We study two dimensional tactoids in nematic liquid crystals by using a Q -tensor representation. A bulk free energy of the Maier–Saupe form with eigenvalue constraints on Q , plus elastic terms up to cubic order in Q are used to understand the effects of anisotropic anchoring and Frank–Oseen elasticity on the morphology of nematic–isotropic domains. Further, a volume constraint is introduced to stabilize tactoids of any size at coexistence. We find that anisotropic anchoring results in differences in interface thickness depending on the relative orientation of the director at the interface, and that interfaces become biaxial for tangential alignment when anisotropy is introduced. For negative tactoids, surface defects induced by boundary topology become sharper with increasing elastic anisotropy. On the other hand, by parametrically studying their energy landscape, we find that surface defects do not represent the minimum energy configuration in positive tactoids. Instead, the interplay between Frank–Oseen elasticity in the bulk, and anisotropic anchoring yields semi-bipolar director configurations with non-circular interface morphology. Finally, we find that for growing tactoids the evolution of the director configuration is highly sensitive to the anisotropic term included in the free energy, and that minimum energy configurations may not be representative of kinetically obtained tactoids at long times.
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A novel Landau-de Gennes model with quartic elastic terms
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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- Award ID(s):
- 1729538
- NSF-PAR ID:
- 10310882
- Date Published:
- Journal Name:
- European Journal of Applied Mathematics
- Volume:
- 32
- Issue:
- 1
- ISSN:
- 0956-7925
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/S095679252000008X
