A molecular-statistical theory of the orientational elasticity of nematic liquid crystals has been developed employing the orientational deformation tensor which describes the rotation of the director. An explicit expression for the general elasticity tensor of the nematic phase has been obtained and the Frank elastic constants are expressed in terms of the three independent parameters of this tensor. Explicit expressions for the Frank elastic constants have been derived in the molecular field approximation in terms of the orientational order parameters and the corresponding coefficients of expansion of the intermolecular potential in spherical invariants. Frank elastic constants have been calculated numerically for nematic liquid crystals composed of both polar and nonpolar molecules together with the orientational order parameters using the classical Gay-Berne model interaction potential and the two of its popular modifications. The polarity of the uniaxial molecular shape has been directly introduced into the model potential by modifying the distance of closest approach. The elastic constants are presented as functions of temperature for different values of the molecular elongation, the anisotropy of the potential well and the molecular shape polarity. It has been shown that the elastic constants are much more sensitive to the details of the intermolecular interaction potential in comparison with the orientational order parameters. In particular, a relatively weak polarity of the molecular shape may result in an unusual decrease of the splay constant $K_{11}$ which may vanish at some temperature leading to the instability of the homogeneous nematic phase. This may represent a mechanism of the formation of the splay-bend phase.

中文翻译：

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由极性和非极性分子组成的向列液晶的分子统计弹性理论

向列型液晶的取向弹性的分子统计理论已经被开发出来，它利用了取向指向量的张量来描述指向矢的旋转。已经获得了向列相的一般弹性张量的显式表达式，并且用该张量的三个独立参数表示了Frank弹性常数。在分子场近似中，已根据方向性阶数参数和球形不变量中分子间电势的相应膨胀系数，得出了Frank弹性常数的明确表达式。已经使用经典的盖伊-伯恩模型相互作用势及其两种流行的修改方法，对由极性和非极性分子组成的向列液晶以及取向顺序参数进行了数值计算，得出了弗兰克弹性常数。通过修改最接近的距离，将单轴分子形状的极性直接引入到模型电势中。对于不同的分子伸长率，势阱的各向异性和分子形状极性，弹性常数作为温度的函数表示。已经表明，与取向顺序参数相比，弹性常数对分子间相互作用电位的细节更为敏感。特别是，${ķ}_{11}$它在某些温度下可能消失，导致均相向列相的不稳定性。这可以代表张开弯曲阶段形成的机制。