Enhanced Landau–de Gennes potential for nematic liquid crystals from a systematic coarse-graining procedure

The macroscopic theory of nematics is conveniently described in terms of the phenomenological Landau–de Gennes free energy. Here we show how such an effective free energy can be obtained explicitly from a microscopic model via the help of a systematic coarse-graining procedure. We test our approach for the two- and three-dimensional Lebwohl-Lasher model of nematics. The effective free energy that we obtain is consistent with the phenomenological Landau–de Gennes form for weak orientational ordering and the Maier-Saupe theory of the isotropic-nematic transition. For strong orientational ordering, however, the effective free energy increases rapidly and diverges logarithmically near the fully oriented state. The explicit form for the regularized Landau–de Gennes potential proposed here restricts the order parameter to physical admissible values and reproduces our numerical data accurately.

Figure 1

The order parameter $Q_{11}$ of the Lebwohl-Lasher model on a square lattice as a function of the Lagrange multiplier $\lambda$ for $\beta J=1.0$ (top) and $0.125$ (bottom). A uniaxial form of the Lagrange multiplier is chosen. Symbols denote the numerical values from Monte Carlo simulations, and the lines are obtained from fits to Eq. (12). Inset: The susceptibility $\chi =1/c_1$ between $\mathbf{Q}$ and $\mathit{\Lambda }$ [Eq. (C3)] as a function of temperature. The symbols denote the numerical values obtained from linear regression, for values $|\mathit{\Lambda }/N|<0.1$. The broken line is the mean-field prediction from Eq. (C7), and the full line is the fit mentioned in the text.

Figure 2

Dimensionless difference $\Delta f=\Delta F/k_{\mathrm{B}}T$ between the effective free energy (6) and the mean effective interaction energy defined in the text as a function of the Maier-Saupe order parameter $S_2$. Different symbols correspond to different lattice symmetries. The data nicely collapse onto a single master curve. The thick solid line represents the scaled ideal orientational entropy $-S^{\mathrm{id}}/N{k}_{\mathrm{B}}$, which describes the numerically obtained master curve very well.

Figure 3

The dimensionless effective free energy $\mathcal{F}/N{k}_{\mathrm{B}}T$ as a function of the Maier-Saupe order parameter $S_2$ for the planar Lebwohl-Lasher model on a square lattice. The reduced temperature $k_{\mathrm{B}}T/J$ is decreasing from top to bottom as 8.0, 1.0, 0.7, 0.566. Symbols denote the numerical values while the lines are obtained from Eq. (11). Inset: The quantity $-\ln p\left(\mathbf{Q}\right)$ defined in the text is shown as a function of $S_2$ for reduced temperatures, from left to right, 8.0, 0.566, 0.45, 0.40, 0.38, 0.35, 0.30.

Figure 4

Bifurcation diagram shows the Maier-Saupe order parameter $S_2$ as a function of $T_c/T$. Solid lines are the solution of Eq. (13). Symbols denote the results of Monte Carlo simulations for different lattice symmetries.

Figure 5

The Maier-Saupe orientational order parameter $S_2$ as a function of the scaled temperature $T/T_c$ for the three-dimensional Lebwohl-Lasher model. Circles are the result of extensive, but standard Monte Carlo simulations [37], and squares denote the result of our Monte Carlo simulations in the generalized canonical ensemble (2), extrapolated for $\mathit{\Lambda }\to 0$. The solid line is the theoretical prediction from the solution to Eq. (13), and the broken line is $S_2^{\mathrm{D}}$ from Ref. [13] defined in the text.

Figure 6

The Maier-Saupe orientational order parameter $S_2$ as a function of the dimensionless elongation rate $\tau \dot{\epsilon }$. From bottom to top, the interaction strength is increasing as $z^{*}\beta J=0,2,4,6$. Solid lines show the prediction of Eq. (D2), and broken lines correspond to the Doi-Edwards form (D4). Open and full symbols denote the result of Monte Carlo simulations of the two- and three-dimensional Lebwohl-Lasher model, respectively.