Ericksens Transversely Isotropic Fluid

In the absence of elastic stresses, the stress tensor for a flowing nematic is given by (Ericksen 1960, 1961) 

Since the director n can be influenced by the flow, an additional dynamic equation must 

There are four such steady-state solutions in the (n, 3 ) plane, as depicted in the Ericksen diagram shown in Fig. 10-5. Since nematics are nonpolar (i.e., the head of the director is indistinguishable from the tail ), two of the four solutions shown in Fig. 10-5 are redundant of the remaining two solutions, one is unstable and the other stable. For A 1 and a shear rate y that is positive, the stable solution is the one with a positive sign in Eq. (10-4). Equation (10-4) tells us that if A 1, the stable orientation angle 9 approaches 45°, while if A is near unity, 9 approaches zero that is, the director becomes parallel with the flow direction. 

Note that the normal stress differences Ni and N2 are linear in the shear rate y, as is the shear stress. In contrast, for isotropic viscoelastic fluids, Ni x N2 oc. at low shear rates (see Section 1.4.3). 

If A 1, Eq. (10-4) has no solutions in the shearing plane. This means that the director rotates endlessly in the deformation plane. The period of this rotation—that is, the time it takes to rotate through an angle of tt, is 

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J.L. Ericksen, Transversely Isotropic Fluids, Kolloid-Zeit, 173, 117-122 (1960). 

Comparison with Ericksen s Transversely Isotropic Fluid Theory... 

Ericksen proposed a theory for fluids such as nematic liquid crystals which could become anisotropic during flow. By assuming symmetry around the director, the expression for the stress tensor was somewhat simplified. We compare here briefly Ericksen s transversely isotropic fluid theory with the transient behavior observed for thermotropic copolyesters of PHB/PET. 

Fig. 10. Prediction of the shear stress using Ericksen s transversely isotropic fluid theory. The domains were assumed to be initially oriented along 2-direction (i.e. perpendicular to the shear surface).

The shear stress growth on the inception of shear flow may reflect the orientation of the liquid crystalline domains. Orientation seems to occur within less than 2 strain units in shear flow. This primary normal stress difference can exhibit different phenomena from the shear stress response. In particular for the 60 mole % PHB/PET system, values of N are positive and rise gradually to the equilibrium values whereas the 80 mole % PHB/PET system can exhibit negative values of N. Ericksen s transversely isotropic fluid theory can qualitatively handle some of the observed phenomena. Further studies which couple the transient flow behavior to the orientation and morphology need to be carried out. 

Ericksen (1960) proposed a model for the transversely isotropic fluid (TIF), given by ... 

The first rheological theory for Uquid crystals was developed by LesUe and Ericksen, building on Ericksen s earUer transversely isotropic fluid. The theory is formulated in terms of a director field n, and it is similar to the fiber theory in the preceding section, except that it includes a contribution to the free energy from an interactive potential that causes the molecules to align at rest. The usual form of the free energy F resulting from distortions of the director field is... 

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