Effect of long range order on sheared liquid crystalline materials Part 1: compatibility between tumbling behavior and fixed anchoring

Abstract

A viscoelastic constitutive equation for liquid crystalline flows is presented in terms of tensor order parameter dynamics. The presented equation takes into account short range order, long range order (Frank elasticity), and flow effects, and captures the spatio-temporal evolution of the tensor order parameter. The dimensionless parameters arising in the theory are the Deborah (De) number and Ericksen (Er) number, and their ratio is a new dimensionless number R which represent the ratio of short range elasticity to long range elasticity. At Er → ∞ the model converges to the Doi equation, and for De → 0 to the Leslie-Ericksen director equation; the consistency with the Doi theory is shown by the classical shear flow solutions: tumbling, oscillating, and steady state. The model is used to analyze rectilinear simple shear flow subjected to strong anchoring conditions. For the range of governing parameters studied in this paper it is found that for small Er, a spatially nonhomogenous steady state of the tensor order parameters exists. For De corresponding to the tumbling solutions of the Doi equation, the present theory predicts the presence of a composite three layer structure: (a) a center bulk region characterized by director tumbling; and (b) two director oscillating regions next to the bounding surfaces. The amplitude of the director oscillations decrease from a maximum at the boundaries between tumbling-oscillating modes to zero at the bounding surfaces. The boundary between the tumbling-oscillating mode is characterized by the time periodic emergence of an abnormal nematic state, which is characterized by a tensor order parameter with two equal eigenvalues on the shear plane. The emergence of the abnormal nematic state provides for a compatibility mechanism between oscillations and rotations. Thus, we have found through the use of computational modeling a new mechanism that explains the compatibility of Doi's tumbling solutions and fixed surface anchoring conditions. The new flow mechanism is independent of the type of strong anchoring conditions, and arises with planar, homeotropic, and any arbitrary director anchoring.