Splay and bend elasticity in two-dimensional nematic liquid crystals

We present a molecular dynamics simulation study of the Frank orientational elastic constants, namely, splay (K₁) and bend (K₃), in a two-dimensional (2D) thermotropic nematic liquid crystal composed of Gay-Berne ellipses with interaction parameters (3, 5, 2, 1) and aspect ratio 3 : 1. The elastic constants are extracted via the molecular orientation fluctuation method, in which the ensemble-averaged Fourier-transformed orientational order tensor is fitted to a bilinear surface in reciprocal wavevector space within the nematic director frame. A systematic finite-size analysis over five system sizes (N = 10 000 to N = 160 000) identifies N = 160 000 as the optimal system size for convergence toward the thermodynamic limit, with fitting errors below 3.2% for K₁ and below 2.1% for K₃. Both elastic constants decrease monotonically with increasing temperature, reflecting the progressive loss of orientational order as the nematic-to-isotropic transition is approached. Throughout the nematic phase, K₃ > K₁ at all state points, with a bend-to-splay ratio K₃/K₁ ≈ 2 that remains robust across the nematic phase and decreases toward unity only near the transition. The orientational order is further characterized via the orientational correlation function g₂(r), which exhibits algebraic decay g₂(r) ∼ r^-η₂(T) throughout the nematic phase, confirming quasi-long-range orientational order consistent with the Mermin-Wagner theorem. The decay exponent η_2,fit(T) increases monotonically with temperature and reaches ≈1/4 at the nematic-to-isotropic transition, consistent with the universal Kosterlitz-Thouless prediction. A comparison with the Nelson-Halperin formula reveals near-quantitative agreement deep in the nematic phase ((T) = η_2,fit/η_2,NH ≈ 0.880-0.870 at T = 0.90-0.95), providing independent validation of the computed elastic constants, while the systematic decrease of (T) toward higher temperatures signals the growing role of critical fluctuations near the transition.