Flow-alignment parameter

Fig. 9 Critical values as functions of the flow alignment parameter X for various viscosities (a, b) and compressibilities (c, d). In the upper row we plot this dependence for a set of (isotropic) viscosities ranging from v, = 1 (thick solid line,) down to V = 10 3 (thick dashed line,). The lower row illustrates the behavior for varying layer compressibility Bo with Bo 3 f°r the thick solid curve and Bo = 100 for the thick dashed curve. In all plots the thin solid lines give the behavior for some intermediate values. For an interpretation of this behavior see the text...

Out of the five viscosities, only two (V2 and V3) show a significant influence on the critical values. In Fig. 8 we present the dependence of 9C and qc on an assumed isotropic viscosity (upper row) and on these two viscosity coefficients (middle and lower row). Since the flow alignment parameter X has a remarkable influence on these curves we have chosen four different values of X in this figure, namely X = 0.7, X = 1.1, X = 2, and X = 3.5. The curves for X < 1 and X > 3 for an isotropic viscosity tensor are very similar to the corresponding curves where only V2 is varied. In this parameter range the coefficient V2 dominates the behavior. Note that the influence of V3 on the critical values is already much smaller than that of V2. We left out the equivalent graphs for the other viscosity coefficients, because they have almost no effect on the critical values. For further comments on the influence of an anisotropic viscosity tensor see also Sect. 3.4. 

All the parameters we have discussed up to now caused variations in the critical values that did not select specific values of the considered parameter. In this aspect the situation is completely different in the case of the flow alignment parameter X. As shown in Fig. 9 there is a clear change in behavior for X 1 and X 3. The... 

Note that the flow-alignment parameter A = -72/71 is a reversible quantity. For A > 1 the director can align in the flow plane at a fixed angle p with respect to the velocity (x-axis), where = (A-1)/(A + 1) = a3/ 2. For A < 1 there is tumbling [73],... 

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... 

Survey. Next, results are presented for the time-averaged rheological behavior and the time averaged alignment in the nematic phase at a state point where no stable flow alignment is possible. In particular, the temperature = 0 and the parameters Ak = 1.25 and k = 0 are chosen. In Fig. 8 the shear stress and the viscosity are displayed as functions... 

Figure 17. The first and the zz normal stress differences (top row), the flow alignment angle and the in-plane alignment (bottom row) as fimctions of the shear rate for Ak = 1.05 and k = 0.4, at the temperature i = 0. The tumbling parameter is Aeq = 0.833. The total run time t corresponds to the shear deformation -yt = 1500.

Fig. 29 Upper panels. Neutron scattering intensities obtained from a CPCl/Hex nematic micellar solution in the (qv,qvv)-plane at different shear rates (a) j =0.94 s , (b) y=34.5 s ,and (c) 7= 250 s . The nematic phase was made in deuterated water at concentration c = 35.2wt.% and molar ratio [Hex]/[CPC1] = 0.49 [301], Lower panel. Shear rate dependence of the orientational order parameter y (Eq. 5) obtained from the SANS cross-sections shown in the upper panel. The continuous line is a guide for the eyes. The increase of the order parameter was interpreted in terms of a transition between the tumbling and the flow-alignment regimes [290,303]...

An alternative to reorientation of the sample or the magnetic field is the application of shear during the NMR measurement [130-134]. For liquid-crystalline samples with high viscosity, such as liquid crystal polymers, the steady-state director orientation is governed by the competition between magnetic and hydrodynamic torques. Deuteron NMR can be used to measure the director orientation as a function of the applied shear rate and to determine two Leslie coefficients, and aj, of nematic polymers [131,134]. With this experiment, flow-aligning and tumbling nematics can be discriminated. Simultaneous measurement of the apparent shear viscosity as a function of the shear rate makes it possible to determine two more independent viscosity parameters [131, 134]. 

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. 

As demonstrated, Eq. (7) gives complete information on how the weight fraction influences the blend viscosity by taking into account the critical stress ratio A, the viscosity ratio 8, and a parameter K, which involves the influences of the phenomenological interface slip factor a or ao, the interlayer number m, and the d/Ro ratio. It was also assumed in introducing this function that (1) the TLCP phase is well dispersed, fibrillated, aligned, and just forms one interlayer (2) there is no elastic effect (3) there is no phase inversion of any kind (4) A < 1.0 and (5) a steady-state capillary flow under a constant pressure or a constant wall shear stress. 

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