On the elusive saddle-splay and splay-bend elastic constants of nematic liquid crystals

The elastic behavior of nematics is commonly described in terms of the three so-called bulk deformation modes, i.e., splay, twist, and bend. However, the elastic free energy contains also other terms, often denoted as saddle-splay and splay-bend, which contribute, for instance, in confined systems. The role of such terms is controversial, partly because of the difficulty of their experimental determination. The saddle-splay (K24) and splay-bend (K13) elastic constants remain elusive also for theories; indeed, even the possibility of obtaining unambiguous microscopic expressions for these quantities has been questioned. Here, within the framework of Onsager theory with Parsons-Lee correction, we obtain microscopic estimates of the deformation free energy density of hard rod nematics in the presence of different director deformations. In the limit of a slowly changing director, these are directly compared with the macroscopic elastic free energy density. Within the same framework, we derive also closed microscopic expressions for all elastic coefficients of rodlike nematics. We find that the saddle-splay constant K24 is larger than both K11 and K22 over a wide range of particle lengths and densities. Moreover, the K13 contribution comes out to be crucial for the consistency of the results obtained from the analysis of the microscopic deformation free energy density calculated for variants of the splay deformation.