Abstracts
Developing hierarchical finite element methods for modelling bistable liquid crystal devices
Christopher J.P. Newton and Stephen L. Cornford
Bistable liquid crystal devices often rely on defects to stabilise one or both of the stable states and to provide a barrier which prevents one of the, possibly higher energy, states relaxing into the other one. Modelling these devices is a challenge because of the disparate length scales. A device is typically several microns thick, while the stabilising defects are only tens of nanometres in size. Using a fine mesh everywhere is impractical and even refining the mesh near defect regions is a problem, particularly when modelling their switching characteristics as the defects often move during the switching process. Together with colleagues in the Mathematics Department of Strathclyde University we have been developing a hierarchical finite element code to model these devices. The initial assumption when starting this project, that using hierarchical finite elements would allow us to develop simple adaptive methods with system requirements comparable to the widespread adaptive methods based on mesh refinement, has been borne out by our results so far. We report on these results and the challenges that still face us if we are to develop a really efficient three dimensional modelling code.
Simulation studies of structure and organisation in chromonic phases and macromolecular liquid crystals
M. R. Wilson, J. S. Lintuvuori and F. Chami
Molecular order within a chromonic phase is not well understood. From system to system, little is known about how molecules order. Moreover, we know little about the strength of interactions between molecules in stacks or the ordering of water molecules around stacks. We presents new results from an atomistic simulation study designed to answer some of these fundamental questions. Results are presented from atomistic simulations of a chromonic stack for the molecule edicol (sunset yellow) in water, looking at the ordering of edicol molecules in the columns, the structure of the bulk phases and the free energy change on binding to a chromonic stack. Results are also presented for quantum predictions of NMR shielding. Simulation results are linked to recent experimental measurements in solution, which aim to providing a fundamental understanding of how self-assembly occurs in chromonics.
Macromolecular liquid crystals composed of polyphilic segments are of considerable importance in soft matter chemistry. In such systems, unfavourable molecular interactions between different parts of a molecule can lead to nanophase/microphase separation and the formation of a range of fascinating structures. Careful, tuning molecular interactions provides the possibility to produce self-assembled structures for future molecular electronic devices or for biomimetic applications. We present new coarse-grained potentials, which allow us to see molecular self-assembly in complex architectures. We show results for single site systems as well as for a multipedal liquid crystal. For the latter, we see spontaneous self assembly into a lamellar phase, with a coupled change in molecular structure. We show also how the same methodology can be applied to a range of other macromolecular systems.
We show also how our new coarse-grained potentials can be combined with the recently developed technique of Statistical Temperature Molecular Dynamics (STMD). The latter uses the relation between statistical temperature and density of states; effectively combining multicanonical molecular dynamics with Wang-Landau sampling by dynamic update of the statistical temperature estimate. This can be used as a way of sampling a temperature window efficiently to identify different mesophases; and also of providing free energies and entropies, which are difficult to obtain by conventional simulation methodologies.
[1] Lintuvuori, J. S., Wilson M. R., Phys. Chem. Chem. Phys., 2009, 11, 2116.
[2] Lintuvuori, J. S., Wilson, M. R., J. Chem. Phys., 2008, 128, 044906.
[3] Kim, J., Straub, J. E., Keyes, T., Phys. Rev. Lett., 2006, 97, 05061, Kim, J., Straub, J. E., Keyes T., J. Chem. Phys., 2007, 126, 135101, Kim, J., Straub, J. E., Keyes, T., Phys. Rev. E., 2007, 76, 011913.
Flow and Texture Modeling of Liquid Crystalline Materials
Alejandro D. Rey
A review of flow and texture modeling of liquid crystalline materials with emphasis on carbonaceous mesophases is presented. Two models of nematodynamics are presented and discussed in terms of their ability to resolve time and length scales likely to arise in typical rheological and processing flows. Defect physics and rheophysics are integrated with nematodynamics and specific mechanisms of defect nucleation and annihilation are used to derive texture scale power laws. The integrated nematodynamics models specialized to carbonaceous mesophases are used to analyze: (i) linear and nonlinear viscoelasticity, (ii) rheological flows, and (iii) carbon fiber and flow-induced textures. The linear and nonlinear viscoelasticity reveals the essential nature of these materials : coupling between flow-induced orientation and orientation-induced flow , elastic storage through orientation gradients, and anisotropy.
The rheological flow simulations, shown to be in excellent agreement with experimental data, reveal several liquid crystal specific rheological characteristics including shear thinning due to anisotropic viscosities and flow-induced orientation, and negative first normal stress difference due to orientation nonlinearities in the shear stress. Nematodynamic predictions are shown to follows a Carreau-Yasuda liquid crystal equation. Nematodynamics predictions rationalize shear-induced texture refinement in terms of defect nucleation and coarsening mechanisms and are used to derive texture scaling relations in terms of macroscopic, molecular, and flow time scales. This knowledge is then condensed into a generic texture-flow diagram that specifies the required temperature and Deborah number required to produce well oriented monodomain materials. The fine details of mesophase structuring by flow through screens are shown to be captured by nematostatic simulations.
Finally the mechanisms behind the carbon fiber textures produced by melt spinning of carbonaceous mesophases are elucidated. The proven range and predictive accuracy of nematodynamics to simulate flows of textured mesophases and the ever-growing industrial interest in lower cost high performance super-fibers and functional materials will fuel the evolution of liquid crystal rheology and processing science for years to come.
Ferronematic and Ferroelectric colloidal liquid crystals
T.J. Sluckin
Brochard and de Gennes [1] suggested in 1970 that a colloid of magnetic particles suspended in a nematic matrix could dramatically amplify the interaction between a magnetic field and the nematic director. An experimental verification of this phenomenon by Chen and Amer [2] has proved very difficult to reproduce, probably because these suspensions are difficult to stabilise. In recent work [3], we have been able to fit these experimental results phenomenologically, as well as predict a rich spectrum of Frederiks-like phenomena, depending on the magnitude of the magnetic-nematic coupling. More recently there has also been experimental work in a number of laboratories in which repeatable magnetic effects are observable. A related system involves ferroelectric, rather than ferromagnetic, colloidal particles. The addition of colloidal particles in these systems sometimes increases the nematic-isotropic phase transition temperature [4], thus presumably strengthening the effective nematic interaction, although cooperative effects can in principle reduce the Frederiks threshold field [5]. In colloidal systems under consideration the orientational effect of the colloidal particles is sufficiently weak not to cause defect formation, but sufficiently strong that the colloidal particles can affect the local nematic order parameter over length scales long compared to the inter-particle distance. In this talk I shall give a brief review of the current status of these materials, emphasising our new results and open problems.
[1] F. Brochard and P.G. de Gennes, J. Phys. France 31, 691 (1970).
[2] S.H. Chen and N.M. Amer, Phys. Rev. Lett. 51, 2298 (1983).
[3] V. I. Zadorozhnii, A. N. Vasilev, V. Yu. Reshetnyak, K. S. Thomas and T. J. Sluckin, EPL 73, 408 (2006).
[4] F. Li et al, Phys. Rev. Lett. 97, 147801 (2006).
[5] V.Y. Reshetnyak, S.M. Shelestiuk and T.J. Sluckin, Mol. Cryst. Liq. Cryst. 454, 201 (2006).
Continuum theory for biaxial nematics desrcibed by two order parameter tensors
Andre Sonnet
I derive from a variational principle the equations of motion for a biaxial nematic liquid crystal described by two second-rank order parameter tensors. The constitutive ingredients are the densities of free energy and dissipation. With an appropriate choice of Landau-deGennes potential in the bulk free energy, the resulting theory is valid in the uniaxial and biaxial nematic phases and in the isotropic phase. I consider pre-transitional phenomena and the relation two the classical two-director theory.
Steric effects on intermolecular forces
Epifanio Virga
The lecture will review some recent attempts to incorporate in one and the same mathematical model for molecular interactions both the long-range, attractive forces, generally of a dispersion nature, and the short-range, repulsive forces, generally of a steric nature. An appropriately defined steric tensor appears to be a useful tool to gain a unified view on intermolecular forces: to illuminate the theory, the steric tensor will be computed for a number of illustrative cases.
Julia Yeomans
Active systems, such as bacterial swarms and cytoskeletal gels, produce their own energy. The equations that have been used to describe active systems at the continuum level are essentially those of liquid crystal hydrodynamics, together with an additional term in the stress tensor representing the activity. A key prediction of such models is the existence of self-stabilising kink states that spontaneously generate fluid flow in quasi-one dimensional channels [1]. We describe the kink states for both contractile and extensile particles moving in a narrow channel as a function of their flow alignment properties and initial orientation [2]. This provides a framework for unifying many of the results in the literature and for discussing the relation between microscopic and continuum models of active systems.
[1] R. Voituriez, J.F. Joanny and J. Prost, EPL 70 (2005) 404.
[2] S.F. Edwards and J.M. Yeomans, EPL 85 (2009) 18008.
Sidewall Control of Multistable Azimuthal Nematic Liquid Crystal Alignment
C.V. Brown, S. Ladak, C. Evans, A. Davidson, and N.J. Mottram
Bistable alignment of a nematic liquid crystal has previously been achieved using a structured surface morphology on one of more of the substrates that confines the nematic layer [1-6]. If the bounding substrates are flat we have shown that a number of distinct stable azimuthal (in-plane) alignment configurations can be created in a planar nematic layer that is laterally confined in a square shaped well [7-9].
In the current work bistable and multistable nematic liquid crystal alignment configurations have been created in channels of planar nematic liquid crystal that are laterally bounded by one or more in-plane sawtooth grating structures [10]. This in-plane geometry allows the 2-dimensional director configurations to be simply interpreted from their optical textures and provides an excellent model system for studying the stabilising influence of nematic defects. The optical textures of the observed states are in excellent agreement with the predictions of nematic Q-tensor theory.
Preliminary results will be presented from time resolved studies of the switching between optical textures using two types of liquid crystal: (i) standard liquid crystal materials where coupling to the flexoelectric polarisation plays an important role, and (ii) dual frequency liquid crystals in which the sign of the dielectric anisotropy depends on the frequency of an applied a.c. voltage.
[1] R.N. Thurston et. al. IEEE Trans. Elec. Dev. (1980) ED27(11), 2069
[2] R. Barberi et. al. J. Appl. Phys. (1998) 84(3), 1321
[3] C.V. Brown et. al. Liq. Cryst. (2000) 27, 233
[4] S. Kitson and A. Geisow, Appl. Phys. Lett. (2002) 80(19), 3635
[5] A. Majumdar et. al. Phys. Rev. E. (2007) 75, 051703
[6] J-H. Kim et. al. Nature (2002) 420, 159
[7] N.J. Mottram et. al. UK patent GB20040026582, 8/6/2006
[8] C. Tsakonas et. al. Appl. Phys. Lett. (2007) 90, 111913
[9] G.G. Wells and C.V. Brown, Appl. Phys. Lett. (2007) 91, 223506
[10] S. Ladak et. al. J. Phys. D: Appl. Phys. (2009) 42, 85114
Backflow dynamics of nematic defects
Paolo Biscari and Tim Sluckin
We present an asymptotic theory which includes in a perturbative expansion the coupling effects between micro- and macroscopic dynamics in a nematic liquid crystal. Backflow effects are most significant in the presence of defect motion, since in this case the macroscopic motion may strongly reduce the total energy dissipation and, thus, increase the defects' velocity. As an example, we illustrate how backflow influences the speeds of opposite-charged defects.
Modelling Smectic A Liquid Crystals
Iain Stewart
Smectic liquid crystals are anisotropic layered fluids that exhibit elastic properties and are sensitive to applied electric or magnetic fields. A novel nonlinear continuum theory for the statics and dynamics of smectic A liquid crystals will be presented and some applications will be given, e.g. (i) distortions in liquid crystal cells, (ii) the equilibrium structure of families of biological lipid bilayers subject to weak anchoring boundary conditions. The roles of some important material parameters, such as surface tension, compression modulus and weak anchoring strength, will be highlighted. Comparisons with experimental data will also be made.
The Onsager model for biaxial nematics
Jonathan Robbins and Valeriy Slastikov
We discuss the Onsager model for biaxial nematics, and obtain some rigorous results for the phase diagram in the case of the London interaction. These are based in part on results of Fatkullin and Slastikov (2005) for the Maier-Saupe interaction.
Analysis of multiscale phenomena in liquid crystal elastomers
Carme Calderer
We consider boundary value problems arising in mechanical and electrical loading of samples of liquid crystal elastomers. The energy of the problem couples elastic deformation and order tensor gradients with electric elds. We study the equilibrium equations analytically and numerically, and investigate critical geometrical and loading parameters that determine the transition between regular periodic pattern and congurations with isotropic defects and cavities. We discuss modeling problems arising in the coupling of nematic order with electric elds, for dielectric as well as ferroelectric mesogenic side chains, and their possible implications in device manufacturing.
Computer simulations of liquid crystals: displays and beyond
Claudio Zannoni
The development of liquid crystals with specifically tailored functionalities increasingly relies on establishing a robust link between molecular features of the mesogens and macroscopic properties. This implies at first a generic understanding of the effects of molecular shape, elongation, dipoles etc. on the resulting molecular organizations, e.g. calamitic or discotic liquid crystal phases. However, a much more ambitious goal is that of predicting liquid crystal properties from a specific molecular structure. This does not only imply calculating single molecule observables at one generic state point of choice, but rather determining physical properties and morphologies of the materials at various temperatures and working conditions, with the inevitable consequence of extremely demanding computer simulations. In the talk we plan to show some examples of current applications of computer simulations of liquid crystals in various areas [1,2]. At the molecular resolution level we show, in particular, the simulation of a simple twisted nematic device[3]. At atomistic level, we demonstrate the current possibilities of predicting the organization and properties of low molecular mass liquid crystals starting from their molecular structure and in particular the prediction of transition temperatures and observable properties (e.g. dielectric constant, NMR dipolar couplings) for nematics [4]. We also examine some recent applications of methodologies originated from liquid crystals to organic electronics [5] and proteins [6].
[1] R. Stannarius, More Than Display Fillings, Nature Mater. 8, 617 (2009)
[2] S. Sergeyev, W. Pisula, Y. H. Geerts, Discotic Liquid Crystals: A New Generation of Organic Semiconductors, Chem. Soc. Rev. 36, 1902 (2007)
[3] M. Ricci, M. Mazzeo, R. Berardi, P. Pasini, C. Zannoni, A molecular level simulation of a twisted nematic cell , Faraday Discussions, published on line (2009). DOI: 10.1039/b901784d
[4] G. Tiberio, L. Muccioli, R. Berardi, C. Zannoni, Towards in silico liquid crystals. Realistic transition temperatures and physical properties for n-cyanobiphenyls via molecular dynamics simulations, ChemPhysChem 10, 125 (2009)
[5] Y. Olivier, L. Muccioli, V. Lemaur, Y.H. Geerts, C.Zannoni, J. Cornil, Theoretical Characterization of the Molecular and Hole Transport Dynamics in Liquid-Crystalline Phthalocyanine Stacks, J. Phys. Chem., accepted, (2009)
[6] A. Pietropaolo, L. Muccioli, R. Berardi, C. Zannoni, A chirality index for investigating protein secondary structures and their time evolution, Proteins 70, 667 (2008); A. Pietropaolo, L. Muccioli, C. Zannoni, E. Rizzarelli, Conformational preferences of the full chicken prion protein in solution and its differences with respect to mammals, ChemPhysChem, 10, 1500 (2009)
Acknowledgments: this work was partly supported by EU project BIND FP7-216025.
Mesh-free methods for liquid crystal simulation
Doug Cleaver
We present here a study that relates to the development of novel mesoscopic off-lattice simulation techniques for ordered fluids, in particular thermotropic LCs, to be applied in both static and flow regimes.
At first we considered dissipative particle dynamics (DPD) as a method of choice due to its simplicity and popularity. DPD [1,2] is a promising mesoscopic simulation technique which, over the last decade, has become a popular method for simulating dynamical and rheological properties of both simple and complex fluids. Usefully, Ellero et al [3], following ten Bosch [4], have investigated the behaviour of DPD particles invested with an additional vector degree of freedom, culminating in the development of a fluid particle model for viscoelastic flows. We have extended the ideas of Ellero et al by associating a traceless, symmetric, order tensor, Q, with each DPD particle and incorporating Q-tensor dependence into the interparticle forces and torques. The resulting method is very efficient and offers qualitative insight into some complex LC behaviours. However, it also suffers significant deficiencies which limit its practical utility.
In the light of this, we have alternatively considered LC simulations base don the class of top-down approaches termed "meshfree methods". In this, we have taken a continuum LC description and, through appropriate discretisation, determined the interaction terms that are required to yield a particle-based simulation which has a consistent description at the mesoscale. This approach follows the general methodologies of Modified Smoothed Particle Hydrodynamics (MSPH) [5,6] which allow one to solve partial differential equations on a set of randomly distributed interpolation points. The result is a continuum solver which can readily be applied in any number of dimensions and is inherently multi-scale. It is, thus, particularly effective at modelling LC systems involving defects and complex surfaces [7].
References:
[1] P. J. Hoogerbrugge and J. M. V. A. Koelman, Europhysics Letters, 19, 155 (1992)
[2] P. Espanol and P. B. Warren, Europhysics Letters, 30, 191 (1995)
[3] M. Ellero, P. Espanol, E.G. Flekkoy, Physical Review E, 68, 041504 (2003)
[4] B.I.M. ten Bosch, Journal of Non-Newtonian Fluid Mechanics, 83, 231 (1999)
[5] M.B. Liu, W.P. Xie and G.R. Liu, Applied mathematical modelling, 29, 1252 (2005)
[6] G.M. Zhang and R.C. Batra, Computational mechanics, 34, 137 (2004)
[7] M.V. Yakutovich, C.J.P. Newton, and D.J. Cleaver, Molecular Crystals and Liquid Crystals 502, 245 (2009)
Steric contributions to ordering potentials in nematic liquid crystals
Fulvio Bisi
Steric effects appear to play a key role in inducing ordered states; for example, purely polar steric interactions have been shown to be capable of inducing unexpected orientationally ordered states [1]. In addition to that, it is known that the quadrupolar approximation to the excluded-volume interaction between hard spherocuboids can be written precisely as the superposition of two London interactions: one repulsive and one attractive [2]; such a combination of flavors is also met in a molecular field model for biaxial nematics [3]. Finally, it has been shown that dispersion forces interaction can be combined with hard-core repulsion in a theory based on a steric fourth-rank tensor [4]. The lecture will show some recent achievements in this field.
The excluded volume for non-convex molecules -modelled as chains of tangent hard spheres- can be computed analytically as a function of the relative orientation; we show how to compute such function and expand it over a set of WSAFs (Wigner's Symmetry Adapted Functions), to obtain a viable approximation to the pure excluded volume potential.
Furthermore, we describe a pair Hamiltonian incorporating the average electric dipolar energy exchanged between molecules with the same excluded region; such a potential is used to build a mean-field model. A numerical bifurcation analysis is performed, and we discuss the dependence of the stability of different phases upon two interaction parameters, one for the degree of intrinsic biaxiality in the shape tensor and one for the relative orientation of the electric dipole within each molecule.
[1] Bisi, F.; Rosso, R.; Virga, E.G.; Durand, G.E. Phys. Rev. E 2008, 78, 011705.
[2] Rosso, R.; Virga, E.G. Phys. Rev. E 2006 74 , 021712.
[3] Bisi, F.; Virga, E.G.; Gartland Jr, E.C.; De Matteis, G.; Sonnet, A.M.; Durand G.E. Phys. Rev. E 2006, 73 , 051709.
[4] Sonnet, A.M.; Virga, E.G. Phys. Rev. E 2008, 77, 031704.
Energetic Variational Principle in Complex Fluids
Chun Liu
Complex fluids are prevalent in many important physical, biological and engineering applications: industrial macroscale flows such as polymer extrusion, small scale flows such as those found in and around cells in systems biology, geophysical flows as in
subsurface contaminant transport, carbon sequestration and chemical injection for oil recovery and multi-phase (component) flows such as combustion of powders, solid (coal) particles in flames, and droplets in air (clouds). The most common origin and manifestation of anomalous phenomena in complex fluids are different "elastic" effects. The different rheological and hydrodynamic properties can be attributed to the special coupling between the transportation of the internal variable and the induced elastic stress. In our energetic formulation, this represents a competition between the kinetic energy and the elastic
energy. I will go over the the modeling of some of these examples. We will discuss the common feature of these models and also the specific difficulties associated with different applications.
3D assemblies of colloidal particles in homogeneous and inhomogeneous nematics
S. Žumer
Our recent modeling of assemblies of colloidal particles dispersed in homogenous and inhomogenous nematic phases is described. The approach is based on phenomenological description and topological theory. In solvents where nematic order is present an effective anisotropic long range inter-particle coupling appears. It together with topological constrains leads to numerous organizations of colloidal particles not present in simple liquids [1-4]. Particularly interesting are situations where particles do not shear only areas of deformed order but also disclination lines that leads to a string-like coupling [2,4]. Recent developments oriented toward three dimensional structures will be illustrated on the assembling of 1D and 2D structures in 3D lattice and on possible selfassembly of colloidal particles in intrinsically inhomogeneous blue phase structures [5]. We expect that nematic colloids will open new ways toward the assembling of complex structures needed for photonics and metamaterials.
References:
(1) I. Muševič, M. Škarabot, U. Tkalec, M. Ravnik, S. Žumer, Two-dimensional nematic colloidal crystals self-assembled by topological defects, Science 18, 954-958 (2006)
(2) M. Ravnik, M. Škarabot, S. Žumer, U. Tkalec, I. Poberaj, D. Babič, N. Osterman, and I. Muševič, Entangled Nematic Colloidal Dimers and Wires, Phys. Rev. Lett 99, 247801 (2007)
(3) U. Tkalec, M. Ravnik, S. Žumer and I. Muševič, Vortexlike Topological Defects in Nematic Colloids: Chiral Colloidal Dimers and 2D Crystals, Phys. Rev. Lett. 103, 127801 (2009).
(4) M. Ravnik and S. Žumer, Nematic colloids entangled by topological defects, Soft Matter 5, 269 (2009).
(5) G. P. Alexander and J. M. Yeomans, Stabilizing the blue phases, Phys. Rev. E 74, 061706 (2006)
Computing the wavespeed of Soliton-like solutions in SmC* Liquid Crystals
Lawrence Seddon
We present a novel method for numerically computing the wave speed of a soliton-like travelling
wave in SmC* liquid crystals that satisfies a parabolic partial differential equation with a
general nonlinear term. We show by transforming the PDE to a co-moving frame and recasting
the resulting problem in phase-space, that the original PDE can be expressed as an
exceptional nonlinear Volterra-type equation of the second kind which maybe solved
numerically for the wave speed and we demonstrate a technique for doing so.
Ion effects in liquid crystal cells
Andrew Davidson
Much research has been done into manufacturing improved liquid crystal materials with the aim of removing ionic interference to improve switching characteristics. However, in some bistable displays it has been seen experimentally that ions can be used to reduce required switching voltages and may even be the dominant switching force in other devices. Here we show the reaction of ions to AC and DC fields and introduce the premise of a maximum ion density to improve modeling accuracy. We demonstrate how the presence of ions can be used to increase the magnitude of the electric field in the cell during switching and investigate how ions can be used to switch a bistable device.
Effects of flexoelectricity in nematic liquid crystal colloids
Tine Porenta
Nematic liquid crystals have an apolar director. Nevertheless, a macroscopic polarization called flexoelectric polarization can be induced in the medium by splay and bend elastic deformation of the director field. Consequently, it is the largest in the regions surrounding nematic defects and it can become of substantial importance in confined geometries such as nematic colloids. Here we present preliminary results of our theoretical study on the effect of flexoelectricity in nematic liquid crystal colloids. Modelling is based on the numerical minimization of the mesosocopic Landau-de Gennes free energy. In single particles and colloidal dimmers, flexoelectricity is found to determine the local structure of liquid crystalline defect and their spatial conformation. Additionally, flexoelectricity also affects relative (meta)stability of various colloidal structures.
A fast centre manifold based algorithm to determine steady state liquid crystal alignment
Keith Daly
We use a centre manifold based algorithm to solve the normally stiff equations of motion which determine liquid crystal alignment. We assume that the order parameters of the liquid crystal are, to first order, determined entirely by the bulk free energy. The elastic and electrostatic free energy contributions, which are relatively small away from defects, are then treated as a perturbation which drives the dynamics of the liquid crystal alignment. The resulting equations can then be solved relatively quickly using a numerical root finding method on a desktop PC.
Uniqueness in the Freedericksz transition with week anchoring
Fernando Costa
We consider a boundary value problem for a quasilinear pendulum equation with nonlinear boundary conditions that arises in a classical liquid crystals setup, the Freedericksz transition. A change of variables transforms the problem into the equation xtt = -f (x) for t ∈ (-T, T), with boundary conditions xt = ±(β/T)f (x) at t = ∓T , for a convex non-linearity f . By analysing an associated inviscid Burgers' equation, we prove uniqueness of monotone solutions in the original non-linear boundary value problem.
This result has been for many years conjectured in the liquid crystals literature, e.g. in [E.G. Virga, Variational Theories for Liquid Crystals, Appl. Math. Math. Comput., vol. 8, Chapman & Hall, London, 1994] and in [I.W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction, Taylor & Francis, London, 2003].
Biaxial Nematics formed from Flexible Molecules
Tung To
The early theories and so predictions for biaxial nematics were based on the assumption that the constituent molecules were rigid and possess D2h symmetry. Neither assumption is true for real mesogenic molecules; invariably the molecules contain flexible alkyl chains which cause the molecular shape, symmetry and biaxiality to change with conformation. This is especially true when the alkyl chains link the mesogenic groups as in liquid crystal dimers and tetrapodes [1], both of which have been claimed to form a biaxial nematic phase [2, 3].
The influence of the flexibility on the phase behaviour is known to be profound for uniaxial nematic phases because of the coupling between the conformational and orientational order. In particular, the elongated conformers are stabilised in the uniaxial nematic phase at the expense of the bent. This results in the classic odd-even effects exhibited by liquid crystal dimers and the prediction of a nematic-nematic transition for odd but not even dimers [4]. It is to be expected that such effects should also influence the stability of the biaxial nematic phase which could favour the bent conformers with respect to the linear. To explore these intriguing possibilities we have used a generic model of liquid crystal dimers in which the various conformers are represented by just two, one linear and the other bent form [4].
The model has been investigated using a molecular field approach. In this the two arms of the dimers are taken to be equivalent so that the biaxiality of the bent form depends solely on the interarm angle. In fact, when this is tetrahedral (cos-1(-1/3)) the molecular biaxiality is maximal and a system of such bent conformers is found to undergo a weak transition directly from the isotropic to the biaxial nematic phase [5, 6]. We have studied the predicted phase behaviour of a mixture of exchanging bent and linear conformers as a function of the internal energy difference between them. The model system is found to exhibit a fascinating rich phase behaviour including a uniaxial nematic made of linear conformers, another made of bent conformers and a biaxial nematic formed from bent conformers. In a previous molecular field calculation an assumption was made that the Boltzmann factor governing the relative amount of the two conformers is constant with temperature [7]. By allowing this Boltzmann factor to vary with temperature we determined a more physical phase diagram relating the temperatures and the conformational energy difference between the linear and bent conformers. A similar phase behaviour was found although the biaxial nematic phase is an island surrounded by a uniaxial nematic.
References:
[1] K. Merkel, A. Kocot, J. K. Vij, R. Korlacki, G. H. Mehl, and T. Meyer. Phys. Rev. Lett., 93,
237801, 2004.
[2] M. J. Freiser. Phys. Rev. Lett., 24, 1041, 1970.
[3] H. Toriumi. Private communication.
[4] A. Ferrarini, G. R. Luckhurst, P. L. Nordio, and S. J. Roskilly. Liq. Cryst., 21, 373, 1996.
[5] F. Biscarini, C. Chiccoli, P. Pasini, F. Semeria, and C. Zannoni. Phys. Rev. Lett., 75, 9,
1803, 1995.
[6] M. A. Bates and G. R. Luckhurst. Phys. Rev. E, 72 051702, 2005.
[7] A. Ferrarini, G. R. Luckhurst, P. L. Nordio, and S. J. Roskilly. Chem. Phys. Letts., 214,
409, 1993.
Fast-switching flexoelectric display with high contrast
Flynn Castles
The flexoelectro-optic effect provides a fast switching mechanism (0.01 - 0.1 ms), suitable for use in field- sequential -color, full motion video displays. An in- plane electric field is applied to a short pitch chiral nematic liquid crystal aligned in the Uniform Standing Helix (or Grandjean) texture. The switching mechanism is experimentally demonstrated, and the display performance is theoretically investigated as a function of device parameters. A contrast ratio of 2000:1 is predicted.
Numerical investigation of frustrated nematic liquid crystal system by using Q tensor model
Giuseppe Lombardo
Nematic liquid crystals are aggregates of calamitic molecules and most related experimental phenomena are well described by their mean molecular orientation, i.e. by the director, and by the scalar order parameter, considering a perfect uniaxial symmetry. However, when the nematic distortion is very strong and it occurs over a length scale comparable with the nematic coherence length, the molecular order may be significantly altered, as in the case of the core of a defect or in the case of highly frustrated nematic systems. Such systems, where spatial and/or temporal changes of the nematic order are relevant, require a full Landau-de Gennes Q-tensor description.
In this work, we present the implementation of the Q-tensor model by using one/bi-dimensional finite element methods that allow to describe the dynamical evolution of the nematic phase. The main numerical results obtained by nano-confining or by elctrical stressing nematic material will be discussed and compared with experimental measurements.
Inverse Problems of Liquid Crystals
Basang Tsering Xiao
It is interesting to investigate the optical tomography of liquid crystals. Mathematically we call it inverse problem and the goal is to reconstruct the
permittivity from the measured data. For solving the inverse problem we need to solve the forward problem first which is to prove the existence and uniqueness
solution of Maxwell's Equations. In this poster, we showed some part of our work on both forward and inverse problem.
Defects in nematic liquid crystals - uniaxiality versus biaxiality
Apala Majumdar
We study equilibrium configurations in a spherical droplet with homeotropic boundary conditions. We carry out an analytic study of the uniaxial radial hedgehog configuration. In particular, we construct inner and outer approximations to the uniaxial radial hedgehog solution and make predictions about its energy. We compare the uniaxial radial hedgehog configuration to the biaxial torus configuration and make predictions about the stability and instability of the uniaxial radial hedgehog configuration, as a function of the ball radius and the absolute temperature.