Multiscale approach to nematic liquid crystals via statistical field theory

We propose an approach to a multiscale problem in the theory of thermotropic uniaxial nematics based on the method of statistical field theory. This approach enables us to relate the coefficients $A$, $B$, $C$, $L_1$, and $L_2$ of the Landau-de Gennes free energy for the isotropic-nematic phase transition to the parameters of a molecular model of uniaxial nematics, which we take to be a lattice gas model of nematogenic molecules interacting via a short-ranged potential. We obtain general constraints on the temperature and volume fraction of nematogens for the Landau-de Gennes theory to be stable against molecular orientation fluctuations at quartic order. In particular, for the case of a fully occupied lattice, we compute the values of the isotropic-nematic transition temperature and the order parameter discontinuity predicted by (i) a continuum approximation of the nearest-neighbor Lebwohl-Lasher model and (ii) a Lebwohl-Lasher-type model with a nematogenic interaction of finite range. We find that the predictions of (i) are in reasonably good agreement with known results of Monte Carlo simulation.

Figure 1

Modified Lebwohl-Lasher model: a two-dimensional slice of the three-dimensional cubic lattice on which nematogens (represented by orange-colored rods) reside. The $j\mathrm{th}$ rod has a direction parallel to ${\mathbf{n}}_j$ and a position vector ${\mathbf{r}}_j$, inhabiting a cell that has a linear dimension $a$. A cell can either be occupied by a nematogen, or unoccupied. The circle (colored blue, radius $b$) indicates the effective range of the nematogen-nematogen interaction potential $v\left(\mathbf{r}\right)$.

Figure 2

Behavior of the coefficient $C$ of the Landau-de Gennes theory as a function of $\varphi$, the fraction of lattice cells occupied by nematogens. $C$ is positive for $\varphi >50/63$.