Tumbling parameter

If the molecular field h is zero, then Eq. (10-13) can be reduced to Ericksen s equation (10-3), where the tumbling parameter X is now given by... 

In Eqs. (10-20), the six Leslie viscosities are given in terms of the characteristic viscosities 1 and ao (described below), the tumbling parameter k, the second and fourth moments... 

The tumbling parameter X can be obtained by a numerical solution of the Smoluchowski equation, or analytically using the following simple approximate form (Stepanov 1983 -Kroger and Sellers 1995 Archer and Larson 1995) ... 

Figure 10.10 Tumbling parameter X versus reduced temperature Tr = T/Tj i for the various liquid crystals listed. The broken line is the exact prediction of the Smoluchowski equation for large aspect ratio p, and the solid and dot-dashed lines are from the approximate expression (10-24) with p large and p = 5. (From Archer and Larson, reprinted with permission from J. Chem. Phys. 103 3108, Copyright 1995, American Institute of Physics.)...

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) ... 

Figure 11.18 Predictions of the tumbling parameter A as a function of reduced concentration C/C2 from the Smoluchowski equation for hard rods with the Onsager potential. The exact result from the spherical-harmonic expansion is shown, compared to approximate results from an analytic formula and from the perturbation expansion of Kuzuu and Doi. The open circles (O) are estimates from the periods of shear stress oscillations in transient shearing flows for PEG solutions (see Walker et al. 1995), and the closed circle ( ) is from a direct conoscopic measurement of Muller et al. (1994).

The quantity K(a) is near unity for the physical range of a. The tumbling parameter, A, in a simple sheering flow is related to the flow alignment, or Leslie angle 6 by... 

Figure 3. The tumbling parameter as function of the temperature or concentration variable d for Ak = 1.45,1.25 and k = 0 (upper and lower thin curves) as well as Ak = 1-25,1.05 and = 0.4 (upper and lower thick curves). The dashed horizontal line marks the limit between the flow aligned (Aeql) and the tumbling (Aeq < 1)...

A stable flow alignment, at small shear rates, exists for Aeq l only. For Aeq < 1 tumbling and an even more complex time dependent behavior of the orientation occur. The quantity Aeq - 1 can change sign as function of the variable cf. Fig. 4. For Aeq < 1 and in the limit of small shear rates 7, the Jeffrey tumbling period [18] is related to the Ericksen-Leslie tumbling parameter Agq by... 

Figure 16. The shear stress versus the shear rate for Ak = 1.25 aad

Director tumbling is predicted from molecular theories of rigid rod systems subject to shear flow and depends critically on the characteristics of the microscopic order parameters S2 and S4. The equation below shows the relationship for the tumbling parameter, p, proposed by Larson (20). 

When comparing the values of constitutive parameters found for Titan and Zenith 600, one should recall that Titan is a random copolyester of ethylene-terephthalate and hydroxybenzoic acid with two methylene flexible spacers, whereas Zenith 6000 is a fully aromatic copolyester with kinks. Therefore, Zenith 6000 is expected to be much more rigid than Titan. Our simulations confirmed this. Th demonstrate that the values of viscoelastic parameters Go and t)o, parameter of viscous tumbling ky. and anisotropy parameters a, ri, and T2 found for Titan are smaller than those of Zenith 6000, while the values of elastic tumbling parameter and the anisotropic parameter p for Titan are greater than those for Zenith 6000. 

The Larson-Doi theory predicts qualitatively the essential features of transient shear flow of some model TLCPs (Ugaz 1999). It is, however, not clear to what extent the Larson-Doi theory can describe the dynamics of TLCPs that do not exhibit tumbling. This is because the Larson-Doi theory is based on the Ericksen-Leslie theory, which determines structural responses through the tumbling parameter A,. As will be presented later in this chapter, the experimental data available to date suggest that TLCPs are flow aligning. It is fair to state that the theoretical attempts reported thus far contain, understandably, many crude approximations, and so do not warrant quantitative comparison with experimental results for textured LCPs, particularly TLCPs. Thus the development of a molecular viscoelastic theory for textured LCPs is still in its infancy. 

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