Seminars (MLC)

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 two-dimensional curved manifolds will be described, and an "augmented" theory will be presente d for nematic shells, which to the in-plane orientational order adds information on the escape of molecules along the shell's normal.

Maier-Saupe 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 Maier-Saupe theory, then extend it to biaxial LC molecules, to inhomogeneous LCs, and to higher spatial dimensions, and discuss the relation between Maier-Saupe and other main theories of LCs.

I will attempt to traverse the road from basic equations of molecular dynamics at the microscopic level to PDE dynamics at the level of coarse-grained 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.

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 so-called 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 (splay-bend 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 ill-posed, 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.

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.

Symmetry considerations, as well as compatibility with the Oseen-Frank theory, require the presence of a cubic term (involving spatial derivatives) in the Q-tensor 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 Q-tensors (without using the Ball-Majumdar singular potential).

This is joint work with G. Iyer and X. Xu (Carnegie-Mellon).

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) shear-induced dynamics of nematic polymers revealed by numerical solutions -- of various second-moment tensor approximations, and of the fully resolved Doi-Hess-Smoluchowski 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 non-stationary behavior in large regions of parameter space. Phase diagrams analogous to the monodomain problem show transitions between stationary and non-stationary 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 follow-up discussion during the program.

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 [2-4].

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.

Pattern-forming (Turing) instabilities are observed sometimes to generate patches of periodic structure rather than domain-filling 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.

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.

Three-wave interactions form the basis of our understanding of many pattern-forming 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 three-wave 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 self-assembled colloidal systems is more speculative.

We discuss how Onsager theory with the Maier-Saupe interaction leads naturally to a bulk free energy depending on the $Q$-tensor that blows up as the minimum eigenvalue $\lambda_{\rm min}(Q)\rightarrow -1/3$, using methods closely related to those of Katriel, Kventsel, Luckhurst and Sluckin (1986). With this bulk energy, and in the one constant approximation for the elastic energy, it is shown that for suitable boundary conditions, minimizers $Q$ of the total free energy for a nematic liquid crystal filling a region $\Omega$ satisfy the physical requirement that $\inf_{x\in\Omega}\lambda_{\rm min}(Q(x))>-1/3$.

Hard-body models for lyotropic liquid crystalline phases date back to Onsager (1949) who showed that a fluid of hard rods can exhibit a transition from an isotropic to a nematic phase that is driven purely by entropy. Onsager’s treatment is based on a second-virial description of the free energy that is accurate in the (Onsager) limit of very long thin rods (spherocylinders). For shorter spherocylinders and for smectic and crystalline phases, as well as for treating inhomogeneous fluids, e.g. situations arising at interfaces between phases and in anchoring and wetting at substrates, it is necessary to develop theories in which the ensemble averaged one-body particle density depends on both the orientation and the position of the particles. Density Functional Theory (DFT), developed first for simple fluids with spherical particles, is one such theory and it has emerged as powerful means of tackling phase transitions and the structure and thermodynamics of inhomogeneous fl uids. This lecture will provide an overview of the basics of DFT before focusing on the successful geometry-based Fundamental Measure Theory (FMT) approach introduced originally by Rosenfeld (1989) for hard-sphere mixtures. FMT for spheres has as its starting point the incorporation of the exact second virial contribution into the free energy functional. Attempts to extend the ideas of FMT to hard bodies of arbitrary shape were made by Rosenfeld (1994, 1995). These failed to yield a stable nematic phase for spherocylinders, partly because they did not include the correct Onsager limit. In recent years there has been renewed effort to develop improved FMT that go towards capturing this limit. I shall describe progress for a variety of model colloidal liquid crystalline fluids including hard spherocylinders, mixtures of hard spheres and rods, and hard thin platelets. If time permits I shall mention some recent applications of Dynamical DFT to non-equilibrium properties.

I will report theory and simulations of the continuum equations for a droplet of active polar liquid crystal. These equations offers a simple representation of a ``cell extract", such a droplet of actomyosin solution, in which myosin motors moving on actin filaments create internal stresses as a result of biological activity. (This system can in turn be viewed as a stripped-down representation of the cytoskeleton which causes locomotion of eukarotic cells.) Actomyosin is an active liquid crystal whose polarity describes the mean sense of alignment of actin fibres. In the absence of polymerization and depolymerization processes (`treadmilling') which arise respectively at the plus and minus ends of the filaments, the active dynamics should be unchanged when polarity is reversed. Our results suggest that, contrary to most literature opinion, locomotion can arise in the absence of treadmilling, by spontaneous symmetry breaking (SSB) of polarity inversion symmetry.

A fascinating way to improve the present-day liquid crystal technology consists of imagining to use new liquid crystal phases with "exotic" properties, like the biaxial nematic phase. However, as an essential step in this direction one has to establish the conditions under which such a phase is thermodynamically stable. Inspired by a recent experiment on a colloidal suspension of mineral goethite particles [1], we use a mean field theory to investigate the phase behavior of boardlike particles. By modelling these “nanoscopic bricks” as cuboids with a hard-body interaction, we analyze the conditions of stability of the long-searched biaxial-nematic phase. We show that under specific conditions size-polydispersity, a common property in most colloids, can increase appreciably the stability of this liquid crystal phase [2]. Moreover, we deduce that this effect can be interpreted in terms of an effective attraction, and therefore that a similar stability could be induced by a non-adsorbing depletant, like a polymeric solution [3]. [1] E. van den Pol et al., Phys. Rev. Lett. 103, 055901 (2009) [2] S. Belli, A. Patti, M. Dijkstra and R. van Roij, Phys. Rev. Lett. 107, 148303 (2011) [3] S. Belli, M. Dijkstra and R. van Roij, J. Phys.: Condens. Matter 24, 284128 (2012)

This talk focuses on the development, analysis and numerical implementation of mathematical models for a planar bistable nematic device reported in a paper by Tsakonas, Davidson, Brown and Mottram. We model this device within a continuum Landau-de Gennes framework and investigate the cases of strong and weak anchoring separately. In both cases, we find six distinct states and compute bifurcation diagrams as a function of the anchoring strength. We introduce the concept of an optimal boundary condition that prescribes the optimal interpolation between defects at the vertices. We develop a parallel lattice-based Landau-de Gennes interaction potential, by analogy with the Lebwohl-Lasher lattice-based model and study multistability within this discrete framework too by means of Monte Carlo methods. We also use the off-lattice based Gay Berne model to study the structure of the stable states. The different numerical approaches are compared and we discuss their relative strengths a nd shortcomings. We conclude by a brief discussion on a multiscale modelling approach wherein we can couple a lattice-based interaction potential to a conventional continuum model. This is joint work with Chong Luo and Radek Erban.

A dangerous over-confidence now prevails in the assumption that detailed all-atom or united-atom models which are used to represent the properties of fluid molecules (e.g. the OPLS-type potentials) are sufficient to describe molecular systems with a precision that supplements experiments. More than 1% of all recent articles published in the open science and engineering community deal with molecular simulations at this level and in some cases the accuracy of the results is taken for granted. The fitting of parameter of the force fields is, however, still rather unsophisticated as compared to other aspects of computer modelling. Common practice is to hand fit a few parameters to a few experimental data points (e.g., a radial distribution function, solubility data and/or enthalpies at a given temperature or phase state). In this contribution we propose a new way of obtaining the required force field parameters. In our methodology one requires access to a physical-based equation of state that describes the complete Helmholtz free energy in closed algebraic form, i.e., an equation of state (EoS) that is based on a defined intermolecular potential. Such an equation can then be used to explore a very large parameter space to estimate the locally optimal parameter set that provides an optimal description of the available macroscopical experimental data. This parameter set represents not just a unique fit to a single temperature or density, but rather an over-arching average. If the equation of state is expressed in terms of the free energy of the system for a well defined intermolecular potential, it can be used to develop a “top-down averaged” intermolecular potential. Here we follow this line of thought and present a proof-of-concept of such methodology, employing a recently developed EoS of the Statistical Associating Fluid Theory (SAFT) family using the so-called Mie intermolecular potential.

The behaviour of axially symmetric particles has been well-investigated by both theory and computer simulation. Colloidal particles with such shapes have also been studied experimentally. For one component systems, nematic and smectic phases have been observed for rod-like particles, while discotic nematic and columnar phases have been noted for discs. If, however, one considers less symmetrical particles, then other phases become possible. Freiser (1970) showed theoretically that such a system might form a biaxial nematic phase, in which all particle axes are partially aligned, as opposed to a normal, uniaxial nematic where only one axis is ordered. Experimentally, a biaxial nematic phase has been reported for lyotropic systems (Yu & Saupe, 1980) and for suspensions of board-like goethite particles (van den Pol et al., 2009). There also exist reports of thermotropic biaxial nematic phases, though debate still continues as to whether these really exist for these systems. I would like to present simulation results (and hopefully some simple theory) on two types of model particle which might show biaxial behaviour. The first model is of V-shaped particles (also called boomerangs, bananas and bent-cores), while the second is closely related to the board-like shapes of goethite mentioned above. In both cases the particles interact via repulsive interactions only. Both models have received previous theoretical and simulation attention, but hopefully a little extra investigation will not come amiss.

In both cases we used constant pressure, and sometimes constant stress, molecular dynamics simulations, compressing the system from an initial isotropic phase. For relatively straight V-shped particles, the simulations are straightforward and result in uniaxial nematic and biaxial smectic phases. For a bond angle of less than ca. 130 degrees, however, the system tends to jam on compression and equilibration becomes problematic. We therefore investigated mixtures of V-shapes to see whether mixing suppressed the smectic phases, giving room for a biaxial nematic phase to form. While, at least to date, this hope was not fulfilled, we still observed some effects that we believe are of interest.

The other system studied is of fused hexagons – a model related to hard boards. The phase behaviour observed here was rather rich. Depending on geometry we found rod-like, discotic and biaxial nematic phases. Rod-like particles formed both uniaxial and biaxial smectic A and C phases. Disc-like particles formed not only columnar phases but also lamellar phases. Hopefully we will be able to rationalise the presence of at least some of these structures using simple-minded stability analysis.

Since the proposal for a remarkably simple theory of ferromagnetism made by Weiss in 1906, under the assumption that each molecule suffered an effective magnetic field (le champ intérieur, in Weiss' words) mimicking the average action of all other molecules, the notion of mean field has grown and acquired a life of its own. The most striking application to liquid crystal science of the mean-field formalism is perhaps the Maier-Saupe theory for the nematic phase. Many other models and approximations are comprised under the general heading of mean-field theory, though often one may hardly find any trace of an average, collective field there, its place being taken instead by a generalized order field. Some theories in this ample catalogue are variational, while others are not. All feature a key self-consistency condition, which may involve a probability distribution density as well as an order field. The lecture will attempt to justify the key self-consistency equatio n in a rigorous way for the different flavours that theory has taken.

The recent increase in computer speed has determined unprecedented possibilities of modelling physical and chemical processes “in silico”. This is most true for liquid crystals, as the large system sizes and long time scales necessary for reliable predictions of their self-assembly are now becoming affordable, and where atomistic molecular dynamics simulations have proved that an accurate but classical description of intermolecular forces is adequate for obtaining a quantitative agreement with experiments for nematics and discotics. In this context, in Bologna we developed a force field for n-alkyl cyanobiphenyls (nCBs)able to reproduce their experimental phase transition temperatures within a few degrees. The choice of nCBs as prototypical liquid crystal systems opens the way to an informative cross comparison between experiments, simulations, and theory. In fact, the abundance of experimental studies provides a rich database of almost any possible physical property, which serves as a stringent test for simulation predictions, and is able to reveal weaknesses and strengths of the microscopic model. Once the model has been validated, simulations can be considered superior to theoretical predictions, because they rely on a much lower number of assumptions. It becomes then possible to “revisit” and validate existing and maybe even very successful theories, not only on the basis of their predictions (comparison with the experiment) but also on their physical foundations (comparison with simulation s). This presentation will cover all the stages of this “virtuous” exercise, including: the derivation of the force field; II) the calculation of macroscopic observables III) the comparison with mean field descriptions for the nematic and smectic phases; IV) new attempts of addressing continuum theories for liquid crystal alignment. To conclude, a personal perspective of where theory could help the simulation and of future applications will be given.

Structuring of liquid crystalline fluids allows for various exciting material mechanisms such as self-assembly [1], memory effects [2], entanglement [3], nonlinear electrophoresis [4], nonlinear rotary dynamics [5], and nanoscopic surface shape changing [6]. Here, we present strategies for creating colloidal and bulk liquid crystal superstructures, in 2D and 3D, using nematic, twisted nematic, and cholesteric blue phases. Our work is based on the numerical minimization of the phenomenological Landau-de Gennes free energy and solving hybrid Lattice Boltzmann algorithm for Beris-Edwards nematodynamics model, with full link to experiments. We show that 3D colloidal crystals can be assembled from elastic dipoles of spherical beads in nematic liquid crystals or via inherently inhomogeneous order profiles in cholesteric blue phases [7]. By using colloidal platelets, we show that crystalline [8] and quasi-crystalline symmetry can be imprinted into the structures. Topological defects are manipulated into structures of knots and links using various colloidal arrays [9]. Finally, passive and active material flow is used to produce distinct backflow generated complex nematic profiles in microfluidic channels.

[1] P. Poulin, H. Stark, T. C. Lubensky, D. A. Weitz, Science 275, 1770 (1997). [2] T. Araki, M. Buscaglia, T. Bellini, and H. Tanaka, Nature Materials 10, 303 (2011). [3] M. Ravnik, et al, Phys. Rev. Lett. 99, 247801 (2007). [4] O. D. Lavrentovich, I. Lazo and O. P. Pishnyak, Nature 467, 947 (2010). [5] J. S. Lintuvuori, K. Stratford, M. E. Cates, D. Marenduzzo, Phys. Rev. Lett. 107, 267802 (2012). [6] D. Vanzo, M. Ricci, R. Berardi and C. Zannoni, Soft Matter 8, 11790 (2012). [7] M. Ravnik, G. P. Alexander, J. M. Yeomans, and S. Zumer, Proc. Natl. Acad. Sci. USA 108, 5188 (2011). [8] J. Dontabhaktuni, M. Ravnik and S. Zumer, Soft Matter 8, 1657 (2012). [9] U. Tkalec, M. Ravnik, S. Copar, S. Zumer and I. Musevic, Science 333, 62 (2011).

Multicomponent thermotropic liquid crystal mixtures are widely used in the display industry to obtain desired material properties. Usually, the behavior of such mixtures differs little from that of pure materials. In other systems, such as lyotropic, chromonic, colloidal and elastomeric liquid crystals, the behavior can be dramatically different. Standard mean field theories assume a single component, and do not provide an adequate description for such systems. I will discuss strategies to incorporate both attractive and repulsive interactions of multicomponent systems into mean field models, and present some results.

Liquid crystal elastomers (LCE) are functional materials consisting of weakly crosslinked polymer networks with embedded liquid crystalline (mesogenic) molecules. Consequently, LCE are characterized by a pronounced coupling between macroscopic strain and orientational mesogenic order. As the latter can be controlled by external stimuli such as temperature, electric field, or ultraviolet light, LCE have great potential for application as sensors and actuators.

Here large-scale molecular simulations of swollen main-chain LCE will be presented. The simulated experiments include temperature scans, stress-strain runs, and the application of an external electric field. Our isostress Monte Carlo simulations are capable of reproducing isotropic, nematic and smectic phases, as well as a stress-induced isotropic-to-nematic transition. Moreover, a transversal electric field is seen to induce nematic director rotation resulting in orientational stripe domains. The role of sample swelling has been explored as well.

The simulation output has also been used to connect to typical experimental observables, such as sample dimensions, specific heat, deuterium magnetic resonance spectra, and scattered X-ray patterns.

This paper presents theory and modeling of structure and dynamics of three representative anisotropic soft matter materials :(i) confined nematics ; (ii) membranes and surfactant-laden interfaces, and (iii) fiber-filled soft membranes, highlighting the interactions between geometry, order parameters, and material anisotropy. (i)Confined nematics are described using nematodynamics in the bulk, at surfaces and contact lines and used to analyze cholesteric collagen solutions under shear and in film casting processes and demonstrate how liquid crystalline polymer models are able to resolve experimentally observed flow-alignment, banded textures, and free surface undulations. (ii) Membranes are described using membrano-dynamics, which extends the Helfrich-Boussinesq-Scriven model by accounting for bending and torsion dissipation, and is used to establish direct connections between membrane shape and rheology. Lastly we describe (iii) fiber-filled membranes using the integration of nemato-dynamics and membrano-dynamics and apply the theory to plant cell walls, where the paranematic order of the cellulose semiflexible fibrils is coupled to the soft pectin-based membrane curvature, as reported experimentally.

Liquid crystals (LC) continue to offer a fascinating variety of fundamental physics problems related to molecular organizations in the bulk and their modifications close to interfaces [1]. Here we plan to show some recent results for the simulation of these organizations at molecular and atomistic resolution. At molecular resolution we employ Gay-Berne models and report results for the shape, internal order and chirality of freely suspended low molar mass nematic nanodroplets [2]. We also show that systems as complex as swollen LC elastomers and their deformation in response to the application of an electric field can be simulated [3]. At atomistic level molecular dynamics simulations [4] can now predict actual morphologies and properties in the bulk [4,5] from a specific molecular structure and they can also be useful to analyse NMR data [6,7]. Having validated simulations in the bulk, we also investigate LC close to selected interfaces like hydrogen terminated silicon [1] and crystalline and glassy silica with controlled roughness [8], trying to show how orientational anchoring can be introduced from the microscopic point of view probing the limits of continuum theory on the nanoscale [9].

1 A. Pizzirusso, R. Berardi, L. Muccioli, M. Ricci, C. Zannoni, Chemical Science,3,573(2012) 2 D. Vanzo, M.Ricci, R. Berardi, C.Zannoni, Soft Matter, 8, 11790(2012) 3 G. Skacej, C.Zannoni, PNAS,109,10193(2012) 4 G. Tiberio, L. Muccioli, R. Berardi, C. Zannoni, ChemPhysChem, 10, 125(2009) 5 M.F. Palermo, A. Pizzirusso, L. Muccioli, C. Zannoni, to be submitted (2013) 6 A.Pizzirusso, M.B. Di Cicco, G. Tiberio, L. Muccioli, R. Berardi, C. Zannoni, J.Phys.Chem.B 116,3760(2012) 7 A. C. J. Weber, A. Pizzirusso, L. Muccioli, C. Zannoni, W. L. Meerts, C. A. de Lange, E.E. Burnell, J. Chem. Phys. 136, 174506 (2012) 8 O. M. Roscioni, L. Muccioli, R. G. Della Valle, A. Pizzirusso, M. Ricci, C. Zannoni, to be submitted (2012) 9 M. Ruths, B. Zappone, Langmuir, 28, 8371 (2012).

The nematic phase is normally uniaxial – i.e. there is just one distinct optical axis. In such a phase one set of molecular axes are aligned but the other axes are orientationally disordered. In 1970, however, Freiser showed that a biaxial nematic phase was theoretically possible, in which all three molecular axes are aligned. This phase has since been observed experimentally, predicted to exist theoretically for various particle models and has been seen in simulation studies. Such a phase would have three distinct optical axis and there are possible applications to liquid crystal displays, should a suitable material be found. In this talk I would like to present the results of simulation studies on purely repulsive bent-core models (V-shaped particles). In the limit of very long, thin arms, such shapes have been predicted to exhibit a biaxial nematic phase for bend angles in the region of 110o. In order to test these predictions and also to explore the system’s pha se behaviour at pressures at which the nematic phase is unstable, we have run molecular dynamics simulations for one-component systems, binary mixtures and higher order mixtures (4 – 6 components). In all case the runs started from the isotopic phase and the system, was then slowly compressed, so any phase observed had formed spontaneously. We explored the effects of varying the arm lengths and bend angles of these particles on the phase behaviour. For a one-component system, only particles with bend angles greater than ca. 130o spontaneously formed ordered, equilibrated phases. Typically the phase sequence was isotropic →uniaxial nematic→biaxial smectic A (except for very straight particles in which the smectic phase was uniaxial). No biaxial nematic phase was observed for the arm-lengths simulated. It is possible that if the smectic phase could be destabilized, a biaxial nematic might form in its place. Such destabilization might occur in mixtures. Binary mixtures of bent cores, however, were found.

Random packing of mixtures of hard spherocylinders and hard spheres are studied for $d \approx L$, where $L$ is the length of the spherocylinder and $d$ the diameter of the spheres. Packing fractions of mixtures of hard spherocylindres with aspect ratios $ 4

References [1] Z. X. Zhang and J. S. van Duijneveldt, J. Chem. Phys., 124, 154910 (2006). [2] Henk N.W. Lekkerkerker and Remco Tuinier, Colloids and the Depletion Interaction, Springer, 2011. [3] N. Yasarawan and J.S. van Duijneveldt, Soft Matter, 6, 353-362 (2010).

Polymer vesicles, fluid filled polymer sacs, have attracted much attention for applications such as drug delivery vehicles, miniature chemical reactors, or as synthetic, minimal cells. In these applications the vehicles may undergo significant changes in osmotic pressure, pH, or concentration, which may lead to vesicle rupture or collapse. In order to avoid this a number of possible strategies may be used to stabilise vesicles against changes in external environment. In this talk I will discuss some recent simulation work studying two of these - armouring and swelling.

Recently it has been shown that polymer vesicles may be coated with a layer of colloidal particles that armour these, in a similar manner to some biological systems. Simple Monte Carlo simulations were used to reproduce the packing patterns seen in these experimental systems and to study the effect of surface charge density on the self-assembly [1]. Dissipative particle dynamics simulations were used to study the swelling of a polymer bilayer when exposed to small hydrophobic molecules. Above a critical density of hydrophobic molecules the bilayer undergoes a morphological transition characterised by the formation of a bud within the bilayer, consistent with experimental observations of polymer vesicles [2].

[1] R Chen, DJG Pearce, S Fortuna, DL Cheung, and SAF Bon, J Am Chem Soc, 133, 2151 (2011) [2] CDJ Parmenter, R Chen, DL Cheung, and SAF Bon, Soft Matter, in press

We study the self-assembly of mixed monolayers of hydrophobic and hydrophilic colloidal particles adsorbed at oil/water interfaces both experimentally and theoretically. Experimentally, we find that by tuning the interactions, composition and packing geometry of the mixed monolayer, a rich variety of two-dimensional super-lattice and cluster structures are formed which are stabilised by strong electrostatic interactions mediated through the oil phase [1,2]. The 2D structures obtained are in excellent agreement with zero temperature lattice sum calculations, indicating that the self-assembly process can be effectively controlled [1-3]. Monte Carlo simulations further reveal that the melting behaviour of these super-lattice structures proceeds via a multi-stage process, with melting temperatures that have a very strong and non-monotonic dependence on composition [3].

[1] A.D. Law, D.M.A. Buzza, T.S. Horozov, Phys. Rev. Lett., 106, 128302 (2011) [2] A.D. Law, M. Auriol, D. Smith, T.S. Horozov, D.M.A. Buzza, submitted [2] A.D. Law, T.S. Horozov, D.M.A. Buzza, Soft Matter, 7, 8923 (2011)

Video Conference Seminar from Japan.

We describe mathematical results and techniques of analysis on the structure of defects in thin nematic liquid crystals described by minimizers of the Landau-de Gennes energy involving a tensor-valued order parameter with Dirichlet boundary conditions of nonzero degree. We prove that as the coefficient of the elasticity term tends to zero, a limiting uniaxial texture forms with a finite number of defects, all of degree 1/2 or -1/2. We also describe the location of defects and the limiting energy.

If the Q-tensor order parameter is interpreted as a normalised matrix of second moments of a probability measure on the unit sphere, its eigenvalues are bounded below by -1/3 and above by 2/3. This constraint raises questions regarding the physical predictions of theories which employ the Q-tensor; it also raises analytical issues in both static and dynamic Q-tensor theories of nematic liquid crystals. John Ball and Apala Majumdar recently constructed a singular map on traceless, symmetric matrices that penalises unphysical Q-tensors by giving them an infinite energy cost. In this talk, I shall discuss some mathematical results for a modified Beris-Edwards model of nematic dynamics into which this map is built, including the existence, regularity and so-called `strict physicality' of its weak solutions.

We present a rigorous derivation of the Ericksen-Leslie equation starting from the Doi-Onsager equation. As in the fluid dynamic limit of the Boltzmann equation, we first make the Hilbert expansion for the solution of the Doi-Onsager equation. The existence of the Hilbert expansion is connected to an open question whether the energy of the Ericksen-Leslie equation is dissipated. We show that the energy is dissipated for the Ericksen-Leslie equation derived from the Doi-Onsager equation. The most difficult step is to prove a uniform bound for the remainder in the Hilbert expansion. This question is connected to the spectral stability of the linearized Doi-Onsager operator around a critical point. By introducing two important auxiliary operators, the detailed spectral information is obtained for the linearized operator around all critical points. However, these are not enough to justify the small Deborah number limit for the inhomogeneous Doi-Onsager equation, since the elastic stress in the velocity equation is also strongly singular. For this, we need to establish a precise lower bound for a bilinear form associated with the linearized operator. In the bilinear form, the interactions between the part inside the kernel and the part outside the kernel of the linearized operator are very complicated. We find a coordinate transform and introduce a five dimensional space called the Maier-Saupe space such that the interactions between two parts can been seen explicitly by a delicate argument of completing the square. However, the lower bound is very weak for the part inside the Maier-Saupe space. In order to apply them to the error estimates, we have to analyze the structure of the singular terms and introduce a suitable energy functional. Furthermore, we prove the local well-posedness of the Ericksen-Leslie system, and the global well-posednss for small initial data under the physical constrain condition on the Leslie coefficients, which ensures that the energy of the system is dissipated. Instead of the Ginzburg-Landau approximation, we construct an approximate system with the dissipated energy based on a new formulation of the system.

We investigate stability of radially symmetric solutions in the context of Landau - de Gennes theory. It is well known that radial hedgehog is an unstable solution for low enough temperatures. We show that radial hedgehog is locally stable solution for temperatures close to isotropic-nematic phase transition temperature.

In this work, we show the existence and uniqueness of local strong solution for a coupled Navier-Stokes/Q-tensor system on a bounded domain $\Omega\subset\mathbb{R}^3$ with Dirichlet boundary condition. One of the novelties brought in with respect to the existing literature consists in the fact that we deal with Navier-Stokes equation with variable viscosity. Concerning the methodology, we use an approximation method to handle the linearized system and the existence of solution to the nonlinear system is proved via a Banach's fixed point argument, based on the estimates on the lower order terms.

This work is motivated by a polarimetric experiment (performed in HP Labs) were a thin slab of nematic liquid crystal was placed on a cylindrical mount, and illuminated by a focused polarized laser beam. As the slab is rotated and the polarimetric measurement data (the so-called Stokes parameters) varies with the angle of incidence. The object of this work was to determine what information on the dielectric permittivity of the liquid crystal could be retrieved from this data.

As early as 1972, Mullen and coworkers showed experimentally that the director alignment of a nematic liquid crystal induces an anisotropic, frequency dependent sound speed in nematic liquid crystals. Similarly, Selinger and co-workers have studied a liquid crystal cell where the nematic molecules can be realigned by an ultrasonic wave, leading to a change in the optical transmission through the cell. The existing theoretical models for this acousto-optic effect propose a free energy that depends on the density gradient thus describing the nematic liquid crystal as a compressible second grade fluid. In this talk we will show that that the angular dependence of the sound speed can be easily reproduced by introducing a simple anisotropic term in the stress tensor, thus providing a simpler first-grade model for the acousto-optic effect. The simplest term is non-hyperelastic, but we show that it can be interpreted as the quasi-incompressible approximation of an elastic term which couples the director orientation with the strain. More interestingly, the frequency dependence of the anisotropic sound speed can be recovered by assuming an irreversible relaxation of the reference configuration with respect to which the strain is measured.

The Ericksen Leslie equations model the motion of nematic liquid crytaline fluids. The equations comprise the linear and angular momentum equations with non-convex constraints on the kinematic variables. These equations possess a Hamiltonian structure which reveals the subtle coupling of the two equations, and a delicate balance between inertia, transport, and dissipation. While a complete theory for the full nonlinear system is not yet available, many interesting sub-cases have been analyzed.

This talk will focus on the development and analysis of numerical schemes which inherit the Hamiltonian structure, and hence stability, of the continuous problem. In certain situations compactness properties of the discrete solutions can be established which guarantee convergence of schemes.

The Ericksen-Leslie theory describes the dynamic flow of liquid crystals. In this talk, we will discuss global existence of solutions of the Ericksen-Leslie system for a general Oseen-Frank model in 2D. We also discuss some new results on the local existence, uniqueness and the blow up criterions of strong solutions to the Ericksen-Leslie system in 3D for the general Oseen-Frank model in 3D.

In this talk, we will present recent theoretical and numerical results for models of nematic elastomers within the small strain approach.

While strains exhibited by nematic elastomers are usually large, there are cases where this is not so, and the early modeling approaches were using this framework. In fact, the main reason for the developing small strain theories for nematic elastomers is the clear geometric structure of the resulting energy landscape.

We will exploit this structure to discuss material instabilities and stress-strain diagrams, and to suggest possible generalizations to more realistic models.

Smectic C* liquid crystal films are modeled with a relaxed Frank energy, \begin{equation*} \int_\Omega\Big( k_s(\text{div}\, u)^2 + k_b(\text{curl}\, u)^2 + \frac{1}{2\epsilon^2}(1 - |u|^2)^2 \Big)\, dx . \end{equation*} Here $k_s$ and $k_b$ represent the two dimensional splay and bend moduli for the film respectively with $k_s, k_b > 0$, $\Omega$ is a planar domain, and $u$ is an $\mathbb{R}^2$-valued vector field with fixed boundary data having degree $d>0$. We study the limiting pattern for a sequence of minimizers $\{u_\epsilon\}$ as $\epsilon\to 0$. We prove that the pattern contains $d$, degree one defects and that it has a either a radial or circular asymptotic form near each defect depending on the relative values of $k_s$ and $k_b$. We further characterize a renormalized energy for the problem and show that it is minimized by the limit. This is joint work with Sean Colbert-Kelly.

Polar smectic liquid crystals exhibit a variety of phases with multilayer ordering. Recent progress in experimental technique and discovery of several ferrielectric and antiferroelectric phases has put forward a number of fundamental problems: whether the known set of structures is exhaustive or other polar smectic phases exist? Is there an interconnection between azimuthal ordering of molecules in smectic layers and other degrees of freedom such as polar orientation of molecules? We employed the discrete Landau model of phase transitions with two-component order parameter to calculate the structures and phase diagrams of polar smectic liquid crystals. Structures commensurate and incommensurate with layer spacing are formed due to frustrating next-nearest interlayer interaction. Sequences of phases on temperature and the dependence of the phase sequences on model parameters are studied. Influence of different interlayer interactions on the topology of the phase diagrams is analyzed. The calculated phase diagrams enable to describe formation of various polar phases and their temperature sequences, including the unusual reversed phase sequence and the polar phase with six-layer periodicity. Our calculations also predict the existence of a reentrant phase sequence with two incommensurate phases. The results of our calculations demonstrate that discrete Landau model of phase transitions can be successfully used to describe the manifold of polar smectic structures observed in experiment.

Co-authors: P.V.Dolganov (Institute of Solid State Physics, Chernogolvka, Russia), V.M.Zhilin (nstitute of Solid State Physics, Chernogolvka, Russia), V.K.Dolganov (nstitute of Solid State Physics, Chernogolvka, Russia)

The $Q$-tensor description of the nematic director field is very powerful as it can cope with defects and complicated geometries. On the other hand, it is also computationally very expensive because the equations for the tensor components are stiff: the contain time scales that differ by six to eight orders of magnitude.

In this talk I will discuss how we can use the stiffness of the director equations to our advantage: we can separate the two time scales, eliminate the fast one and reduce the dynamics of interest to a slow manifold, where the equations are no longer stiff. The ensuing approximate model is both accurate and fast.

Of course, there is a price to be paid: the approximation is valid only away from defects. I will conclude the talk discussing how we are attempting to overcome this restriction.

Co-author Keith Daly (University of Southampton)

I describe the theory for two LC-based motors for directly converting light and heat to electrical work. Underpinning these new photo-voltaic strategies is the reduction in LC order by light via photo-isomerisation of dye-rods.

The first motor rests on optically reducing the dielectric susceptibility, the second on reducing the ferro-electric polarisation. Non-linear absorption determines the dynamics by bleaching front propagation.

Motor 3 is an optical-to-mechanical converter, relying on a photo-mechanical force cycle or turbine. It is still work in progress and I only sketch the basic ideas.

In all cases the fundamental efficiency limits (Carnot?) still require evaluation, but it is possible to calculate the efficiencies of the motors as designed. They depend on material parameters, once geometry is optimised. In motors 1 and 2, they could range from about 2 - 7% for existing materials.

Computer simulations (Monte Carlo and Molecular Dynamics) provide a powerful tool for obtaining molecular organizations starting from microscopic interactions. These interactions could be generic and very simple (based on arbitrarily chosen shapes and attraction laws) or realistic, atomistic, based on quantum mechanically generated representations of the constituent molecules. In all cases simulations provide the equivalent of computer experiments that offer terms of comparison for mathematical theories, with a level of detail and of possibilities for virtual experiments (e.g. the switching on and off of some of the interactions) not available in real life. A number of examples will be shown for liquid crystal systems in the bulk and close to interfaces, where recent results hint that the type of boundary conditions currently used in some mathematical theories may have to be extended.

There is a growing technology-driven interest in using external forces to move or shape small quantities of liquids. One existing technique, electrowetting, involves the application of an electric field to a conducting liquid. A disadvantage of this technique is that the liquid must remain in contact with the electrodes. However, this is not the case in liquid dielectrophoresis, where a dielectric (i.e. non-conducting) liquid is used. A common aspect to both these techniques is that electrical surface stresses at liquid-air or liquid-liquid interfaces play an important role.

In this work we consider a layer of dielectric liquid of non-dimensional depth $h(x,t)$ wetting a horizontal electrode with a hydrodynamically passive dielectric fluid (e.g. air) above. A second electrode is located at a distance $d>h$ above the lower substrate. When the applied voltage is increased past a critical value, an instability occurs on the free surface of the liquid. We investigate how the material and cell geometry parameters affect the critical applied voltage and the form of the instability. Using linear stability analysis, we find that there exists a critical spacing $d_{c}$ above which the fastest growing unstable mode has a non-zero wave number, so that undulations (``wrinkles'') form on the free surface. Below this critical spacing, the fastest growing unstable mode has a zero wave number, so that wrinkles do not form. In general, we also find that higher values of the inverse Bond number $\tau$ (proportional to the surface tension) lead to a stab ilisation of the zero wave number mode, i.e. higher values of $d$ are required for wrinkling to occur. Furthermore, if the inverse Bond number is sufficiently low, a second critical spacing $\tilde{d}_c

Generally chiral nematic liquid crystals have been seen as an aside to nematics, and have been studied less as a result. However they can exhibit many and varied patterns in the form of cholesteric fingers or filaments. If these behaviours were understood and controllable it could prove to be a valuable advance in industry devices. The notion of cholesteric frustration appears to be what drives the existence of the complicated minima, as a result boundary conditions, cell geometry and surface energies are all extremely important in the creation of a tractable problem, but so too is the function space in which we choose to minimise. More generally it might be possible, in some cases, to see different theories (Oseen-Frank, Ericksen, Q-Tensor) as merely minimisations in different function spaces.

We consider a nematic liquid crystal, in coexistence with its isotropic phase, in contact with a substrate patterned with rectangular grooves. In such a system, the nematic phase may fill the grooves without the occurrence of complete wetting. There may exist multiple (meta)stable filled states, each characterised by the type of distortion (bend or splay) in each corner of the groove and by the shape of the nematic-isotropic interface, and additionally the plateaux that separate the grooves may be either dry or wet with a thin layer of nematic. Using numerical simulations, we analyse the dynamical response of the system to an externally-applied electric field, with the aim of identifying switching transitions between these filled states. We find that order-electric coupling between the fluid and the field provides a means of switching between states where the plateaux between grooves are dry and states where they are wet by a nematic layer, without affecting the configu ration of the nematic within the groove. We find that flexoelectric coupling may change the nematic texture in the groove, provided that the flexoelectric coupling differentiates between the types of distortion at the corners of the substrate. We identify intermediate stages of the transitions, and the role played by the motion of the nematic-isotropic interface. We determine quantitatively the field magnitudes and orientations required to effect each type of transition.

We develop a systematic asymptotic description of domain wall motion in a magnetic nanowire. The Landau--Lifshitz--Gilbert equation is linearized about a static solution and the magnetization dynamics investigated via a perturbation expansion. We compute leading order behaviour, propagation velocities, and first order corrections of both travelling waves and oscillatory solutions, and find bifurcation points between these two types of solutions.

In this talk I will give a brief overview of the work I have undertaken in the first few months of my PhD. Starting with a review of some classic thermally-driven instabilities in Newtonian liquids, including Rayleigh-Benard convection and the Marangoni effect, I then consider instabilities within liquid crystal systems. Using an extension of Ericksen-Leslie theory to include thermal effects, Rayleigh-Benard convection in a nematic is considered. Possible extensions of these previously known results are then discussed in the context of the microfluidics of liquid crystal materials.

A ferrroelectric nematic phase is desirable both from a practical and theoretical viewpoint, but experimentally it has yet to be observed. Mathematically, such a phase in polar mesogens could be described in a similar way to the Landau-De Gennes Q-tensor theory, but importantly disrespecting the typical head-to-tail symmetry, so that a second order parameter representing the polarisation of the state enters into the model. This talk is concerned with designing a model in terms of these moments that respects the physicality of order parameters.

Sm-A LCEs are known to exhibit very different responses to stretching parallel and perpendicular to the smectic layer normal.

Side-chain Sm-A LCE films in particular have been extensively studied both theoretically and experimentally since the work of Nishikawa and Finkelmann in 1999. Typically a drastic drop in the elastic modulus after a critical stress can be observed when stretching parallel to the layer normal. Adams and Warner (2005) derived a model for Sm-A LCEs that explains this softening behaviour by the onset of microstructure also theoretically. The simplest of these microstructures consists in a fine scale buckling of layers. However, more complex microstructures are possible, as was shown in (Adams, Conti, DeSimone, & Dolzmann, 2008).

Our aim is to adapt this theory to try to understand the recent experimental results of Komp and Finkelmann (2007), whose Sm-A LCE films show different optical behaviour and behaviour of the order parameter on stretching parallel to the director when compared with the response of the original Nishikawa and Finkelmann LCE. We believe that the difference may be related to the presence of defects in the sample, as suggested by Komp and Finkelmann: the difficulty lies in how to model this situation.

I will talk a bit about the approach to equilibrium in liquid crystal materials. And I shall talk about viscoelasticity equation and the connection of it to liquid crystal problems. The Ericksen-Leslie system plays an important row in describing the flow of nematic liquid crystal materials. Under Parodi's relation, we have the global well-posedness and Lyapunov stability for the system near local energy minimizers. There also exist some similar results in Q-tensor theory and I am interested to find the relation between the two theories. Further, there shares some common traits in the Ericksen-Leslie system with the problem of the approach to equilibrium in nonlinear quasistatic viscoelasticity system. I will try to compare the two systems and list the difficulties that need to be solved.

We demonstrate that experimental data concerning the main phase transition in membranes can be rationalized in the framework of a theoretical scheme constructed in the spirit of the so-called weak crystallization theory. The scheme enables to explain a combination of seemingly contradictory features of the main phase transition: its first order nature and a well pronounced pre-transitional critical behaviour of various physical quantities.

Self-assembled Smectic-A monolayers composed of chiral fd viruses in the presence of a polymer depletant have been shown by the Dogic group to exhibit a variety of novel structures as a function of temperature and polymer concentration. After reviewing some of the experimental findings I will discuss a phenomenological theoretical model of this system based on the Helfrich membrane energy and the de Gennes theory of smectics. I will show how this model can be used to understand some of the fascinating experimental observations of fd self-assembly.

Defects in liquid crystals present an extremely vast set of situations, depending on the nature of the order parameter, elasticity properties, boundary conditions. One distinguishes : defects characteristic of liquid crystalline media with one or two discrete repeat distances (smectic and columnar phases) which can be investigated by pure geometric methods ; defects characteristic of a director order parameter, which employ the Volterra process (for one dimensional defects) and topological methods, those latter providing scope for defects of various dimensionalities ; finally continuous defects whose consideration is essential for a description of defect physical properties. These topics will be confronted with an historical point of view and possible future developments.

We present our numerical results on the structures of a chiral nematic liquid crystal when it is confined in a thin planar cell with strong anchoring on its surfaces. We are interested in the cases where the liquid crystal exhibit a cholesteric blue phase under no confinement, and the cell thickness is of the order of or smaller than the cholesteric pitch. We show various stable exotic defect structures depending on temperature, cell thickness, and the type of anchoring. They include a hexagonal lattice of Skyrmion excitations, and a regular array of ring defects. These exotic defect structures can be regarded as a result of frustrations between bulk blue phase ordering and surface anchoring that does not allow it.

Nematic disclinations can form stable braids when they are stabilized by a confining geometry, chirality, or by interplay of both effects. These are stable or metastable topologically diverse defect structures in the nematic ordering field. Based on the synergy of our theoretical and numerical, approaches we are able to characterize geometries and properties of disclination loops forming braids by winding numbers, lengths, knot or link types, and self-linking numbers. We focus our attention to selected nematic braids of the lowest complexity: knotted 2D colloidal crystals, opal structures permeated by nematics, and knots in cholesteic drops. With this overview I would like to show how topology and geometry enables the assembling of complex soft materials. [1] S. ?opar and S. Žumer, Nematic Braids: Topological Invariants and Rewiring of Disclinations, Phys. Rev. Lett. 106, 177801 (2011). [2] S. ?opar, and S. Žumer, Quaternions and hybrid nematic disclinations, Proc. R. Soc. A 469, 1471 (2013). [3] U. Tkalec, M. Ravnik, S. ?opar, S. Žumer and I. Muševi?, Reconfigurable Knots and Links in Chiral Nematic Colloids, Science 333, 62 (2011). [4] S. ?opar, N. A. Clark, M. Ravnik and S. Žumer, Elementary building blocks of nematic disclination networks in densely packed 3D colloidal lattices, Soft Matter, DOI: 10.1039/C3SM50475A (2013).

We fabricate colloidal particles with nontrivial surface topology, including colloidal handlebodies, knots, links, Mobius strings, and Seifert surfaces. When introduced into a nematic liquid crystal with a uniform far-field director, these particles induce three-dimensional director fields and topological defects around them. We use director switching by electric fields, laser tweezing of defects, and local photothermal melting of the liquid crystal to promote transformations among many stable and metastable particle-induced director configurations and then explore them by means of a direct label-free three-dimensional nonlinear optical imaging. This reveals many peculiarities of the interplay between topologies of colloidal surfaces, director fields, and defects, which we find being in agreement with Gauss-Bonnet and Poincaré-Hopf index theorems. The lecture will conclude with a brief discussion of how these findings may lay the groundwork for new types of topology-di ctated elastic self-assembly in liquid crystals and experimental study of low-dimensional topology.

We present a lattice Boltzmann study of the dynamics of an actomyosin droplet, described in terms of a continuum model which follows the time evolution of actomyosin density, actin polarisation and flow. This analysis offers a simple representation of a “cell extract”, which is a highly simplified system used in vitro to understand cell dynamics, and which essentially only comprises the actin cytoskeleton and an enclosing cell membrane. In the absence of polymerization and depolymerization processes (‘treadmilling’), the dynamics of our actomyosin droplet arises solely from the contractile motion of myosin motors; this should be unchanged when polarity is inverted. Our results suggest that motility can arise in the absence of treadmilling, by spontaneous symmetry breaking (SSB) of polarity inversion symmetry. This motility mode driven by myosin contractility alone may be relevant to cell motion in three dimensions, where frictional forces, which are crucial to convert actin polymerisatino into motion, are likely to be modest. We also show of active droplets crawling on a substrate, when both treadmilling and contractility are taken into account. Our droplets can adopt a number of morphologies and motility modes found experimentally in cells, such as lamellipodia, pseudopodia and oscillatory cell motion.

Active systems, such as the cytoskeleton and bacterial suspensions, provide their own energy and hence operate out of thermodynamic equilibrium. Continuum models describing active systems are closely related to those describing liquid crystal hydrodynamics, together with an additional ‘active’ stress term. We discuss how the behaviour of the active continuum models depends on model parameters, such as the strength of the activity and the liquid crystal tumbling parameter, and we compare our results to recent experiments on cytoskeletal gels.

We present a hydrodynamic theory of polar active smectics, by which we mean active striped systemsactive systems, both with and without number conservation. For the latter, we find quasi long-ranged smectic order in $d=2$ and long-ranged smectic order in $d=3$. In $d=2$ there is a Kosterlitz-Thouless type phase transition from the smectic phase to the ordered fluid phase driven by increasing the noise strength. For the number conserving case, we find that giant number fluctuations are greatly suppressed by the smectic order; that smectic order is long-ranged in $d=3$; and that nonlinear effects become important in $d=2$.

Fluid interfaces, such as soap films, liquid droplets, or lipid membranes, are known to give rise to several special geometries, whose complexity and beauty continue to fascinate us, as observers of the natural world, and challenge us as scientists. Here I show that a special class of surfaces of constant negative Gaussian curvature can be obtained in fluid interfaces equipped with an orientational ordered phase. These arise in various soft and biological materials, such as nematic liquid crystals, cytoskeletal assemblies, or hexatic colloidal suspensions. The purely hyperbolic morphology originates from the competition between surface tension, that reduces the area of the interface at the expense of increasing its Gaussian curvature, and the orientational elasticity of the ordered phase, that in turn suffers for the distortion induced by the underlying curvature.

Nematic shells are thin films of nematic liquid crystal deposited on rigid colloidal particles, which can be manufactured in different shapes and guises. The two-dimensional order tensor that describes the local organization of liquid crystal molecules, which tend to lie parallel to the colloids' surface, vanishes whenever no orientation is prevailing on average. The points where this takes place are called defect, as they lack order. The lectuer will review recent work concerned with the interaction between defects and the underlying surface. In particular, arguments will be offered that identify appropiate geometric potentials, depending on the shell's curvatures, which either promote or hamper defects, attract or repel them.

We review our experimental studies on the stretching driven director-rotation of polydomain nematic elastomers (PNEs) and monodomain NEs (MNEs). The texture of PNEs primarily depends on the preparation state, i.e., whether the cross-links are introduced in the high-temperature isotropic or low-temperature nematic state. The director in the isotropic-genesis PNEs can be rotated at unusually low energy cost by external fields. As a result, the isotropic-genesis PNEs can be largely deformed at small tensile force and moderate electric field strength. In contrast, the nematic genesis PNEs do not show such soft response due to the memory of the initial random director configuration. We observe the remanent of the initial director configuration even in the highly stretched state. The formation of stripe patterns has been known as a typical phenomenon of mechanical instability for the MNEs stretched normally to the initial global director. We introduce a different type of mechanical instability observed in our recent experiments for monodomain NEs.

We review the structure and mechanical properties of topological defects in nematic elastomers and membranes. Nematic elastomers crosslinked in the isotropic phase (isotropic-genesis polydomains) exhibit anomalously soft mechanical response, which is explained by anisotropic orientational correlation in the polydomain state. In contrast, nematic-genesis polydomains show large mechanical stress due to memory of initial defect distributions. Orientation-curvature coupling in nematic membranes causes a correlation pattern similar to that of isotropic-genesis polydomains, which results in low-energy defected states and anomalously slow ordering kinetics. The notion of relative orientational correlation function is introduced to characterize the multi-defect states in these and other materials.

Liquid crystal elastomers (LCEs) are rubbery materials that composed of liquid crystalline polymers (LCPs) crosslinked into a network. The rod-like mesogens incorporated into the LCPs are have random orientations in the high temperature isotropic phase, but can adopt the canonical liquid crystalline phases as the temperature is lowered. In this talk I will describe some modelling work of the layered smectic phase of LCEs.

Smectic liquid crystal elastomers have highly anisotropic mechanical behaviour. This arises in side chain smectic-A systems because the smectic layers behave as if they are embedded in the rubber matrix [1] (the same cannot be said of main chain smectic systems). The macroscopic mechanical behaviour of these solids is sensitive to the buckling of the layers, so it is a multiscale problem. A coarse grained free energy that includes the fine-scale buckling of the layers has been developed [2], which enables continuum modelling of these systems. I will describe how this continuum model, when augmented with an additional energy term describing layer buckling and other effects such as finite chain extension, can be used to model deformation of smectic-A elastomers in different experimentally accessible geometries.

Modelling smectic-C elastomers, with their tilted director, present a bigger challenge to calculating their coarse grained energy. The constraint placed on the director by the layer normal results in some unusual properties of their soft modes such as negative Poisson ratio. I will describe the geometry of these deformation modes in smectic-C elastomers [3].

[1] C. M. Spillmann et al, Phys. Rev. E 82, 031705, (2010). [2] J. Adams, S. Conti and A. DeSimone, Mathematical Models and methods in Applied Sciences, 18, 1 (2008). [3] A. W. Brown and J. M. Adams, Phys. Rev. E, 85, 011703 (2012) .

Reconfigurable assemblies of active, auto-chemotactic gels
Balazs, A (University of Pittsburgh)
Thursday 27 June 2013, 14:45-15:30
http://www.newton.ac.uk/programmes/CFM/seminars/2013062714454.html

Liquid crystalline materials allow for formation of complex optical and photonic patterns –the superstructures- within the bulk of the material or by adding colloidal inclusions. Memory [1], self-assembly [2], topology [3], and material flow [4] can serve as guiding and controlling mechanism of the structures, offering possible competitive advantages to be used in complex optics and photonics. Here, we present three dimensional colloidal and bulk liquid crystal superstructures, as recently achieved by numerical modelling and experiments. Central to the superstructures are complex conformations of topological defects, as they can bind, stabilise, or distort the structure. We show that 3D colloidal crystals can be assembled from elastic dipoles of spherical beads in nematic liquid crystals [5] or via inherently inhomogeneous order profiles in bulk and confined cholesteric blue phases [6]. Colloidal crystals are generalised to close-packed colloidal lattices, which we show can serve as natural templates for defect networks [7]. Finally, photonic bands are calculated for selected structures and possible defects in the structure are discussed [8].

[1] T. Araki, M. Buscaglia, T. Bellini, H. Tanaka, Nature Materials 10, 303 (2011). [2] P. Poulin, H. Stark, T. C. Lubensky, D. A. Weitz, Science 275, 1770 (1997). [3] U. Tkalec, M. Ravnik, S. Copar, S. Zumer I. Musevic, Science 333, 62 (2011). [4] A. Sengupta, U. Tkalec, M. Ravnik, J.M. Yeomans, C. Bahr, S. Herminghaus, Phys. Rev. Lett. 110, 048303 (2013). [5] A. Nych, U. Ognysta, M. Škarabot, M. Ravnik, S. Žumer, I. Muševič, Nature Comm. 4, 1489 (2013). [6] M. Ravnik, G. P. Alexander, J. M. Yeomans, S. Zumer, Proc. Natl. Acad. Sci. USA 108, 5188 (2011). [7] S. Copar, N.A. Clark, M. Ravnik, S. Zumer, Elementary building blocks of densely packed 3D colloidal lattice entangled by nematic disclinations, accepted in Soft Matter. [8] M. Stimulak and M. Ravnik, Photonic bands in blue phase colloidal crystals, to be submitted;

In-situ photopolymerization of liquid crystalline (LC) monomers has proven to be a valuable technique for the formation of well-ordered polymer networks. Their anisotropic properties led to a variety of applications in optics, electronics and mechanics. The use of light to initiate polymerization enables lithographic approaches for patterning. The LC behaviour enables formation of complex morphologies on molecular level. Controlling the director profile of an LC network film in transversal direction gives geometrical morphing upon minor changes in order parameter. Examples of suited profiles of molecular orientation are twisted or splayed director configurations tied up in the polymer network. Reversible order parameter changes can be induced by a variety of means. It can be simply induced by temperature changes resulting in gradients in thermal expansion over the cross-section of the film. But more sophisticated and of interest for applications is a light induced change as a result of the E-Z isomerization of a built-in azo group. When the LC polymer networks are confined in the x-y plane of the film, e.g. by strong adhesion to a high modulus substrate, deformation takes place into the third dimension forming surface topographies.

Rather than composing the netyworks of covalent bonds alone, one can chose to replace some bonds by secondary interactions such as hydrogen bridges, thus providing responsive molecularly organized hydrogels. We applied the H-bridge based dimerization of benzoic acid to form nematic LC acrylate monomers. By a controlled and reversible rupture of the H-bridges mechanical responses can be initiated. Photopolymerization of smectic LC monomers lock in structures of different length scales. The first length scale is the resolution of lithography, and goes down to a few micrometers. The second length scale is set by the spacing of smectic layers and is typically a few nanometers. The third length scale is the intermolecular distance in the layers, usually around 1 nm or below. By modifying smectic molecules with H-bridges the smectic periodicity can break-up into separated layers with a well-defined spacing, also in the nanometer range. The nanopores form by breaking the H-bridges at elevated temperatures or by contact with an alkaline solution. The integrity of the film is maintained by copolymerizing with fully covalent smectic crosslinkers. By making this crosslinker photosensitive the pore size can be regulated by UV light.

Liquid crystal elastomers are solid liquid crystals; they combine elasticity with orientational order. Mechanical strain therefore changes liquid crystalline order and the optical properties of these materials. Conversely, light can change the orientational order, and give rise to mechanical forces and changes in shape. Light-matter interactions in LCEs therefore involve a broad range of unusual phenomena, which raise a number of intriguing questions. I will present the results of some experiments probing such phenomena, and discuss the underlying physics.