Symmetry, Bifurcation and Order Parameters
Monday 7th January 2013 to Friday 11th January 2013
09:00 to 09:45  Registration  
09:45 to 10:00  Welcome by Director, Professor John Toland  INI 1  
10:00 to 11:00 
Fluids with attitude
This talk includes: an account of the early development of liquid crystal science (c18881940), a history of key mathematical ideas, a discussion of devices in liquid crystals, and finally a personal view of future perspectives.
Friedrich Reinitzer (1888, Prague) found that cholesteryl benzoate exhibited two fluid phases, one of which was cloudy. Otto Lehmann (Karlsruhe) studied similar compounds, which seemed both liquid and crystal, hence the term “liquid crystal”. In France Georges Friedel (1922) realised that liquid crystals were instead orientationally ordered fluids, inventing the terms “nematic”, “smectic” and “cholesteric”. The first (“swarm”) theory was due to Emil Bose (Danzig, 1908). More successful was the distinguished Swedish theoretical physicist Carl Wilhelm Oseen, who constructed a hydrostatic theory of liquid crystals (192244). Oseen’s theory explained Frederiks’s results from the USSR on threshold fields, but his Ph.D. student Adolf Anzelius was not able to build a consistent dynamical theory.
The first statistical mechanical theory was a mean field picture based on CurieWeiss magnetism, due to François Grandjean (France, 1917). His work was ignored. The later 1958 MaierSaupe theory is essentially identical. The liquid crystal order parameter is due to the Russian physicist Victor Tsvetkov in 1941.
Liquid crystal devices emerged in the USA in the 1960s. The TN cell was patented in 1970. There remains dispute over its discovery. Device development was accompanied by enormous theoretical and mathematical activity. The current hydrodynamic theory due to Ericksen and Leslie in 1966, and the Landau theory due to de Gennes in 1970, have been immensely influential. The importance of lcd’s has focussed mathematical interest in liquid crystals in recent years, but in the future these ideas may be even more important in understanding processes in living cells.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Liquid crystals, liquid crystal droplets and colloids: basic properties
The review presents basic physical properties of liquid crystals: orientational and translational elasticity (OseenFrank formalism), surface anchoring, and topological defects. A special emphasis is on liquid crystals in form of drops and on liquid crystals containing small particles in the bulk. The liquid crystal colloids enable new physical effects, such as nonlinear electrophoresis. The work presented is based on research sponsored by the National Science Foundation, USA.

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
EG Virga (Università degli Studi di Pavia) Order tensors of equilibrium phases
Order in fluids is described by means of tensorial measures which have in general a statistical derivation, as appropriate moments of a molecular distribution function. Their very definition places order tensors at the crossroad between microscopic and macroscopic approaches, in that land of middle where "mesoscopic" theories flourish. We shall give specific examples of how different order tensors can describe the equilibrium phases of different ordered fluids. Both uniaxial and biaxial nematic liquid crystals will be covered, as well as smectic liquid crystals, thus showing how both orientational and spatial orderings can be represented in one and the same setting. Our presentation will not be limited to phases in bulk; ordering at interfaces will also be considered along with the ways it can be affected by the surface curvature. Similarly, ordering of fluids on twodimensional curved manifolds will be described, and an "augmented" theory will be presente d for nematic shells, which to the inplane orientational order adds information on the escape of molecules along the shell's normal.

INI 1  
15:00 to 15:30  Afternoon Tea  
15:30 to 16:30 
I Stewart (University of Strathclyde) EricksenLeslie theory for nematic liquid crystals and its developments
A review of the EricksenLeslie theory for the dynamics of nematic liquid crystals will be presented. Its developments related to smectic and other liquid crystals phases will also be discussed. Applications to 'switching phenomena' in liquid crystals will be mentioned.

INI 1  
16:30 to 17:30  T.B.A.  INI 1  
17:30 to 18:00  Welcome Wine Reception 
09:00 to 09:40 
An Overview of the OseenFrank elastic model and some symmetry aspects of the Straley MeanField model for Biaxial Nematic liquid crystals
In this (mostly) expository talk, we will present a brief overview of two independent topics: (1) the OseenFrank model for the spatially varying orientational properties of confined uniaxial nematic liquid crystals and (2) the meanfield model of Straley for the bulk phase behavior of biaxial nematic liquid crystals. The OseenFrank elastic model is a phenomenological variational model for equilibrium orientational properties characterized by a unitlength vector field. It is a macroscopic continuum model that has been very successful in modeling liquid crystals at the typical scales of experiments and devices. We will discuss the development of the model, its range of applicability, its relation to other models, and its strengths and weaknesses. The meanfield model of Straley was put forward almost 40 years ago as an attempt to describe the bulk phases and transitions for molecules of an architecture that would promote the development of spontaneous biaxial order. The model has received an intensive reexamination in recent years. Some of the symmetry properties of the model arise in unconventional ways, such as through degeneracies in the representations of the states of the system. We will discuss the symmetries of this model and the multiple symmetrybreaking bifurcations encountered in the numerical exploration of its equilibrium phases.

INI 1  
09:40 to 10:20 
X Zheng (Kent State University) On the MaierSaupe theory of Nematic liquid crystals
MaierSaupe theory, the first successful theoretical model of thermotropic nematic liquid crystals (LCs), is a mean field description of a system of cylindrically symmetric particles interacting via London dispersion forces. The theory predicts a uniaxial nematic phase at low temperatures, and a first order phase transition to an isotropic fluid phase as the temperature is increased. In this talk, I will first give a brief introduction of the canonical MaierSaupe theory, then extend it to biaxial LC molecules, to inhomogeneous LCs, and to higher spatial dimensions, and discuss the relation between MaierSaupe and other main theories of LCs.

INI 1  
10:20 to 11:00 
L Longa (Uniwersytet Jagiellonski) The Landau theory of liquid crystals and its molecular interpretation
A purpose of this review is to give a systematic exposition of the Landau theory of phase transitions (LTPT). The construction of the Landau free energy expansion and the methods to identify the equilibrium structures will be discussed and illustrated with examples of phase transitions involving nematic and smectic phases. Then it will be demonstrated
how the Landau theory can be derived in a systematic way from (molecular) density functional theories, like mean field. This connection is especially attractive for it allows to view the LTPT as a simple tool to study bifurcation mechanisms, leading to phase transitions at the level of molecular modeling.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:10 
Onsager  type theorias, where to from here?
I will attempt to traverse the road from basic equations of molecular dynamics at the microscopic level to PDE dynamics at the level of coarsegrained order parameters, trying to clearly capture approximations and simplifications made along the way. My primary goal is to understand the relation of various mathematical frameworks to real liquid crystalline systems and to establish quantities most suitable for analysis of critical phenomena, defects, and symmetries in liquid crystalline systems.

INI 1  
12:15 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
Solid liquid crystals
Liquid crystals can become solids when they form glasses or when liquid crystalline polymers are crosslinked to form rubber. They can show properties richer than solids and liquid crystals separately. Glasses have high moduli and their directors are not mobile with respect to the solid matrix. Rubber maintains the molecular mobility of a liquid; it can be deformed hugely, has a low modulus, and its director is mobile. Both can have their order reduced by heat, light and solvent, and then mechanically contract along the director by a few percent (glasses) and by 100s% (rubber). Topological defects in their director fields means that such mechanical response generates Gaussian curvature or topology changes.
Mobile directors respond to imposed strains by reapportioning natural length in directions required by distortions, rendering their energetic cost zero or very small. If necessary textures of such low cost deformations are required to comply with boundary conditions in much the same way as in Martensite. Indeed such techniques of quasiconvexification have been extended by DeSimone et al to complete and generalise the soft mechanics discovered theoretically and experimentally by physicists and chemists.
I shall sketch some of these phenomena and present recent results on the mechanical and topological effect of disclinations in nematic solids, and on how polydomain nematic solids can be supersoft.

INI 1  
15:00 to 15:30  Afternoon Tea  
15:30 to 16:30 
Molecular Origin of K13 Revisited
Expanding a distortion free energy in terms of spatial gradients of density is a standard approach to construct an elastic theory of condensed matter. When applied to nematic liquid crystals, this socalled gradient expansion leads to the celebrated Frank theory that serves as the sound basis on which to analyze the response of nematic liquid crystals to electric fields and boundary constraints. The success of the theory has been thoroughly proven except the anomalous surface contribution associated with K13 (splaybend elasticity). The K13 term involves a gradient of the nematic director normal to the boundary, and hence the straightforward minimization of free energy under a given boundary condition becomes mathematically illposed, running into various unphysical behaviors.
More than a decade ago, I showed (at least I think I showed) that K13 is an artifact of gradient expansion as applied to a nonlocal interaction free energy by way of the density functional theory. When consistently done, the gradient expansion always results in K13=0 eliminating all the lingering problems that K13 has created.
The purpose of this talk is to revisit this issue. Close look at the K13 issue at the molecular level not only solves its own problem but also gives us a chance to shed new light on such fundamental structural characteristics of liquid crystals as chirality, flexoelectricity, and more.

INI 1  
16:30 to 17:30 
EG Virga (Università degli Studi di Pavia) What an old problem has still to say: the infamous K13case
Since the introduction in the freeenergy density of the K13elastic term proposed by Nehring and Saupe at the beginning of the 70's on the basis of a molecular theory, objections have been raised as to both its variational compatibility and statistical consistency. Often, the misbehaviours introduced by this elastic term, which appears as a bulk energy but is indeed a surface energy, are also threateningly referred to as paradoxes. Famous among these are, in particular, OldanoBarbero's and SomozaTarazona's. These paradoxes have been variously resolved, but the K13problem is still intriguing. One reason is that yes, the K13 term is a surface energy, but by no means is it a null Lagrangian. Another reason is that its very motivation, being statistical in nature, is rooted in the molecular interaction thought of as responsible for liquid crystallinity in the first place. Combining these reasons, one readily identifies two interwoven themes of current research: co ntinuum limits and surface properties. The lecture will attempt to put the history of the K13problem into the perspective of these avenues.

INI 1  
17:30 to 18:30 
AM Sonnet (University of Strathclyde) Hard core effects in mean field theories
Classically, there have been two different ways to obtain mean field theories for liquid crystals. One is based on short range repulsive forces and the other on long range electrostatic forces. In the former approach, it is the anisotropic shape of the molecules that leads to the anisotropic interaction, and in the latter it is the anisotropy of the molecular charge distribution. In real molecules, both causes of anisotropy will be present and can be expected to contribute to the effective interaction. It is thus desirable to assess the combined effect of anisotropic long range attraction and short range repulsion.
Starting from a long range intermolecular interaction energy, a mean field pair potential can be obtained by averaging over all possible relative positions of two molecules in a fixed relative orientation. The effects of hard core repulsion can be taken into account by an appropriate choice of the domain of integration for the averaging. This involves determining an excluded region, the region that one molecule cannot penetrate due to the hard core repulsion exerted by the other.

INI 1 
09:00 to 10:00 
J Ball (University of Oxford) Function spaces and liquid crystals
Function spaces are an essential part of mathematical models of nature, specifying allowed singularities in solutions. The talk will discuss the definition and roles of function spaces appropriate for liquid crystals, how they influence the description of defects, and their interaction with boundary conditions and topology.

INI 1  
10:00 to 10:40 
On the cubic instability in the Qtensor theory of nematics
Symmetry considerations, as well as compatibility with the OseenFrank theory, require the presence of a cubic term (involving spatial derivatives) in the Qtensor energy functional used for describing variationally the nematics. However the presence of the cubic term makes the energy functional unbounded from below.
We propose a dynamical approach for addressing this issue, namely to consider the L^2 gradient flow generated by the energy functional and show that the energy is dynamically bounded, namely if one starts with a bounded, suitable, energy then the energy stays bounded in time. We discuss notions of suitability which are related to the preservation of a physical constraint on the eigenvalues of the Qtensors (without using the BallMajumdar singular potential). This is joint work with G. Iyer and X. Xu (CarnegieMellon). 
INI 1  
10:40 to 11:20 
Problems related to defects in nematic liquid crystals
We describe several problems involving defects in liquid crystals and a mathematical framework in terms of tensors or directors that allows us to analyze the nature of defects in energy minimizing configurations.

INI 1  
11:20 to 11:50  Morning Coffee  
11:50 to 12:30 
Defects in nematic polymer hydrodynamics: survey of our work over the past 10 years
The organizers, Peter and David, asked me to give a basic introduction to defects in nematic hydrodynamics and to cover whatever aspects of our group's work I felt appropriate for a kickoff workshop. The lecture touches on highlights of work with Qi Wang and Xiaofeng Yang at South Carolina and Ruhai Zhou at Old Dominion University. The first topic covered is the remarkable (to us) shearinduced dynamics of nematic polymers revealed by numerical solutions  of various secondmoment tensor approximations, and of the fully resolved DoiHessSmoluchowski equation for the orientational PDF. Next we study nematic distortions arising due to physical boundary conditions at stationary and moving boundaries, in both 1D and 2D simulations, again revealing persistent nonstationary behavior in large regions of parameter space. Phase diagrams analogous to the monodomain problem show transitions between stationary and nonstationary attractors. What is the role of defects? We employ defect core detection and tracking diagnostics that apply independent of space dimension, and in dimensions 2 or higher we assess topology only after a positive test for defect cores. The fundamental role of oblate defect cores is illustrated in both 1D and 2D unsteady attractors. In each topic, open problems potentially of interest for rigorous mathematics are raised, with the hope of followup discussion during the program.

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
13:30 to 14:30 
Onsager's Variational Principle in Soft Matter Dynamics
In the celebrated paper on the reciprocal relation for the kinetic coefficients in irreversible processes, Onsager extended Rayleigh's principle of the least energy dissipation to general irreversible processes. In this presentation, I shall show that this variational principle is very convenient in considering the nonlinear and nonequilibrium phenomena in soft matter. I will discuss this using the examples of (i) viscoelasticity of colloidal suspensions and polymer solutions, (ii)diffusionmechanical coupling in polymer solutions and gels, and (iii) nematohydrodynamics etc.

INI 1  
14:30 to 15:30 
Describing and determining the order of liquid crystals in the bulk and close to interfaces
The definition and actual determination of orientational and positional order parameters plays a key role in describing the variety of molecular organizations of liquid crystals and in quantifying their changes, e.g. at phase transitions or when approaching an interface. The classical description of order in terms of a single parameter, which implicitly assumes a simple rigid and uniaxial molecular shape, has to be refined in a number of ways, e.g. to allow for biaxiality [1] or internal flexibility [2,3] or inhomogeneities in thin films [4] or nanodroplets [5]. This increased complication has become essential in view of progresses in experimental and simulation techniques, now offering an unprecedented level of detail and of possibilities of putting theories of liquid crystals and ultimately our understanding of anisotropic phases to the test. In the talk we present some examples of order and structure determinations from coarse grained [1,5] and atomistic simulations [24].

INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 17:00  Geometric theories of conservative liquid crystal dynamics  INI 1 
09:00 to 10:00  When shapes collide: finding order in disorder  INI 1  
10:00 to 11:00 
Liquid crystals and what they have taught us
Broken symmetries and conservation laws lie at the very heart of our understanding of the physical world. They control the physical properties and phenomena ranging from the masses of elementary particles to the flow to the oceans. Liquid crystals provide an almost ideal laboratory for studying phenomena  lowenergy elasticity and hydrodynamics, topological defects, and critical fluctuations  associated with broken symmetries. This talk will review some of the important things we have learned from liquid crystals including how to describe broken rotational symmetries, what crystals really are and how to describe their elasticity and dynamics, and how fluctuations and spatial dimension affect the existence of order.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
R Kamien (University of Pennsylvania) Topological defects and the ground state manifold
Do homotopy groups completely characterize topological defects? In this presentation, I will describe the pitfalls to this notion and describe some progress in working around the ambiguities and puzzles that arise in such descriptions.

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
P Chossat ([CNRS  Université de Nice]) The mathematics of pattern formation: a modern view
After the seminal contributions of A. Turing, G.I. Taylor, L. Landau, L. Michel in the first half of the past century, new mathematical methods have emerged to study the phenomenon of spontaneous pattern formation. Decisive progress was made using geometrical methods (M. Golubitsky, I. Stewart) and analytical tools (G. Iooss, K. Kirchgässner). I shall show on examples how this theory, known as Equivariant Bifurcation Theory, applies to a variety of problems, including liquid crystals. I shall also quote some open questions which are still under investigation by mathematicians working in these topics.

INI 1  
15:00 to 15:30  Afternoon Tea  
15:30 to 16:30 
J Dawes (University of Bath) Localised pattern formation
Patternforming (Turing) instabilities are observed sometimes to generate patches of periodic structure rather than domainfilling patterns. I will present physical examples and toy model PDEs, and outline the bifurcation theory that provides an explanation for this, at least in one spatial dimension.

INI 1  
19:30 to 22:00  Conference Dinner at Emmanuel College 
09:00 to 09:40 
P Zhang (Peking University) From molecular symmetry to order parameters
We propose a systematic molecular modeling of liquid crystals, the models can be used to depict isotropic, nematic, smectic, columnar, cholesterics and blue phases. The tensor model can be reduced from molecular model using closure and Taylor expansion, the vector model can be reduced from tensor or molecular model using axial symmetry assumption. Using Newton mechanic and virial expansion, we build a generic molecular model to describe phase behaviors of rigid molecules of arbitrary shape. The system is characterized by a pairwise kernel function. The kernel function can be simplified by the molecular symmetry. Onsager potential is a leading order for rodlike molecular using hard core potential, and MaierSaupe potential is a good second approximation using Lennard Jones potential. The new models including simplified kernel functions are proposed for bentcore molecules and other shape molecules. We also clarify the criteria of choosing order parameters, both from theoretical aspects and from results of experiments and simulations. According to these criteria, we explain why the eigenvalue of second moment is chosen to describe spatially homogeneous phases of rodlike molecules, and predict the choice of order parameters for bentcore molecules and other molecules of different symmetries. The rigorous analysis for choosing the order parameter will be given for MaierSaupe model of rodlike molecular.

INI 1  
09:40 to 10:20 
Invariant theory: a useful tool for equivariant dynamics and the mathematics of liquid crystals
In this talk we will present some classical and modern results from invariant theory and and apply them to problems in equivariant dynamics and bifurcation and also to some problems in the theory of liquid crystals. We hope to identify and explain some problems where similar ideas are useful.

INI 1  
10:20 to 11:00 
Free energy according to Poincare' and Landau
In the Landau theory of phase transitions one considers an effective potential U whose symmetry group G and degree d depend on the system under consideration; generally speaking, U is the most general Ginvariant polynomial of degree d. When such a U turns out to be too complicate for a direct analysis, it is essential to be able to drop unessential terms, i.e., to apply a simplifying criterion. Criteria based on singularity theory exist and have a rigorous foundation, but are often very difficult to apply in practice. Here we consider a simplifying criterion and rigorously justify it on the basis of classical LiePoincare theory; this builds on (and justifies) a proposal by Gufan.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:10 
Threewave interactions in problems with two length scales: Faraday waves and softmatter quasicrystals
Threewave interactions form the basis of our understanding of many patternforming systems because they encapsulate the most basic nonlinear interactions. In problems with two comparable length scales, it is possible for two waves of the shorter wavelength to interact with one wave of the longer, as well as for two waves of the longer wavelength to interact with one wave of the shorter. Consideration of both types of threewave interactions can generically explain the presence of complex patterns, such as quasipatterns, and spatiotemporal chaos. Two length scales arise naturally in some examples of polymer micelles and in the Faraday wave experiment, where a viscous fluid is subjected to vertical vibration. Our results enable some previously unexplained experimental observations of spatiotemporal chaos in the Faraday wave experiment to be interpreted in a new light; application to quasicrystals recently observed in selfassembled colloidal systems is more speculative.

INI 1  
12:15 to 13:30  Lunch at Wolfson Court  
13:30 to 14:00 
M Warner & M Osipov & J Ball ([Cambridge/Strathclyde/Oxford]) Overviews of other MLC workshops
Liquid crystal (LC) phases were first identified and even named by their topological defects. The rich interplay between geometry, topology and optics is ubiquitous through all liquid crystals. Workshop 4 (WS4) addresses the characterisation of defects, their essential appearance in complex systems such as colloidal liquid crystals, how they template complex structures, and their special character in nonsimple spaces (such as those with Gaussian curvature). WS4 is concerned with solid liquid crystals, both elastomers where the director remains mobile, and glasses where the director is pinned to the material frame. The unique mechanics of solid liquid crystals leads to new phenomena, some of which are described by techniques of quasiconvexification first exploited in Martensites. Their mechanics connects with the defects theme since topological defects in LC solids, on illumination or temperature change, cause changes in Gaussian curvature or topology. WS4 also pursues active nematics, their connections with solid nematics, and the role of defects in active systems.
I shall review the themes of the workshop, concentrating on those less familiar to liquid crystal specialists.

INI 1  
14:00 to 14:15  Closing remarks  INI 1 