An Efficient $\mathcal{Q}$-Tensor-Based Algorithm for Liquid Crystal Alignment away from Defects

Abstract

Abstract. We develop a fast and accurate approximation of the normally stiff equations which minimize the Landau-de Gennes free energy of a nematic liquid crystal. The resulting equations are suitable for all configurations in which defects are not present, making them ideal for device simulation. Specifically they offer an increase in computational efficiency by a factor of 100 while maintaining an error of order (10 −4 ) when compared to the full stiff equations. As this approximation is based on a Q-tensor formalism, the sign reversal symmetry of the liquid crystal is respected. In this paper we derive these equations for a simple two-dimensional case, where the director is restricted to a plane, and also for the full three-dimensional case. An approximation of the error in the perturbation scheme is derived in terms of the first order correction, and a comparison to the full stiff equations is given.

Multiscale models of colloidal dispersion of particles in nematic liquid crystals

“…Conversely, the thermotropic term M(a)a is O(1), indicating that the liquid crystal dynamics, Eq. (10a), has at least two different timescales [32]. Evolution on the fast timescale is determined by the thermotropic term M(a)a which drives the Q tensor into the uniaxial state [32].…”

“…(10a), has at least two different timescales [32]. Evolution on the fast timescale is determined by the thermotropic term M(a)a which drives the Q tensor into the uniaxial state [32]. We are interested in the reorientation of the liquid crystal due to the elastic and electrostatic energies.…”

“…(A19) and span the tangent space of the manifold of uniaxial Q tensors [32]. These equations are solved subject to the constraint that the scalar order parameter is a constant determined by the temperature of the nematic and the biaxiality is zero.…”

“…To solve the homogenized equations we used a pseudospectral code as described in Ref. [32] and we solved the cell problems (20) and (21) (21) for a range of values of Q 0 and interpolating. We also computed the response of an undoped cell for comparison by using a spectral method.…”

“…In the case of the nematic liquid crystal these vectors have the additional properties of spanning the kernel of the Hessian of the thermotropic part of the free energy density and being tangent to the solution manifold defined by Eq. (A12a) [32]. The solvability condition reads…”