# Workshop Programme

## for period 18 - 22 March 2013

### Analytical and Computational Paths from Molecular Foundations to Continuum Descriptions

18 - 22 March 2013

Timetable

 Monday 18 March 08:30-08.50 Registration 08:50-09:00 Welcome by Director, Professor John Toland 09:00-09:50 Ball, J (University of Oxford) Satisfaction of the eigenvalue constraints on the $Q$-tensor Sem 1 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 $\inf_{x\in\Omega}\lambda_{\rm min}(Q(x))>-1/3$. 09:50-10:40 Frenkel, D (University of Cambridge) The physics of unphysical simulations Sem 1 Simulations are patient. In particular, they can be used to model systems that are interesting but unrealistic'. In fact, from the time of Onsager onwards, unphysical limits have played a key role in our understanding of lyotropic liquid crystals. Interestingly, simulations allow us to probe interesting limits that are inaccessible to experiments. 10:40-11:00 Morning Coffee 11:00-11:50 Evans, B (University of Bristol) Density Functional Theory for Hard-Body Models of Liquid Crystals Sem 1 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. 11:50-12:40 Berardi, R (Università di Bologna) Coarse-grained modelling and computer simulations of liquid crystals Sem 1 Coarse-grained models for liquid crystals are typically based on pair potentials where an entire mesogenic molecule is represented by one (or a few) anisotropic geometrical object (e.g. a spherocylinder, or an ellipsoid) with either purely repulsive or attractive-repulsive interactions. Computer simulations relying on these simple off-lattice models are able to reproduce the experimental phase sequences and order parameters of thermotropic mesogens and are useful for studying the relationship between specific molecular properties (e.g. shape or interaction anisotropies) and macroscopic liquid crystalline behaviour. We will review the principal coarse-grained level models currently used in computer simulations of liquid crystals and discuss their advantages and shortcomings using the results for selected cases. 12:40-13:30 Lunch at Wolfson Court 14:00-14:50 Cates, M (University of Edinburgh) Active liquid crystals and the origins of cellular locomotion Sem 1 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. 14:50-15:40 Vanakaras, A (University of Patras) Biaxial Nematics: Symmetry and Hierarchical Domain Structure Sem 1 We present theoretical and computer simulation studies on the structure of nematic liquid crystals formed by bent-core mesogens (BCM) and by board-like colloids (BLC). The presence of local orientational and/or positional ordering is a key feature for the interpretation of the biaxial nematic ordering observed in these systems. In the first part we present the full phase diagram, calculated from MC molecular simulations, of sterically interacting BLC, for a range of experimentally accessible molecular dimensions/anisometries of colloids of this shape. New classes of phase transition sequences such as nematic-nematic and, for the first time, a direct transition from a discotic and a biaxial nematic to an orthogonal smectic-A phase have been identified. We demonstrate rigorously the formation of supramolecular entities and explain the observed phase transitions in terms of the "shape anisotropy" of these entropy driven supramolecular assemblies. In the second part the structure of nematic liquid crystals formed by bent-core mesogens is studied in the context of MC simulations of a simple molecular model that captures the symmetry, shape, and flexibility of achiral BCMs. Our results indicate the formation of (i) clusters exhibiting local smectic order, orthogonal or tilted, with strong in-layer polar correlations and antiferroelectric juxtaposition of successive layers and (ii) large homochiral domains through the helical arrangement of the tilted smectic clusters, while the orthogonal clusters produce achiral (untwisted) nematic states. The results of our work offers a deeper understanding of the nematic-nematic transitions and, ultimately, of the nematic phase and can serve as a comprehensive guide to experiment, towards the design of anisotropic liquids with the desired functionality, as well as to theory for testing and improving analytical molecular models using simple intermolecular potentials. 15:40-16:00 Afternoon Tea 16:00-16:50 Belli, S (Utrecht University) Brick-by-brick stabilizing the Biaxial Nematic Phase Sem 1 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) 16:50-17:50 Welcome wine reception
 Tuesday 19 March 09:00-09:50 Gartland Jr., EC (Kent State University) Liquid Crystal Director Models with Coupled Electric Fields Sem 1 Historically, many liquid-crystal devices and experiments have involved low-molecular-weight nematic liquid crystals, in supra-micron-size confinements, with coupled electric fields. In such settings, equilibrium orientational properties can be modeled most effectively using the Oseen-Frank elastic theory coupled with the equations of electrostatics. In this (mostly) expository talk, we will discuss some of the issues that arise in the mathematical and numerical treatment of such classical models. These issues include the intrinsic minimax nature of such models, which arises from the negative-definite way in which the electrostatic potential enters the free energy functional and which can also arise when Lagrange multipliers are used to enforce the pointwise unit-vector constraints on the liquid-crystal director field, as well as the complications this indefiniteness adds to the assessment of local stability of equilibria. We will also discuss the anomalous behavior that can be exhibited at the thresholds of certain electric-field-induced instabilities because of the nature of the coupling between the director field and the electric field. In addition, we will contrast the macroscopic Oseen-Frank model with the mesoscopic Landau-de Gennes model in such contexts. 09:50-10:40 Majumdar, A (University of Bath) Modelling a planar bistable device on different scales Sem 1 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. 10:40-11:00 Morning Coffee 11:00-11:50 Mottram, N (University of Strathclyde) Modelling planar bistable devices: from Q-tensor to director models Sem 1 In this talk we continue the theme of bistable devices from the previous presentation. Here we compare the Q-tensor model to a director based model, i.e. moving from a mesoscopic approach to a macroscopic approach, in a number of practical examples. We consider the polygonal confinement considered in the previous talk and extend to more general cases of bistability and multistability arising from the morphology of bounding substrates. Bifurcation diagrams of stable states, as parameters such as anchoring strength are varied, are computed. We review some of the advantages and disadvantages of the two modelling approaches and compare results to experimental measurements. Although a director based model is inherently restricted in its inability to model effects such as surface melting and disclination lines, we find that all stable states are in fact reproduced. Additionally we find that, rather surprisingly, the dynamics of switching in the bistable devices can be modelled accu rately using a director theory. 11:50-12:40 Muller, EA (Imperial College London) SAFT force fields for coarse-grained MD simulations Sem 1 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. 12:40-13:30 Lunch at Wolfson Court 13:30-17:00 Free Afternoon
 Wednesday 20 March 09:00-09:50 Dijkstra, M (Universiteit Utrecht) Glassy dynamics, spinodal fluctuations, and nucleation in suspensions of colloidal hard rods and plates Sem 1 Using computer simulations we study nucleation in a colloidal supension of hard rods. We study the kinetic pathways for the isotropic-to-nematic transition in a fluid of long hard rods, and find spinodal decomposition as well as nucleation and growth depending on the supersaturation [1]. In supersaturated isotropic fluid states of short hard rods, we observe nucleation of multilayered crystalline clusters. At sufficiently high supersaturations, we find that the nucleation is hampered by glassy dynamics. For intermediat rods, we find that the formation of the (stable) smectic phase out of a supersaturated isotropic state is strongly suppressed by an isotropic-nematic spinodal instability that causes huge spinodal-like orientation fluctuations with nematic clusters diverging in size [2]. In suspensions of colloidal platelets, we find that the cubatic phase is metastable, and that perpendicularly oriented particle stacks in the isotropic fluid phase inhibits the formation of the columnar phase [3]. [1] A. Cuetos and M. Dijkstra, Physical Review Letters 98, 095701 (2007). [2] R. Ni, S. Belli, R. van Roij, and M. Dijkstra, Physical Review Letters 105, 088302 (2010). [3] M. Marechal, A. Patti, M. Dennison, and M. Dijkstra, Physical Review Letters 108, 206101 (2012). 09:50-10:40 Masters, A (University of Manchester) Simulations of model biaxial particles Sem 1 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. 10:40-11:00 Morning Coffee 11:00-11:50 Galindo, A (Imperial College London) Phase behaviour in mixtures of unixial hard particles: biaxiality and confinement Sem 1 The nematic biaxial phase has remained a key challenge in the science of liquid crystals since it was first proposed. Recently the first experimental evidence of stable biaxial nematic phases has been obtained in thermotropic liquid crystals of single component biaxial mesogens by Madsen et al., and others. Still elusive however is the possibility of stabilizing biaxial nematic phases in mixtures of uniaxial particles. This avenue has been explored in some detail using theory and computer simulation, but leads one to the conclusion that, at least in the case of mixtures of hard particles, the nematic biaxial phase is thermodynamically unstable with respect to demixing into two uniaxial phases. Theoretical calculations have, however, pointed out that with an appropriate attractive unlike interaction, a homogeneous biaxial nematic phase could be stabilized. Experimental work on mixtures of rod and disc-like molecules has tended to confirm the view that such a system would favou r phase separation, until the recent studies of Apreutesei and Mehl. In this contribution, we use canonical Monte Carlo molecular simulations to study model mixtures of rodlike and disklike molecules interacting through two intermolecular potential models: one incorporating spherically symmetric (isotropic) attractive interactions; another with anisotropic attractive interactions. These models exhibit nematic and smectic biaxial phases. In the final part of the talk, if time allows, I will briefly discuss the changes in the phase behavior that occur when uniaxial disc-like particles are placed in confinement between parallel walls and consider the surface ordering and capillary phenomena in this system. 11:50-12:40 Virga, E (Università degli Studi di Pavia) Different flavours of the mean-field theory Sem 1 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. 12:40-13:30 Lunch at Wolfson Court 14:00-14:50 Longa, L (Uniwersytet Jagiellonski) Lebwohl-Lasher models of liquid crystals: from quadrupolar to spontaneously induced chiral order Sem 1 In 1972 P. A. Lebwohl and Q. Lasher (LL) (PRA 6, 426 (1972)) have carried out standard Monte Carlo simulations on the lattice version of the Maier-Saupe (MS) model to test predictions of the MS mean-field calculations. Preserving uniaxial symmetry ($D_{\infty h}$)for nematics they assumed liquid crystalline molecules to occupy the sites of a three dimensional cubic lattice subjected to periodic boundary conditions. Pair interaction potential, limited to nearest-neighbor molecules, was given by the second Legendre polynomial of the relative angle between the molecular long axes. The simulations showed that the LL lattice model undergoes a weak first-order phase transition between isotropic and uniaxial nematic order, in qualitative agreement with MS predictions. Since the model has proved to correctly account for the essential symmetry of liquid crystalline orientational order a large amount of work has been and is currently devoted to generalizations of the LL model to more complex situations. They involve, without trying to be exhaustive, (a) investigation of the nematic ordering in confined geometries, subject to different surface anchoring fields, (b) effect of an external field on the isotropic - nematic phase transition(s), (c) simulations of electro-optical devices, (d) simulation of chiral liquid crystal phases, (e) orientational properties of elastomers and (f) physics of two-dimensional systems. In this talk, after a brief review of properties and generalizations of the LL model, I will concentrate on simple versions of this model that can be useful in investigating spontaneous formation of macroscopic chiral domains of opposite handednesses observed in bent-core, dimer and ferrocene mesogens. More specifically, I will discuss properties of the LL model with quadrupolar and octupolar pair-interactions. The model will be shown to generate long-range biaxial order along with ambidextrous twist deformations. A possibility of generating nonzero splay and bent configurations will also be discussed. The class of LL models is generic in the sense that only symmetry allowed terms are retained in the interaction potential. Hence, orientational structures identified not only characterize nematic-like states but can also coexist with a long-range positional order, characteristic of smectic, columnar or crystalline phases. 14:50-15:40 Muccioli, L (Università di Bologna) Atomistic molecular dynamics simulations of cyanobiphenyls: A test bench for liquid crystal theories Sem 1 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. 15:40-16:00 Afternoon Tea 16:00-16:50 Ilnytskyi, J (National Academy of Sciences of Ukraine) Microphase separation driven transitions in macromolecular liquid crystals by computer simulations Sem 1 We present the results of some recent simulations of macromolecular liquid crystal systems that undergo order-disorder transitions driven by a microphase separation. Molecular dynamics simulations are performed to study a liquid crystal elastomer of a side-chain architecture crosslinked in the SmA phase. Several effects have been observed: (i) the increase of the SmA-I transition temperature as the result of crosslinking; (ii) memory effects in liquid crystallinity and shape when the elastomer is driven through the Sm-I transition; (iii) both cases of homogeneous director reorientation and stripe formation when the load is applied along the nematic director [1]. In another set of results we consider bulk self-assembly of liquid crystal dendrimers studied by means of coarse-grained molecular dynamics simulations. We discuss the details of the modelling and its application to polymer-modified gold nanoparticles. The particular model dendrimer being studied demonstrates conforma tional bistability, with both rod-like and disc-like conformations stable at lower temperatures. Each conformation can be induced by the external field of appropriate symmetry, promoting further self-assembly of macromolecules into a bulk monodomain SmA or a columnar phase, respectively [2]. The domains of both phases are found to coexist and influence the system properties in a broad temperature interval including transition to the macroscopically isotropic phase. We also discuss the effect of surface anchoring on the self-assembly of these macromolecules [3]. [1] J.M.Ilnytskyi, M.Saphiannikova, D.Neher, M.P.Allen, Soft Matter (2012), DOI: 10.1039/c2sm26499d [2] J.M.Ilnytskyi, J.S.Lintuvuori, M.R.Wilson, Condens. Matter Phys. 13, 33001 (2010). [3] J.M.Ilnytskyi, M.Schoen, M.R.Wilson, in preparation.