Splay-bend surface elastic constant of nematic liquid crystals: A solution of the Somoza-Tarazona paradox

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The Nehring-Saupe [J. Chem. Phys. 54, 337 (1971); 56, 5527 (1972)] elastic free energy of nematic liquid crystals (NLCs) contains the splay-bend elastic constant ${K}_{13},$ which affects only the elastic surface free energy. Several years ago, Somoza and Tarazona [Mol. Phys. 72, 991 (1991)] showed that the value of ${K}_{13}$ depends on the nonlocal to local mapping that is used to define the local elastic free energy. Then they concluded that the splay-bend constant is not a well-defined physical parameter. In the present paper we show that the Somoza-Tarazona result comes from an inconsistent treatment of the boundary effects. If all the boundary effects are correctly taken into account in an elastic approach, the elastic surface free energy contains an effective elastic constant ${K}_{13}^{\mathrm{eff}}$ that is mapping independent. ${K}_{13}^{\mathrm{eff}}$ is the sum of three different constants: the classical Nehring-Saupe bulk constant ${K}_{13}$ and two specific interfacial constants ${K}_{1}$ and ${K}_{h}.$ While each surface constant (${K}_{13},$ ${K}_{1},$ and ${K}_{h}$) depends on the kind of nonlocal to local mapping, the resulting surface constant ${K}_{13}^{\mathrm{eff}}{=K}_{13}{+K}_{1}{+K}_{h}$ is mapping independent. Using a simple molecular model of the intermolecular interactions, we obtain explicit expressions of ${K}_{13}^{\mathrm{eff}}$ in terms of the characteristic parameters of the intermolecular energy. In the final part of this paper we discuss the meaning and the physical consequences of the elastic surface free energy ${F}_{s}.$ We show that ${F}_{s}$ is a semimacroscopic parameter that provides an approximate elastic description of the interfacial layer. Furthermore, we point out that the elastic surface free energy should not be confused with the thermodynamic surface free energy that appears in a consistent continuum theory of NLCs.