Abstract
We study global minimizers of the Landau–de Gennes (LdG) energy functional for nematic liquid crystals, on arbitrary three-dimensional simply connected geometries with topologically non-trivial and physically relevant Dirichlet boundary conditions. Our results are specific to an asymptotic limit coined in terms of a dimensionless temperature and material-dependent parameter, t and some constraints on the material parameters, and we work in the trightarrow infty limit that captures features of the widely used Lyuksyutov constraint (Kralj and Virga in J Phys A 34:829–838, 2001). We prove (i) that (re-scaled) global LdG minimizers converge uniformly to a (minimizing) limiting harmonic map, away from the singular set of the limiting map; (ii) we have points of maximal biaxiality and uniaxiality near each singular point of the limiting map; (iii) estimates for the size of “strongly biaxial” regions in terms of the parameter t. We further show that global LdG minimizers in the restricted class of uniaxial mathbf{Q}-tensors cannot be stable critical points of the LdG energy in this limit.
Highlights
Nematic liquid crystals (LCs) are anisotropic liquids with long-range orientational order i.e. the constituent rod-like molecules have full translational freedom but align along cer-Communicated by J
Mathematicians have turned to the analysis of the celebrated Landau–de Gennes (LdG) theory for nematic liquid crystals, in various asymptotic limits: see for example [8,10,15,24] which is not an exhaustive list but are directly relevant to our paper
We rigorously prove the co-existence of both maximal biaxiality and uniaxiality near each singular point of a global LdG minimizer on an arbitrary three-dimensional domain with uniaxial Dirichlet boundary conditions, and whilst this does not recover the structure of the biaxial torus solution, this is consistent with the picture of a torus of maximal biaxiality that contains a uniaxial ring of negative scalar order parameter
Summary
55 Page 2 of 22 tain locally preferred directions [11,38]. The existence of distinguished directions renders nematics sensitive to light and external fields leading to unique electromagnetic, optical and rheological properties [11,16,30]. The bulk potential, fB , is minimized by uniaxial tensors of a constant norm for low temperatures i.e. for. This assumption can be true for some commonly used liquid crystal materials [18] This assumption forms the basis of the popular Lyuksyutov constraint in the literature where authors work with Q-tensors of a fixed norm [17,18] i.e. We rigorously prove the co-existence of both maximal biaxiality and uniaxiality near each singular point (to be interpreted appropriately) of a global LdG minimizer on an arbitrary three-dimensional domain with uniaxial Dirichlet boundary conditions, and whilst this does not recover the structure of the biaxial torus solution, this is consistent with the picture of a torus of maximal biaxiality that contains a uniaxial ring of negative scalar order parameter. We note that the eigenvalues of the minimizers of the bulk potential fB in (2) respect physical bounds if and only if [22,38]
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