Molecular and Gradient Elasticities

Because nematic liquid-crystalline polymers by definition are both anisotropic and polymeric, they show elastic effects of at least two different kinds. They have director gradient elasticity because they are nematic, and they have molecular elasticity because they are polymeric. As discussed in Section 10.2.2, Frank gradient elastic forces are produced when flow creates inhomogeneities or gradients in the continuum director field. Molecular elasticity, on the other hand, is generated when the flow is strong enough to shift the molecular order parameter S = S2 from its equilibrium value 5 . (Microcrystallites, if present, can produce a third type of elasticity see Section 11.3.6.) 

Significant shifts in S are not expected to occur in small-molecule nematics unless the shear rate is extraordinarily high. For polymeric nematics, however, molecular relaxation times T are typically 0.001-10 sec, or even higher, and therefore molecular elastic effects are produced at shear rates y r = 0.1-1000 sec L Thus, the order parameter S is significantly distorted away from that of equilibrium when the Deborah number De (discussed in Section 3.6.2.1.1) is of order unity or greater, where 

The relaxation time r is given by r = ij6Dr, where Dy is the rotational diffusivity of the molecules in the nematic phase. 

The dimensionless Ericksen number that characterizes the strength of the viscous forces compared to the Frank gradient elastic forces is defined in Section 10.2.5 [Eq. (10-27)] as 

When De 1 and molecular elasticity is negligible, the flow properties of polymeric nematics can, in principle, be described by the Leslie-Ericksen equations (see Section 10.2.3). However, at moderate and high De, the Leslie-Ericksen continuum theory fails, and a molecular theory is required to describe the effect of flow on the distribution of molecular orientations. 

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