Non-local q-tensor approach for description of elastic deformations of nematic liquid crystals at sub-micron scale

Within the framework of phenomenological approach, a generalized concept of a non-local order parameter, that have the form of a traceless correlation tensor function or a tensor integral operator, is introduced. The main relationships, which determine the equilibrium orientational states and the phase transitions of a deformed nematic liquid crystal, are described. The resulting expression for the free energy is quadratic with respect to the gradients of the order parameter tensor. The limiting transition of the non-local order parameter to the local one leads to an expression that complements the classical Oseen-Frank theory of elasticity. The considered model includes eight independent elastic constants. Three of them relate to "bulk" constants analogous to Frank constants, another three ones relate to "surface" constants, and the remaining two constants determine the anisotropic effect of the gradient of the scalar order parameter on the director field. The constants are necessary for the consideration of a liquid crystal state near defects, as well as for the description of orientation effects caused by the inhomogeneity of order parameter. It is shown that in the free energy expression, the "surface" terms contribute to the director orientation in the case of a non-constant scalar order parameter, even in the approximation of rigid boundary conditions.