Time correlation functions in the Lebwohl-Lasher model of liquid crystals

Time correlation functions in the Lebwohl-Lasher model of nematic liquid crystals are studied using theory and molecular dynamics simulations. In particular, the autocorrelation functions of angular momentum and nematic director fluctuations are calculated in the long-wavelength limit. The constitutive relations for the hydrodynamic currents are derived using a standard procedure based on non-negativity of the entropy production. The continuity equations are then linearized and solved to calculate the correlation functions. We find that the transverse angular momentum fluctuations are coupled to the director fluctuations and are both propagative. The propagative nature of the fluctuations suppresses the anticipated hydrodynamic long-time tails in the single-particle autocorrelation functions. The fluctuations in the isotropic phase are, however, diffusive, leading to $t^{-d/2}$ long-time tails in $d$ spatial dimensions. The Frank elastic constant measured using the time correlation functions are in good agreement with previously reported results.

Figure 1

Angular momentum autocorrelation functions (normalized) as defined in Eq. (26) for components perpendicular to the nematic director for different wave vectors. The dashed lines correspond to the simulation results and the solid lines are fits to the theoretical expression in Eq. (29). Parameters are chosen as $T=1.0$, $N={32}^3$, and $k=2\pi /32$, $4\pi /32$, and $6\pi /32$ (top to bottom at $t=2.5$).

Figure 2

Temperature dependence of spin viscosity $\nu$ in the isotropic phase. Symbols denote the simulation results and the line corresponds to the fitting function given by Eq. (42). System size: $N={32}^3$.

Figure 3

Single-particle angular momentum autocorrelation functions in the real space in the isotropic phase. Simulation results are represented by symbols [circles, two-dimensional (2D); squares, three-dimensional (3D)] and theoretical results [Eq. (45)] by solid lines. For 2D systems, while the position coordinates of the rotors are confined to a plane, the orientation vectors point on a 3D unit sphere. Parameters: $N={64}^3$, $T=2.63$ for 3D systems and $N={100}^2$, $T=1.50$ for 2D systems.