Nematic Liquid Crystals (NLCs) find widespread industrial use in Liquid Crystal Display (LCD) devices, and so there is considerable interest in furthering knowledge of how they behave, both within such devices, but also in situations where they flow (e.g. spreading and coating flows). Under NSF funding (DMS 1211713), we are investigating both scenarios.

NLCs typically consist of polar, rod-like molecules which, due to electrostatic interactions, tend to align with their neighbors. This imparts short-range orientational order to the material, giving some elastic character, and a viscosity that depends on the direction of shear. In addition, the molecules also have a preferred orientation at a bounding surface, a phenomenon known as anchoring (and the details of which depend on the interactions between the NLC and the surface, which can be altered by suitable surface treatments). These features can be modeled by a director field for the NLC; a unit vector field that represents the local average direction of the long axis of the molecules.

In our work we use the Leslie-Ericksen formulation: an energy equation, coupled to a momentum equation plus incompressibility (analogous to the Navier-Stokes equations). The energy equation minimizes the net free energy of the NLC, comprising bulk and surface contributions (the latter model the anchoring). In each of the problems we consider we exploit asymptotic simplifications, due either to a small geometrical aspect ratio or to highly disparate timescales.

Our work on free surface flows has focused on so-called "thin film" spreading flow, where the height of a spreading film of NLC is everywhere much smaller than typical lateral length scales. As the film flows, the molecular orientation of the NLC has to adapt to the changing geometry. In most situations of interest, the timescale of spreading is much longer than that of the elastic reorientation response of the molecules, which allows the energy equation to be decoupled from the flow momentum equation. A 4th order nonlinear PDE for the free surface height may be derived, the stability properties of which depend strongly on the anchoring conditions imposed at the underlying substrate and the free surface. Our work to date on this problem is described in our publications below; ongoing efforts are focused on the influence of van der Waals' type interactions, relevant at very small length scales (in so-called "ultra-thin" films); and on the effect of external forcing supplied, e.g., by an applied electric field.

Our work relevant to LCD devices has focused on (i) exploring the potential for bistable devices that operate with greatly reduced power consumption relative to conventional LCDs; and (ii) formulating mathematical models to describe the phenomenon of director gliding, which can occur when an NLC is adjacent to a polymeric bounding surface (such considerations may be important in the design of flexible polymer-based display devices). LCDs rely on the birefringence principle; the ability of NLCs to rotate the plane of polarized light, to an extent that depends on the molecular configuration. A pixel of a conventional LCD consists of a thin layer of NLC sandwiched between parallel transparent plates (which act as electrodes). The sandwich is placed between crossed polarizers. In the simplest scenario, the bounding electrodes can be treated to impart planar anchoring, such that the NLC molecules prefer to lie parallel to the plates and in a specified direction. Choosing the top plate anchoring direction to be orthogonal to that of the lower plate, the NLC molecules will then undergo a "twist" deformation across the layer in the absence of other forces. Light passing through one polarizer will then enter the NLC layer as a polarized beam, and its plane of polarization will be rotated by the layer so that it can pass through the second (crossed) polarizer, giving a bright pixel. However, when an electric field is applied across the electrodes, the "twist" deformation of the NLC molecules is disrupted; the molecules instead align with the electric field, perpendicular to the electrodes, and in this configuration cannot rotate the plane of the polarized light beam. The light is then blocked by the second polarizer, giving a dark pixel.

Application of a field (or not) to individual pixels in a LCD can then be used as the basis for creating contrast in a display. This is energetically expensive however, since contrast can be maintained only by sustained application of an electric field. There is thus industrial interest in designing "bistable" devices: NLC sandwiches that, by suitable treatment of the bounding plates (to tune the anchoring conditions) allow them to sustain two stable steady states in the absence of an applied electric field (in the scenario outlined above, the simple "twisted" state is the unique field-free state). A display could then retain its configuration without any applied power, with power required only to switch the display to a new state. Finding anchoring conditions that permit bistability of pixels is not difficult; more challenging is to find bistable scenarios that allow for reversible switching between the two stable states (via transient application of an electric field). Our work on bistable LCDs has focused on determining "optimal" anchoring conditions on the bounding plates that allow for bistability, with maximal optical contrast between the two states, and with reversible, fast, switching between the states, at moderate applied fields (to minimize power consumption).

The "director gliding" phenomenon referred to above occurs when a layer of NLC is adjacent to a polymeric bounding surface, with some bulk force (due, e.g. to an electric field) applied to the molecules of the NLC causing them to be pulled away from the preferred anchoring orientation (the so-called "easy axis" associated with the anchoring) at the polymeric boundary. Due to the interactions between the NLC molecules and those of the polymeric bounding surface, such a force exerts a torque on the polymer molecules, which can lead to a slow reorientation of the easy axis. The anchoring properties of the surface can thus change slowly in time, with potential consequences for the optical properties of a polymer-based display. We have derived mathematical models (described in our publications) to represent such dynamic "gliding" behavior. Current efforts are focused on obtaining quantitative agreement with experimental data.

We present the results of large scale simulations of 4th order nonlinear partial differential equations of diffusion type that are typically encountered when modeling dynamics of thin fluid films on substrates. The simulations are based on the alternate direction implicit (ADI) method, with the main part of the computational work carried out in the GPU computing environment. Efficient and accurate computations allow for simulations on large computational domains in three spatial dimensions (3D) and for long computational times. We apply the methods developed to the particular problem of instabilities of thin fluid films of nanoscale thickness. The large scale of the simulations minimizes the effects of boundaries, and also allows for simulating domains of the size encountered in published experiments. As an outcome, we can analyze the development of instabilities with an unprecedented level of detail. A particular focus is on analyzing the manner in which instability develops, in particular regarding differences between spinodal and nucleation types of dewetting for linearly unstable films, as well as instabilities of metastable films. Simulations in 3D allow for consideration of some recent results that were previously obtained in the 2D geometry [28]. Some of the new results include using Fourier transforms as well as topological invariants (Betti numbers) to distinguish the outcomes of spinodal and nucleation types of instabilities, describing in precise terms the complex processes that lead to the formation of satellite drops, as well as distinguishing the shape of the evolving film front in linearly unstable and metastable regimes. We also discuss direct comparison between simulations and available experimental results for nematic liquid crystal and polymer films.

The interaction between nematic liquid crystals and polymer-coated substrates may lead to slow reorientation of the easy axis (so-called “director gliding”) when a prolonged external field is applied. We consider the experimental evidence of zenithal gliding observed by Joly et al. [Phys. Rev. E 70, 050701 (2004)] and Buluy et al. [J. Soc. Inf. Disp. 14, 603 (2006)] as well as azimuthal gliding observed by S. Faetti and P. Marianelli [Liq. Cryst. 33, 327 (2006)], and we present a simple, physically motivated model that captures the slow dynamics of gliding, both in the presence of an electric field and after the electric field is turned off. We make a quantitative comparison of our model results and the experimental data and conclude that our model explains the gliding evolution very well.

We discuss instabilities of fluid films of nanoscale thickness, with a particular focus on films where the destabilising mechanism allows for linear instability, metastability, and absolute stability, depending on the mean film thickness. Our study is motivated by nematic liquid crystal films; however, we note that similar instability mechanisms, and forms of the effective disjoining pressure, appear in other contexts, such as the well-studied problem of polymeric films on two-layered substrates. The analysis is carried out within the framework of the long-wave approximation, which leads to a fourth-order nonlinear partial differential equation for the film thickness. Within the considered formulation, the nematic character of the film leads to an additional contribution to the disjoining pressure, changing its functional form. This effective disjoining pressure is characterised by the presence of a local maximum for non-vanishing film thickness. Such a form leads to complicated instability evolution that we study by analytical means, including the application of marginal stability criteria, and by extensive numerical simulations that help us develop a better understanding of instability evolution in the nonlinear regime. This combination of analytical and computational techniques allows us to reach novel understanding of relevant instability mechanisms, and of their influence on transient and fully developed fluid film morphologies. In particular, we discuss in detail the patterns of drops that form as a result of instability, and how the properties of these patterns are related to the instability mechanisms.

We consider a mathematical model that consists of a nematic liquid crystal layer sandwiched between two parallel bounding plates, across which an external field is applied. We investigate how the number and type of solutions for the director orientation within the layer change as the field strength, anchoring conditions, and material properties of the nematic liquid crystal layer vary. In particular, we focus on how the inclusion of flexoelectric effects alters the Freedericksz and saturation thresholds.

In describing the physics of living organisms, a mathematical theory that captures the generic ordering principles of intracellular and multicellular dynamics is essential for distinguishing between universal and system-specific features. Here, we compare two recently proposed nonlinear high-order continuum models for active polar and nematic suspensions, which aim to describe collective migration in dense cell assemblies and the ordering processes in ATP-driven microtubule-kinesin networks, respectively. We discuss the phase diagrams of the two models and relate their predictions to recent experiments. The satisfactory agreement with existing experimental data lends support to the hypothesis that non-equilibrium pattern formation phenomena in a wide range of active systems can be described within the same class of higher-order partial differential equations.

ATP-driven microtubule-kinesin bundles can self-assemble into two-dimensional active liquid crystals (ALCs) that exhibit a rich creation and annihilation dynamics of topological defects, reminiscent of particle-pair production processes in quantum systems. This recent discovery has sparked considerable interest but a quantitative theoretical description is still lacking. We present and validate a minimal continuum theory for this new class of active matter systems by generalizing the classical Landau–de Gennes free-energy to account for the experimentally observed spontaneous buckling of motor-driven extensile microtubule bundles. The resulting model agrees with recently published data and predicts a regime of antipolar order. Our analysis implies that ALCs are governed by the same generic ordering principles that determine the non-equilibrium dynamics of dense bacterial suspensions and elastic bilayer materials. Moreover, the theory manifests an energetic analogy with strongly interacting quantum gases. Generally, our results suggest that complex nonequilibrium pattern-formation phenomena might be predictable from a few fundamental symmetry-breaking and scale-selection principles.

We consider the interaction between nematic liquid crystals (NLCs) and polymer substrates. Such substrates can interact with NLCs, exhibiting a phenomenon known as director gliding: the preferred orientation of the NLC molecules at the interface changes on time scales that are slow relative to the elastic relaxation time scale of the NLC. We present two models for gliding, inspired by experiments that investigate the interaction between the NLC and a polymer substrate. These models, though simple, lead to nontrivial results, including loss of bistability under gliding. Perhaps surprisingly, we find that externally imposed switching between the steady states of a bistable system may reverse the effect of gliding, preventing loss of bistability if switching is sufficiently frequent. Our findings may be of relevance to a variety of technological applications involving liquid crystal devices, and particularly to a new generation of flexible liquid crystal displays that implement polymeric substrates.

Recent experiments by Sengupta et al. (Phys. Rev. Lett. 2013) [9] revealed interesting transitions that can occur in flow of nematic liquid crystal under carefully controlled conditions within a long microfluidic channel of width much larger than height, and homeotropic anchoring at the walls. At low flow rates the director field of the nematic adopts a configuration that is dominated by the surface anchoring, being nearly parallel to the channel height direction over most of the cross-section; but at high flow rates there is a transition to a flow-dominated state, where the director configuration at the channel centerline is aligned with the flow (perpendicular to the channel height direction). We analyze simple channel-flow solutions to the Leslie–Ericksen model for nematics. We demonstrate that two solutions exist, at all flow rates, but that there is a transition between the elastic free energies of these solutions: the anchoring-dominated solution has the lowest energy at low flow rates, and the flow-dominated solution has lowest energy at high flow rates.

We consider a coating flow of nematic liquid crystal (NLC) fluid film on an inclined substrate. Exploiting the small aspect ratio in the geometry of interest, a fourth-order nonlinear partial differential equation is used to model the free surface evolution. Particular attention is paid to the interplay between the bulk elasticity and the anchoring conditions at the substrate and free surface. Previous results have shown that there exist two-dimensional travelling wave solutions that translate down the substrate. In contrast to the analogous Newtonian flow, such solutions may be unstable to streamwise perturbations. Extending well-known results for Newtonian flow, we analyse the stability of the front with respect to transverse perturbations. Using full numerical simulations, we validate the linear stability theory and present examples of downslope flow of nematic liquid crystal in the presence of both transverse and streamwise instabilities.

The flow of nematic liquid crystals down an inclined substrate is studied. Under the usual long wave approximation, a fourth-order nonlinear parabolic partial differential equation of the diffusion type is derived for the free surface height. The model accounts for elastic distortions of the director field due to different anchoring conditions at the substrate and the free surface. The partial differential equation we derive admits 2D travelingwave solutions, which may translate stably or exhibit instabilities in the flat film behind the traveling front. These instabilities, which are distinct from the usual transverse instability of downslope flow, may be analyzed and explained by linear stability analysis of a flat translating film. Intriguing parallels are found with the instabilities exhibited by Newtonian fluid flowing on an inverted substrate and Newtonian fluid flow outside a vertical cylinder.

A thin layer of nematic liquid crystal (NLC) across which an electric field is applied is a setup of great industrial importance in liquid crystal display devices. There is thus a large literature modeling this situation and related scenarios. A commonly used assumption is that an electric field generated by electrodes at the two bounding surfaces of the layer will produce a field that is uniform: that is, the presence of NLC does not affect the electric field. In this paper, we use calculus of variations to derive the equations coupling the electric potential to the orientation of the NLC’s director field, and use a simple one-dimensional model to investigate the limitations of the uniform field assumption in the case of a steady applied field. The extension of the model to the unsteady case is also briefly discussed.

Systematic asymptotic methods are used to formulate a model for the extensional flow of a thin sheet of nematic liquid crystal. With no external body forces applied, the model is found to be equivalent to the so-called Trouton model for Newtonian sheets (and fibres), albeit with a modified ‘Trouton ratio’. However, with a symmetry-breaking electric field gradient applied, behaviour deviates from the Newtonian case, and the sheet can undergo finite-time breakup if a suitable destabilizing field is applied. Some simple exact solutions are presented to illustrate the results in certain idealized limits, as well as sample numerical results to the full model equations.

Bistable liquid crystal displays (LCDs) offer the potential for considerable power savings compared with conventional (monostable) LCDs. The existence of two (or more) stable field-free states that are optically distinct means that contrast can be maintained in a display without an externally applied electric field. An applied field is required only to switch the device from one state to the other, as needed. In this paper we examine the basic physical principles involved in generating multiple stable states and the switching between these states. We consider a two-dimensional geometry in which variable surface anchoring conditions are used to control the steady-state solutions and explore how different anchoring conditions can influence the number and type of solutions and whether or not switching is possible between the states.We find a wide range of possible behaviors, including bistability, tristability, and tetrastability, and investigate how the solution landscape changes as the boundary conditions are tuned.

We study spreading dynamics of nematic liquid crystal droplets within the framework of the long-wave approximation. A fourth-order nonlinear parabolic partial differential equation governing the free surface evolution is derived. The influence of elastic distortion energy and of imposed anchoring variations at the substrate are explored through linear stability analysis and scaling arguments, which yield useful insight and predictions for the behaviour of spreading droplets. This behaviour is captured by fully nonlinear time-dependent simulations of three-dimensional droplets spreading in the presence of anchoring variations that model simple defects in the nematic orientation at the substrate.

Bistable liquid crystal displays offer the potential for considerable power savings compared with conventional (monostable) LCDs. The existence of two stable field-free states that are optically distinct means that contrast can be maintained in a display without an externally applied electric field. An applied field is required only to switch the device from one state to the other, as needed. In this paper we examine a theoretical model of a possible bistable device, originally proposed by Cummings and Richardson (Euro J Appl Math 17:435–463 2006), and explore means by which it may be optimized, in terms of optical contrast, manufacturing considerations, switching field strength, and switching times. The compromises inherent in these conflicting design criteria are discussed.

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