Tumbling nematic

Fig. 2.8.16 Director orientation, 0, as a function of shear rate for both flow aligning (solid squares) and tumbling (open squares 325 K, solid circles 328 K and open circles 333 K) nematic polymers. (From Siebert et al. [10].)...

Larson s results [154] are divided into the three shear rate regimes - tumbling, wagging, and steady-state - as explained below. He chose the strength of mean-field potential 2L2dc in Eq. (41) to be 10.67, which corresponds to the concentration cA of the nematic phase coexisting with the isotropic phase (in the second virial approximation), and expressed the shear rate in terms of T defined by... 

Tumbling regime At very low shear rates, the birefringence axis (or the director) of the nematic solution tumbles continuously up to a reduced shear rate T < 9.5. While the time for complete rotation stays approximately equal to that calculated from Eq. (85), the scalar order parameter S,dy) oscillates around its equilibrium value S. Maximum positive departures of S(dy) from S occur at 0 n/4 and — 3n/4, and maximum negative departures at 0 x — k/4 and — 5it/4, while the amplitude of oscillation increases with increasing T. 

Note that the period is inversely proportional to shear rate y hence, the strain period Py is independent of shear rate. When A < 1 the nematic is called a tumbling nematic, while when A > 1, the nematic is flow-aligning. As discussed in Sections 10.2.5 and 10.2.6, both cases (tumbling and flow-aligning) can occur in small-molecule liquid crystals. 

Figure 10.15 Damping of shear stress and reversal of damping as a function of time after startup of shearing in a cone-and-plate rheometer at y = 8 seer for a tumbling nematic, 8CB. In (a) the shearing direction is reversed after imposition of 170 strain units the damping of the stress nscillatinns is rp.vp.rsed. In (h), shear reversal occurs after imposition of 480 strain iiniK fhp damping is not reversed. (From Gu et al. 1993, with permission from the Journal of Rheology.)...

If a homeotropically aligned tumbling nematic is sheared at small Er = yxVh/K, then the director at the midplane of the sample rotates modestly toward the flow direction, until it reaches a steady state where the viscous forces driving the director rotation are balanced by the Frank stresses (see Fig. 10-16 at Er 1). As Er increases in small increments, so that a steady state is attained between each increment in Er, the steady-state director at the midplane is rotated to a greater and greater extent (see Fig. 10-16 at Er 10). 

Figure 10.19 Velocity field (a) and director pattern (b) in roll cells that form in a tumbling nematic initially oriented in the vorticity direction of a shearing flow. (From Larson 1993, with permission from the Journal of Rheology.)...

Problem 10.3(a) (Worked Example) From the Leslie-Ericksen equations at high shear rate, derive Eq. (10-29) for the director angle 6 in a shearing flow of a tumbling nematic. 

From slow-shear-rate solutions of the Smoluchowski equation, Eq. (11-3), with the Onsager potential, Semenov (1987) and Kuzuu and Doi (1983, 1984) computed the theoretical Leslie-Ericksen viscosities. They predicted that ai/a2 < 0 (i.e., tumbling behavior) for all concentrations in the nematic state. The ratio jai is directly related to the tumbling parameter X by X = (1 -h a3/a2)/(l — aj/aa). Note the tumbling parameter X is not to be confused with the persistence length Xp.) Thus, X < I whenever ai/a2 < 0. As discussed in Section 10.2.4.1, an approximate solution of Eq. (11-3) predicts that for long, thin, stiff molecules, X is related to the second and fourth moments Sa and S4 of the molecular orientational distribution function (Stepanov 1983 Kroger and Sellers 1995 Archer and Larson 1995) ... 

Most of the discussion in this section on Region III rheology has focused on HPC and PBG solutions, because for these LCPs, Region III behavior can clearly be identified. Many other LCPs, lyotropic and thermotropic, show no steady-state negative N, perhaps because they are not tumbling nematics or because for them Region III is not reached even at the... 

D. F. Gu and A. M. Jamieson, Shear Deformation of Homeotropic Monodomains Temperature Dependence of Stress Response for Flow-Aligning and Tumbling Nematics, J. Rheol. 38 (1994) 555 ... 

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