Leslie viscosity parameters

FIGURE 2 Canonical sets of (left) viscosity coefficiaits and (right) Leslie viscosity parameters of MBBA in units of Pa s [23,25]. 

The nematic liquid crystal is orientationally soft, since restoring forces associated with deformation in the director field are very weak. This softness makes alignment of n in bulk samples occur even in very weak external magnetic or electric fields, F s H or E, or by interaction with boundary surfaces and flows in the liquid. This softness also allows for long wavelength thermal fluctuations in the director field. The Leslie viscosity parameters rather than the viscosity coefficients are the more natural quantities of interest for those methods that monitor the viscoelastic response of the nematic to director field modulations. Modulation of n in space and time manifests itself in variations of many bulk properties, e.g. the refractive index [27-37,41-44,48,51,94-106], electric susceptibility [38,39,107-110], or magnetic resonance spectra [40,45-47,111-113]. However, only a limited number of the viscosity parameters/coefficients can be precisely determined by these methods. 

When the deforming field is rapidly switched on and off, the transient behaviour of n that follows is determined by the viscoelastic properties of the sample, the boundary conditions, and the initial and final states of the director pattern. Such experiments typically provide the most reliable information on the rotational viscosity coefficient. In order to model transient behaviour in a particular geometry a set of the Leslie equations of motion is solved. This solution gives the time evolution of the azimuthal, 9(t,r), and polar, S(t,r), angles describing the orientation of n with respect to some reference frame at any given arbitrary position r in the sample. These functions are parametrised by the Leslie viscosity parameters and the elasticity constants. 

Xe (fi 0 is directly related to the director pattern variations in space and time, in an analogous way to that of NMR [113]. Extraction of the Leslie viscosity parameters requires numerical solution of the nematic equations of motion, calculating XE(fit>z)> integrating over the sample volume to get Xe> and fitting the latter to the experimentally observed transient behaviour. Fitted paramet include the set of the Leslie parameters [109]. 

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... 

Formulas for the Leslie viscosities, in turn, were derived from the Smoluchowski equation for hard rods by Kuzuu and Doi (1983,1984 Semenov 1987), and are given in Eqs. (10-20) with ao = 0- These formulas require as inputs values of 2,54, k, and Dr, which are functions of polymer concentration C. Reasonably reliable analytic functions for these dependencies were obtained by Kuzuu and Doi using a perturbation expansion for large order parameter, yielding... 

Few other sets of viscosities exist for polymeric nematics. Yang and Shine (1993) obtained three of the Leslie viscosities for monodomains of poly(n-hexyl isocyanate) (PHIC) from rheological measurements in the presence of an electric field, and they obtained values reasonably consistent with the predictions of the Kuzuu-Doi expressions. From monodomains of the polyion PBZT, poly(l,4-phenylene-2,6-benzobisthiazole) in methane sulfonic acid, some of the Leslie-Ericksen parameters have been extracted via light-scattering and magnetic-field-reorientation studies (Berry 198S Srinivasarao and... 

Using scattering and spectroscopy experiments, it has been shown that the physical quantities characterizing the wormlike micellar nematics such as the order parameter, Leslie viscosities ratio, or alignment angles can be determined. The main result of this section is the analogy between the wormlike micelles and the liquid-crystalline polymers, as far as their nematic states are concerned. Because these... 

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]. 

To describe the hydrotfynamics of the nematic phase five independent viscosity parameters (dynamic behaviour) and three elastic constants (static bdiaviour) are necessary [10-17]. These viscosity parameters cannot be identified with the expoimoital coefficioits directly, but with cotain linear combinations of them. Following the most widely accqrted nonatodynamics of Leslie [4] we have ... 

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) ... 

The detail of the Doi theory can be refered to in the book by Doi and Edwards The Theory of Polymer Dynamics (1986) in which the Leslie coefficients are related to the order parameter and the isotropic viscosity fj as follows... 

It would be very instructive to relate the experimental (Miesowicz) and theoretical (Leslie) coefficients of viscosity. Our task now is to use the viscous tensor (9.20) and find the relationships between the coefficients for each of the three basic orientations of the director, namely = 1, iiy = 1, or = 1. At first, we shall prepare some combinations of parameters useful in all the geometries mentioned ... 

The shear viscosity coefficients t], 1)2, V3 and 77i2 and the rotational viscosity coefficient 7i form a complete set of independent coefficients from which the Leslie coefficients can be determined with the help of the Parodi equation. The corresponding equations are given in Chap. VII, Sec. 8.1 of Vol. 1. Figure 24 [74] shows the Leslie coefficients for MBBA as a function of temperature. Due to the different dependence on the order parameter (see Chap. VII, Sec. 8.1 of Vol. 1 of this Handbook), the coefficients exhibit different bending above the clearing point. The temperature dependence of differs greatly from that of the other coefficients, as it is not a real viscosity. 

Apart from basic viscosity experiments, there is an increasing number of contemporary methods monitoring the director field modulations, i.e. the splay, twist and bend deformations [19]. Flows related to these deformations are characterised by viscosity coefficients rispiay, r tvrist> and ribeod respectively. These viscosities can be conveniently expressed via combinations of the Miesowicz viscosity coefficients and/or the Leslie parameters [20,21,26] ... 

---