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 -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 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.

中文翻译：

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Γ-将手性掺杂向列液晶的平均场模型收敛到胆甾醇的 Oseen-Frank 描述

掺有少量缺乏镜像对称性的分子的细长分子系统可以形成宏观扭曲的胆甾型液晶相。这项工作的目的是从涉及成对分子相互作用的更基本模型中严格推导出胆甾醇的 Oseen-Frank 模型。提出了两种向列型主体/手性掺杂剂混合物的非局部平均场模型，表明胆甾型液晶的 Oseen-Frank 弹性自由能可以同时在大畴和稀释掺杂剂中获得渐进制度。通过-收敛技术，表明在渐近极限下，掺杂剂-掺杂剂相互作用可以忽略不计，弗兰克常数和向列主序参数不受掺杂剂存在的干扰，但是掺杂剂-主体相互作用的镜像不对称导致宏观扭曲的基态。螺旋波数和掺杂剂浓度之间的比例常数，螺旋扭转功率 (HTP)，可以通过这样的分析明确地找到，具有非线性温度依赖性。根据主体-主体和主体-掺杂剂相互作用的相对强度，表明 HTP 可能随温度增加或减少。