Density functional approach to study the elastic constants of biaxial nematic liquid crystals

Abstract

A density functional theory for bulk and surface elastic constants of biaxial nematic liquid crystals is developed. It is based on a functional Taylor expansion of the free energy of a distorted biaxial nematic with respect to the one-particle distribution function. Detailed microscopic expressions for the biaxial elastic constants of bulk and surface deformations are derived by expanding further the distribution functions into symmetry-adapted Wigner matrices. The final expressions depend on generalized orientational order parameters characterizing the biaxial nematic and on expansion coefficients of the direct pair correlation function. The case where the expansions are truncated at the lowest nontrivial order with respect to the momentum index of the Wigner matrices is analyzed in detail. It gives only six distinct, nonzero bulk elastic constants. The mixed elastic constants, which measure distortions of more than one director, vanish within this approximation. As in the uniaxial case, a splay-bend degeneracy for all directors is apparent. The theory is next applied to the biaxial nematic phase recently studied by Biscarini et al. [Phys. Rev. Lett. 75, 1803 (1995)] providing numerical estimates of biaxial elastic constants for the case of thermodynamically stable biaxial ordering. It is shown that the values of the elastic constants connected with secondary directors are much lower than those associated with the primary one.

Field response and switching times in biaxial nematics

“…Theoretical investigations are also far from simple. For instance, the number of elastic constants for a BXN is over a dozen [13][14][15] and a similar proliferation of specific material constants occurs for other properties, e.g., rheological. 16,17 Apart from widening the gap between molecular and material properties, this causes obvious computational difficulties in setting up, parametrizing, and solving even simple models for studying the response of a BXN to an external field.…”

Poisson-bracket approach to the dynamics of bent-core molecules

“…The last three terms have no simple analogues in uniaxial elasticity. Theoretical calculations based on the microscopic approach [22,23,24] have been done to determine the magnitudes of these elastic constants from first principles, as the results strongly suggested a match to what one would intuitively expect that these "cross"-terms should take on smaller magnitudes than the other nine elastic constants. The existence of polarity allows for terms that break the p → −p in the elastic free energy.…”

“…The effect of the anisotropy in the length of the two arms as well as the angle between them has been investigated in a more recent Monte Carlo simulation [34]. Analytically, a few other theoretical issues have also been looked at, including static properties such as stability of phases [16,17,18,19,20,21], elasticity and viscosity [22,23,24,25], and critical behavior [26]. The dynamical properties [27,28,29,30,31,33] of biaxial nematic systems have also been considered.…”

Statistical Theory of Biaxial Nematic and Cholesteric Phases

“…There will be some deformation of the alignment and the order parameter tensor Q(r) will vary from point to point. Generally, both eigenvectors and eigenvalues of Q depend on r. But in the hydrodynamic limit the eigenvalues are nearly constant and only the directors depend on position [86]. The inhomogeneity of the director field results in a distortion (elastic) free energy.…”