Tumbling and Flow-Aligning

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)  

Nearly exact numerical solutions of the Smoluchowski equation show that for the Maier-Saupe potential, A 1 when S = S2 0.524. For the Onsager potential, A 1 for all values of the order parameter within the nematic range. Values of A for the Onsager potential are plotted in Fig. 11-18. 

Direct experimental determinations of the sign of a2/a in LCPs by optical conoscopy or similar direct methods are rare, mainly because the optical observations must be made 

For PEG, Eq. (11-17) gives the values of k plotted in Fig. 11-18. These values are in reasonable agreement with the predictions of the approximate solution of the Smoluchowski equation. 

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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. 

As shown in Fig. 9, the stress relaxation curves of all AEC/AA solutions collapse into one curve when the solutions were presheared with the same rate. Because the stress relaxation is at the molecular level and the chiro-optical properties reflect the suprastructural level, it is expected that the lyotropic solutions with different chiro-optical properties have the same stress relaxation behavior in both the tumbling and flow-align regions. 

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