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A novel Landau-de Gennes model with quartic elastic terms

Part of: Foundations, constitutive equations, rheology Optimality conditions Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  24 March 2020

DMITRY GOLOVATY Open the ORCID record for DMITRY GOLOVATY [Opens in a new window] ,
MICHAEL NOVACK  and
PETER STERNBERG
DMITRY GOLOVATY
Affiliation:
Department of Mathematics, University of Akron, Akron, OH44325, USA email: dmitry@uakron.edu
MICHAEL NOVACK
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN 47405, USA emails: mrnovack@indiana.edu; sternber@indiana.edu
PETER STERNBERG
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN 47405, USA emails: mrnovack@indiana.edu; sternber@indiana.edu
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Abstract

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.

Keywords

Nematic liquid crystalGamma-convergenceLandau-de Gennes theoryQ-tensordisparate elastic constantsnematic-to-isotropic phase transition

MSC classification

Primary: 35J50: Variational methods for elliptic systems
Secondary: 35B36: Pattern formation 49K20: Problems involving partial differential equations 76A15: Liquid crystals
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Issue 1 , February 2021 , pp. 177 - 198
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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Footnotes

†

D. G. acknowledges the support from NSF DMS-1729538. M. N. and P. S. acknowledge the support from a Simons Collaboration grant 585520.

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