Journal articles: 'Oseen Frank model'

1

GOLOVATY, DMITRY, MICHAEL NOVACK, and PETER STERNBERG. "A novel Landau-de Gennes model with quartic elastic terms." European Journal of Applied Mathematics 32, no. 1 (March 24, 2020): 177–98. http://dx.doi.org/10.1017/s095679252000008x.

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Within the framework of the generalised Landau-de Gennes theory, we identify a Q-tensor-based energy that reduces to the four-constant Oseen–Frank energy when it is considered over orientable uniaxial nematic states. Although the commonly considered version of the Landau-de Gennes theory has an elastic contribution that is at most cubic in components of the Q-tensor and their derivatives, the alternative offered here is quartic in these variables. One clear advantage of our approach over the cubic theory is that the associated minimisation problem is well-posed for a significantly wider choice of elastic constants. In particular, this quartic energy can be used to model nematic-to-isotropic phase transitions for highly disparate elastic constants. In addition to proving well-posedness of the proposed version of the Landau-de Gennes theory, we establish a rigorous connection between this theory and its Oseen–Frank counterpart via a Г-convergence argument in the limit of vanishing nematic correlation length. We also prove strong convergence of the associated minimisers.

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Taylor, Jamie M. "Oseen–Frank-type theories of ordered media as the Γ-limit of a non-local mean-field free energy." Mathematical Models and Methods in Applied Sciences 28, no. 04 (April 2018): 615–57. http://dx.doi.org/10.1142/s0218202518500161.

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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.

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Li, Jinkai, Edriss S. Titi, and Zhouping Xin. "On the uniqueness of weak solutions to the Ericksen–Leslie liquid crystal model in ℝ2." Mathematical Models and Methods in Applied Sciences 26, no. 04 (February 16, 2016): 803–22. http://dx.doi.org/10.1142/s0218202516500184.

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This paper concerns the uniqueness of weak solutions to the Cauchy problem to the Ericksen–Leslie system of liquid crystal models in [Formula: see text], with both general Leslie stress tensors and general Oseen–Frank density. It is shown here that such a system admits a unique weak solution provided that the Frank coefficients are close to some positive constant. One of the main ideas of our proof is to perform suitable energy estimates at the level one order lower than the natural basic energy estimates for the Ericksen–Leslie system.

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Xia, Jingmin, Patrick E. Farrell, and Florian Wechsung. "Augmented Lagrangian preconditioners for the Oseen–Frank model of nematic and cholesteric liquid crystals." BIT Numerical Mathematics 61, no. 2 (March 23, 2021): 607–44. http://dx.doi.org/10.1007/s10543-020-00838-9.

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AbstractWe propose a robust and efficient augmented Lagrangian-type preconditioner for solving linearizations of the Oseen–Frank model arising in nematic and cholesteric liquid crystals. By applying the augmented Lagrangian method, the Schur complement of the director block can be better approximated by the weighted mass matrix of the Lagrange multiplier, at the cost of making the augmented director block harder to solve. In order to solve the augmented director block, we develop a robust multigrid algorithm which includes an additive Schwarz relaxation that captures a pointwise version of the kernel of the semi-definite term. Furthermore, we prove that the augmented Lagrangian term improves the discrete enforcement of the unit-length constraint. Numerical experiments verify the efficiency of the algorithm and its robustness with respect to problem-related parameters (Frank constants and cholesteric pitch) and the mesh size.

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Emerson, D. B. "A posteriori error estimators for the Frank-Oseen model of liquid crystals." Journal of Coupled Systems and Multiscale Dynamics 5, no. 2 (December 1, 2017): 95–110. http://dx.doi.org/10.1166/jcsmd.2017.1125.

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6

Lasarzik, Robert. "Measure-valued solutions to the Ericksen–Leslie model equipped with the Oseen–Frank energy." Nonlinear Analysis 179 (February 2019): 146–83. http://dx.doi.org/10.1016/j.na.2018.08.013.

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7

Orquín-Serrano, I., J. Vijande, F. R. Villatoro, A. Ferrando, P. Fernández de Córdoba, and H. Michinel. "Transparent Boundary Condition for Oseen-Frank Model. Application for NLC Cells With Patterned Electrodes." Journal of Physics: Conference Series 605 (April 28, 2015): 012028. http://dx.doi.org/10.1088/1742-6596/605/1/012028.

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8

Greco, Carlo. "Uniqueness of director configuration states for liquid crystals in the case of weak anchoring." IMA Journal of Applied Mathematics 84, no. 4 (July 23, 2019): 748–62. http://dx.doi.org/10.1093/imamat/hxz013.

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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.

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Adler, J. H., T. J. Atherton, D. B. Emerson, and S. P. MacLachlan. "An Energy-Minimization Finite-Element Approach for the Frank--Oseen Model of Nematic Liquid Crystals." SIAM Journal on Numerical Analysis 53, no. 5 (January 2015): 2226–54. http://dx.doi.org/10.1137/140956567.

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10

Hong, Min-Chun, and Zhouping Xin. "Global existence of solutions of the liquid crystal flow for the Oseen–Frank model in R2." Advances in Mathematics 231, no. 3-4 (October 2012): 1364–400. http://dx.doi.org/10.1016/j.aim.2012.06.009.

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11

Sala, Filip A., Marzena M. Sala-Tefelska, and Maksymilian J. Bujok. "Influence of temperature diffusion on molecular reorientation in nematic liquid crystals." Journal of Nonlinear Optical Physics & Materials 27, no. 01 (March 2018): 1850011. http://dx.doi.org/10.1142/s021886351850011x.

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In this paper, the theoretical analyses of optically induced molecular reorientation are presented in a nematic liquid crystal cell. Due to the absorption of the light beam, the temperature of the cell increases, which induces change of anisotropy and the Frank elastic coefficients. In our analyses, a simple model is used. It is based on the Frank–Oseen elastic theory to describe molecular reorientation and on a heat diffusion equation to model thermal effects. Molecular reorientation, its width and maximum value, and extraordinary refractive index are analyzed versus various values of absorption coefficient, rate of change of elastic coefficient, and thermal conductance. It appears that thermal effects do not influence the width of the molecular reorientation. The nonlocality is nearly unaffected by the increase of temperature. However, the maximum molecular reorientation might increase with temperature, in most cases, the extraordinary refractive index decreases with temperature. Only for unlikely high rates of change of elastic coefficients, the refractive index increases with temperature. Moreover, the transition between liquid crystal and isotropic phases is also discussed. The range of input beam width for which the material lasts in a liquid crystal state is also presented. Simulations are performed for parameters corresponding to two liquid crystals: low birefringent 1550 and 6CHBT of a medium birefringence.

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Hu, Yucheng, Yang Qu, and Pingwen Zhang. "On the Disclination Lines of Nematic Liquid Crystals." Communications in Computational Physics 19, no. 2 (February 2016): 354–79. http://dx.doi.org/10.4208/cicp.210115.180515a.

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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.

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13

Lasarzik, Robert. "Weak-strong uniqueness for measure-valued solutions to the Ericksen–Leslie model equipped with the Oseen–Frank free energy." Journal of Mathematical Analysis and Applications 470, no. 1 (February 2019): 36–90. http://dx.doi.org/10.1016/j.jmaa.2018.09.051.

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14

Spencer, Russell K. W., Bae-Yeun Ha, and Nima Saeidi. "Self-consistent field theory of chiral nematic worm-like chains." Journal of Chemical Physics 156, no. 11 (March 21, 2022): 114902. http://dx.doi.org/10.1063/5.0078937.

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Many macromolecules of biological and technological interest are both chiral and semi-flexible. DNA and collagen are good examples. Such molecules often form chiral nematic (or cholesteric) phases, as is well-documented in collagen and chitin. This work presents a method for studying cholesteric phases in the highly successful self-consistent field theory of worm-like chains, offering a new way of studying many biologically relevant molecules. The method involves an effective Hamiltonian with a chiral term inspired by the Oseen–Frank (OF) model of liquid crystals. This method is then used to examine the formation of cholesteric phases in chiral-nematic worm-like chains as a function of polymer flexibility, as well as the optimal cholesteric pitch and distribution of polymer segment orientations. Our approach not only allows for the determination of the isotropic–cholesteric transition and segment distributions, beyond what the OF model promises, but also explicitly incorporates polymer flexibility into the study of the cholesteric phase, offering a more complete understanding of the behavior of semiflexible chiral-nematic polymers.

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15

Mora-Corral, Carlos, and Marcos Oliva. "Relaxation of nonlinear elastic energies involving the deformed configuration and applications to nematic elastomers." ESAIM: Control, Optimisation and Calculus of Variations 25 (2019): 19. http://dx.doi.org/10.1051/cocv/2018005.

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We start from a variational model for nematic elastomers that involves two energies: mechanical and nematic. The first one consists of a nonlinear elastic energy which is influenced by the orientation of the molecules of the nematic elastomer. The nematic energy is an Oseen–Frank energy in the deformed configuration. The constraint of the positivity of the determinant of the deformation gradient is imposed. The functionals are not assumed to have the usual polyconvexity or quasiconvexity assumptions to be lower semicontinuous. We instead compute its relaxation, that is, the lower semicontinuous envelope, which turns out to be the quasiconvexification of the mechanical term plus the tangential quasiconvexification of the nematic term. The main assumptions are that the quasiconvexification of the mechanical term is polyconvex and that the deformation is in the Sobolev space W1,p (with p > n − 1 and n the dimension of the space) and does not present cavitation.

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16

Taylor, Jamie M. "Γ-convergence of a mean-field model of a chiral doped nematic liquid crystal to the Oseen–Frank description of cholesterics." Nonlinearity 33, no. 6 (May 5, 2020): 3062–102. http://dx.doi.org/10.1088/1361-6544/ab74f5.

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17

LUO, C., and M. C. CALDERER. "Numerical study of liquid crystal elastomers by a mixed finite element method." European Journal of Applied Mathematics 23, no. 1 (August 22, 2011): 121–54. http://dx.doi.org/10.1017/s0956792511000313.

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Liquid crystal elastomers present features not found in ordinary elastic materials, such as semi-soft elasticity and the related stripe domain phenomenon. In this paper, the two-dimensional Bladon–Terentjev–Warner model and the one-constant Oseen–Frank energy expression are combined to study the liquid crystal elastomer. We also impose two material constraints, the incompressibility of the elastomer and the unit director norm of the liquid crystal. We prove existence of minimiser of the energy for the proposed model. Next we formulate the discrete model, and also prove that it possesses a minimiser of the energy. The inf-sup values of the discrete linearised system are then related to the smallest singular values of certain matrices. Next the existence and uniqueness of the Lagrange multipliers associated with the two material constraints are proved under the assumption that the inf-sup conditions hold. Finally numerical simulations of the clamped-pulling experiment are presented for elastomer samples with aspect ratio 1 or 3. The semi-soft elasticity is successfully recovered in both cases. The stripe domain phenomenon, however, is not observed, which might be due to the relative coarse mesh employed in the numerical experiment. Possible improvements are discussed that might lead to the recovery of the stripe domain phenomenon.

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18

Sala, Filip. "Beam splitting in chiral nematic liquid crystals." Photonics Letters of Poland 10, no. 4 (December 31, 2018): 109. http://dx.doi.org/10.4302/plp.v10i4.867.

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By lunching the beam into the chiral nematic liquid crystals it is possible to achieve a non-diffractive beam similar to a soliton. This effect is caused by the molecular reorientation i.e. nonlinear response of the material forming the areas of higher refractive index. Diffraction is suppressed by the focusing effect. For appropriate launching conditions it is also possible to achieve a beam which splits into two or more separate beams. Such phenomenon is discussed in this article and analyzed theoretical. To model this effect Fully Vectorial Beam Propagation Method coupled with the Frank-Oseen elastic theory is used. Simulations are performed for various input beam powers, widths, polarization angles and launching positions. Full Text: PDF ReferencesG. Assanto and M. A. Karpierz, "Nematicons: self-localised beams in nematic liquid crystals", Liq. Cryst. 36, 1161–1172 (2009) CrossRef G. Assanto, Nematicons: Spatial Optical Solitons in Nematic Liquid Crystals, John Wiley & Sons Inc. Hoboken, New Jersey (2013) DirectLink A. Piccardi, A. Alberucci, U. Bortolozzo, S. Residori, and G. Assanto, "Soliton gating and switching in liquid crystal light valve", Appl. Phys. Lett. 96, 071104 (2010). CrossRef D. Melo, I. Fernandes, F. Moraes, S. Fumeron, and E. Pereira, "Thermal diode made by nematic liquid crystal", Phys. Lett. A 380, 3121 – 3127 (2016). CrossRef U. Laudyn, M. Kwaśny, F. A. Sala, M. A. Karpierz, N. F. Smyth, G. Assanto, "Curved optical solitons subject to transverse acceleration in reorientational soft matter", Sci. Rep. 7, 12385 (2017) CrossRef M. Kwaśny, U. A. Laudyn, F. A. Sala, A. Alberucci, M. A. Karpierz, G. Assanto, "Self-guided beams in low-birefringence nematic liquid crystals", Phys. Rev. A 86, 013824 (2012) CrossRef F. A. Sala, M. M. Sala-Tefelska, "Optical steering of mutual capacitance in a nematic liquid crystal cell", J. Opt. Soc. Am. B. 35, 133-139 (2018) CrossRef U. A. Laudyn, A. Piccardi, M. Kwasny, M. A. Karpierz, G. Assanto, "Thermo-optic soliton routing in nematic liquid crystals", Opt. Lett. 43, 2296-2299 (2018) CrossRef F. A. Sala, M. M. Sala-Tefelska, M. J. Bujok, J. "Influence of temperature diffusion on molecular reorientation in nematic liquid crystals", Nonlinear Opt. Phys. Mater. 27, 1850011 (2018) CrossRef I-C Khoo Liquid crystals John Wiley & Sons, Inc (2007) DirectLink P. G. de Gennes, J. Prost, The Physics of Liquid Crystals, Clarendon Press (1995) DirectLink U. A. Laudyn, P. S. Jung, M. A. Karpierz, G. Assanto, "Quasi two-dimensional astigmatic solitons in soft chiral metastructures", Sci. Rep. 6, 22923 (2016) CrossRef J. Beeckman, A. Madani, P. J. M. Vanbrabant, P. Henneaux, S-P. Gorza, M. Haelterman, "Switching and intrinsic position bistability of soliton beams in chiral nematic liquid crystals", Phys. Rev. A 83, 033832 (2011) CrossRef A. Madani, J. Beeckman, K. Neyts, "An experimental observation of a spatial optical soliton beam and self splitting of beam into two soliton beams in chiral nematic liquid crystal", Opt. Commun. 298–299, 222-226, (2013) CrossRef G. D. Ziogos, E. E. Kriezis, "Modeling light propagation in liquid crystal devices with a 3-D full-vector finite-element beam propagation method", Opt. Quant. Electron 40, 10 (2008) CrossRef F. A. Sala, M. A. Karpierz, "Chiral and nonchiral nematic liquid-crystal reorientation induced by inhomogeneous electric fields", J. Opt. Soc. Am. B 29, 1465-1472 (2012) CrossRef F. A. Sala, M. A. Karpierz, "Modeling of molecular reorientation and beam propagation in chiral and non-chiral nematic liquid crystals", Opt. Express 20, 13923-13938 (2012) CrossRef F. A. Sala, "Design of false color palettes for grayscale reproduction", Displays, 46, 9-15 (2017) CrossRef

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19

Sierakowski, Marek, and Małgorzata Teterycz. "Possible limitations of the classical model of orientational optical nonlinearity in nematics." Open Physics 6, no. 3 (January 1, 2008). http://dx.doi.org/10.2478/s11534-008-0079-0.

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AbstractOrientational 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.

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20

Sadovskaya, O. V., and V. M. Sadovskii. "ANALYSIS OF THE UNSTABLE STATE OF A LIQUID CRYSTAL BASED ON THE OSEEN-FRANK MODEL." Вестник Башкирского университета, 2022, 525. http://dx.doi.org/10.33184/bulletin-bsu-2022.3.6.

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21

Allen, Aaron C., and Xiaoyu Zheng. "Equilibrium configurations in a nematic pi-cell under an electric field." Frontiers in Soft Matter 2 (September 2, 2022). http://dx.doi.org/10.3389/frsfm.2022.984400.

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In this work, we study the equilibrium configurations in a nematic pi-cell under an electric field using the one-dimensional Oseen-Frank director model. The equilibrium orientational configurations that can be attained in a pi-cell are known as splay, bend, and twist. Among those, bend and twist are topologically equivalent and can be transitioned into one another as voltage varies. The transition can be continuous or abrupt depending on the material parameters. On the other hand, the splay configuration becomes asymmetric for sufficiently high voltages if the liquid crystal has a positive dielectric anisotropy. We determine those threshold voltages and characterize the order of transitions in terms of the elastic constants and the pretilt angle at the boundary.

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Journal articles on the topic 'Oseen Frank model'

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