Elastic continuum theory of biaxial nematics

Abstract

The elastic-distortion free-energy density of biaxial nematics is derived with use of the formalism of tensor analysis. The macroscopic description of biaxial nematics involves 12 bulk elastic constants. The appearance of chirality introduces an additional number of 5 twist terms.

Temperature Dependence of the Elastic Constants for Biaxial Nematic Liquid Crystals

“…The three chiral contributions k 1 (l ·V X fj, km(m ·V X m), and kn(n ·V X n) appear naturally in this description and must be included in the generalized free energy for the hard biaxial nematic phase, in accord with the derivation by methods of invariant theory [8,10,11]. It would be a highly singular, nongeneric case, if out of the three chiral wavevectors k 1 fK 12 , km/Km 2 , kn/Kn 2 one would vanish and the other two would become identical, as requested by Brand and Pleiner [12] for chiral biaxial nematic liquid crystals (this requirement is denied in [7] and [13]).…”

“…Starting from a general elastic LGdG free energy (10) and reducing the order parameter space of symmetric and traceless tensors to an order parameter space whose elements satisfy the condition of "hard biaxiality" ( 16), we derived the elastic theory for biaxial nematic liquid crystals.…”

“…The phase, in which the Goldstone variables { l(r), m(r), n(1')} vary in space, but the order parameters and the degree of biaxiality T are constant, is denoted "hard biaxial nematic phase" [7]. If ansatz (16) is inserted into the extended Landau-Ginzburg-de Gennes (LGdG) elastic energy density /ei (10), then a generalized Frank-Oseen elastic energy for hard biaxial nematic liquid crystals results. This energy was derived by Trebin [8], Brand and Pleiner [9], and Govers and Vertogen [10,11] by symmetry considerations and has the form (according to [11])…”

Landau–de Gennes model for nonuniform configurations in nematic liquid crystalline elastomers

“…(26) or (30) to Eq. (31), and with the details of the calculations, can be found, for example, in [13] and [14].…”

Statistical Theory of Biaxial Nematic and Cholesteric Phases

“…The inhomogeneity of the director field results in a distortion (elastic) free energy. In a continuum approach this energy is obtained as an expansion about an undistorted reference state with respect to gradients of the tensor order parameter Q [87][88][89] or gradients of the directors [90]. There are many equivalent forms of the biaxial elastic free energy and it is possible to transform one to another after making use of relations that follow from constraints imposed by the orthonormality of the directors.…”