Scalings and Limits of Landau-de Gennes Models for Liquid Crystals: A Comment on Some Recent Analytical Papers

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.