in the Landau-de Gennes Theory In the Landau-de Gennes theory the anchoring energy density is written as a function of the Q-tensor. This means that order variations also affect the surface energy. Perhaps the simplest way of approximating the anchoring effect of an aligning surface is by means of a penalty type expression [70, 93]: F pen = W Tr ( (Q Q 0 ) 2), (4.10) where Q 0 is the preferred easy Q-tensor. Clearly the energy density is minimised when Q = Q 0. Expression (4.10) shows a sin 2 variation with respect to angular 53

71 departures from the easy direction [93]. However, similarly to the original Rapini- Papoular expression (4.5), the penalty anchoring does not distinguish between polar and azimuthal anchoring strengths. Another expression for the surface energy density in the Landau de-gennes theory describes the effect of an isotropic surface on a LC material, i.e. a surface giving degenerate alignment, where only the director tilt is constrained. This is a Landau power series expansion on the surface normal unit vector ˆv and Q [94]: F exp = c 1 (ˆv Q ˆv) + c 2 Tr(Q 2 ) + c 3 (ˆv Q ˆv) 2 + c 4 (ˆv Q 2 ˆv). (4.11) Here, c i are scalar coefficients that determine the preferred tilt angle and surface order. Expression 4.11 has been used e.g. in [95, 96] to study anchoring transitions. Slow convergence (order of 1000 Newton iterations) of numerical schemes with (4.11) as a surface energy term has been reported in [95], making the expression computationally too expensive for the modelling of device dynamics. An expression for anisotropic anchoring, linear in Q, has been studied in [97]: f s = Tr(H Q), (4.12) where H is a symmetric traceless tensor describing the symmetry of the surface. However, since this expression is linear there is no control over the surface order parameter which tends to either positive or negative infinity depending on the exact form of H and the anchoring strength. 54

72 4.4 The Anchoring Energy Density of an Anisotropic Surface in the Landau-de Gennes Theory A generalisation with a reduction in symmetry as compared to (4.11), that can be written as a power expansion truncated to 2 nd order on the Q-tensor and two orthogonal unit vectors whose directions are determined by the surface treatment is presented here: F s = a s Tr(Q 2 ) + + W 1 (ˆv 1 Q ˆv 1 ) + W 2 (ˆv 2 Q ˆv 2 ) + W 3 (ˆv 1 Q ˆv 2 ) + X 1 (ˆv 1 Q ˆv 1 ) 2 + X 2 (ˆv 2 Q ˆv 2 ) 2 + X 3 (ˆv 1 Q ˆv 2 ) 2 + X 4 (ˆv 1 Q 2 ˆv 1 ) + X 5 (ˆv 2 Q 2 ˆv 2 ) + X 6 (ˆv 1 Q 2 ˆv 2 ) + X 7 (ˆv 1 Q ˆv 1 )(ˆv 2 Q ˆv 2 ) + X 8 (ˆv 1 Q ˆv 2 )(ˆv 1 Q ˆv 2 ) + X 9 (ˆv 1 Q ˆv 2 )(ˆv 1 Q ˆv 1 ) + X 10 (ˆv 1 Q ˆv 2 )(ˆv 2 Q ˆv 2 ), (4.13) where W i and X i are anchoring strength coefficients. The simplest case that still allows for anisotropic anchoring with a preferred order parameter is when the scalar coefficients W 3 and X i are zero. In this case the surface anchoring energy reduces to: F s = a s Tr(Q 2 ) + W 1 (ˆv 1 Q ˆv 1 ) + W 2 (ˆv 2 Q ˆv 2 ). (4.14) The principal axes of anchoring (ê, ˆv 1, ˆv 2 ) are the easy direction and two mutually orthogonal unit vectors respectively, so that ê = ˆv 1 ˆv 2. Equation (4.14) can be directly discretized for implementation, but is here expanded in an analytical form in order to show how meaningful values can be assigned to the scalar coefficients a s, W 1 and W 2. ê is the easy direction only when both W 1 and W 2 are positive scalars. If W i = 0 and W j > 0, the anchoring becomes degenerate in the (ê, ˆv i )-plane. Setting 55

73 W 1 or W 2 to a negative value minimises F s in the direction of ˆv 1 or ˆv 2, and ê loses its physical meaning as the easy direction. The types of anchoring achieved by using negative coefficients are equivalent to a rotation of the principal axes when using positive W 1 and W 2. For this reason, only cases of non-negative anchoring strength coefficients are considered in what follows. Without loss of generality, the geometry can be defined locally: (ê, ˆv 1, ˆv 2 ) are chosen to coincide with the (x, y, z) coordinates. The traceless Q-tensor, when including biaxiality of LCs, is written as: Q ij = S 2 (3n in j δ ij ) + P (k i k j l i l j ), (4.15) where S is the scalar order parameter and P the biaxiality parameter. n, k and l, are the director and two vectors that define the direction of nematic order in three dimensions and δ ij is the Kronecker delta. The three orthogonal unit vectors n, k and l can be written in terms of the three angles α, β and γ, where α is the angular deviation of n from the (ê, ˆv 2 ) plane (local twist), β is the angular deviation of n from the (ê, ˆv 1 ) plane (local tilt) and γ is a rotation of k and l around n determining the orientation of the plane of biaxial order: n = cos(α) cos(β) sin(α) cos(β) sin(β), (4.16) k = sin(α) cos(γ) + cos(α) sin(β) sin(γ) cos(α) cos(γ) sin(α) sin(β) sin(γ) cos(β) sin(γ), (4.17) 56

74 l = sin(α) sin(γ) cos(α) sin(β) cos(γ) cos(α) sin(γ) + sin(α) sin(β) cos(γ) cos(β) cos(γ), (4.18) so that when α = β = γ = 0, ( n, k, l) = (ê, ˆv 1, ˆv 2 ). Equation (4.14) can then be written in terms of S, P, α, β and γ as: F s = ( ) 3 a s 2 S2 + 2P 2 + W 1 {F 1S (S, α, β) + F 1P (P, α, β, γ)} + W 2 {F 2S (S, β) + F 2P (P, β, γ)}, (4.19) where F 1S and F 2S are: F 1S (S, α, β) = S 2 ( 3 sin 2 α cos 2 β 1 ) = 3S 2 ( n ˆv 1) 2 S 2, (4.20) and F 2S (S, β) = S 2 ( 3 sin 2 β 1 ) = 3S 2 ( n ˆv 2) 2 S 2. (4.21) In the limit of constant uniaxial order F 1P, F 2P and the isotropic part of F s are constants and can be ignored, and (4.14) reduces to the sum of (4.20) and (4.21) multiplied by W 1 and W 2 respectively. In this case (4.14) is equivalent to expression (4.9) of Zhao et al., with anchoring strength coefficients related by a factor of 3S/2. Figure 4.2 shows the angular variation of the anchoring energy density for different values of the polar to azimuthal anchoring ratio, R = W 2 /W 1 when order variations 57

75 (a) (b) (c) (d) Figure 4.2: Normalised anisotropic parts of the anchoring energy density for a surface with ê = [1, 0, 0], ˆv 1 = [0, 1, 0] and ˆv 2 = [0, 0, 1]. (a) R = 1. (b) R = 3. (c) R = 0. (d) R =. (R = W 2 /W 1 ) are not considered Determining Values for the Anchoring Energy Coefficients Without the simplification of constant uniaxial order, the preferred surface order and biaxiality parameters S e and P e, that minimise (4.14), are determined by the relative values of W 1, W 2 and a s. The two constants W 1 and W 2 define the anisotropic azimuthal and polar anchoring strengths and the value of a s determines the resulting 58

76 easy surface order. The preferred surface order occurs when n = ê, i.e. α = β = 0. Equation (4.19) then simplifies to: F s = a s ( 3 2 S2 + 2P 2 ) + W 1 { (2 cos 2 γ 1 ) P 1 2 S } + W 2 { ( 2 cos 2 γ + 1 ) P 1 2 S }. (4.22) The value of a s which minimises F s for a given value of the surface order parameter, S e, can be found by minimising (4.22) w.r.t. S, giving: a s = W 1 + W 2 6S e. (4.23) The resulting biaxiality parameter distribution as function of γ in the plane of l and k is found in a similar fashion by minimising (4.22) with respect to P and substituting a s from (4.23) giving: P e = 1 R { 1 2 cos 2 γ } R 2 S e. (4.24) Alternatively, in terms of the three eigenvalues of Q, expression (4.14) is minimised when the eigenvalue in the direction of ê is λê = S e, and the difference between the two remaining eigenvalues is λˆv1 λˆv2 = 2P e. Figure 4.3 shows the three eigenvalues of a Q that minimises the surface energy density of (4.14) as a function of R, normalised for S e = 1. Two cases can be identified from the figure. 1. R = 1, the two anchoring strength coefficients are equal, (W 1 = W 2 ) and λˆv1 = λˆv2 = λê/2, so that Q at the surface is uniaxial with a positive order parameter S = λê = S e and n = ê. 2. R < 1 or R > 1, the two anchoring strength coefficients are not equal. As R varies from 1 to 0 or from 1 to, the surface order undergoes a transition from a 59

77 Eigenvalues λ ê λ 1 λ ˆv ˆv R Figure 4.3: Eigenvalues of a Q-tensor that minimises the surface energy density as a function of R, when S e is unity. positive uniaxial order to a negative uniaxial order through a biaxial state. In the limits of R = 0 or R =, when either W 1 or W 2 is zero, the anchoring is planar degenerate with a uniaxial negative scalar order parameter of value S = 2S e, with n parallel to the unit vector corresponding to the non-zero anchoring coefficient. However, a more complete description of the surface order needs to include the bulk energy density terms, which in the standard Landau-de Gennes theory for nematic liquid crystals favour a uniaxial Q-tensor with a positive scalar order parameter S = S 0. The resulting Q at the surface then describes a state that minimises the combination of the surface and bulk terms. Figures 4.4 and 4.5 show the calculated variation in order for various anchoring conditions when the bulk thermotropic coefficients for the 5CB liquid crystal (see appendix A) are used with the single elastic coefficient approximation and K = 5pN. When the anchoring energy is low the bulk terms dominate and Q at the surface is close to the bulk equilibrium value for all R. Figures 4.4a and 4.4b show the variation of the order parameter (S = λ e ) and biaxiality parameter (P = (λˆv1 λˆv2 )/2) with R and the distance to the surface when W J/m 2. A small degree of biaxial order is induced at the surface when R > 1, resulting in a decrease in S. The variations in order are contained within about a ten nanometre thick transition region 60

78 S P R Dist R Dist (a) (b) Figure 4.4: (a) Scalar order parameter S and (b) biaxiality parameter P as functions of the distance from the surface (in µm) and the ratio R between W 2 and W 1. near the surface. Figure 4.5 shows the eigenvalues of Q at the surface, normalised by S 0, as functions of W 2 for R = 1, 3 and. For comparison, the eigenvalues corresponding to a linear surface energy density (a = 0) when R = 1 are also shown (marked with circles). The influence of increased anchoring strengths can be observed in the eigenvalues. The surface energy becomes comparable to the bulk energy in the region around W 2 = 10 3 to 10 1 J/m 2, where a reduction in λ e can be observed. As the anchoring strength is further increased, the surface anchoring becomes the dominant energy term, and the eigenvalues converge towards those that minimise the surface energy as shown in figure Numerical Results Results of numerical simulations using the weak anchoring expression of (4.14) are presented next. First, results of simulations of the switching of a twisted nematic cell using the Landau-de Gennes and the Oseen-Frank theories with weak anchoring are shown. Then, the effect of anchoring induced biaxiality and order variations on the effective anchoring strength is investigated in the Landau-de Gennes theory. The numerical simulations are performed using the finite elements discretisation 61

79 R Eigenvalues R 1 R 1.5 λ e λ 1 λ W 2 [J/m 2 ] Figure 4.5: Normalised eigenvalues of Q at the surface as a function of W 2 for R = 1, 3 and, when a is set according to expression 4.23 (no markers) and for the linear case a s = 0 and R = 1 (circles). of the Landau-de Gennes theory described in chapter 5 and a previously developed finite elements implementation of the Oseen-Frank theory [98]. In both cases the weak surface anchoring energy densities are modelled by (4.14) and (4.9) respectively. The simulations are performed using a finite elements mesh of dimensions µm., with periodic x and y side boundaries. In practice this is equivalent to a one dimensional case. The values of the thermotropic energy coefficients are for 5CB in both cases (see Appendix A), with (T T ) = 4 giving an equilibrium order parameter S Comparison between the Landau-de Gennes and Oseen- Frank Models Two cases are considered for the comparison between the Oseen-Frank and Landaude Gennes models. First, the switching of a twisted nematic cell (with 90 twist throughout and 5 pre-tilt) is compared for a constant ratio of the polar and azimuthal anchoring strengths, with R = 3, as a function of the applied voltage. Both the midplane and surface tilt, and the surface twist angles are obtained using both theories and plotted in figures 4.6a 4.6c. Then, a constant 1.5 V is applied, but R is varied 62

80 from 1 to Again, the mid-plane and surface tilts and the surface twist angles are recorded and plotted in figures 4.6d 4.6f. In both cases the polar anchoring strengths are kept constant at B 2 = J/m 2 and W 2 = 2B 2 /(3S 0 ), whereas the azimuthal anchoring strengths are set as B 1 = B 2 /R and W 1 = W 2 /R. Furthermore, in the Landau-de Gennes theory, the isotropic surface energy density coefficient a is determined by equation (4.23), assuming S e = S 0. Values for the three elastic coefficients and dielectric anisotropy for the 5CB liquid crystal are used (see. appendix A). The two simulations yield slightly different results, but this is to be expected since the Zhao et al. expression does not allow for order variations occurring both at the surfaces due to the anchoring and close to the surfaces where the director field undergoes rapid distortions due to the electric field Effect of Order Variations on the Effective Anchoring Strength In section 4.4.1, a proportionality relationship with a factor of 3S/2 between the anchoring strength coefficients W i of (4.14) and B i of (4.9) was established in the limit of constant uniaxial order. However, when R 1 this assumption is not true implying that the anchoring energy density will be different from (4.9), and the actual effective anchoring strength, W eff, acting on the director will differ from the expected value of W i used in expression (4.14). In order to investigate this, the torque balance [90, 91] method described earlier in section is used in conjunction with modelling results of the Q-tensor distribution [99] to find W eff acting on the director. The azimuthal anchoring strength is found by considering a twisted cell with zero tilt (90 twist, 0 tilt). The polar anchoring strength is found by considering a cell with equal but opposite amount of pre-tilt on both surfaces (±45 tilt) without twist. The latter configuration produces a constant splay deformation through the cell. 63

81 Mid plane tilt angle, degrees LdG OF Voltage (a) Surface tilt angle, degrees LdG OF Voltage (b) Surface twist angle, degrees LdG OF Surface tilt angle, degrees LdG OF Voltage (c) Polar to azimuthal anchoring strength ratio, R. (d) Mid plane tilt angle, degrees LdG OF Surface twist angle, degrees LdG OF Polar to azimuthal anchoring strength ratio, R. (e) Polar to azimuthal anchoring strength ratio, R. (f) Figure 4.6: (a) (c) Tilt and twist angles as a function of V, with a constant R = 1 3. (d) (f) Tilt and twist angles as a function of R, with a constant applied voltage V =

82 Equation (4.3) is modified by including the proportionality factor of 3S 0 /2 as explained in section 4.4, to give the effective azimuthal and polar anchoring strengths: W 1eff = 2Kφ 3S 0 d sin(2 φ), (4.25) and W 2eff = 2Kθ 3S 0 d sin(2 θ). (4.26) In (4.25) and (4.26) φ, θ, φ, θ and d have the same meaning as defined earlier in section A single elastic coefficient approximation K = K 11 = K 22 = K 33 = 7pN. is used in both cases. The thermotropic coefficients for 5CB (see Appendix A) are used. For both cells, starting with values of W 1 and W 2, the distribution of Q over the complete cell can be found by modelling using the Landau-de Gennes theory. Then, using expressions (4.25) and (4.26), the effective anchoring strength coefficients are calculated from the director profile obtained from the tensor field. The ratio between W ieff and W i is plotted in figure 4.7. The azimuthal and polar anchoring strengths were set as W 1 = J/m 2, W 2 = W 1 /R for R > 1 and W 2 = J/m 2, W 1 = W 2 R for R < 1. When R is close to 1 and the order at the surfaces is uniaxial a good agreement between W i and W ieff is found. As R departs from 1, the effective anchoring strength in the plane of increased biaxial order is reduced, whereas anchoring to the same plane is increased. That is, when R < 1, W 2eff < W 2 and when R > 1, W 1eff < W 1. It is then possible to define an effective anchoring anisotropy, R eff = W 2eff /W 1eff, which is greater than R when R > 1 and smaller than R, when R < 1. In general, the difference between W ieff and W i depends on the degree of surface biaxiality and order parameter variation, so that the effective anchoring strength is a function of 65

83 1.1 Relative Anchoring Strength W 2eff /W 2 W 1eff /W R Figure 4.7: Ratio of the effective azimuthal anchoring strength coefficient and W 1 as a function of R both bulk and surface terms. 4.6 Discussion and Conclusions A power series expansion in terms of the Q-tensor and two mutually orthogonal unit vectors has been used to describe the anchoring energy density at the interface between a solid surface and a liquid crystal in the Landau-de Gennes theory. This expression allows for practical and flexible modelling of various weak anchoring types, ranging from isotropic through anisotropic to degenerate anchoring. The lower order terms of the expansion have been considered, resulting in a simple expression with three coefficients, which in the limit of constant uniaxial order reduces to the well-known anisotropic generalisation of the Rapini-Papoular anchoring expression of Zhao, Wu and Iwamoto [17, 18]. This allows the assignment of numerical values with a physical meaning to the scalar coefficients of the expression. Both the polar and azimuthal anchoring strengths can be independently defined, as well as the value of the easy surface order parameter. Inclusion of higher order terms may allow for an improved description of variations in order or the anchoring energy when the angle between the director and the easy direction is large, but this would introduce the disadvantage of added coefficients 66

84 (material parameters) whose values need to be known. Furthermore, simulations using higher order expansions dramatically reduced the convergence of the numerical scheme used. This is in accordance with [95], where it is reported that typically more than thousand iterations of the Newton-Raphson method were needed to achieve convergence using equation (4.11) as a surface term in a Landau-de Gennes model. On the contrary, here, using the lower order terms of expression (4.14), the rate of convergence of the numerical scheme is practically unaffected by including the surface energy term to the model as compared to strong anchoring conditions where the Q- tensor is simply fixed at the surfaces. Results of numerical simulations of the switching characteristics of a twisted test cell under various anchoring conditions and applied electric fields, using a finite element discretisation of (4.14) in the Landau-de Gennes theory, compare well with those using (4.9) in the Oseen-Frank model. The resultant tilt and twist angles differ typically by less than 2 and this can be explained by the fact that biaxiality and order variations are not considered in the Oseen-Frank formulation. The effect of varying the anisotropy of the anchoring and the magnitude of the anchoring strength are also investigated. As the anisotropy of the surface anchoring is increased from R = 1 to R =, the surface order undergoes a transition from a uniaxial positive ordering to a uniaxial negative order through a state of biaxial order (see figure 4.3). The tendency for this to happen depends on the relative magnitudes of the anchoring energy and the thermotropic energy of the Landau-de Gennes theory, which favours a positive uniaxial order (see figure 4.5). It was found by applying the torque balance method to results of simulations that the anchoring induced order variations at the surfaces also change the effective anchoring strengths. As the surface becomes biaxial, the effective anchoring strength is increased to the plane of biaxial order and decreased in the same plane. In other words, surface biaxiality induced by the anisotropy of the anchoring energy density 67

85 further increases the anisotropy of the anchoring, so that R eff > R. The practical implications of this in a simulation is that the ratio of W 2 and W 1 can be underestimated to achieve a desired effective anchoring anisotropy. However, in order to do this accurately it may be necessary to measure the effective values of the anchoring strengths (e.g. by simulating the torque balance method, as done here) since these also depend on the properties of the bulk thermotropic energy. 68

86 Chapter 5 Finite Elements Implementation 69

87 5.1 Introduction In this chapter, the methods used in this work for obtaining a numerical solution to the coupled equations governing the liquid crystal physics and the electrostatic potential are presented. The equations to be solved are partial differential equations (PDE) that must in practice be solved numerically due to the complexity of the problem. A number of different methods for solving PDEs on a computer exist, e.g. the finite differences, finite volumes, finite elements and various mesh free methods. The finite elements method is chosen for this work for three reasons: 1. Complex geometries pose no problems for the method. 2. Unstructured meshes allow for local refinement of the spatial discretisation making accurate three dimensional modelling of LC devices with defects computationally feasible. 3. Implementation of boundary conditions is efficient and relatively straightforward. Broadly speaking, two different situations are considered: The solution sought describes either the LC dynamics or the steady state. The dynamic case describes the time evolution of the LC orientation and order distribution. This can be used e.g. for describing the switching between on and off states of a pixel in a LC display device. The dynamic behaviour is found by repeatedly solving the equations (3.47) and (3.48), giving the time rate of change of the Q-tensor and updating it accordingly. The steady state situation describes the static LC orientation and order distribution when time. This solution corresponds to the case when the Euler-Lagrange equations (3.22) and (3.23) are satisfied. It is possible to obtain this solution by simply performing a sufficiently long dynamic simulation (in practice, it is not necessary to simulate until time, but some tens of milliseconds usually suffice). However, other computationally more efficient methods can be used for solving the Euler-Lagrange equations in cases where only the final LC configuration is of interest. 70

88 5.2 The Finite Element Method In the finite differences method, variables and their spatial derivatives are represented by interpolation of values on a (usually regular) grid. These can then be directly substituted into the PDEs that are to be solved. This is known as the strong solution. In the finite elements method, however, an indirect approach of seeking a solution satisfying some conditions which simultaneously satisfy the original problem is taken. The solution obtained in this way is known as the weak solution (but despite its name it is by no means less correct). In order to obtain the weak solution, the strong form of the problem (the PDEs) must be re-written in a weak form. Two commonly used methods for obtaining the weak formulations of a problem are the weighted residuals method and the variational method. When the problem is self-adjoint, the two approaches result in identical FE formulations. Before describing the procedure of obtaining a weak formulation, some definitions that are needed in the process are presented. The Boundary Value Problem In general, the problem that is to be solved using the FE method is defined within a region Ω with boundaries Γ. This can be written in terms of PDEs as: Lu(x) = s(x) in Ω (5.1) Bu(x) = t(x) on Γ (5.2) where L and B are linear operators, u(x) is the unknown sought function of spatial coordinate x and s(x) and t(x) are some known functions. Boundary conditions (5.2) must be imposed in order for a unique solution to exist. Different types of boundary terms exist, e.g. B = 1 results in fixed or Dirichlet 71

89 boundaries, where the value of u is known on Γ, and B = ˆη in Neumann boundaries where the gradient of u normal to the boundary is known. Inner Product The inner product of two functions f(x) and g(x) is defined as: f, g = f(x)g(x) dx. (5.3) When f, g = 0 for any and all choices of g, it must follow that f = 0. This property is used later in the FE formulation to minimise an error residual. Spatial Discretisation The finite element method is a technique for obtaining a numerical approximation to some unknown function u(x). The exact function is approximated by forming the expansion: u(x) ũ(x) = n u j b j (x), (5.4) j=1 where b j (x) are known basis functions (e.g. sinusoidals or polynomials) and u j are scalar coefficients. The task of trying to find the exact function u(x) in an infinite dimensional search space is then reduced to calculating n discrete values that produce the best approximation of the solution. This process is explained sections and The accuracy of the approximation depends on the form of the chosen basis functions b j (x), and the number of terms used in the expansion. In general, as n, ũ(x) u(x). 72

90 5.2.1 Weighted Residuals Method The weighted residuals method is a systematic method for obtaining the weak form of PDEs. Starting from the general PDE given in (5.1), an error residual r(x) can be defined as: r(x) = Lu(x) s(x). (5.5) The task is then to find u(x), such that the error is zero everywhere. This is equivalent to requiring that the inner product between r(x) and any possible test function h(x) vanishes, i.e.: r(x), h(x) = Lu(x) s(x), h(x) = 0. (5.6) The test function h can now be approximated by the expansion: h(x) = n c i w(x), (5.7) i=1 and substituted into (5.6) giving: r(x), h(x) = n c i r(x), w i (x) = 0. (5.8) i=1 Since 5.6 has to be satisfied for any h, it is sufficient to write r(x), h(x) = r(x), w i (x) = 0 for i = 1...n (5.9) 73

91 Similarly, expanding u(x) in terms of basis functions gives: r(x), w i (x) = Lu(x) s(x), w i (x) N = L u j b j (x) s(x), w i (x) = j=1 N u j Lb j (x), w i s(x), w i (x) = 0 for i = 1...n. (5.10) j=1 In (5.10), only the values of the coefficients u j are unknown, and the expression can be written in matrix form as: Au = f, (5.11) where A ij = { w i (x), Lb j (x) }, u j = {u j } and f i = { s(x), w i (x) } for i = 1...n. Many standard methods for finding the solution vector u on a computer exists. These are outlined in section The basis and weight functions have not yet been defined, and different choices are possible (see e.g. [100] p. 46). The Galerkin approach where the weighting functions are chosen as the same set of functions used to expand the desired function u is common in the FE method, i.e. b i (x) = w i (x) Variational Method Another way of obtaining a weak solution of (5.1) is using an appropriate variational form. This is an integral expression Π that maps the sought function u(x) to a scalar (i.e. it is a functional), and is stationary with respect to small variations δu when 74

92 u(x) is the solution to the problem: Π = Ω L(u, u,...) dx + B(u, u,...) dx. (5.12) x Γ x The solution to the problem is then obtained by requiring that the first variation of Π vanishes: δπ = 0 (5.13) Enforcing stationarity (5.13) implies the satisfaction of a partial differential equation, known as the Euler equation for the variational form and some boundary condition, known as natural boundary condition. So, if the form is chosen such that its Euler equation and the natural boundary condition correspond to (5.1) and (5.2), the desired solution is found by enforcing (5.13). It is possible to construct a variational expression in a systematic fashion starting from the differential equations (5.1) and (5.2) (see e.g. [101, 102]). Alternatively, a variational expression can be identified from the physics describing the problem. The integral expression can e.g. be the total energy of the system that is modelled, and is minimised for the correct solution u. After a variational expression has been established, u can be approximated by the expansion: u ũ = n u j b j (x) (5.14) j=1 The sought approximation to the solution is then given by the set of discrete coefficient values u i that render Π stationary, that is: δπ = Π u i = 0 for all i = 1...n, (5.15) 75

93 which is a system of n equations. The process of seeking stationarity of the variational expression with respect to the scalar coefficients of the expansion (5.14) is commonly known as the Rayleigh-Ritz procedure. If the functional Π does not contain terms higher than quadratic in u and its derivatives, (5.15) results in a system of n linear equations and can be written in matrix form as: Ku = f. (5.16) If the resulting equations are not linear in u i, some linearisation technique (e.g. the Newton s method described in section 5.6.1) can be used Enforcing Constraints and Boundary Conditions In some cases the natural boundary condition is adequate and no action is required, the function u that satisfies (5.13) will satisfy the desired boundary conditions. If the natural boundary condition is not adequate, it is often possible to modify the functional (variational form) to change this. If this is still not adequate and a boundary condition or another constrain must be enforced explicitly, there are various techniques to do so in the finite elements method. Some of these are described next. In addition to boundary conditions (5.2) which must be satisfied, other constraints may have to be imposed on a system. In this work, for example, the incompressibility of the LC material must be maintained. In general, the constraint which limits the unknown function can be written as an additional differential relationship C(u) = 0. The equations can then be supplemented using this relationship as a Lagrange multiplier or as a penalty term [100]. 76

94 Lagrange Multipliers When constraints are enforced using Lagrange multipliers, the supplemented functional describing the problem is written as: Π(u, λ) = Π(u) + λc(u) dx, (5.17) Ω where λ is the Lagrange multiplier enforcing the constraint C(u) = 0. The final discretized system of equations can be written in matrix form as: K = K C C T 0 u λ = f 1 f 2. (5.18) This approach increases the number of unknowns to be solved since in the finite element method λ is discretized and its value must be found at each node where the constraint is enforced. Furthermore, zeros are introduced along the diagonal increasing the condition number of the matrix K, which may complicate the matrix solution process. Penalty Terms Alternatively, it is possible to enforce constraints by the addition of penalty functions to the original equations. The functional can then be written as: Π = Π + α C(u)C(u) dx, (5.19) Ω where α is a positive penalty coefficient. The resulting matrix system after FE discretisation can be written as: Ku = (K + αk C )u = f, (5.20) 77

95 where K C contains the terms corresponding to the penalty functional. The value of α determines the degree to which the constraint is enforced; the larger α is, the more stringently the constraint is enforced. However if α is chosen too large (5.20) will differ too much from the actual problem defined by (5.1) and (5.2). Direct Enforcement of Boundary Conditions In addition to using supplementary Lagrange multipliers and penalty terms, it is possible to enforce some boundary conditions directly on the variables once the matrix problem is assembled. This normally results in a rearrangement and elimination of terms from the matrix system. The advantage with this approach is that the boundary conditions are exactly enforced and the number of unknowns that are solved is reduced without affecting the condition number of the matrix. If the values of u are known at the nodes k, and unknown elsewhere, the (now known) terms K ik u k can be passed to the right hand side, resulting in a transformation of f i into f i K ik u k and the elimination of the rows and columns k (since u k are not unknown, there is no need to establish those equations). The unknown nodal values are found from the solution to the reduced system Kũ = f, where: K = {K ij }, ũ = {u j }, f = {fi }, with i, j k. (5.21) Periodic boundary conditions or any other situation where the value of u is constrained to be equal but free for a set of nodes, e.g. the electric potential on a disconnected electrode that is left floating, can easily be enforced when the nodal equivalencies are known, such that u l = u k. 78

96 In this case multiple nodal values are effectively represented by a single degree of freedom in the matrix system. In order to take into account the contributions of the nodal values u k, the matrix entries located at these rows and columns are added to the corresponding rows and columns l and eliminated from the system. After the reduced system is solved, the values u k are recovered from u k = u l Solution Process The coupled equations presented in chapter 3 governing the LC physics consist of both linear and nonlinear equations. The Euler-Lagrange equations for the Q-tensor are nonlinear while the electrostatic potential is described by the linear Poisson equation. Obtaining a numerical solution to nonlinear simultaneous equations typically consists of an iterative linearisation process (see section 5.6.1) which involves solving linear systems multiple times. Whether the problem is linear or nonlinear, it is necessary to solve linear matrix systems of the form: Ku = f, (5.22) where K is known as the stiffness matrix, u is the solution coefficient vector consisting of the unknown nodal values and f is the source vector. One way of solving (5.22) would be to invert the matrix K, and write: u = K 1 f. (5.23) However, this is in general impractical for large, sparse systems, and much faster algorithms such as Gaussian elimination, LU-decomposition or some variant of Krylov subspace methods are used in practice (see e.g. [103], for details on these). Typically, matrix solver routines can be categorised into direct and iterative methods. Direct methods are less affected by the matrix conditioning than iterative ones, 79

97 but they also require more computer memory, limiting the size of the problems that can be solved. In this work, the solutions are obtained using routines included in the MATLAB software package. 5.3 Shape Functions Linear (first order) tetrahedral shape functions are used for the spatial interpolation of the variables of interest in three dimensions and two dimensional linear triangles for the surface terms. Other types of shape functions are possible, but tetrahedral and triangular elements are in general better suited than e.g. quadrilaterals for the meshing of complex geometries. Higher order shape functions provide a more rapid convergence of the solution, but introduce other difficulties: First of all, the programming of the finite element implementation is more complex, especially when mesh adaptation is used (see chapter 6). Secondly, the resulting matrix bandwidth is increased due to the higher number of interconnected nodes, making the matrix solution process slower. Thirdly, the matrix assembly time is greatly increased due to the larger number of Gaussian quadrature points needed for the exact evaluation of the integrals (see section 5.3.1). Four shape functions, N i, i = 1...4, one for each corner, are needed for first order tetrahedral elements. For the purpose of simplifying the integrals that are essential to the finite elements method, it is more convenient to express these in terms of local 80

98 coordinates r, s and t ranging from 0 to 1: N 1 = r N 2 = s N 3 = t N 4 = 1 r s t (5.24) The physical meaning of the local coordinates can be understood in terms of a ratio of volumes. For example, the value of N 1 for the tetrahedron shown in figure 5.1 at any location P inside the element is ([100] p.187): N 1 = r = Volume(P,2,3,4) Total Element Volume. (5.25) The value of a variable u (or a global x, y or z coordinate) is interpolated within the tetrahedron by: u = N 1 u 1 + N 2 u 2 + N 3 u 3 + N 4 u 4, (5.26) where u i are the four nodal values of u. Gradients of the shape functions also need to be evaluated for the spatial derivatives involved in the PDEs. In local coordinates this can be achieved by considering the Jacobian matrix for the coordinate transformation between the Cartesian and local coordinates: J = x r x s x t y r y s y t z r z s z t, (5.27) 81

99 so that the derivatives can locally be expressed as: N i x N i y N i z = J 1 N i r N i s N i t (5.28) Figure 5.1: Local coordinates of a tetrahedron Analytic and Numerical Integration of Shape Functions The finite element method relies on writing the equations in a form which involves integrals over the domain Ω. This is performed on an element by element basis, taking advantage of local element coordinates: Ω e f(x, y, z) dx dy dz = J 1 1 t 1 t s f(r, s, t) dr ds dt, (5.29) where, in the case of linear tetrahedra, J equals six times the volume of the element e over which the integration is performed. It is possible to evaluate (5.29) either analytically or using numerical integration techniques. However, the complexity of implementing analytic integration increases 82

100 with the number and order of terms that need to be evaluated. This is because terms of equal order in the shape functions need to be collected and grouped together, requiring extensive manipulation of the equation of the weak form. For this reason, numerical Gaussian Quadrature integration whose complexity does not increase with the equations is used in this work. In Gaussian Quadrature, the integrals are evaluated by forming a weighted sum of values of f at discrete sampling points: 1 1 t 1 t s f(r, s, t) dr ds dt n w i f(r i, s i, t i ). (5.30) i=1 Provided that a sufficiently large number, n, of sample points is used, the integrals can be evaluated exactly. For example, if it is known that the value of a variable changes in a linear fashion within an element, only a single Gauss point located at the centre of the element is needed, i.e. n = 1, w 1 = 1, r 1 = s 1 = t 1 = 1. Similarly, if 4 the value is known to vary quadratically, four points are needed and so on. The values of the weights and the locations of the integration points can be found tabulated in many standard textbooks on the finite elements method and applied mathematics, e.g [100, 104]. 5.4 General Overview of the Program Three sets of coupled PDEs are solved for the dynamic case and two for the steady state. The steady state case requires solutions to the electric potential and the Q- tensor field. In dynamic simulations, it is additionally possible to include the flow field of the liquid crystal material and its effect on the Q-tensor field. The general approach to solving these equations is given next. Figure 5.2 shows a flowchart describing the basic structure of the solution process for the dynamic case. Each time step involves finding an electric potential distribution 83

101 consistent with the Q-tensor field, and an optional flow solution. The flow field is assumed to follow the liquid crystal [12], and is updated after the potential and Q- tensor solutions for the time step are found. The Q-tensor dynamics is calculated using an iterative Crank-Nicolson time stepping scheme described in more detail in section This is indicated in figure 5.2 by the Newton Iterations loop arrow. In practice, the execution time of this loop takes up a major portion of the total running time of the program. The finite element mesh may be refined at the end of each time step if necessary (see chapter 6 for more details). 5.5 Electrostatic Potential Externally applied electric fields are used for the switching of LC devices. The electric field is given by the negative gradient of the electric potential φ which satisfies the Poisson s equation: ε 0 ( ε φ) = ρ, (5.31) where ε 0 and ε are the permittivity of free space and the relative permittivity tensor respectively and ρ is a charge density. Inside the LC material, ε is defined in terms of the Q-tensor as: ( 2 ε ij = ε δ ij + ε Q ij + 1 ) 3S 0 3 δ ij. (5.32) The charge density ρ may be due to ions in the LC material (not considered in this work) or due to the flexoelectrically induced polarisation (see section 3.4.3). The electric potential is approximated using the expansion φ φ j N j and an 84

102 Figure 5.2: Flowchart of the program execution. inner product of expression (5.31) and Galerkin weight functions N i is formed: φ j Ω N i ( ε N j ) dω = N i ρ dω. (5.33) Ω 85

103 Integrating (5.33) by parts gives: φ j Ω N i ( ε N j ) dω + φ j Γ N i ( ε N j ) ˆη dγ = N i ρ dω, (5.34) Ω where ˆη is a unit vector normal to each element face. The boundary term reduces to zero in internal elements that have no faces on external boundaries of the FE mesh due to cancellation of the opposing directions of ˆη in neighbouring elements, and can be ignored. However, it must be taken into account in elements where the Neumann boundary condition φ ˆη = 0 is required: φ j Ω N i ( ε N j ) dω+φ j ˆη N i ( ε N j + N j ) dγ = N i ρ dω. (5.35) Γ N Ω Here, the surface integral only need to be performed over Neumann boundaries Γ N. The resulting matrix is: K ij = ε k αβ Ω N i N j N k dω + ˆη α N i ( ε k x α x αβn k N j + N j ) dγ, (5.36) β Γ N x β x α and the source vector is given by: f i = N i ρ dω. (5.37) Ω The Greek subscripts α and β refer to the Cartesian coordinates x, y and z. The permittivity tensor is discretized as ε ε k N k. The FE discretisation of the Poisson s equations results in a linear system of equations, which is solved as described in section (5.2.4). 86

104 5.6 Q-Tensor Implementation In order to solve the Euler-Lagrange equations that minimise the LC free energy, the symmetry and tracelessness of the Q-tensor must be maintained. When these conditions are satisfied, the Q-tensor represents five independent degrees of freedom: Three rotational degrees of freedom and two for the LC order distribution. It is possible to solve for each of the 9 tensor components while enforcing symmetry and tracelessness using Lagrange multipliers. However, it is computationally more efficient to write Q in a five dimensional subspace [105] as: Q = 5 q i T i, i=1 (5.38) where T 1 = (3ê z ê z I)/ 6, T 2 = (ê x ê x ê y ê y )/ 2, T 3 = (ê x ê y + ê y ê x )/ 2, (5.39) T 4 = (ê x ê z + ê z ê x )/ 2, T 5 = (ê y ê z + ê z ê y )/ 2, where ê x, ê x and ê x are unit vectors in the x, y, and z directions respectively. The free energy described in section 3.4 is then written in terms of the modified tensor Q. This results in five Euler-Lagrange equations that satisfy the tracelessness and symmetry properties of the Q-tensor: f i = F q i k F q i,k. (5.40) 87

105 Equations (5.40) are discretised using the weighted residuals approach with Galerkin weight functions to obtain the weak form for the FE formulation. The resulting expressions are lengthy and in order to avoid human errors in the programming of these, the symbolic algebra software Maple is used to generate the code Newton s Method Newton s method is a well known iterative scheme for finding roots of nonlinear equations (see e.g. [106] p. 270). It is based on a Taylor expansion of a function f(u): f(u + u) f(u) + f (u) u + O(h 2 ). (5.41) Requiring that f(u + u) = 0 and rearranging gives u, which is used to update the value of u: u m+1 = u m + u m = u m f(um ) f (u m ), (5.42) where m is the Newton iteration number. Repeating the process in an iterative fashion converges to the value of u that satisfies f(u) = 0, provided that the initial value u 0 is sufficiently close to the solution. When solving for the Q-tensor field that minimises the free energy, f(u) is replaced by the vector obtained from the finite element discretisation of the five Euler-Lagrange equations f = {f 1, f 2, f 3, f 4, f 5 } T and f (u) by the Jacobian matrix J: J = f 1 q f 5 q 1 f 1 q 5 f 5 q 5. (5.43) 88

106 The nonlinear equations are then solved by successively solving the linear system J m q m = f m and updating q m+1 = q m + q m, until q is smaller than some tolerance value Time Integration Time integration is needed for simulating the dynamics of a LC device. This is performed using the finite differences method in time. Explicit Time Stepping A simple explicit time stepping algorithm giving the time evolution of the Q tensor can be constructed by considering: q t+ t = q t + t q t, (5.44) where the subscript denotes the time, q is the time derivative of the Q tensor and t is the size of the time step. As described in chapter 3, q is obtained from equation (3.52) or (3.47). A finite element discretisation of this then results in the matrix equation: M q t = f t, (5.45) where M is the mass matrix Ω N i N j, and f t is the right hand side vector resulting from the discretised Euler-Lagrange equations. It is then possible to find the LC dynamics by evaluating equations (5.44) and (5.45) successively. However, although this approach is relatively simple, it is only first order accurate and also very unstable: The size of the time step is limited by the Courant-Friedrichs-Lewy condition which relates the maximum time step to the spatial discretisation [107]. 89

107 Implicit Time Stepping An improved time integration scheme can be devised by approximating the time derivative using central differences in time and representing nonlinearities by r: M q t+ t/2 + f t+ t/2 = r. (5.46) This scheme is known as the Crank-Nicolson time integration, and is unconditionally stable for linear systems [108]. Nonlinearities, represented by r, in the time derivatives can be taken into account by performing Newton iterations within each time step (this is shown as the Newton Iterations loop in figure (5.2)). The central differences are written as: M q t+ t/2 = 1 t M(q t+ t q t ), (5.47) and f t+ t/2 = 1 2 (f t + f t+ t ) = 1 2 (A tq t + A t+ t q t+ t ) (g t + g t+ t ), (5.48) where A and g correspond to non-linear and linear terms respectively in the free energy. Using (5.47) and (5.48), expression (5.46) can be re-written as: { M t + A } { t+ t At q t+ t M } q t + 1 t 2 (g t + g t+ t ) = r. (5.49) The goal is then to find q t+ t such that r = 0. This can be achieved using Newton s method by writing: K m q m t+ t = r m, (5.50) 90

108 and q m+1 t+ t = qm t+ t + q m t+ t, (5.51) where the superscripts denote the Newton iteration number and K is the Jacobian matrix: K m = rm q m t+ t = { } M t + Jm t+ t, (5.52) 2 and J is as defined in (5.43). Iterations within each time step are performed until q m t+ t is deemed to be sufficiently small. Additional loops may be needed in order to make sure that the electric potential is consistent with the Q-tensor field both before and after the time step (see figure 5.2 ). This could be avoided by solving for the potential simultaneously with the Q-tensor, but the solution vector would then be extended by the number of nodes. Variable Time Step The ability to automatically adapt the size of t results in savings in computation time: Longer time steps can be taken when the Q-tensor is changing slowly and shorter steps when Q is changing rapidly. This can be achieved e.g. by writing [109]: ( ) k tolerance t new = t old, (5.53) error where tolerance and k are user defined values (e.g. tolerance = 10 3 and k = 3) and error is an error estimate on the time derivative. The error estimate can be calculated in various ways, but in general it is related to the magnitudes of the corrections made to q t+ t during the Newton iterations in the Crank-Nicolson scheme. 91

109 5.7 Implementation of the Hydrodynamics It has been previously explained in section 3.6 how the incompressible flow of the LC material may be described by the generalised Navier-Stokes equations: ρ dv dt v = 0, = σ p, (5.54) where ρ is the LC density, σ is the stress tensor consisting of viscous and elastic contributions and p is the hydrostatic pressure. The time derivative is the material time derivative: dv dt = v + v v (5.55) t In the case of slow elasticity driven flow of LCs, it is possible to make two simplifications to equations (5.54): The steadiness approximation and the low Reynolds number approximation. The steadiness approximation [12] is based on the assumption that changes in the flow field are much more rapid than changes in the Q-tensor field. When this is true, the partial time derivative in (5.55) can be ignored, and the flow is assumed to follow the changes in the LC orientation. The validity of this assumption can be checked by verifying that the characteristic times τ Q and τ v for the Q-tensor and the flow fields respectively satisfy τ v τ Q, where [12, 9]: τ Q = µ 1 ξ 2 L 1, τ v = ρ H2 α 4, (5.56) (5.57) where L 1 is an elastic constant, α 4, µ 1 and ρ are LC viscosities and density respectively, H is a characteristic length of the LC cell or container and ξ is the characteristic 92

110 length of the Q-tensor: ξ = 27CL1 B 2. (5.58) Typically ξ is in the order of a few nanometres resulting in τ Q 10ns, whereas τ v may be a few orders of magnitude smaller. In cases when τ Q τ v the time derivative cannot be ignored and time stepping for the flow field should be performed. The Reynolds Number Re is a dimensionless parameter relating the inertial and viscous forces of a flow (e.g. [110] p. 301): Re = v H ν, (5.59) where H is again a measure or characteristic length of the container size and ν is the kinematic fluid viscosity (dynamic viscosity divided by the density). When Re is low, the nonlinear convective term in (5.55) is negligible, rendering the Navier-Stokes equations linear. The flow of the LC material can then be estimated at any instant in time (in practice, after each time step) by solving the steady state incompressible Stokes equations: σ p = 0 v = 0 (5.60) Enforcement of Incompressibility In the incompressible Stokes equations, the hydrostatic pressure acts as a Lagrange multiplier to enforce the non-divergence of the flow field. However, it is a well known problem in the field of computational fluid dynamics that a straightforward FE discretisation of the equations (5.60) results in numerical difficulties. These appear as spurious pressure solutions, where the pressure field is oscillatory and the incompress- 93

111 ibility of the flow field is not satisfied [14]. Mixed Interpolation Different approaches to overcome this problem exist. One possibility is to use the socalled mixed formulation approach with higher order shape functions for interpolating the flow solution than those used for the pressure. It is, for example, possible to use second order functions for the flow and linear elements for the pressure. This is a popular approach in two dimensional problems, where the number of degrees of freedom is usually relatively small [10, 11, 14]. However, in three dimensions this approach often results in prohibitively large systems due to the additional nodes needed for the higher order elements. This was found to be especially true in this work, bearing in mind that the flow solution is updated at the end of each time step. Pressure Penalty Alternatively, it is possible to enforce the incompressibility by the pressure penalty formulation. In this approach, the continuity equation v = 0 is replaced by [14]: ε v = p, (5.61) where ε is a user defined large positive scalar coefficient. It is then possible to eliminate the pressure from the equations by substituting (5.61) into (5.60). Although this approach reduces the number of degrees of freedom that need to be found, the resulting system of equations becomes poorly conditioned due to the large value of ε. This means that iterative Krylov sub-space solvers often do not converge to a solution. 94

112 Pressure Stabilisation An alternative approach, taken here, allowing for equal order interpolation functions is to use the so-called Brezzi-Pitkäranta stabilisation technique [13]. This method relies on introducing a perturbation to the continuity equation: v = ɛh 2 e 2 p, (5.62) where ɛ is a user defined small positive scalar coefficient and h e is the local mesh size of element e. The effect of the right hand side in (5.62) is to smooth the pressure solution. A FE discretisation of the stabilised Stokes equations gives rise to the following matrix system to be solved: D C T C T v p = f 1 f 2, (5.63) where the sub-matrices are D and C arise from the Stokes equations given in (5.60) and T from the added stabilisation term. An advantage of the stabilisation method is that the condition of the matrix is improved due to the non-zero terms on the matrix diagonal due to the addition of T. The condition of the matrix is further improved by introducing scaled shape functions for the pressure, so that components of D and T are of comparable magnitude. The stabilised formulation is tested on a container with a 90 bend, as shown in figure 5.3, using different values for the stabilisation coefficient ɛ. In this test, σ is taken to be that for an ordinary isotropic liquid (i.e only the viscous coefficient α 4 0). The flow magnitude at the inflow is fixed to take a quadratic form, while no-slip boundary conditions ( v = 0)are applied to the side boundaries. The pressure is fixed to zero at the outflow boundary. Figure (5.4) shows the magnitudes of the flow and pressures on a plane through 95

113 Figure 5.3: Container with 90 bend for testing the stabilised Stokes flow. ɛ = 10 4 ɛ = 10 6 ɛ = 10 9 Figure 5.4: Flow magnitude (top row) and pressure (bottom row) solutions obtained using three different values the stabilisation parameter ɛ = 10 4, 10 6 and 10 9 ). 96

114 the centre of the mesh for three different values of ɛ. The effect of over stabilisation (ɛ = 10 4 ) can be seen in the first column where the flow field is not divergence free. Similarly, in the last column, the effect of under stabilisation can be observed as spurious pressure oscillations start to appear when the stabilisation parameter is reduced to ɛ = It was found that ɛ typically result in non divergent flow without introducing pressure oscillations. 97

115 Chapter 6 Mesh Adaptation 98

116 6.1 Introduction The dimensions of some of the geometric features in an LC device may be very small compared to the overall size of the device. For example, the bistable LC device modelled in chapter 8, contains three dimensional posts with corners that are rounded to correspond to arcs with radii in the order of tens of nanometres, whereas the thickness of the cell is several microns. Similarly, spatial variations in the orientation of the director field can be gradual throughout most of a device, but very high in the vicinity of defects or aligning surfaces. In the Landau-de Gennes theory this results in a Q-tensor field with a low gradient throughout most of a device and a high gradient localised near regions of large distortions. Typically the diameters of defect cores are, depending of the exact values of the material parameters an the temperature, in the order of tens of nanometres or less. Often it is exactly these small scale structural features and defects or disclinations in the director field that are of interest. Consequently the spatial discretisation should be sufficiently accurate in these regions in order to describe the geometry and to capture the variations of the Q-tensor field. Accurate representation of small scale geometrical features relies on the finite element mesh provided by the user. The density of this mesh should be sufficiently high in these regions in order to properly describe the geometry. In this work the commercial GiD [111] program is used in the mesh generation. However, the regions where the Q-tensor varies rapidly are not fixed and defect movement is possible during the operation of a device. Instead of having a dense mesh throughout the whole structure, it is often more efficient (especially in three dimensions) to increase the mesh density locally in regions of rapid distortions of the Q-tensor and decrease it in regions of slowly varying Q during the simulation. This is known as mesh adaptation. A mesh adaptation algorithm for three dimensional tetrahedral meshes has been implemented to be used in conjunction with the finite elements discretisation of the 99

117 Landau-de Gennes theory described in chapter 5. This algorithm performs mesh refinement based on user specified criteria in regions where the accuracy of the interpolation is considered insufficient for describing the Q-tensor field. A brief overview of finite element mesh adaptation is given in section 6.2. Then, in section 6.3, the algorithm implemented as part of this work is described and example results are shown in section 6.4. An alternative way of adapting the accuracy of the interpolation is described and results for a reduced one dimensional problem are presented in section 6.5. Finally, possible future developments are suggested in section Mesh Adaptation In general, a mesh adaptation algorithm consists of two stages: (1) Assessment of the local error in a trial solution and (2) adaptation of the spatial discretisation to improve the interpolation of the solution. After this, a new solution can obtained on the improved mesh Assessment of the Error In the so-called a posteriori error analysis a previously obtained solution is analysed in order to find regions in the finite element mesh where the accuracy of the interpolation should be improved for increased accuracy and can be worsened for higher computational efficiency. The error assessment stage can in general be classified either as error estimation or error indication [112]. The error estimation method is based on defining an approximation of an error measure within each element as the norm: e i = ũ i u i, (6.1) where e is the error within element i, ũ i is the approximated solution and u i, the 100

118 exact solution. In most cases the exact solution u is not known, but in general it is possible to provide a local estimate which is more accurate than ũ, so that an approximation of the the error can be calculated. This estimate can for example be formed by recovering a smoothed solution over a patch of elements using interpolation functions of higher order than that used for ũ or on a finer mesh [100]. Error indicators are based on heuristic considerations where a readily available quantity, specific to the problem at hand, is chosen as an error indicator [112]. This can for example be a gradient of the sought solution or some other physics-based quantity. In this work two different error indicators are considered. Firstly, the free energy within each tetrahedron can be calculated, and elements where the total energy is above some threshold value are chosen for refinement [10]. A second, simpler approach considers only the value of the scalar order parameter. Elements that contain regions where the order parameter is outside some user specified range of values are chosen for refinement. In practise, the performance of the two error indicators is found to be identical, provided the threshold values are chosen appropriately Adapting the Spatial Discretisation Different methods for changing the interpolation exist. Three general schemes for adapting the spatial discretisation can be classified: The h, p and r-methods (see e.g. [100, 113, 114, 115] and references therein). In the h-method, the number of nodes is locally changed. This can be achieved by splitting or recombination of existing elements or by complete or partial re-meshing of the domain. In the p-method the order of the interpolation polynomials is locally changed. Use of hierarchical elements allows for addition or removal of higher order polynomials without changing the shape functions of the lower order interpolants. In the r-method only the nodal positions are relocated without changing the number of elements or the order of interpolation 101

119 functions. The advantage of this method is that the computational load remains constant throughout the simulation, but often an additional system of equations needs to be solved for determining the new node locations. Different combinations of these three are also possible. In this work, the h-method is implemented in three dimensions and a simple one dimensional test of the p-method is presented. Local h-refinement can be achieved in various ways, but the resulting mesh must be conforming (i.e. no hanging nodes may exist), and the mesh quality should not degrade as a result of successive refinements. A tetrahedral mesh is said to be of good quality when the elements are (nearly) equilateral. Low quality meshes may interpolate poorly and the condition of the stiffness matrix tends to be worsened [116, 117]. It is possible to obtain an improved mesh by complete remeshing the domain of interest while ensuring that the new mesh density is appropriately changed from the previous mesh. However, in three dimensions this process may be computationally too expensive. Furthermore, programming a three dimensional mesh generator is no easy task. Instead, the density of the existing mesh may be changed locally by insertion of new nodes or removal of existing ones. In [118, 119], refinement of tetrahedral meshes by bisection of a single element edge has been described. Provided the edge to be bisected is chosen appropriately, a degradation of the mesh quality is bounded below the initial mesh by a positive constant. An alternative approach, taken here, is to subdivide elements selected for refinement into eight sub tetrahedrons. This is a generalisation of a two dimensional red green refinement for triangular meshes (see e.g. [120]) into three dimensions [121, 122]. The elements selected for refinement during the error estimation stage are termed red, whereas transitional green elements need to be refined to ensure mesh conformity. Figures 6.1 (a) (f) show the possible ways an unrefined tetrahedron, fig. 6.1 (a), may be divided. For the red elements, fig. 6.1 (b), new nodes are added 102

120 at the mid sides of each edge resulting in a subdivision into eight smaller tetrahedra. Additionally, if an element shares more than three edges with previously selected red tetrahedra it is included in the list of red elements. However, if an element shares three or fewer edges with the red tetrahedra (but at least one), different subdivision possibilities exist: These are the various green elements, shown in figures 6.1 (c)-(f). D D cd bd C ad C bc B ac B ab A A (a) Unrefined tetrahedron. D (b) Red D C C B ac B ab ab A A (c) Green1 D (d) Green2a D cd C C bc B ac B ab ab A A (e) Green2b (f) Green3 Figure 6.1: Element refinement by the red-green method. Bisected edges are drawn in bold. Original nodes are labelled with capital letters whereas new nodes resulting from edge bisection are labelled using lower case letters. 103

121 6.3 Overview of the Mesh Adaption Algorithm Mesh adaptation algorithms that include both mesh refinement and de-refinement sometimes employ special tree-like data structures to represent the hierarchy of refined and unrefined elements in a mesh and use recursive algorithms in the refinement of neighbouring elements, e.g. [120, 123, 124]. However, the algorithm developed here is required to work with the mesh represented by simple array data structures, as this is the format used for the finite element program described in chapter 5. A benefit of this is that the algorithm developed here is general and can be included in other finite element programs with only small modifications to the code. The mesh adaption algorithm works by starting the refinement process from a copy of the initial user created mesh, making explicit de-refinement unnecessary and thus reducing the complexity of the algorithm. Errors are estimated on a mesh from a previous solution, which may or may not already be refined. Elements in the original mesh that contain regions of high error are selected for refinement. This process is repeated until no more refinable elements are found, or for a user defined number of iterations. The steps taken in each refinement iteration are listed below, and explained in more detail. 1. Choose refinable elements in mesh. 2. Expand region(s) of refinement if necessary. 3. Identify green elements. 4. Identify red and green surface elements. 5. Create new nodes and new elements. 6. Remove old elements chosen for refinement. 7. Interpolate Q-tensor field onto new mesh. 104

122 8. Repeat from step 1 or exit refinement algorithm. 1. A list of red tetrahedra is constructed. Two different criteria that can be used either separately or in combination for finding these elements are implemented: Firstly, the total free energy of the LC material is calculated within each element. The free energy density is higher in regions with large distortions in the Q-tensor field, such as in the vicinity of defects. If the integral of this energy density over the volume of an element is above some user defined threshold, the element is marked as refinable. Secondly, elements that contain nodes from the previous result where the scalar order parameter is outside a user defined range can be marked for refinement. 2. It is often necessary to include elements that were not selected in step 1 to the list of refinable elements. This may be due to two reasons: Sometimes an element not previously marked red may be neighbouring several red elements (it shares four or more of its edges with the elements already in the list of red elements). In this case that element is also added to the list of red tetrahedra. When a structure with periodic boundaries is simulated, it may be necessary to refine a region near one of the boundaries. The periodicity must be maintained and it is then necessary to extend the region of red elements to the opposite side of the mesh. 3. The tetrahedral elements of the mesh can be subdivided in different ways depending on how many of its edges are bisected. In order to ensure conformity, transitional green elements must also be created by subdivision into two or three smaller tetrahedra. At this stage, lists of green tetrahedra are created depending on the number of edges shared with any red elements selected in steps 1 and Two-dimensional triangular elements are used to represent alignment surfaces. These must also to be refined when a red or green tetrahedron is located at the surface. Red and green triangles are identified depending on the number of edges to be bisected. Red triangles are divided into four and green triangles into two smaller triangles. 105

123 5. Edge elements consisting of two nodes are created from the lists of red and green tetrahedra. New node coordinates located at the centres of each edge are created. Then, new two and three dimensional elements are created by dividing the red and green triangles and tetrahedra. 6. All the red and green elements are removed from the mesh data structure and are replaced by the newly formed smaller elements. This is necessary in order to avoid overlapping elements. 7. After the new mesh is created, variables from the previous result are interpolated onto the new mesh. The Q-tensor field is interpolated in terms of (θ, φ, ψ, λ 1, λ 2 ), where θ, φ and ψ are Euler angles of the eigenvectors and λ 1 and λ 2 of the eigenvalues of the Q-tensor. The reason for performing the interpolation in terms of these derived values instead of the actual Q-tensor components used in the calculations is that the physical meaning of direction and degree of order are maintained. Figure 6.2 shows the effect of interpolation of the individual Q-tensor components within a one dimensional linear element extending from x = 0 to x = 100. The eigenvalues resulting from interpolation of the Q-tensor components are plotted between the two nodes. The eigenvalues of the tensor are of equal magnitudes at both of the nodes, but the orientation of the director changes by an arbitrary angle (B-A) through the element. The interpolated Q-tensor at x = 50 then appears to be biaxial and with a reduced order parameter. 7. Finally, the newly created mesh may be either further refined by repeating steps 1 to 6, or the mesh adaptation algorithm may be exited and the simulation can be continued using the new adapted mesh. 106

124 Figure 6.2: Example of error introduced by linear interpolation of the components of a Q-tensor field representing a rotation of the director field of a constant order. Black dots represent the original nodes and gray dots the new added node. 6.4 Example Defect Movement in a Confined Nematic Liquid Crystal Droplet The switching dynamics of a spherical liquid crystal droplet was used in the testing and development of the mesh adaptation algorithm. The simulated structure consists of a spherical liquid crystal region of 1µm diameter immersed in a cube of solid isotropic dielectric material. The anchoring of the LC material is assumed planar degenerate, resulting in a pair of point defects located at opposing sides of the sphere. A slight asymmetry is introduced by scaling two of the dimensions of the structure by a few percent in order to ensure the existence of a unique LC configuration that minimises the total free energy. Electrodes are placed at the top and bottom surfaces of the cube containing the LC sphere. A part of the initial unrefined mesh for the structure is shown in figure 6.3. The size of the initial mesh is tetrahedra and 5717 nodes. The mesh adaptation algorithm is chosen to perform three refinement iterations every seven time steps on the initial mesh based on the value of the scalar order 107

125 parameter. Refinement of a tetrahedron is performed when the value of the order parameter within that element is below 75%, 30% and 1.5% of the equilibrium order parameter S 0. Typically, the resulting meshes consist of approximately tetrahedra and 9000 nodes. Results showing the director field during the switching process are shown with the corresponding meshes in figures 6.4 (a) (f). After the applied potential is removed, the director field relaxes back to the inital configuration shown in figure 6.4 (a) due to the slight asymmetry of the structure. Figure 6.3: Partial 3-Dimensional view of initial unrefined mesh for LC droplet inside a cube of fixed isotropic dielectric material. Approximately a quarter of the dielectric region (coloured white) and half of the liquid crystal (coloured grey) are shown. 6.5 Hierarchical p-refinement An alternative to the h-method implemented in three dimensions is the p-method, where the order of the interpolation functions is increased locally to improve the accuracy. One way of doing this is by use of hierarchical elements [100]. These are higher order polynomials that can be added to elements without effecting the lower order shape functions already present. 108

126 a b c d e f Figure 6.4: (a), (c), (d) 2-Dimensional slices through the centre of a nematic droplet during switching by an external electric field. Director colour indicates scalar order parameter and background electric potential. (b), (d), (e) 3-Dimensional views of corresponding meshes. 109

127 Using the hierarchical approach, the discretised approximation of function u is written in each element as the sum: u n u i Ni u 1 N 1 + u 2 N 2 + u 3 N 3 + u 4 N u n N n, (6.2) i=1 where u i are the discretised values of u and N i the corresponding spatial interpolation functions or shape functions. In a one dimensional system i = 1, 2 correspond to the nodal degrees of freedom, whereas i > 2 correspond to higher order internal (bubble) degrees of freedom. Figure 6.5 (a) shows second, third and fourth order one dimensional hierarchical shape functions plotted against the local element coordinate r. The superposition of standard linear elements and a second order hierarchical element is shown in figure 6.5 (b). N 3 N Displacement Linear Linear + O2 hierarchical u 3 u 2 N u Local Coordinate, r (a) Local Coordinate, r (b) Figure 6.5: (a) Second, third and fourth order hierarchical shape functions for a one dimensional finite elements implementation. (b) Example of superposition of first and second order hierarchical element shape functions. Linear element (dashed line) is p-refined by the addition of a second order (solid line) shape function. In a single dimension, the shape functions can be written as functions of the local 110

128 coordinate r ranging from 0 to 1 as: N 1 = (1 r) N 2 = r N 3 = 4(1 r)r N 4 = 36 3 (1 r)(1/2 r)r N 5 = 81(1 r)(1/3 r)(2/3 r)r In order to test the algorithm, a one dimensional finite elements discretisation of the equations of the Landau-de Gennes theory using hierarchical elements has been written. The convergence of this interpolation scheme was studied for a test case where the director is fixed homeotropic on one surface of a thin cell and planar on the other. No initial tilt bias is given, so that a region of melting from horizontal to vertical orientation of the director occurs at the centre of the cell. Although no real defects can be considered to exist in a one dimensional geometry, the configuration described here corresponds to the director and order parameter profile through the centre of a 1 2 defect in two dimensions, as shown in figure 6.6 (a). The resulting eigenvalues of the Q-tensor are plotted in figure 6.6 (b). The thickness of the cell is taken as 0.1 µm., a single elastic constant approximation with K = 5pN/m 2 is used and the thermotropic coefficients are for the 5CB LC material at (T T ) = 4 K (see appendix A). The mesh density is uniform throughout the domain but the number of elements is varied, i.e. no local h refinement is considered. The convergence of the scheme using different orders of interpolation functions can be seen in figure 6.7 (a), where the free energy of the system is plotted as a function of the element size. It can be seen that adding the third order shape functions, O3, improves the accuracy of the scheme considerably more than the second and fourth order functions, O2 and O4. 111

129 Eigenvalues λ x λ y λ z 0.2 (a) Z / µ m (b) Figure 6.6: (a) 1 defect in two dimensions (left) and the one dimensional director profile through the centre (right). (b) Eigenvalues of the Q-tensor in the one 2 dimensional case plotted against the z dimension. This is due to the fact that both the solution (see eigenvalues λ x and λ z in figure 6.6 (b)) and the third order functions are odd functions in the spatial coordinate z, whereas the second and fourth order polynomials are even. In this test the p-refinement is not local. That is, all the elements in the mesh contain the same number of degrees of freedom. However, only a fraction of the higher order degrees of freedom are in fact needed to describe the solution. This can be seen in figure 6.7 (b), where the effective number of degrees of freedom (degrees of freedom with displacement magnitudes larger than 10 6 ) are plotted against the element size. The effective higher order degrees of freedom are concentrated at the centre of the structure where the gradient of the solution is high (see figure 6.8). This means that in a two or three dimensional implementation, where efficiency is more important, higher order polynomials only need to be added locally to elements containing defects. 6.6 Discussion A three dimensional mesh adaptation algorithm has been developed and implemented. The algorithm performs local mesh h-refinement in regions selected using an empirical 112

130 error indicator, making modelling of three dimensional defect dynamics feasible on a standard PC workstation. The performance of a hierarchical p-refinement scheme was tested in a simplified one dimensional case. A three dimensional implementation of the p-refinement scheme is more complicated than the simple one dimensional described here. This is because higher order degrees of freedom must be assigned, in addition to the internal element volumes, also to edges and faces separating neighbouring elements. Then, for example in the case of a tetrahedron, addition of a higher order hierarchical polynomial introduces a total of 11 new degrees of freedom (four faces, six edges and one volume). Care must be taken with the ordering of the element node numbering in neighbouring elements to ensure continuity of the higher order face and edge polynomials. A full three dimensional hp-refinement scheme is left as future work. In this work it is assumed that any solid surface-liquid crystal interfaces are static and do not change during the simulation, so that the initial meshing should satisfy the requirements of mesh density for proper description of the geometry of the LC device. This is true in the case of most optoelectronic devices. However, nematic liquid crystals find new applications as solvents for microemulsions and particle dispersions, in e.g. biomolecular sensors [1] or in the self-assembly of crystal structures [2]. The LC material interacts with the immersed nano- or micro-scale particles affecting their position and orientation due to the elastic forces of the director field, so that the LCparticle interfaces can no longer be considered static. Moving boundaries are possible using the finite elements method, and are in fact extensively used e.g. in finite elements models for structural mechanics [100] or Stefan problems [115] (a problem where a phase boundary can move with time). However, as mentioned before, mesh generation and remeshing is a cumbersome task, especially when a good quality mesh without inverted or degenerate elements is needed. An alternative, more recently developed method could be extending the finite element method with discontinuous 113

131 elements using the XFEM method [125]. In this method, moving discontinuities are represented by additional discontinuous shape functions superpositioned on the underlying standard finite element mesh eliminating the need for mesh adaption. The XFEM method can be added on top of existing finite element code, and has been used in e.g simulation of elastic fracture mechanics, multi-phase flow and representation of microstructures [125, 126, 127]. Free Energy [Arbitrary Units] O1 O1+O2 O1+O2+O3 O1+O2+O3+O Element Size / µ m. (a) Effective Number of Degrees of Freedom O1 O1+O2 O1+O2+O3 O1+O2+O3+O Element Size / µ m. (b) Figure 6.7: Comparison between results obtained using hierarchical elements of different order. (a) Total free energy as a function of element size. (b) The effective number of degrees of freedom as a function of element size u 3 u 4 u 5 Magnitude Z / µ m Figure 6.8: Magnitudes of higher order hierarchical degrees of freedom as a function of the z-dimension. The number of 1-D elements is 50, resulting in an element size of 2 nm. 114

132 Chapter 7 Validation and Examples 115

133 7.1 Introduction In this chapter, examples of results obtained using the Q-tensor LC modelling software developed for this work are presented. When possible, these are compared with previously published work or with results obtained using other modelling methods. First, defect-free cases where the elastic distortion energy dominates and the Oseen-Frank theory is expected to give similar results to the Landau-de Gennes theory are compared. Then, cases where order variations and defects play an important role are considered. Finally, results for a zenithally bistable device, modelled in three dimensions for the first time, are presented. 7.2 Three Elastic Constant Formulation The dynamic three elastic coefficient formulation on the Landau-de Gennes energy is validated by simulating the switching dynamics of a twisted nematic cell and comparing the results with predictions obtained using an established finite elements implementation of the Oseen-Frank energy developed earlier at UCL [98]. Material parameters for the 5CB LC material are used (see appendix A). The cell thickness is chosen as 1 µm. The anchoring is strong on both surfaces with 5 pre tilt and 90 twist through the cell. Cases with weak anchoring in the two theories are compared in chapter 4. Starting from uniform director configurations at time = 0, a 2V potential is applied across the cells for a duration of 3 ms, after which the director fields are allowed to relax for a further 7 ms. The tilt angles at z = 0.5µm are recorded and are plotted in figure (7.1). The results show good agreement with only small observed differences. These can be attributed to order parameter variations that occur near aligning surfaces where the elastic distortion is high and to differences in the implementation of the two algorithms. 116

134 OF LdG Tilt [degrees] Time [ms] Figure 7.1: Comparison of tilt angles at z = 0.5µm as a function of time using the Oseen-Frank (dashed line) and the Landau-de Gennes (solid line) theories. 7.3 Switching Dynamics of a TN-Cell, with Back flow When a large holding voltage is removed from a twisted nematic cell, an optical bounce in the transmitted light can be observed. The reason for this has been shown long ago to be the director at the mid plane of the cell momentarily tipping over due to shear flow, also known as back flow, of the LC material [128, 129]. The dynamics of a one micron twisted nematic cell with 5 degrees pre-tilt is simulated with and without taking into account the effect of flow of the LC material. The anchoring is assumed strong on both surfaces and the material parameters for the 5CB liquid crystal material are used (see appendix A). A 3V potential difference is applied across the cell for the duration of 2ms., after which the potential is removed. The mid plane tilt angle is recorded and plotted versus time in figure

135 Midplane Tilt Angle [Degrees] No Flow Flow Time [ms] Figure 7.2: Switching dynamics of a twisted nematic cell, with and without flow of the LC material. 7.4 Defect Dynamics The dynamics of defects are validated by studying the annihilation of a ± 1 2 line defect pair. This has previously been examined theoretically in [12, 63, 11], using two dimensional discretisations. In [12] and [11], the process is modelled using finite differences and finite elements implementations of the Qian-Sheng equations respectively. In [63], a finite differences implementation of the Berris-Edwards equations is used. A mesh of nm dimensions with periodic boundary conditions for the (x, z)planes at y = 0 and y = 4nm is used. This is comparable to a 2D discretisation where the defect lines are assumed to extend to infinity in the y-dimension. The material parameters used are for the MBBA LC material (see appendix A). Two distinct initial director configuration are considered: (a) Starting from a director configuration where the tilt angle is set to θ a (x, z) = ( ) ( 1 ( 2 tan 1 z x d tan z x+d)), and (b) starting from the initial configuration θ b (x, z) = θ a (x, z) + π. In both (a) and 2 (b) the director is in the (x, z) plane resulting in zero twist throughout the cell. The coefficient d is the distance between the defects and the centre of the cell, d = 50nm is used. Figures 7.3 (a) and (b) show the initial director fields for the two cases. Dynamic simulations are performed both with and without the effect of flow of the 118

136 LC material. The positions of the defects are recorded at each time step by finding the locations of the mesh nodes that correspond to minima in the order parameter. The variation of defect positions with respect to time can be seen plotted in figure 7.4, while the resulting flow fields at time = 20µm are shown in 7.3 (c) and (d). The z [µm] z [µm] x [µm] (a) x [µm] (c) z [µm] z [µm] x [µm] (b) x [µm] (d) Figure 7.3: The two initial director configurations for the defect annihilation cases (a) and (b), and the corresponding flow solutions (c) and (d) at time = 20 µs. effect of including the flow is to speed up the defect movement, with the positive defect accelerated more than the negative one. When the effect of flow is ignored the two defects move at the same speeds, yielding in identical results in both cases (a) and (b). The flow field is found to be sensitive to both the defect separation as well as the spatial discretisation. Nevertheless, good agreement is found between the results 119

137 X [µ m] (a) (b) (a) and (b), no flow Time [ms] Figure 7.4: Defect positions with respect to time for the two initial configurations, with and without flow. In both cases when flow is ignored, identical results are obtained. The solid line represents the position of the positive defect and the dashed line the position of the negative defect. obtained here and those published earlier. 120

138 7.5 Defect Loops in the Zenithally Bistable Device The work presented in this section shows a more substantial example of the modelling capabilities of the tools presented in the earlier chapters: A periodic grating structure known as the Zenithally Bistable Device (ZBD) is investigated. The device has previously been modelled considering only two dimensions [130, 131], but here results from three dimensional analysis are presented. Conventional liquid crystal devices are usually monostable, that is, the liquid crystal director field always relaxes to the same configuration after applied voltages are removed. In contrast, bistable LC devices have two distinct stable configurations or states to which the director field may relax, and in which they remain without applied holding voltages. Advantages of bistability include lower power consumption and the possibility of passive addressing of high resolution LC devices. Stable states of the director field correspond to minima in the free energy of the LC material. In monostable devices only one minimum is used in the operation of the device, whereas in bi- and multistable devices two or more local minima separated by barriers of higher free energy can be reached. Bistability in nematic LC devices can be achieved in a number of ways including the use of surface anchoring exhibiting bistability [134, 135, 136, 69], non-uniform surface alignment patterns [137, 138] or surfaces shaped as two- or three-dimensional micropatterns [19, 20, 130, 139], as is the case with the ZBD. In the ZBD, the two states are knownas the continuous (C state) and the discontinuous or defect states (D state). In the C state the director field undergoes continuous distortions whereas the D state is characterised by the presence of ± 1 2 defect line pairs running along the peaks and troughs of the grating structure (the positive defect along the troughs and the negative along the peaks). Previously, simulations of the structure have been carried in two dimensions [130, 131]. The first of these [130] shows the switching process, back and forth, between the 121

139 two stable states under several simplifying assumptions; strong anchoring, a single elastic constant approximation and a negligible dielectric anisotropy. The second study [131] drops these assumptions, but concentrates on the defect movement during the annihilation process from the D to the C state. Switching between the stable states is controlled by the sign of the applied voltage, via the flexoelectric effect. In two dimensional cases the grating is assumed to extend to infinity. However, in reality the lengths of the gratings are finite and a 180 degree shift in the grating structure has been found experimentally to stabilise the defect loops that form [132, 133]. The grating is now a three-dimensional structure that contains slips. The effect of the slip region on the static defect structures has been modelled in three dimensions and the results for this are presented next. A more comprehensive analysis taking into account the time dynamics is necessary in order to study the role of the slips on the switching dynamics and the stabilising effect they have on the defect loops. This work is currently under way and results will be reported elsewhere. The ZBD Grating Profile in Three Dimensions In two dimensions, the surface profile has been previously represented in [131] using the function: Z(x) = H 2 sin ( 2πx P x + α sin 2πx ), (7.1) P x where P x is the grating pitch in the x direction, H is the grating height and α is a scalar coefficient used for determining the asymmetry of the grating profile. Figures (7.5) (a) and (b) show the two dimensional director fields for the C and D states for a grating structure described by expression (7.1), with P x = 1µm, H = 0.65µm and α = 0.5. In three dimensions the surface profile is also a function of the y coordinate, 122

140 (a) (b) Figure 7.5: The continuous (a) and discontinuous (b) states found in the two dimensional representation of the ZBD grating structure. 123

141 and is here written as: Z(x, y) = H { Υ 1 (y) Z 1 (x) + Υ 2 (y) Z 2 (x) + hυ 3 (y) }, (7.2) where Z 1 (x) = 1 { [ 2πx 1 sin + α sin 2πx ]}, 2 P x P x Z 2 (x) = 1 { [ ( )]} 2πx 2πx 1 sin + φ + α sin + φ, 2 P x P x { ([ Υ 1 (y) = 1 + exp y + w ] 1 s)}, (7.3) 2 { ([ Υ 2 (y) = exp y w ] 1 s)}, 2 Υ 3 (y) = 1 { Υ 1 (y) + Υ 2 (y) }. In equations (7.2) and (7.3) Z 1 and Z 2 describe the the two grating profiles in the x-direction. These are essentially the same as in (7.1), but the additional parameter φ determines the phase difference between the two gratings. The functions Υ 1, Υ 2 and Υ 3, based on sigmoid functions, are used to specify the surface profile in the y- direction. The parameter s determines the steepness of the transition from a grating to the slip region with larger value of s resulting in a more rapid transition, and w is the width of the slip. Finally, h and H determine the relative height of the slip region and the heights of the grating ridges respectively. Three different grating profiles with different slip heights are considered. In all cases H = 0.65µm, P x = 1µm, α = 0.5, φ = 180, s = 20 and w = 0.5µm are used, but h is chosen as 0, 0.5 or 1. The resulting surface profiles are plotted in figures (7.6) (a), (b) and (c). The cell gap is chosen as 2.5 µm in each case and periodic boundary conditions are enforced along the (y, z) faces and Neumann boundaries along the (x, z) faces. The anchoring on all alignment surfaces is homeotropic. 124

142 Z [µ m] X [µ m] Y [µ m] Z [µ m] X [µ m] Y [µ m] Z [µ m] X [µ m] Y [µ m] (a) (b) (c) Figure 7.6: Three different surface profiles for the ZBD structure, with the height of the slip region set to 0, 0.5 and 1 times the ridge height in (a), (b) and (c) respectively. Modelling and Results For simplicity, a single elastic coefficient approximation with K = 15pN is assumed. For reasons of computational efficiency the thermotropic coefficients are set to A = 0Nm 2, B = Nm 2 and C = Nm 2, resulting in slightly larger defect core sizes than when using experimentally measured values. Since the emphasis here is on the defect structures and not the switching between between the two states, the effect of electric fields is not considered. In order to model the defect state of the device, the initial director field is set horizontal along the x direction within the troughs. The LC is then allowed to relax to the nearest stable state, which in this case is the D state. The C state can be obtained in a similar fashion by starting from a vertical director profile within the troughs. At a distance from the slip region, the director field is contained in the (x, z) plane as shown in figures (7.5) (a) and (b). Closer to the slip, the director twists into the y direction in order to satisfy the homeotropic boundary condition imposed by the vertical surfaces around the slip. Iso-surfaces of reduced order parameter due to the ± 1 2 defect pairs are shown in figures (7.7)(a), (b) and (c) for the three surface profiles. Closed defect loops are formed due to a continuous transition from positive to negative defect through a twisting of the director parallel to the axis of the defect line (marked with circles in the figures). The negative defect line is pinned to the convexly shaped 125

143 portions of the surface whereas the positive one follows the concave portions. The strength of the LC anchoring to the grating affects the distance between the surface and the defect line. Weaker anchoring allows the defect closer to the surface whereas stronger anchoring expels the defect further into the bulk of the LC. Further study of the ZBD geometry with an emphasis on the slip region is planned. In particular, the role of the slip region as a possible defect nucleation site in the switching process will be investigated. 7.6 Discussion and Conclusions Results obtained using the modelling tools developed for this work have been presented. These were compared to previously published data or to results obtained using other established methods in order to verify the correctness of the implementation. The process of verification was started with simple cases that take into account only a few LC characteristics at a time and then progressed to more complex situations. First, defect free cases where the elastic distortions dominate were considered in order to validate the response to electric fields and the three elastic coefficient formulation. This was then taken further by additionally solving for the flow of the LC material and observing the induced backflow after the release of a holding voltage. Then, the dynamics of pair annihilating half integer defect lines were modelled. This was done twice, first taking into account the flow of the LC and then without the flow. As expected, including the effect of flow favoured the movement of the positive defect, whereas when ignoring it the rate of movement of both defects was equal. Finally, a larger problem was considered in order to demonstrate the scope of the type of problems that can be tackled using the modelling software: The static defect line configurations in the slip region separating the ends of grating structures in a Zenithally Bistable Device was modelled in three dimensions. It is possible that this 126

144 region plays an important role in the switching dynamics of the device and further investigation is currently under way. 127

145 Iso-surfaces of reduced order Iso-surfaces of reduced order (a) (b) Iso-surfaces of reduced order (c) Figure 7.7: Iso-surfaces of reduced order parameter showing the locations of the defect lines. Circles are drawn to indicate the regions of the ± 1 defect transitions

146 Chapter 8 Modelling of the Post Aligned Bistable Nematic Liquid Crystal Structure 129

147 8.1 Introduction In this chapter the operation of a bistable device, the post aligned bistable nematic (PABN) [140] liquid crystal device is modelled. This is another example of bistable devices whose operation relies on a structured solid surface in contact with the LC material (see section 7.5 for more information on bistable technologies). Due to the geometry of the PABN structure, its operation cannot be fully described in two dimensions. In addition, defect dynamics of the director field is important in the switching process. For this reason, the 3D finite element discretisation of the Landau-de Gennes free energy described earlier in this thesis is used to model the dynamic behaviour of the device. The device geometry is explained and previously published information is introduced in section 8.2. The approach taken to modelling is explained in section 8.3 and results are given starting from section Overwiew of the The PABN Device The PABN device is a bistable liquid crystal device under development at the Hewlett- Packard laboratories. The bistability of the PABN device is achieved by sandwiching nematic LC material between two different surfaces. One of the bounding surfaces contains an array or grating of microscopic posts, whereas the other surface is flat. The anchoring on the flat surface (called from now on the top surface) is homeotropic. The bottom surface, including the surfaces of the posts, is untreated and imposes planar degenerate anchoring on the director. The result of this is that two distinct stable director configurations exist. Between crossed polarisers, one of these appers bright and the other dark [20]. The posts may be fabricated of photoresist on glass surfaces using photolitographic techniques, or directly of the substrate itself, which may e.g. be a flexible plastic [20]. 130

148 Various post shapes and sizes are reported to be possible, with dimensions ranging from 0.1 3µm and cross sectional shapes including circles, ovals, squares and diamond shapes. The distances between the posts and the cell gaps are also reported to be of similar magnitudes [140]. Previous theoretical predictions of the two stable states obtained using the Oseen- Frank theory in [19, 20, 140, 21] suggest that distinct director configurations with different levels of tilt angle exist. These are known as the planar and the tilted states. The predicted director fields in [19, 20, 140] suggest that the planar state is characterized by a pair of 1 2 defect lines along the vertical edges of the posts. It is suggested that a balance between the energies of the defects and the flat top surfaces of the posts result in the stability of the planar state. The tilted state is argued to be stable due to the absence of the defect lines of high energy. This argument is supported by experimental evidence of the planar state becoming unstable when the height of the post is increased sufficiently (resulting in longer line defects of higher total energy) with respect to the cell gap. More recently, in [21], four distinct stable configurations of the director field were modelled. None of these states corresponds to the previously suggested planar states, but the tilted states were topologically identical. The difference between the models is that in [21], the director field is fixed along the edges of the posts, whereas in the earlier publications this is not the case. 8.3 Modelling the PABN Device Before the operation of the full device is modelled, a single corner of a single post is considered. This is useful, since as will be shown later, the director configurations found around the single corner are found to be essential for the operation of the complete structure. 131

149 The material parameters used in the modelling are as follows: The single elastic constant approximation K = K 11 = K 22 = K 33 = 7pN was used, resulting in L Thermotropic coefficients for the 5CB material at 4 K below the nematic-isotropic transition (see appendix A) and the modified coefficients (A = 0Nm 2 K 1, B = Nm 2 and C = Nm 2 ) were used. Both sets of coefficients were found to result in qualitatively identical results, but using the modified parameters, the computational load is significantly reduced. A material with negative dielectric anisotropy with values ε = 8 and ε = 3 is chosen. The flexoelectric polarisation is represented using an expression linear in the gradient of the Q-tensor, which is the special case when the splay and bend coefficients are equal e 11 = e 33 = 3S 0 2 e (see section (2.5)). The value of e is chosen as Cm 1, which is comparable to both experimentally measured values [29] and theoretical predictions [30]. The anchoring on the bottom surface, including the surfaces of the post, is assumed planar degenerate and the anchoring energy density is written as: f s = a s Tr(Q 2 ) + W (ˆv i Q ijˆv j ), (8.1) where ˆv is the local surface normal unit vector and the coefficients a s and W have the same meaning as described in section (4.4.1). The value of the anchoring coefficient W is chosen large enugh to prevent topological changes through breaking of anchoring. It was found that W J/m 2 is sufficient. The value of a s is set to a s = W 6S 0. The anchoring of the top surface is assumed strong homeotropic The Geometry of the Modelling Window Two differerent modelling windows are needed, and can be seen in Fig Fig. 8.1a shows a cell containing a full post in the periodic structure while the calculation cell in Fig. 8.1b contains anly one corner (a quarter of the cross-section) of a post and a 132

150 fraction of the total height of the structure. The grating structure consisting of microscopic posts is assumed periodic (although it does not need to be [140]) allowing the modelling of a structure extending to infinity by considering a single cell with periodic boundary conditions. The external dimensions of the modelling window are µm, and the boundary conditions on the (x, z) and (y, z) planes are periodic. For the separated corner structure shown in figure 8.1 (b), the external boundary conditions are left free, corresponding to Neumann boundaries in the single elastic coefficient approximation. The base of the structure, including the post consists of isotropic dielectric material. The corners and edges of the post are rounded to a radius of 20 nm. The top surface of the base (including the post and its sides) are LC-solid-surface-interfaces, where planar degenerate anchoring conditions are applied. Strong homeotropic anchoring is applied at the top surface at z = 3.05µm. Planar electrodes are placed at z = 0 and z = 3.05µm. In the actual device, some degree of asymmetry is introduced to the geometry in order to ensure that a preferred alignment in the azimuthal direction exists [19, 20]. Here, this is achieved by the choice of the initial director field orientation, so that a main diagonal (x = y) can be identified along which on average the director is aligned. 8.4 Modelling Results A Topological Study of a Single Corner First, the stable configurations for the single corner are modelled by minimising the total free energy in the absence of electric fields. Three topologically distinct configurations are found by starting the minimisation process from different initial director configurations. These will be referred to as the horizontal, continuous vertical and discontinuous vertical states, after the orientations and distortions of the director 133

151 (a) (b) Figure 8.1: The geometries of the 3-D modelling windows (a) for the full device, and (b) for the isolated corner. fields found in the states. Furthermore, a defect state is modelled by minimising the free energy in the presence of an externally applied electric field. The horizontal state is characterised by the director field lying nearly parallel to the bottom surface (in the (x, y) plane) of the modelling window. The director field bends in a continuous fashion aroud the corner. Figure 8.2a shows the director field on a slice in the (x, y) plane at z = 0.3µm through the isolated corner structure. In the continuous and discontinuous vertical configurations, the bulk xy-components of the director field is approximately perpendicular to those of the the horizontal state (see Fig. 8.2b). However, the director field can be described as flowing over the corner, rather than bending around it, so the z-components of the director field is significant near the vertical surfaces of the corner. The difference between the continuous and discontinuous vertical states can be seen in figure 8.3a and b, where the director field is displayed on a vertical slice through the diagonal of the isolated corner structure, the (x = y, z) plane. Defects in the director field can be seen near the top and bottom corners of the structure in the discontinuous vertical configuration, whereas 134

152 (a) Horizontal Configuration (b) Continuous Vertical Configuration Figure 8.2: Director profiles for the horizontal (a) and continuous vertical (b) states on a regular grid along the (x, y) plane through the centre of the isolated corner structure at z = 0.3µm. The discontinuous vertical state is not shown, as it appears nearly identical to the continuous vertical state from this point of view. in the continuous vertical configuration these are not present. (a) Continuous Vertical (b) Discontinuous Vertical Figure 8.3: The director field on a regular grid along the diagonal (x = y, z) plane through the separated corner structure. (a) Stable continuous vertical configuration, (b) stable discontinuous vertical configuration In the defect configuration the director field lies in the (x, y) plane, with a 1 2 defect line extending along the edge of the post (see Figs. 8.4a and b). This state 135

153 is found to be unstable unless an externally applied electric field is present and the flexoelectric coefficient e is zero, and is a transitional configuration separating the two topologically nonequivalent stable vertical states. The defect configuration can be achieved by applying an electric field which due to the negative dielectric anisotropy aligns the director field along the horizontal (x, y) plane (to demonstrate the topology of the defect structure, e has been set to zero in figure 8.4). The effect of a non-zero flexoelectric coefficient e is to counteract the horizontally aligning dielectric response and to cause a director field deviation into the z direction. The degree and direction of this deviation depends on the magnitudes and signs/directions of e and the electric field. This vertical deviation is necessary for switching between the two stable states. After removal of the electric field, the director field relaxes to the vertical state that is closer to the angular deviation caused by the flexoelectric effect. If e is chosen as zero, the director field always relaxes to the continuous vertical state due to the geometry of the corner and the aligning effect it has on the director field. (a) (b) Figure 8.4: Defect line along a post edge during switching. (a) a magnified view of (x, y) plane at z = 0.3 µm cutting through the post. Darker background colour indicates a reduction in the order parameter near the defect core. (b) 3-D view of same post edge with a dark iso-surface for the order parameter showing the extent of the line defect. 136