Γ-convergence of a mean-field model of a chiral doped nematic liquid crystal to the Oseen–Frank description of cholesterics

Abstract

Systems of elongated molecules, doped with small amounts of molecules lacking mirror symmetry can form macroscopically twisted cholesteric liquid crystal phases. The aim of this work is to rigorously derive the Oseen-Frank model of cholesterics from a more fundamental model concerned with pairwise molecular interactions. A non-local mean-field model of the two-species nematic host/chiral dopant mixture is proposed, and it is shown that Oseen-Frank's elastic free energy for cholesteric liquid crystals can be obtained in a simultaneously large-domain and dilute-dopant asymptotic regime. By techniques of Γ-convergence, it is shown that in the asymptotic limit, dopantdopant interactions are negligable, the Frank constants and nematic host order parameter are unperturbed by the presence of dopant, but the mirror asymmetry of the dopant-host interaction leads to a macroscopically twisted ground state. The constant of proportionality between the helical wavenumber and dopant concentration, the helical twisting power (HTP), can be explicitly found through such an analysis, with a nonlinear temperature dependence. Depending on the relative strengths of the host-host and host-dopant interactions, it is shown that HTP may increase or decrease with temperature.

Mathematical problems of nematic liquid crystals: between dynamical and stationary problems

“…The study of the similar limit in the dynamical problems has generated several notable results such as [52] (QS to EL), [106] (LDG to EL), [107] (LDG to harmonic maps), [103,[108][109][110][111] (molecular theories to OF). However, the presence of the flow adds formidable difficulties to the study of this limit for weak solution, see in particular in the relaxed EL problem context the Open Problem 2.4 in [58].…”

Hölder regularity and convergence for a non-local model of nematic liquid crystals in the large-domain limit

“…Justification by formal gradient expansions leaves open the question as to the consistency minimisers of the original free energy with minimisers of its approximation. To this end, recent work has been focused on rigorous asymptotic analysis of non-local free energies, which similarly produce the Oseen-Frank model in a large-domain limit [31,32,42,43]. These approaches "bypass" the intermediate and non-rigorous derivation of Landau-de Gennes.…”

A One-Dimensional Variational Problem for Cholesteric Liquid Crystals with Disparate Elastic Constants

“…Our purpose in this article is to continue the analysis of a family of models with disparate elastic constants arising in the mathematics of liquid crystals [5,6,7,8]. In particular, the problem (1.1) can be viewed as a highly simplified, relaxed version of the Oseen-Frank model for cholesteric liquid crystals, [2,13,20,21,22,23] based on the elastic deformations of an S 1 -or S 2 -valued director n, cf. [24].…”