Sawtooth profile in smectic A liquid crystals (Q2789364)

As is known that in the liquid crystal confined between flat plates, an instability will occur above a threshold magnetic field (Helfrich-Hurault effect), where layer undulation will appear. With increasing the magnetic field above this critical value, the sinusoidal shape of the smectic layer will change into a chevron (zigzag) pattern. The paper is devoted to study of a high-field regime and its influence on the chevron profile. First, it is introduced the de Gennes model and the scaling regime of the problem. To rigorously study the zigzag pattern in full generality, it is analyzed via \(\Gamma\)-convergence a 2D de Gennes energy functional without identifying the director with the layer normal. The de Gennes free energy density includes nematic, smectic A and magnetostatic contributions. Nondimensionalization leads to the identification of a small parameter \(\varepsilon\). The paper considers regimes, where the field strength is of order \(O(1/\varepsilon)\). At refolmulating the de Gennes free energy, it is captured a double well potential having two minimum states for the director on the sphere and a Modica-Mortola type inequality is used on the sphere equipped with a new metric associated with the double well potential. The authors consider periodic boundary conditions and adapt to the problem for vector fields the variational approach on the flat torus. Then, the results are proved for the Chen-Lubensky (CL) model, which includes a second order gradient term for the smectic order parameter. The role of the coefficient of the second order gradient term in the chevron formation is studied. The \(\Gamma\)-convergence analysis for the 2D de Gennes energy in the flat torus setting applies in a straightforward manner to the 2D CL energy and indicates that the \(\Gamma\)-limit for CL provides a lower energy than the \(\Gamma\)-limit for de Gennes energy. The present study in 2D suggests that the chevron structure is essentially 1D, hence to capture the fundamental features of the vertical stripes, it is derived also a \(\Gamma\)-convergence result for the 1D CL energy on the interval with periodic boundary conditions. Numerical simulations in 3D are carried out to illustrate the saw-tooth profiles of undulations by solving the gradient flow equations. The molecular alignment and layer structure confirm the mathematical analysis. The numerics show that the evolution from the sinusoidal perturbation at the onset of undulations to the chevron pattern occurs with an increase of the wavelength.