In Silico Measurement of Elastic Moduli of Nematic Liquid Crystals

Experiments on confined droplets of the nematic liquid crystal 5CB have questioned long-established bounds imposed on the elastic free energy of nematic systems. This elasticity, which derives from molecular alignment within nematic systems, is quantified through a set of moduli which can be difficult to measure experimentally and, in some cases, can only be probed indirectly. This is particularly true of the surfacelike saddle-splay elastic term, for which the available experimental data indicate values on the cusp of stability, often with large uncertainties. Here, we demonstrate that all nematic elastic moduli, including the saddle-splay elastic constant $k_{24}$, may be calculated directly from atomistic molecular simulations. Importantly, results obtained through in silico measurements of the 5CB elastic properties demonstrate unambiguously that saddle-splay elasticity alone is unable to describe the observed confined morphologies.

Figure 1

The top row shows idealized bulk elastic modes (a) splay, (b) twist, and (c) bend, which can be directly probed in the experiment. The bottom row shows the 5CB molecule (d) and cylindrical twist deformations, which rely on the saddle-splay elastic constant $k_{24}$, in stable (e) and unstable (f) configurations under conditions of degenerate planar anchoring representative of the commonly studied 5CB-water interface. Saddle splay is not directly measurable through experiment but can be inferred indirectly. The positive definiteness of the elastic free energy expressed through the Ericksen bound $k_{22}-k_{24}\ge 0$ is thought to be violated for 5CB, though experiments are not conclusive.

Figure 2

(a) A harmonic restraint is applied to the periodic edges of a simulation box in order to align the molecules in the $\widehat{\mathbf{z}}$ direction. (b) Molecule orientations in the central region of the simulation box are biased using basis function sampling [43] according to the appropriate order parameter to excite the desired elastic mode. Shown here are arrows representing splay deformations from the nonperturbed state. (c) Over the course of a simulation, molecules enter and exit the respective regions. Only those molecules which lie within the regions shown in purple (edges) and orange (center) are biased. A gradient is produced across the box dimension as a result of the sampling and the corresponding free energy is calculated. The resulting bulk elastic coefficients ($k_{ii}$) for 5CB (d) are compared to experimental data from Madhusudana and Pratibha [45] (squares) and Chen and co-workers [46] (triangles). Connected circles represent elastic constant calculations using the methodology outlined in this work. Uncertainties are calculated using 1500 bootstrap cycles on the collected decorrelated samples.

Figure 3

Snapshot of 5CB cylindrical system (a) with solvent removed. Because of finite anchoring-induced ordering within the cylinder, the transition temperature $T_{\mathrm{NI}}$ is shifted slightly (b) by $\approx 5\mathrm{K}$. The calculated saddle-splay surfacelike elastic constant ($k_{24}$) for 5CB (c) shows no violation of the Ericksen bound, delineated by $k_{22}-k_{24}\ge 0$. To validate $k_{24}$ stability, we test the unbiased director probability distribution $p(n_{\theta })$ against the normal distribution (d) using a Kolmogorov-Smirnov test (representative data at 296 K shown; distributions obtained at other temperatures are plotted in Fig. S2 of the Supplemental Material [26]). Uncertainties in the elastic measurements are estimated using 1500 bootstrap cycles on the collected decorrelated samples, yielding error bars comparable to the size of the points. Roughness in trendlines is not due to a statistical uncertainty of each measurement, but instead due to the underlying fluctuations in nematic order and volume that arise under $NPT$ preparation.