The Well Order Reconstruction Solution for three-dimensional wells, in the Landau–de Gennes theory

Canevari, Giacomo; Harris, Joseph; Majumdar, Apala; Wang, Yiwei

doi:10.1016/j.ijnonlinmec.2019.103342

Cited by 31 publications

(31 citation statements)

References 26 publications

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“…In order to compute these locally stable points, we use spectral methods to discretize the order parameter Q. Spectral methods are efficient numerical methods with high accuracy [74]. Several previous studies have shown that the spectral method is a powerful tool to numerical study LdG free energy [60,[75][76][77][78]. The key idea to apply the spectral method to our system is to use a bispherical polar coordinate system (ξ, μ, ϕ) [60], which is given by Then, we can expand the tensor function Q(r) in terms of special functions: real spherical harmonics of (μ, ϕ) and Legendre polynomials of ζ [60].…”

Section: Appendix B: Numerical Methodsmentioning

confidence: 99%

Dynamic tuning of the director field in liquid crystal shells using block copolymers

Noh

Wang

Liang

et al. 2020

Phys. Rev. Research

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When an orientationally ordered system, like a nematic liquid crystal (LC), is confined on a self-closing spherical shell, topological constraints arise with intriguing consequences that depend critically on how the LC is aligned in the shell. We demonstrate reversible dynamic tuning of the alignment, and thereby the topology, of nematic LC shells stabilized by the nonionic amphiphilic block copolymer Pluronic F127. Deep in the nematic phase, the director (the average molecule orientation) is tangential to the interface, but upon approaching the temperature T NI of the nematic-isotropic transition, the director realigns to normal. We link this to a delicate interplay between an interfacial tension that is nearly independent of director orientation, and the configuration-dependent elastic deformation energy of an LC confined in a shell. The process is primarily triggered by the heating-induced reduction of the nematic order parameter, hence realignment temperatures differ by several tens of degrees between LCs with high and low T NI , respectively. The temperature of realignment is always lower on the positive-curved shell outside than at the negative-curved inside, yielding a complex topological reconfiguration on heating. Complementing experimental investigations with mathematical modeling and computer simulations, we identify and investigate three different trajectories, distinguished by their configurations of topological defects in the initial tangential-aligned shell. Our results uncover a new aspect of the complex response of LCs to curved confinement, demonstrating that the order of the LC itself can influence the alignment and thereby the topology of the system. They also reveal the potential of amphiphilic block copolymer stabilizers for enabling continuous tunability of LC shell configuration, opening doors for in-depth studies of topological dynamics as well as novel applications in, e.g., sensing and programed soft actuators.

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Section: Appendix B: Numerical Methodsmentioning

confidence: 99%

Dynamic tuning of the director field in liquid crystal shells using block copolymers

Noh

Wang

Liang

et al. 2020

Phys. Rev. Research

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“…Furthermore, for A=−false(B2/3Cfalse), q3 is constant for all physically relevant critical points of the form (2.9). In a 3D setting, the constancy of q3 is only true for the special temperature A=−B2/false(3Cfalse) [43]. While we conjecture that some qualitative solution properties are universal for A<0, a non-constant q3 profile would introduce new technical difficulties that would distract from the main message.…”

Section: Modelsmentioning

confidence: 94%

Solution landscapes of the simplified Ericksen–Leslie model and its comparisonwith the reduced Landau–deGennes model

Han

Yin

et al. 2021

Proc. R. Soc. A.

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We investigate the solution landscapes of a simplified Ericksen–Leslie (sEL) vector model for nematic liquid crystals, confined in a two-dimensional square domain with tangent boundary conditions. An efficient numerical algorithm is developed to construct the solution landscapes by utilizing the symmetry properties of the model and the domain. Since the sEL model and the reduced Landau–de Gennes (rLdG) models can be viewed as Ginzburg–Landau functionals, we systematically compute the solution landscapes of the sEL model, for different domain sizes, and compare them with the solution landscapes of the corresponding rLdG model. There are many similarities, including the stable diagonal and rotated states, bifurcation behaviours and sub-solution landscapes with low-index saddle solutions. Significant disparities also exist between the two models. The sEL vector model exhibits the stable solution C ± with interior defects, high-index ‘fake defect’ solutions, novel tessellating solutions and certain types of distinctive dynamical pathways. The solution landscape approach provides a comprehensive and efficient way for model comparison and is applicable to a wide range of mathematical models in physics.

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“…In other words, only three degrees of freedom out of five remain in a 2D framework. Further, in [28], the authors show that for A = − B 2 3C , q 3 is a constant for all physically relevant solutions of (6) of the form (7), subject to Dirichlet uniaxial tangent boundary conditions on the domain 3C , we have a reduced description in terms of a reduced LdG tensor, P, with only two degrees of freedom such that…”

Section: Landau-de Gennes Theorymentioning

confidence: 99%

“…Such reduced descriptions have been hugely successful for 2D systems or thin three-dimensional (3D) systems, both for capturing the qualitative properties of physically relevant solutions and for probing into defect cores [29][30][31][32][33]. In a fully 3D system, there may be other classes of physically relevant solutions, such as escaped solutions, with additional degrees of freedom [17,28] and this will be pursued in further work.…”

Section: Landau-de Gennes Theorymentioning

confidence: 99%

Solution landscape of a reduced Landau–de Gennes model on a hexagon

et al. 2021

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We investigate the solution landscape of a reduced Landau-de Gennes model for nematic liquid crystals on a two-dimensional hexagon at a fixed temperature, as a function of λ-the edge length. This is a generic example for reduced approaches on regular polygons. We apply the high-index optimization-based shrinking dimer method to systematically construct the solution landscape consisting of multiple solutions, with different defect configurations, and relationships between them. We report a new stable T state with index-0 that has an interior −1/2 defect; new classes of highindex saddle points with multiple interior defects referred to as H-class and TD-class saddle points; changes in the Morse index of saddle points as λ 2 increases and novel pathways mediated by highindex saddle points that can control and steer dynamical pathways on the solution landscape. The range of topological degrees, locations and multiplicity of defects offered by these saddle points can be used to navigate the complex solution landscapes of nematic liquid crystals and other related soft matter systems.

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The Well Order Reconstruction Solution for three-dimensional wells, in the Landau–de Gennes theory

Cited by 31 publications

References 26 publications

Dynamic tuning of the director field in liquid crystal shells using block copolymers

Dynamic tuning of the director field in liquid crystal shells using block copolymers

Solution landscapes of the simplified Ericksen–Leslie model and its comparisonwith the reduced Landau–deGennes model

Solution landscape of a reduced Landau–de Gennes model on a hexagon

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