A Nonhomogeneous Kinetic Model of Liquid Crystal Polymers and Its Thermodynamic Closure Approximation

A general nonhomogeneous extension of the Doi’s kinetic theory with trans-lational diffusion and nonlocal potential is proposed to describe the microstructuresand defect dynamics of Liquid Crystal Polymer (LCP) solutions. The long-range elas-ticity of polymer molecules is depicted by a kernel type potential, from which onecan derive the well-known Marrucci-Greco potential with weak spatial distortion as-sumption. Applying quasi-equilibrium closure approximation, we get a second-ordermoment model for isotropic long-range elasticity, and this reduced moment modelmaintains the energy dissipation. Implemented by the invariant-based fitting method,the moment model is a decent tool for numerical simulations of defect dynamics andtexture evolution in LCP solutions. The numerical results of in-plane rotational caseshow that the reduced second-order moment model qualitatively predicts complicatednonhomogeneous director dynamics under moderate nematic potential strength, andthe translational diffusion plays an important role in defect dynamics.

Data-Driven Upscaling of Orientation Kinematics in Suspensions of Rigid Fibres

Describing the orientation state of the particles is often critical in fibre suspension applications.Macroscopic descriptors,the so-called second-order orientation tensor(or moment)leading the way,are often preferred due to their low computational cost.Closure problems however arise when evolution equations for the moments are derived from the orientation distribution functions and the impact of the chosen closure is often unpredictable.In this work,our aim is to provide macroscopic simulations of orientation that are cheap,accurate and closure-free.To this end,we propose an innovative data-based approach to the upscaling of orientation kinematics in the context of fibre suspensions.Since the physics at the microscopic scale can be modelled reasonably enough,the idea is to conduct accurate offline direct numerical simulations at that scale and to extract the corresponding macroscopic descriptors in order to build a database of scenarios.During the online stage,the macroscopic descriptors can then be updated quickly by combining adequately the items from the database instead of relying on an imprecise macroscopic model.This methodology is presented in the well-known case of dilute fibre suspensions(where it can be compared against closure-based macroscopic models)and in the case of suspensions of confined or electrically-charged fibres,for which state-of-the-art closures proved to be inadequate or simply do not exist.

A tensor model for liquid crystals on a spherical surface

Rod-like molecules confined on a spherical surface can organize themselves into nematic liquid crystal phases. This can give rise to novel textures displayed on the surface, which has been observed in experiments. An important theoretical question is how to find and predict these textures. Mathematically, a stable configuration of the nematic fluid corresponds to a local minimum in the free energy landscape. By applying Taylor expansion and Bingham approximation to a general molecular model, we obtain a closed-form tensor model, which gives a free energy form that is different from the classic Landau-de Gennes model. Based on the tensor model, we implement an efficient numerical algorithm to locate the local minimum of the free energy. Our model successfully predicts the splay, tennis-ball and rectangle textures. Among them, the tennis-ball configuration has the lowest free energy.

Low-Dimensional SIR Epidemic Models with Demographics on Heterogeneous Networks

To investigate the impacts of demographics on the spread of infectious diseases, a susceptib- le-infectious-recovered （SIR） pairwise model on heterogeneous networks is established. This model is reduced by using the probability generating function and moment closure approximations. The basic reproduction number of the low-dimensional model is derived to rely on the recruitment and death rate, the first and second moments of newcomers＇ degree distribution. Sensitivity analysis for the basic reproduction number is performed, which indicates that a larger variance of newcomers＇ degrees can lead to an epidemic outbreak with a smaller transmission rate, and contribute to a slight decrease of the final density of infectious nodes with a larger transmission rate. Besides, stochastic simulations indicate that the low-dimensional model based on the log-normal moment closure assumption can well capture important properties of an epidemic. And the authors discover that a larger recruitment rate can inhibit the spread of disease.