Note: Formation of the nematic splay-bend in two-dimensional systems of bow-shaped particles

Recently, Tavarone et al. (J. Chem. Phys. 143, 114505 (2015)) discussed phase behavior of zig-zag and bow-shaped particles composed of three needles. The authors presented very interesting results of extensive Monte Carlo simulations with periodic boundary conditions in the constant-NVT and the constant-NPT ensembles. In addition to isotropic, nematic, and smectic phases, they identified a modulated nematic, which is actually the nematic splay-bend phase ($N_{SB}$), long-anticipated for bent-core systems (Europhys. Lett. 56, 247 (2001)). They also described isotropic-nematic and nematic-smectic transitions using Density Functional Theory in mean-field approximation. The authors, however, did not provided a theoretical description of the $N_{SB}$. Here, we present a simple theory of a phase transition to the $N_{SB}$ phase to fill the gap. In our study, we use Onsager-type Density Functional Theory with perfect order approximation and Meyer parametrization of modulated structures. We present results for arbitrary ratios of the length of central and side segments and opening angles of bow-shaped particles.