Introduction to Q-tensor theory

This paper aims to provide an introduction to a basic form of the ${\bf Q}$-tensor approach to modelling liquid crystals, which has seen increased interest in recent years. The increase in interest in this type of modelling approach has been driven by investigations into the fundamental nature of defects and new applications of liquid crystals such as bistable displays and colloidal systems for which a description of defects and disorder is essential. The work in this paper is not new research, rather it is an introductory guide for anyone wishing to model a system using such a theory. A more complete mathematical description of this theory, including a description of flow effects, can be found in numerous sources but the books by Virga and Sonnet and Virga are recommended. More information can be obtained from the plethora of papers using such approaches, although a general introduction for the novice is lacking. The first few sections of this paper will detail the development of the ${\bf Q}$-tensor approach for nematic liquid crystalline systems and construct the free energy and governing equations for the mesoscopic dependent variables. A number of device surface treatments are considered and theoretical boundary conditions are specified for each instance. Finally, an example of a real device is demonstrated

Analytic solutions of a simple advection-diffusion model of an oxygen transfer device

Artificial blood oxygenation is an essential aspect of cardiopulmonary bypass surgery, maintaining physiological levels of oxygen and carbon dioxide in the blood, and thus temporarily replacing the normal function of the lungs. The blood-gas exchange devices used for such procedures have a long history and have had varying degrees of success. In this paper we consider a simple model of a new approach to enhancing the diffusion of oxygen into the blood while it is contained in the artificial blood oxygenator. We show that using a transverse flow, which may be set up using mixing elements that we have previously shown experimentally to enhance blood oxygenation, will increase the oxygen levels within the blood. This simple model and associated analytic solutions can then be used to aid the optimisation of blood oxygenation devices

Hele-Shaw flow of a nematic liquid crystal

Motivated by the variety of applications in which nematic Hele-Shaw flow occurs, a theoretical model for Hele-Shaw flow of a nematic liquid crystal is formulated and analysed. We derive the thin-film Ericksen-Leslie equations that govern nematic Hele-Shaw flow, and consider two important limiting cases in which we can make significant analytical progress. Firstly, we consider the leading-order problem in the limiting case in which elasticity effects dominate viscous effects, and find that the nematic liquid crystal anchoring on the plates leads to a fixed director field and an anisotropic patterned viscosity that can be used to guide the flow of the nematic. Secondly, we consider the leading-order problem in the opposite limiting case in which viscous effects dominate elasticity effects, and find that the flow is identical to that of an isotropic fluid and the behaviour of the director is determined by the flow. As an example of the insight which can be gained by using the present approach, we then consider the flow of nematic according to a simple model for the squeezing stage of the One Drop Filling method, an important method for the manufacture of Liquid Crystal Displays, in these two limiting cases

A mathematical study of a bistable nematic liquid crystal device

Electronic version of an article published as Mathematical Models and Methods in Applied Sciences Vol. 17, No. 12 (2007). p. 2009–2034. Article DOI No: 10.1142/S0218202507002546. Copyright World Scientific Publishing Company http://www.worldscientific.com/We consider a model of a bistable nematic liquid crystal device based on the Ericksen– Leslie theory. The resulting mathematical object is a parabolic PDE with nonlinear dynamic boundary conditions. We analyze well-posedness of the problem and global existence of solutions using the theory developed by Amann. Furthermore, using phaseplane methods, we give an exhaustive description of the steady state solutions and hence of the switching capabilities of the device.peerreviewe

Mathematical modelling of fluid flow and solute transport to define operating parameters for in vitro perfusion cell culture systems

In recent years, there has been a move away from the use of static in vitro two-dimensional cell culture models for testing the chemical safety and efficacy of drugs. Such models are increasingly being replaced by more physiologically relevant cell culture systems featuring dynamic flow and/or three-dimensional structures of cells. While it is acknowledged that such systems provide a more realistic environment within which to test drugs, progress is being hindered by a lack of understanding of the physical and chemical environment that the cells are exposed to. Mathematical and computational modelling may be exploited in this regard to unravel the dependency of the cell response on spatio-temporal differences in chemical and mechanical cues, thereby assisting with the understanding and design of these systems. In this paper, we present a mathematical modelling framework that characterizes the fluid flow and solute transport in perfusion bioreactors featuring an inlet and an outlet. To demonstrate the utility of our model, we simulated the fluid dynamics and solute concentration profiles for a variety of different flow rates, inlet solute concentrations and cell types within a specific commercial bioreactor chamber. Our subsequent analysis has elucidated the basic relationship between inlet flow rate and cell surface flow speed, shear stress and solute concentrations, allowing us to derive simple but useful relationships that enable prediction of the behaviour of the system under a variety of experimental conditions, prior to experimentation. We describe how the model may used by experimentalists to define operating parameters for their particular perfusion cell culture systems and highlight some operating conditions that should be avoided. Finally, we critically comment on the limitations of mathematical and computational modelling in this field, and the challenges associated with the adoption of such methods

Acrylamide levels in potato crisps in Europe from 2002 to 2016

European manufacturers' data on acrylamide in potato crisps from 2002 to 2016 were analysed. A previous study showed a 53% reduction in mean acrylamide levels from 763 ng/g in 2002 to 358 ng/g in 2011. Analysis of data from the longer period showed that since 2011 there has been a levelling off, with the mean level for 2016 being 412 ng/g (still a 46% reduction from 2002), suggesting that the most effective acrylamide reduction measures had been devised and implemented by 2011. There were similar trends in the 90th and 95th quantile values, with the 90th quantile values being below 1000 ng/g (the European Commission's current 'Indicative Value' for acrylamide in potato crisps) since 2010. The proportion of samples with acrylamide above 2000 ng/g fell from 4.8% in 2002 to 0.6% in 2016. Acrylamide levels showed marked seasonal variability, being highest in the first half of the year when potatoes were being used from storage, and lowest from July to September when potatoes were being harvested. Acrylamide levels were higher in thicker types of crisp in the early years of the study, but this difference disappeared in the later years, suggesting that manufacturers had acted to reduce acrylamide formation in these products. Higher values for acrylamide were recorded in north and east Europe than in the south and west up to 2013. Levels in the north and east declined in recent years, but remained higher in the north than in the other regions. The manufacturers' data were compared with a much smaller dataset provided by the European Food Safety Authority (EFSA). Levels of acrylamide in the EFSA dataset were consistently higher than in the manufacturers' data, possibly due to uneven sampling through the year and the seasonality of acrylamide levels

Uniqueness in the Freedericksz transition with weak anchoring

In this paper we consider a boundary value problem for a quasilinear pendulum equation with nonlinear boundary conditions that arises in a classical liquid crystals setup, the Freedericksz transition, which is the simplest opto-electronic switch, the result of competition between reorienting effects of an applied electric field and the anchoring to the bounding surfaces. A change of variables transforms the problem into the equation x = −f(x) for ∈ (−T, T), with boundary conditions x = ± T f(x) at = ∓T, for a convex nonlinearity f. By analyzing an associated inviscid Burgers' equation, we prove uniqueness of monotone solutions in the original nonlinear boundary value problem. This result has been for many years conjectured in the liquid crystals literature, e. g. in E. G. Virga, Variational Theories for Liquid Crystals,Chapman and Hall, London, 1994 and in I. W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction, Taylor and Francis, London, 2003

Asymptotic and numerical analysis of a simple model for blade coating

Motivated by the industrial process of blade coating, the two-dimensional flow of a thin film of Newtonian fluid on a horizontal substrate moving parallel to itself with constant speed under a fixed blade of finite length in which the flows upstream and downstream of the blade are coupled via the flow under the blade is analysed. A combination of asymptotic and numerical methods is used to investigate the number and nature of the steady solutions that exist. Specially, it is found that in the presence of gravity there is always at least one, and (depending on the parameter values) possibly as many as three, steady solutions, and that when multiple solutions occur they are identical under and downstream of the blade, but differ upstream of it. The stability of these solutions is investigated, and their asymptotic behaviour in the limits of large and small flux and weak and strong gravity effects, respectively, determined

Flexoelectric Polarization in a Nematic Liquid Crystal Enhanced by Dopants with Different Molecular Shape Polarities

Funding Information: We would like to acknowledge the great support which we received from Prof. I. Muševič, JSI, Ljubljana, Slovenia, Dr M. Klasen-Memmer, Merck, Germany, Dr Santanu Kumar Pal and Dr Golam Mohiuddin, Indian Institute of Science Education and Research (IISER) Mohali, India, and Prof. P. Kula and K. Garbat, MUT, Warsaw, Poland. This work has been partially supported by SeeReal Technologies and MUT Research Grants 13-843/WAT/2022.Peer reviewedPublisher PD

Argon behaviour in an inverted Barrovian sequence, Sikkim Himalaya: the consequences of temperature and timescale on ⁴⁰Ar/³⁹Ar mica geochronology

40Ar/39Ar dating of metamorphic rocks sometimes yields complicated datasets which are difficult to interpret in terms of timescales of the metamorphic cycle. Single-grain fusion and step-heating data were obtained for rocks sampled through a major thrust-sense shear zone (the Main Central Thrust) and the associated inverted metamorphic zone in the Sikkim region of the eastern Himalaya. This transect provides a natural laboratory to explore factors influencing apparent 40Ar/39Ar ages in similar lithologies at a variety of metamorphic pressure and temperature (P–T) conditions. The 40Ar/39Ar dataset records progressively younger apparent age populations and a decrease in within-sample dispersion with increasing temperature through the sequence. The white mica populations span ~ 2–9 Ma within each sample in the structurally lower levels (garnet grade) but only ~ 0–3 Ma at structurally higher levels (kyanite-sillimanite grade). Mean white mica single-grain fusion population ages vary from 16.2 ± 3.9 Ma (2σ) to 13.2 ± 1.3 Ma (2σ) from lowest to highest levels. White mica step-heating data from the same samples yields plateau ages from 14.27 ± 0.13 Ma to 12.96 ± 0.05 Ma. Biotite yield older apparent age populations with mean single-grain fusion dates varying from 74.7 ± 11.8 Ma (2σ) at the lowest structural levels to 18.6 ± 4.7 Ma (2σ) at the highest structural levels; the step-heating plateaux are commonly disturbed. Temperatures > 600 °C at pressures of 0.4–0.8 GPa sustained over > 5 Ma, appear to be required for white mica and biotite ages to be consistent with diffusive, open-system cooling. At lower temperatures, and/or over shorter metamorphic timescales, more 40Ar is retained than results from simple diffusion models suggest. Diffusion modelling of Ar in white mica from the highest structural levels suggests that the high-temperature rocks cooled at a rate of ~ 50–80 °C Ma− 1, consistent with rapid thrusting, extrusion and exhumation along the Main Central Thrust during the mid-Miocene