Computational molecular field theory for nematic liquid crystals

Nematic liquid crystals exhibit configurations in which the underlying ordering changes markedly on macroscopic length scales. Such structures include topological defects in the nematic phase and tactoids within nematic-isotropic coexistence. We discuss a computational study of inhomogeneous configurations that is based on a field theory extension of the Maier-Saupe molecular model of a uniaxial, nematic liquid crystal. A tensor order parameter is defined as the second moment of an orientational probability distribution, leading to a free energy that is not convex within the isotropic-nematic coexistence region, and that goes to infinity if the eigenvalues of the order parameter become nonphysical. Computations of the spatial profile of the order parameter are presented for an isotropic-nematic interface in one dimension, a tactoid in two dimensions, and a nematic disclination in two dimensions. We compare our results to those given by the Landau–de Gennes free energy for the same configurations and discuss the advantages of such a model over the latter.

Figure 1

Examples of the probability distribution, $p\left(\mathit{\xi }\right)$ of Eq. (5), on the sphere spanned by $\mathit{\xi }$ for (a) a uniaxial configuration and (b) a biaxial configuration. Note that the probability distribution involves a uniaxial molecule, but a biaxial order parameter can occur for a probability distribution with biaxial second moment. Only northern hemispheres are displayed since the probability distribution is symmetric about the equator due to the symmetry of the molecules. For these plots, (a) $\mathit{\Lambda }=4diag(-1,-1,0.5)$ and (b) $\mathit{\Lambda }=10diag(-0.25,-1,0.25)$.

Figure 2

Equilibrium value of the uniaxial order, $S$, versus the parameter $\alpha /\left(n{k}_{B}T\right)$. At high $T$, the system is in an isotropic phase, while at low $T$ the system is in a uniaxial nematic phase. A first order phase transition occurs at $\alpha /\left(n{k}_{B}T\right)\approx 3.4049$.

Figure 3

Bulk free energy density as a function of the uniaxial order, $S$, for three values of the parameter $\alpha /\left(n{k}_{B}T\right)$. As $\alpha /\left(n{k}_{B}T\right)$ increases, the free energy becomes nonconvex, leading to coexistence between the isotropic and nematic phases.

Figure 4

$S$ as a function of position for a one-dimensional interface. Dirichlet boundary conditions maintain $S=S_N$ at the left boundary while $S=0$ at the right boundary. $L=1$ for this configuration.

Figure 5

(a) Interface width and (b) bulk surface tension versus $\sqrt{L}$. Dots represent the molecular field theory (MFT) computations while the solid lines are derived from the analytical solution for LdG, Eq. (22), with $B/C=9$. Both the interface width and excess free energy (i.e., surface tension) scale linearly with the parameter $\sqrt{L}$, the same scaling relationship as that of Landau–de Gennes.

Figure 6

Plots of $S(x,y)$ for (a) a tactoid with $m=1$ director configuration at the outer boundary and (b) tactoid with $m=-1/2$ director configuration at the outer boundary. The radius in (a) is $R/{\xi }_{MS}=19.92\pm 0.2$ and the radius in (b) is $R/{\xi }_{MS}=4.59\pm 0.2$. The smaller size of the $m=-1/2$ tactoid is due to the director distortion energy's $m^2$ dependence. For both computations $L=1$.

Figure 7

(a) Director profile and (b) radial plots of the uniaxial order $S$ and the biaxial order $P$ for a nematic disclination. The spatial extent of biaxiality is on the order of the radius of the disclination core. Here, $\alpha /\left(n{k}_{B}T\right)=4$ and $L=1$.

Figure 8

Radius of disclinations plotted as a function of temperature for (a) the molecular field theory of Sec. 2 and (b) the Landau–de Gennes model. $T^{*}$ is the temperature where the isotropic phase loses its metastability, while the dotted line on the plots indicates where coexistence between phases is for the respective model. For the molecular field theory we use $L=1$ and for Landau–de Gennes $L=1$ and $B/C=4$ for all simulations.