# Workshop Programme

## for period 21 - 22 May 2013

### Young Researchers Meeting

21 - 22 May 2013

Timetable

 Tuesday 21 May 10:30-11:00 Registration 11:00-11:10 Introduction (John Toland and John Ball) 11:10-11:30 Corson, L (Strathclyde) Liquid dielectrophoresis and wrinkling on the surface of a fluid layer Sem 1 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 11:30-11:50 Lin, T-S (Loughborough) Hydrodynamic description of thin nematic films Sem 1 We discuss the long-wave hydrodynamic model for a thin film of nematic liquid crystal. Firstly, we clarify how the elastic energy enters the evolution equation for the film thickness. We show that the long-wave model derived through an asymptotic expansion of the full nemato-hydrodynamic equations with consistent boundary conditions agrees with the model one obtains by employing a thermodynamically motivated gradient dynamics formulation based on an underlying free energy functional. As a result, we find that in the case of strong anchoring the elastic distortion energy is always stabilising. Secondly, based on a gradient dynamics approach, we propose a film thickness evolution equation that describes a free surface thin film of nematic liquid crystals on a solid substrate under weak anchoring conditions at the free surface. We show that in the intermediate film thickness range anchoring and bulk energies compete what may result in a linear instability of the free surfa ce of the film. 11:50-12:10 Bedford, S (Oxford) Variational problems for cholesteric liquid crystals - Function spaces and competing theories Sem 1 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. 12:10-12:30 Bowick, M (Syracuse) Open Problems Session Sem 1 12:30-13:30 Lunch at Wolfson Court 13:30-13:50 Blow, M (Lisbon) Interfacial motion in flexo- and order-electric switching between nematic filled states Sem 1 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. 13:50-14:10 Nieuwenhuis, M (Oxford) A description of the smectic phase using statistical mechanics Sem 1 Many different interaction potentials have been used in order to model phase transitions for liquid crystals. One of the most fundamental theories used for numerical simulations is the density functional theory. However, the main task in this theory is to find a suitable approximation for the energy functional which is in general very difficult. In this short presentation I would like to give a brief overview over the techniques that can be used in order to capture the mathematical characteristics of the smectic phase. 14:10-14:30 Lund, R (Bristol) Domain wall motion in magnetic nanowires: an asymptotic approach Sem 1 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. 14:30-15:30 Individual discussions 15:30-16:00 Afternoon Tea 16:00-16:20 Foldes, R (Minneapolis) Boundary-roughness effects in nematic liquid crystals Sem 1 Paolo Biscari and Stefano Turzi considered a plate with an undulatory pattern. They replace the corrugation with sinusoidal boundary conditions, and use formal asymptotics for the analysis. I would like to use the method of gamma convergence to determine the effective energy and its minimizers for this problem. 16:20-16:40 Virga, E (Università degli Studi di Pavia) Open Problems Session Sem 1 16:40-17:00 Newton, C (University of Bristol) Open Problems Session Sem 1 17:00-18:00 Reception
 Wednesday 22 May 09:00-09:20 Hair, W (Strathclyde) Thermal effects in liquid crystal layers Sem 1 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. 09:20-09:40 Ferreiro, C (Bristol) Random packing of rods and spheres Sem 1 Random packing of mixtures of hard spherocylinders and hard spheres are studied for spheres with a diameter similar to the length of the spherocylinders. Packing fractions of hard spherocylindres with aspect ratios 0 09:40-10:00 Taylor, J (Oxford) A physical model to predict a ferroelectric nematic phase Sem 1 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. 10:00-10:20 To, T (Southampton) Molecular field theory for biaxial smectic A liquid crystals Sem 1 Stable biaxial nematics (Nb) have been reported in a few experimental systems and the phases are often difficult to prove conclusively; however, stable biaxial smectic A phases (SmAb) have been found in a larger number of systems in which the evident is conclusive. To understand the stability difference between Nb and SmAb, we use a molecular field theory that combines Straley's theory [1] for biaxial nematics and McMillan's theory [2] for uniaxial smectic A phases. To simplify the calculation, we use alternatively the geometric mean [3] and the Sonnet-Virga-Durand [4] approximation to reduce the number of biaxiality parameters to one; in addition, we use the Kventsel-Luckhurst-Zewdie [5] approximation to decouple the orientational and translational distribution functions. Thus our simple theory has one biaxiality parameter and one smecticity parameter; together with three order parameters. The resulting phase diagrams showed that, for a large region of the para meter space, the presence of the smectic A phases disallowed Nb to form. On the other hand, SmAb is always stable at ground state for positive smecticity parameter. Thus this may explain why SmAb has been found more abundant than Nb. [1] J. P. Straley, Phys. Rev. A 10, 1881 (1974). [2] W. L. McMillan, Phys. Rev. A 4, 1238 (1971). [3] G. R. Luckhurst, C. Zannoni, P. L. Nordio, and U. Segre, Mol. Phys. 30, 1345 (1975). [4] A. Sonnet, E. G. Virga, and G. E. Durand, Phys. Rev. E 67, 061701 (2003). [5] G. F. Kventsel, G. R. Luckhurst, and H. B. Zewdie, Mol. Phys. 56, 589 (1985). 10:20-10:50 Coffee 10:50-11:10 Käbisch, S (Surrey) Modeling Sm-A LCEs with defects Sem 1 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. 11:10-11:30 Tjhung, E (Edinburgh) Designing a crawling cell using soft materials Sem 1 Eukaryotic cells have been observed to be able to move in various media. One obvious example is provided by the keratocyte cells which are able to crawl on a 2D substrate (such as glass slides). In this talk, we aim to build a minimal hydrodynamic model of a crawling cell using ideas from soft matter physics such as binary liquid and liquid crystals. The simplest model of a cell is probably just a droplet sitting on a surface. However, a passive droplet will not be able to move on its own. To make it moves, we have to add some non-equilibrium physics into it. This is provided by actin polymerisation and actin-myosin contraction inside the cell cytoskeleton. These two active processes can then be coarse-grained into a set of hydrodynamic equations which are similar to that of active liquid crystals. 11:30-11:50 Liu, L (Oxford) The approach to equilibrium for the Ericksen-Leslie system and related questions of Q-tensor theory Sem 1 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. 11:50-12:50 Individual discussions 12:50-13:30 Lunch at Wolfson Court 14:00-14:20 Sluckin, T (University of Southampton) Open problems session Sem 1 14:20-14:40 Lubensky, T (University of Pennsylvania) Open problems Session Sem 1 14:40-15:00 De Simone, A (SISSA) Open Problems Session Sem 1 15:00-15:05 Closing