Abstract
We investigate prototypical profiles of point defects in two-dimensional liquid crystals within the framework of Landau–de Gennes theory. Using boundary conditions characteristic of defects of index k/2, we find a critical point of the Landau–de Gennes energy that is characterised by a system of ordinary differential equations. In the deep nematic regime, \(b^2\) small, we prove that this critical point is the unique global minimiser of the Landau–de Gennes energy. For the case \(b^2=0\), we investigate in greater detail the regime of vanishing elastic constant \(L \rightarrow 0\), where we obtain three explicit point defect profiles, including the global minimiser.
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Notes
The careful reader will note that \(\mathrm{tr}(Q) = 0\) implies that the eigenvalues cannot all be positive. In order to obtain positive lengths for the axes, we add to each eigenvalue a sufficiently large positive constant (we assume the eigenvalues of Q are bounded).
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Acknowledgments
GDF, JMR, VS would like to acknowledge support from EPSRC Grant EP/K02390X/1. VS also acknowledges support from EPSRC grant EP/I028714/1. AZ gratefully acknowledges the hospitality of the Mathematics Department at the University of Bristol, through EPSRC grants EP/I028714/1 and EP/K02390X/1.
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Communicated by Felix Otto.
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Di Fratta, G., Robbins, J.M., Slastikov, V. et al. Half-Integer Point Defects in the Q-Tensor Theory of Nematic Liquid Crystals. J Nonlinear Sci 26, 121–140 (2016). https://doi.org/10.1007/s00332-015-9271-8
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DOI: https://doi.org/10.1007/s00332-015-9271-8