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

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 dopant-dopant 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 {\it helical twisting power} (HTP), can be explicitly found through such an analysis, with a non-linear 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.