Leslie-Ericksen-Parody theory

Within the framework of the Ericksen, Leslie, and Parodi theory one can obtain the torque acting on a sheared molecule. 

The molecular approach which we will see eventually proved to be most successful in treating negative is based on the work of Doi [23]. Doi noted that the well established phenomenological theories for thermotropes (which he termed TLP for Ericksen, Leslie and Parodi [68]) which is successful in describing many dynamic phenomena in MLC nematics, is limited for polymeric liquid crystals in that it does not predict nonlinear viscoelasticity. Doi s approach determines the phenomenological coefficients from molecular parameters, so that the effects of, for example, molecular weight and concentration can be treated. He considers a single molecule (the test rod ) and notes that as concentration increases, constraints on its motion are imposed by collisions with other rods. This constraint can be modeled as a tube... 

Using again the theory of Ericksen, Leslie, and Parodi together with the momentum conservation we obtain the differential equations that govern the steady state of the system... 

The hydrodynamic theory for uniaxial nematic liquid crystals was developed around 1968 by Leslie [10, 11] and Ericksen [12, 13] (Leslie-Ericksen theory, LE theory). An introduction into this theory is presented by F. M. Leslie (see Chap. Ill, Sec. 1 of this Volume). In 1970 Parodi [14] showed that there are only five independent coefficients among the six coefficients of the original LE theory. This LEP theory has been tested in numerous experiments and has been proved to be valid between the same limits as the Navier-Stokes theory. An alternative derivation of the stress tensor was given by Vertogen [15]. 

The macroscopic theory that takes into accoimt the effect of the orientation order was developed by Ericksen, Leslie and Parodi, and usually is referred as ELP theory. A microscopic theory based on correlation functions, which then were "translated" to macroscopic terms and extended to other mesomorphic phases, was developed by the Harvard group. Although usually tire ELP flieory is accepted, it seems that the two approaches are equivalent. A continuum theory of biaxial nematics was developed by Saupe, who followed the description we give with (4.1)-(4.8). In the uniaxial situation, they reproduce the Leslie-Ericksen and Harvard theories. 

The general theory of shear flow normal to the helical axis has been discussed by Leslie. This basically uses concepts similar to those described by the Ericksen-Leslie-Parodi theory of nematics. The main difference is that the twist term of the deformation free energy will be ... 

The six viscosities in equation (4.161) to (4.164) and (4.166) can now all be given in terms of the five independent Leslie viscosities arising in the Ericksen-Leslie dynamic theory, assuming the widely accepted Parodi relation (4.96) holds. Hence only five of these viscosities are needed to form a canonical set of viscosities. 

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