# American Institute of Mathematical Sciences

ISSN:
1937-1632

eISSN:
1937-1179

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## Discrete & Continuous Dynamical Systems - S

April 2015 , Volume 8 , Issue 2

Issue on mathematical study on liquid crystals and related topics: Statics and dynamics

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2015, 8(2): i-i doi: 10.3934/dcdss.2015.8.2i +[Abstract](2191) +[PDF](88.2KB)
Abstract:
In recent years, materials related to liquid crystals have attracted many scientists due to their fascinating structures for potential applications and many challenging problems. These materials are ubiquitous in our daily life and understanding such materials are crucial to many applied areas such as in technology and gives rise to a variety of mathematical open questions which require new theories to be developed. This special issue of Discrete and Continuous Dynamical Systems is devoted to modelings, numerics and analysis of new developments in the study of the phenomena observed in the systems of these materials.
The Isaac Newton Institute for Mathematical Sciences in the University of Cambridge, UK, ran a special program on the Mathematics of Liquid Crystals from January 7, 2013 to July 5, 2013. Many active researchers related to the topics ranging from computational and theoretical modeling to mathematical analysis took a part in the program, exchanged their knowledge and ideas, and conducted stimulating discussion on challenging issues in both modelings and analysis. New computational and theoretical results for statics and dynamics were reported after the program. This special issue consists of some of these results and papers from invited speakers of a special session in the 9th AIMS Conference on Dynamical Systems and Differential Equations held at Orlando, Florida, USA, July 1-7, 2012. We hope papers in this volume can provide the current issues of mathematical problems in liquid crystals and related topics.
Final acceptance of all the papers in this volume were made by the normal referee procedure and standard practice of AIMS journals. We wish to thank all the referees who generously agreed to devote their time and effort to read and check all the papers carefully and give comments and recommendations. We also express our thanks to AIMS for the opportunity to publish this special issue and for the technical help.
2015, 8(2): 243-257 doi: 10.3934/dcdss.2015.8.243 +[Abstract](2681) +[PDF](398.9KB)
Abstract:
We prove lower semicontinuity and lower bounds for a Chen-Lubensky energy describing nematic/smectic liquid crystals with physically realistic boundary conditions. The Chen-Lubensky energy captures stable phases of the liquid crystal material, ranging from purely nematic or smectic states to coexisting nematic/smectic states. By including appropriate additional terms, the model includes the effects of applied electric or magnetic fields, and/or electrical self-interactions in the case of polarized liquid crystals. As a consequence of our results, we establish existence of minimizers with weak or strong anchoring of the director field (describing molecular orientation) at the boundary, and Dirichlet or Neumann boundary conditions on the smectic order parameter for the liquid crystal material.
2015, 8(2): 259-282 doi: 10.3934/dcdss.2015.8.259 +[Abstract](1953) +[PDF](494.2KB)
Abstract:
We prove weak lower semi-continuity and existence of energy-minimizers for a free energy describing stable deformations and the corresponding director configuration of an incompressible nematic liquid-crystal elastomer subject to physically realistic boundary conditions. The energy is a sum of the trace formula developed by Warner, Terentjev and Bladon (coupling the deformation gradient and the director field) and the Landau-de Gennes energy in terms of the gradient of the director field and the bulk term for the director with coefficients depending on temperature. A key step in our analysis is to prove that the energy density has a convex extension to non-unit length director fields. Our results apply to the setting of physical experiments in which a thin incompressible elastomer in $\mathbb{R}^3$ is clamped on its sides and stretched perpendicular to its initial director field, resulting in shape-changes and director re-orientation.
2015, 8(2): 283-302 doi: 10.3934/dcdss.2015.8.283 +[Abstract](2444) +[PDF](569.6KB)
Abstract:
In this article, we study minimization of the energy of a Landau-de Gennes liquid crystal elastomer. The total energy consists of the sum of the Lagrangian elastic stored energy function of the elastomer and the Eulerian Landau-de Gennes energy of the liquid crystal.
There are two related sources of anisotropy in the model, that of the rigid units represented by the traceless nematic order tensor $Q$, and the positive definite step-length tensor $L$ characterizing the anisotropy of the network. This work is motivated by the study of cytoskeletal networks which can be regarded as consisting of rigid rod units crosslinked into a polymeric-type network. Due to the mixed Eulerian-Lagrangian structure of the energy, it is essential that the deformation maps $\varphi$ be invertible. For this, we require sufficient regularity of the fields $(\varphi, Q)$ of the problem, and that the deformation map satisfies the Ciarlet-Nečas injectivity condition. These, in turn, determine what boundary conditions are admissible, which include the case of Dirichlet conditions on both fields. Alternatively, the approach of including the Rapini-Papoular surface energy for the pull-back tensor $\tilde Q$ is also discussed. The regularity requirements also lead us to consider powers of the gradient of the order tensor $Q$ higher than quadratic in the energy.
We assume polyconvexity of the stored energy function with respect to the effective deformation tensor and apply methods of calculus of variations from isotropic nonlinear elasticity. Recovery of minimizing sequences of deformation gradients from the corresponding sequences of effective deformation tensors requires invertibility of the anisotropic shape tensor $L$. We formulate a necessary and sufficient condition to guarantee this invertibility property in terms of the growth to infinity of the bulk liquid crystal energy $f(Q)$, as the minimum eigenvalue of $Q$ approaches the singular limit of $-\frac{1}{3}$. It turns out that $L$ becomes singular as the minimum eigenvalue of $Q$ reaches $-\frac{1}{3}$. Lower bounds on the eigenvalues of $Q$ are needed to ensure compatibility between the theories of Landau-de Gennes and Maier-Saupe of nematics [5].
2015, 8(2): 303-312 doi: 10.3934/dcdss.2015.8.303 +[Abstract](1967) +[PDF](333.5KB)
Abstract:
We study the propagation of electromagnetic plane waves through a liquid crystal layer paying particular attention to the problem of optimizing the transmitted intensity. The controllable anisotropy of a liquid crystal layer, either through anchoring conditions on supporting glass plates sandwiching the layer or by the imposition of an external electromagnetic field, allows us to tune the orientation of the layer to maximize or minimize the transmitted intensity of a given wavelength through the layer. For a homogeneous liquid crystal orientation field, we find analytical formulas for the orientation that maximizes the transmission and discuss the circumstances under which we can make the layer effectively transparent for a given wavelength and the possibility of multiple maximizing orientations. The minimizing orientation is unique for a given wavelength, and we can define its value implicitly.
2015, 8(2): 313-321 doi: 10.3934/dcdss.2015.8.313 +[Abstract](2230) +[PDF](329.0KB)
Abstract:
In this article, we study Conley's theorem about the chain recurrence in dynamical systems, that is, the chain recurrent set of continuous map $f$ is the complement of union of $B_{U}(A)-A$, where $A$ is an attractor and $B_{U}(A)$ is a basin of $A$. In this paper, we generalize this theorem to dispersive systems on noncompact spaces.
2015, 8(2): 323-340 doi: 10.3934/dcdss.2015.8.323 +[Abstract](1840) +[PDF](471.6KB)
Abstract:
We discuss a dynamical theory for nematic liquid crystals describing the stage of evolution in which the hydrodynamic fluid motion has already equilibrated and the subsequent evolution proceeds via diffusive motion of the orientational degrees of freedom. This diffusion induces a slow motion of singularities of the order parameter field. Using asymptotic methods for gradient flows, we establish a relation between the Doi-Smoluchowski kinetic equation and vortex dynamics in two-dimensional systems. We also discuss moment closures for the kinetic equation and Landau-de Gennes-type free energy dissipation.
2015, 8(2): 341-379 doi: 10.3934/dcdss.2015.8.341 +[Abstract](1885) +[PDF](1579.3KB)
Abstract:
We develop a hydrodynamic theory for flowing inhomogeneous polymer-nanorod composites (PNCs) coupling the Smoluchowski transport equation for the distribution function of the nanorod dispersed in a polymer matrix and the transport equation for the distribution of the polymer in the host matrix. The polymer molecule phase is modeled by bead-spring Rouse chains while the nanorod phase is modeled as semiflexible rods. The polymer-nanorod surface contact interaction and the conformational entropy of semiflexible nanorods are incorporated, resulting in a coupled system of nonlinear, nonlocal Smoluchowski equations for the polymer and nanorod. We then implement the theory to infer rheological properties and predict mesoscale morphologies in fully coupled plane shear flows. Our numerical study focuses on the mesoscale morphology development with respect to the surface contact interaction due to the pretreated surface properties of the nanorods, extending our studies on monodomain polymer-nanorod composites [16]. We find that surface contact interaction dominates the mesoscopic morphology and thereby corresponding rheological properties. When the nanorod favors parallel alignment with the polymer in the host matrix, the only globally stable state is the flow-aligning steady state. When the nanorod prefers to align orthogonally to the polymer in the matrix, however, spatially inhomogeneous structures, time-dependent homogeneous structures, and various spatial-temporal structures emerge in different regimes of the model parameter space and versus strength of the bulk imposed shear. Effective rheological features of the inhomogeneous morphologies are also predicted by the theory.
2015, 8(2): 381-388 doi: 10.3934/dcdss.2015.8.381 +[Abstract](1944) +[PDF](312.1KB)
Abstract:
In this paper, we establish scaling invariant blow-up criteria for a classical solution to simplified version of two and three dimensional Ericksen-Leslie system. We also consider the model replacing the Navier-Stokes equations by Stokes equations in the system and obtain blow-up criterion in three dimensions.

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