Further Theoretical Proofs for the Existence of a Relation Between the Elastic Constants K'₁₁ (K₁₁), (K'₃₃)(K₃₃) and K₁₃ in Nematics: Three- and One-Dimensional Cases

Abstract

The elaboration of the Ericksen's result for the inequalities in nematics has confirmed the proportionality between the second-order splay-bend elastic constant K₁₃ and the difference between the elastic constants of bend K'₃₃ and splay K'₁₁ as well as the sign of K₁₃ in accordance with the relation 2K₁₃ = K'₃₃ - K'₁₁ obtained previously (Mol. Cryst. Liq. Cryst., 168, 7, 1989). On the basis of detailed theoretical calculations for the one-dimensional case it is demonstrated that this relation follows from the require- ment for a positive value of the elastic energy expressed with extremals and gradients of extremals when all possible symmetric and antisymmetric solutions are included.

Key Words:

• second-order splay-bend elasticity
• relation between K13
• K'33 and K'11 extremals
• inequalities in nematics