Miesowicz coefficients

Note 1 The three Miesowicz coefficients (//i, 772, and 773) describe the shear flow of a nematic phase with three different director orientations, (see Fig. 31) namely 771 for the director parallel to the shear-flow axis 772 for the director parallel to the velocity gradient and 773 for the director perpendicular to the shear flow and to the velocity gradient. 

Note 3 The Miesowicz coefficients are related to the Leslie-Ericksen coejficients by the following equations ... 

Figure 6.12. Miesowicz coefficients of liquid crystalline polymers against order parameter. (From Doi and Edwards, 1986.)...

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 ... 

These three configurations have been investigated by Miesowicz, and the rj s are known as Miesowicz coefficients. In the original paper, as well as in the treatment by deGennes, the definitions of r i and ri3 are interchanged. In deGennes notation, in terms of and we have r a = = ri2, and = ri3. The notation used... 

The above three viscous coefficients are defined as the Miesowicz viscous coefficients. 

In reality, it is difficult to measure the six Leslie coefficients. However, it may be easier to measure the Miesowicz viscous coefficients which are related to the order parameter and rj... 

The relations are shown in Figure 6.12 where the Miesowicz viscous coefficients are plotted as a function of the order parameter of long rods in solution. The Doi and Edwards theory gives the order parameter vs. the concentration shown in Figure 6.13, in which the order parameter starts at... 

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 ... 

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 ... 

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. 

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. 

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 ... 

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] ... 

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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