Oseen-Frank Model Research Articles

Dimensional reduction and emergence of defects in the Oseen-Frank model for nematic liquid crystals

Dimensional reduction and emergence of defects in the Oseen-Frank model for nematic liquid crystals

Γ-convergence of a mean-field model of a chiral doped nematic liquid crystal to the Oseen–Frank description of cholesterics

Systems of elongated molecules, doped with small amounts of molecules lacking mirror symmetry can form macroscopically twisted cholesteric liquid crystal phases. The aim of this work is to rigorously derive the Oseen–Frank model of cholesterics from a more fundamental model concerned with pairwise molecular interactions. A non-local mean-field model of the two-species nematic host/chiral dopant mixture is proposed, and it is shown that Oseen–Frank’s elastic free energy for cholesteric liquid crystals can be obtained in a simultaneously large-domain and dilute-dopant asymptotic regime. By techniques of -convergence, it is shown that in the asymptotic limit, dopant–dopant interactions are negligable, the Frank constants and nematic host order parameter are unperturbed by the presence of dopant, but the mirror asymmetry of the dopant–host interaction leads to a macroscopically twisted ground state. The constant of proportionality between the helical wavenumber and dopant concentration, the helical twisting power (HTP), can be explicitly found through such an analysis, with a nonlinear temperature dependence. Depending on the relative strengths of the host–host and host–dopant interactions, it is shown that HTP may increase or decrease with temperature.

Anisotropic nematic liquid crystals in an applied magnetic field

In the paper, we study the effect of an applied magnetic field on nematic liquid crystals such as magnetic field-induced instability in anisotropic liquid crystals. Under a planar field ansatz for the liquid crystal, the time-dependent system of anisotropic liquid crystals for the Oseen–Frank model reduces to a quasilinear parabolic equation for the orientation angle of the liquid crystal involving critical nonlinearity due to elastic anisotropy. We first establish the global existence of weak solutions to the parabolic problem. When the unequal elastic constants are sufficiently close, we prove convergence to an equilibrium of global smooth solutions, and examine the dynamical instability of an aligned state induced by a strong magnetic field.

Uniqueness of director configuration states for liquid crystals in the case of weak anchoring

Abstract In a thin layer of liquid crystal the configuration of the director field depends on the interaction between the elastic properties of the material, the thickness $d$ of the layer, the boundary conditions and the external fields that may have been applied. Suitable combinations of these factors can give rise to distorted configurations (Freedericksz transitions). In this paper we assume the Oseen-Frank model for the energy and that the director field depends only on the direction orthogonal to the layer; we assume also weak anchoring conditions at the two bounding surfaces, and we mainly study the problem of uniqueness of such distorted configurations. More precisely, we first consider the nematic case in the presence of a magnetic field $\mathbf H$, and we prove the uniqueness of the stable configuration provided the magnitude of $\mathbf H$ is between two critical thresholds, simplifying some results already known in the literature, and calculating explicitly the critical thresholds. Then we study the case of a cholesteric liquid crystal without external field. In this case the director field tends to form a right-angle helicoid around a twist axis orthogonal to the layer, and we have distorted configurations (namely oblique helicoid) for suitable value of $d$. Also in this case, with suitable restrictions on the elastic constants in the Oseen-Frank energy, we find two critical thresholds for $d$, and we prove the existence of only one stable director configuration if $d$ is between them.

Well-posedness of the Ericksen–Leslie system with the Oseen–Frank energy in $${{\varvec{L}}}_{{{\varvec{uloc}}}}^{3}({\mathbb R}^3)$$ L uloc 3 ( R 3 )

We investigate the Ericksen-Leslie system for the Oseen-Frank model with unequal Frank elastic constants in $\mathbb{R}^3$. To generalize the result of Hineman-Wang \cite{HW}, we prove existence of solutions to the Ericksen-Leslie system with initial data having small $L^3_{uloc}$-norm. In particular, we use a new idea to obtain a local $L^3$-estimate through interpolation inequalities and a covering argument, which is different from the one in \cite{HW}. Moreover, for uniqueness of solutions, we find a new way to remove the restriction on the Frank elastic constants by using the rotation invariant property of the Oseen-Frank density. We combine this with a method of Li-Titi-Xin \cite{LTX} to prove uniqueness of the $L^3_{uloc}$-solutions of the Ericksen-Leslie system assuming that the initial data has a finite energy.

Oseen–Frank-type theories of ordered media as the Γ-limit of a non-local mean-field free energy

In this work we recover the Oseen–Frank theory of nematic liquid crystals as a [Formula: see text]-limit of a particular mean-field free energy as the sample size becomes infinitely large. The Frank constants are not necessarily all equal. Our analysis takes place in a broader framework however, also providing results for more general systems such as biaxial or polar molecules. We start from a mesoscopic model describing a competition between entropy and a non-local pairwise interaction between molecules. We assume the interaction potential is separable so that the energy can be reduced to a model involving a macroscopic order parameter. We assume only integrability and decay properties of the macroscopic interaction, but no regularity assumptions are required. In particular, singular interactions are permitted. The analysis becomes significantly simpler on a translationally invariant domain, so we first consider periodic domains with increasing domain of periodicity. Then we tackle the problem on a Lipschitz domain with non-local boundary conditions by considering an asymptotically equivalent problem on periodic domains. We conclude by applying the results to a case which reduces to the Oseen–Frank model of elasticity, and give expressions for the Frank constants in terms of integrals of the interaction kernel.

On a bifurcation problem arising in cholesteric liquid crystal theory

In a cholesteric liquid crystal the director field $n(x,y,z)$ tends to form a right-angle helicoid around a twist axis in order to minimize the internal energy; however, a fixed alignment of the director field at the boundary (strong anchoring) can give rise to distorted configurations of the director field, as oblique helicoid, in order to save energy. The transition to this distorted configurations depend on the boundary conditions and on the geometry of the liquid crystal, and it is known as Freedericksz transition (without external fields). We consider the classical situation of a thin layer between two glass sheet assuming the Oseen-Frank model for the energy, and that the director field depend only on the direction $z$ orthogonal to the layer; then we focus on two kinds of boundary conditions: the planar case and the orthogonal case. In the first, we impose that $n(0)=(1,0,0)$, $n(d)=(\cos\alpha,\sin\alpha,0)$ (where $z=0$ and $z=d>0$ are, respectively, the bottom and the top of the layer), and search for the couples $(d,\alpha)$ such that oblique helicoid appear. In the case $K_1>0$, $K_2=K_3=1$ for the elastic constants of the Oseen-Frank energy, we completely characterize these couples. In the second case it is a classical result that oblique helicoid bifurcates from the trivial solution $n(z)=(0,0,1)$ for suitable values of $d$; then we study the exact number of these nontrivial solutions and their stability.

On the Disclination Lines of Nematic Liquid Crystals

AbstractDefects in liquid crystals are of great practical importance and theoretical interest. Despite tremendous efforts, predicting the location and transition of defects under various topological constraint and external field remains to be a challenge. We investigate defect patterns of nematic liquid crystals confined in three-dimensional spherical droplet and two-dimensional disk under different boundary conditions, within the Landau-de Gennes model. We implement a spectral method that numerically solves the Landau-de Gennes model with high accuracy, which allows us to study the detailed static structure of defects. We observe five types of defect structures. Among them the 1/2-disclination lines are the most stable structure at low temperature. Inspired by numerical results, we obtain the profile of disclination lines analytically. Moreover, the connection and difference between defect patterns under the Landau-de Gennes model and the Oseen-Frank model are discussed. Finally, three conjectures are made to summarize some important characteristics of defects in the Landau-de Gennes theory. This work is a continuing effort to deepen our understanding on defect patterns in nematic liquid crystals.

Transparent Boundary Condition for Oseen-Frank Model. Application for NLC Cells With Patterned Electrodes

In the present work a novel application of Transparent Boundary Conditions (TBC) to nematic liquid crystal cells (NLCC) with planar alignment and a patterned electrode is studied. This device is attracting great interest since it allows soliton steering by optically and externally induced waveguides. We employ the continuum Oseen-Frank theory to find the tilt and twist angle distributions in the cell under the one-constant approximation. The electric field distribution takes into account the whole 2D permittivity tensor for the transverse coordinates. Standard finite difference time domain methods together with an iterative method is applied to find an approximate solution to our coupled problem. A novel class of TBC is used to correctly define the boundary for both the distortion angle problem and the electric field distribution when using patterned electrodes. Thus, we achieve an important decrease of computational needs when solving this kind of problems and we are also capable of exploring weak anchoring conditions for NLCC.

From Microscopic Theory to Macroscopic Theory: a Systematic Study on Modeling for Liquid Crystals

In this paper, we propose a systematic way of liquid crystal modeling to build connection between microscopic theory and macroscopic theory. A new Q-tensor theory based on Onsager's molecular theory which leads to liquid crystals with certain shape has been proposed. Making uniaxial assumption, we can recover the Oseen-Frank theory from the derived $Q$-tensor theory, and the Oseen-Frank model coefficients can be examined. In addition, the smectic-A phase can also be characterized by the derived macroscopic model.

Blow-up Criteria of Strong Solutions to the Ericksen-Leslie System in ℝ3

In this paper, we establish the local well-posedness and blow-up criteria of strong solutions to the Ericksen-Leslie system in ℝ3 for the well-known Oseen-Frank model. The local existence of strong solutions to liquid crystal flows is obtained by using the Ginzburg-Landau approximation approach to guarantee the constraint that the direction vector of the fluid is of length one. We establish four kinds of blow-up criteria, including (i) the Serrin type; (ii) the Beal-Kato-Majda type; (iii) the mixed type, i.e., Serrin type condition for one field and Beal-Kato-Majda type condition on the other one; (iv) a new one, which characterizes the maximal existence time of the strong solutions to the Ericksen-Leslie system in terms of Serrin type norms of the strong solutions to the Ginzburg-Landau approximate system. Furthermore, we also prove that the strong solutions of the Ginzburg-Landau approximate system converge to the strong solution of the Ericksen-Leslie system up to the maximal existence time.

Global existence of solutions of the liquid crystal flow for the Oseen–Frank model in [formula omitted

In the first part of this paper, we establish the global existence of solutions of the liquid crystal (gradient) flow for the well-known Oseen–Frank model. The liquid crystal flow is a prototype of equations from the Ericksen–Leslie system in the hydrodynamic theory and generalizes the heat flow for harmonic maps into the 2-sphere. The Ericksen–Leslie system is a system of the Navier–Stokes equations coupled with the liquid crystal flow. In the second part of this paper, we also prove the global existence of solutions of the Ericksen–Leslie system for a general Oseen–Frank model in R2.

Possible limitations of the classical model of orientational optical nonlinearity in nematics

Abstract Orientational nonlinearity is the major mechanism of nonlinear optical phenomena observed in liquidcrystalline phase while it does not appear to such extent in any other materials. It is caused by distortion of initial molecular arrangement of an anisotropic medium induced by optical field. Deformation of the anisotropic structure means spatial changes of refractive index of the medium. This effect has been studied in earnest since the 1980s as its application became more apparent. In this paper, some results of experimental examination of molecular reorientation in nematics by optical field are presented, which are not explained in frame of existing Oseen-Frank model and Erickson-Leslie continuous theory. Possible reasons of this discordance are considered and a way of explanation is suggested.