SCALINGS AND LIMITS OF LANDAU-DE GENNES MODELS FOR LIQUID CRYSTALS: A COMMENT ON SOME RECENT ANALYTICAL PAPERS

Eugene C Gartland Jr

doi:10.3846/mma.2018.025

Abstract

Some recent analytical papers have explored limiting behaviors of Landaude Gennes models for liquid crystals in certain extreme ranges of the model parameters: limits of “vanishing elasticity” (in the language of some of these papers) and “low-temperature limits.” We use simple scaling analysis to show that these limits are properly interpreted as limits in which geometric length scales (such as the size of the domain containing the liquid crystal material) become large compared to intrinsic length scales (such as correlation lengths or coherence lengths, which determine defect core sizes). This represents the natural passage from a mesoscopic model to a macroscopic model and is analogous to a “London limit” in the Ginzburg-Landau theory of superconductivity or a “large-body limit” in the Landau-Lifshitz theory of ferromagnetism. Known relevant length scales in these parameter regimes (nematic correlation length, biaxial coherence length) can be seen to emerge via balances in equilibrium Euler-Lagrange equations associated with well-scaled Landau-de Gennes free-energy functionals.

Highlights

The Landau-de Gennes and Oseen-Frank models are the two most widely used continuum models to characterize equilibrium orientational properties of materials in the nematic liquid crystal phase
Landau-de Gennes models are typically employed in problems in which geometric length scales are not too large compared to intrinsic length scales, while Oseen-Frank models are used when the geometric length scale is much larger than the core size
We show that these limits are properly interpreted not as limits of vanishing elasticity but as limits in which intrinsic length scales become vanishingly small compared to geometric length scales

Summary

Introduction

The Landau-de Gennes and Oseen-Frank models are the two most widely used continuum models to characterize equilibrium orientational properties of materials in the nematic liquid crystal phase. All three obtain (away from a singular set) limiting uniaxial minimizers of the form (1.3), with constant S determined so as to provide a minimum of fb, and with the director field n corresponding to the minimizer of an appropriate Oseen-Frank model We show that these limits are properly interpreted not as limits of vanishing elasticity but as limits in which intrinsic length scales (associated with defect core sizes and such) become vanishingly small compared to geometric length scales (associated with the size of the problem domain Ω). Different length scales are known to be associated with defect core sizes in these two different regimes, and it is shown below how these can be identified via balances in appropriate scalings of the EulerLagrange equations associated with (1.1)

Scaling analysis

Large-body limit

Low-temperature limit

C C ANI C

Biaxiality in the low-temperature regime

Conclusions