A splay-twist phase stabilized by the interaction between the nematic and torsional fields in nematics

A free energy density for the nematic phase with two symmetry elements - the director, n, and the vector defining the helix direction, t - can be constructed as an extension of the Frank free energy. This formulation has already proven effective in demonstrating that the phase transition between the conventional nematic phase and the twist-bend nematic phase is of second order, characterized by a finite wave vector. In this work, we theoretically investigate the possibility that new periodic phases with finite wave vectors may be energetically favored over uniform structures within the framework of this elastic model. We show that splay-twist-like periodic structures naturally emerge from this theoretical approach. Furthermore, we demonstrate that the existence of a critical wave vector, which determines the periodicity of the non-uniform structure, depends on the elastic parameters, the sample thickness, and the anchoring energy strengths. A key role is played by the elastic constant that couples the nematic director to the helical axis: a distinctive feature of these materials. The splay-twist transition from the uniform nematic phase occurs only when the magnitude of the coupling elastic constant exceeds a threshold value. In this study, we specifically treat the case of a sample with symmetrical interfaces.