Viscosity coefficients Miesowicz

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] ... 

Meyer (1982) predicted that as the length of the chain approaches infinity, the Leslie coefficients ai and 2 should tend to —oo and a3 tends to oo, while 04,05 and 06 are of finite values. The Miesowicz viscosities r/a and rjb are finite while r/c tends to infinity because the velocity is perpendicular to the director and the shear flow. 

We shall now discuss the application of the Ericksen-Leslie theory to some practical problems in viscometry. Probably the first precise determination of the anisotropic viscosity of a nematic liquid crystal was by Miesowicz. He oriented the sample by applying a strong magnetic field and measured the viscosity coefficients in the following three geometries using an oscillating plate viscometer ... 

With the Miesowicz technique one can measure three combinations of the Leslie viscosity coefficients from Eqs. (9.25) to (9.27). On account of the Parodi relationship, to find all five coefficients, one needs, at least, two additional measurements. In particular, the ratio of coefficients a3/a2 can be measured by observation of the director field distortion due to capillary flow of a nematic. The last combination yi = as — as can be found from the dynamics of director relaxation. 

Several notations for the shear viscosity coefficients are used. The notation used here stems from Helfrich [25]. A different notation was proposed by Miesowicz [26]. 

Viscosity coefficients measured in these geometries when n is immobilised by boMiesowicz viscosities. (Note, that in the literature a variety of alternative notations are common in particular the definitions of r i and r 2 are frequently interchanged.) If the orientation of n is fixed in an arbitrary direction with respect to v and Vv, then the effective viscosity coefficient is given by a linear combination of the Miesowicz viscosities, and another viscosity constant Tju, which cannot be visualised in a pure shear-flow ... 

Miesowicz distinguished three principal viscosity coefficients, 7/1,772 and 773, that could be independently measured experimentally by considering the orientation of the director n in relation to the flow velocity v. The three basic flow geometries considered by Miesowicz are depicted in Fig. 4.1 and allow the measurement of the viscosities ... 

So far we have introduced four Miesowicz viscosities. Two other viscosities can be proposed by considering the following. The director n in Fig. 4.1(a), if free to move, will rotate due to a viscous torque the viscosity coefficient 71 is introduced to describe this situation and characterises the torque associated with a rotation of n. For this reason 71 is often called the rotational viscosity or twist viscosity. The coefficient 71 generally determines the rate of relaxation of the director. Also, a rotation of n due to body forces will induce a flow. The viscosity coefficient 72 characterises the contribution to the torque due to a shear velocity gradient in the nematic and is sometimes referred to as the torsion coefficient in the velocity gradient it leads to a coupling between the orientation of the director and shear flow. The two viscosities 71 and 72 have no counterpart in isotropic fluids. We therefore have a total of six viscosities four Miesowicz viscosities plus 71 and 72. It turns out, as will be seen in the problems to be discussed in later Sections, that 7i and 72 are precisely the viscosities introduced in the constitutive theory at equations (4.78) and (4.79), namely. 

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 ... 

In the left part of the figure, the helical axis is parallel to the velocity gradient (shear) shown by two arrows. When cell thickness is less than the cholesteric pitch, d < Pq, and the rate of shear is small, then an effective viscosity is given by averaging two Miesowicz coefficients ... 

Leslie, F.M. Introduction to nematodynamics. In Dunmur, D., Fukuda, A., Luckhurst, G., INSPEC (eds.) Physical Properties of Liquid crystals Nematics, pp. 377-386, London (2001). Parodi, O. Stress tensor for nematic liquid crystals. J. Phys. (Paris) 31, 581-584 (1970) Miesowicz, M. The three coefficients of viscosity of anisotropic liquids. Nature 158, 27 (1946) Influence of the magnetic field on the viscosity of para-azoxyanisole. Nature 136, 261 (1936). 

Miesowicz M., (1946). The Three Coefficients of Viscosity of Anisotropic Liquids, Nature Vol. 27, pp. 158. 

A technique pioneoed by Miesowicz is based on the observation of damping of very small oscillations of a thin glass plate cmnpletely immersed in the investigated liquid [5-7]. The oscillation anq>litude. A, decreases with time due to viscous drag on the plate, and the rate of decrease can be related to the viscosity of the liquid. The damping coefficient has the form... 

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