Rheology in Region III The Effect of Molecular Elasticity

The term director is usually only defined in the limit of vanishing shear rate where the order parameter tensor has uniaxial symmetry. Even at finite shear rates, however, the order parameter tensor S remains well-defined it can be represented pictorially by an ellipsoidal shape, whose major axis is the largest eigenvalue of S. If, for example, the major axis of S lies in the x y plane, then its angle 9 measured clockwise with respect to the x direction is given by 

If we now generalize the definition of director to be the unit vector parallel to the major axis of the order parameter tensor, we find that at vanishingly low shear rates, where the order parameter tensor is nearly uniaxial, this definition reduces to the usual meaning of the term director.  

As the shear rate increases, the numerical solutions of the Smoluchowski equation (11-3) begin to show deviations from the predictions of the simple Ericksen theory. Tn particular, the scalar order parameter S begins to oscillate during the tumbling motion of the director (for a discussion of tumbling, see Sections 11.4.4 and 10.2.6). The maxima in the order parameter occur when the director is in the first and third quadrants of the deformation plane i.e., 0 9 — nn it j2, where n is an integer. Minima of S occur in the second and fourth quadrants. The amplitude of the oscillations in S increases as y increases, until S is reduced to only 0.25 or so over part of the tumbling cycle. 

As y is increased still further, the tumbling behavior of the director is replaced by a wagging motion wherein the director oscillates back and forth about the flow direction. The frequency of this wagging motion increases with further increases in y and after another transitional shear rate is exceeded, the wagging motion damps out and a steady-state is attained in both 5 and in the director orientation (Larson 1990). 

This increase in birefringence is qualitatively predicted by the molecular theory but the magnitude of the observed increase is considerably greater than predicted (Hongladarom et al. 1993). 

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