Ericksen-Leslie theory

In a flowing liquid crystal, both the viscous stresses and Frank elastic stresses are normally important. Thus, the Ericksen theory for the viscous stresses, must somehow be combined with the Frank theory for the elastic stresses. This was accomplished by Leslie, who 

Relationships between the Leslie viscosities (the a s) and Ericksen viscosities (the /x s) can be found in Eq. (11-26). 

N is the rotation rate of n relative to that of the background fluid. If the director field is uniform, Eq. (10-3) can be used to express N in terms of D and n, and then Eq. (10-10) reduces to Ericksen s equation (10-2). 

However, if the director field is not uniform, Frank distortional stresses influence the rate of rotation of the director, and a new equation for h (or, equivalently for N) is required to replace Ericksen s equation (10-3). This is obtained from the balance of angular momentum, which gives 

The vector h is the so-called molecular field, it is produced by the same director gradients that produce the distortional energy Wd- Specifically, 

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Another peculiar property of LCPs is shown in Fig. 15.47, where the transient behaviour of the shear stress after start up of steady shear flow is shown for Vectra A900 at 290 °C at two shear rates. We will come back to this behaviour in Chap. 16 for lyotropic systems where this behaviour is quite common and in contradistinction to the transient behaviour of conventional polymers, as presented in Fig. 15.9. This damped oscillatory behaviour is also found for simple rheological models as the Jeffreys model (Te Nijenhuis 2005) and according to Burghardt and Fuller, it is explicable by the classic Leslie-Ericksen theory for the flow of liquid crystals, which tumble, rather than align, in shear flow. Moreover, it is extra complicated due to the interaction between the tumbling of the molecules and the evolving defect density (polynomial structure) of the LCP, which become finer, at start up, or coarser, after cessation of flow. 

The three elastic constants are the Frank elastic constants, called after Frank, who introduced them already in 1958. They originate from the deformation of the director field as shown in Fig. 15.52. A continuous small deformation of an oriented material can be distinguished into three basis distortions splay, twist and bend distortions They are required to describe the resistance offered by the nematic phase to orientational distortions. As an example, values for Miesowicz viscosities and Frank elastic constants are presented in Table 15.10. It should be mentioned that those material constants are not known for many LCs and LCPs. Nevertheless, they have to be substituted in specific rheological constitutive equations in order to describe the rheological peculiarities of LCPs. Accordingly, the viscosity and the dynamic moduli will be functions of the Miesowicz viscosities and/or the Frank elastic constants. Several theories have been presented that are more or less able to explain the rheological peculiarities. Well-known are the Leslie-Ericksen theory and the Larson-Doi theory. It is far beyond the scope of this book to go into detail of these theories. The reader is referred to, e.g. Aciemo and Collyer (General References, 1996). 

The steady state is reached after several oscillations and the time of the minima and maxima may be scaled by qt, where q is the constant shear rate. As already said in Chap. 15, this behaviour is according to Burghardt and Fuller explicable by the classic Leslie-Ericksen theory for the flow of liquid crystals, which tumble, rather than align, in shear flow. Again it is far beyond the scope of this book to go into detail of this theory. 

Latent heat of fusion (crystallisation), 118 Layer thickness, 698, 699 Length of folds in crystal lamellae, 727 Lennard-Jones equation, 658 scaling factors, 658 temperature, 658,661, 662,663 Leslie-Ericksen theory, 585, 587, 641 Leuco-emeraldine, 345,346 Lewis... 

The total stress tensor in the Leslie-Ericksen theory is the sum of the viscous stress of Eq. (10-10), an isotropic pressure, and the Frank distortional stress, given by... 

As Er is increased further to around 10 in 8CB (at 37°C), there is a roll-cell instability involving (a) a periodic modulation of the director field in the vorticity direction and (b) a cellular flow. The rolls cells are parallel to the primary flow direction (Pieranski and Guyon 1974) (see Fig. 10-19). These transitions in the director field have been both predicted from the Leslie-Ericksen theory (Manneville and Dubois-Violette 1976 Larson 1993) and... 

The first normal stress difference exhibits a linear dependency on the shear rate in the region of constant viscosity for the two solutions in Figure 2. This proportionality is predicted by the Doi theory (10) and the Leslie-Ericksen theory (111 although the basic assumption in these theories, i.e. a monodomain structure, is not satisfied. 

The tendency of LCs to resist and recover from distortion to their orientation field bears clear analogy to the tendency of elastic solids to resist and recover from distortion of their shape (strain). Based on this idea, Oseen, Zocher, and Frank established a linear theory for the distortional elasticity of LCs. Ericksen incorporated this into hydrostatic and hydrodynamic theories for nematics, which were further augmented by Leslie with constitutive equations. The Leslie-Ericksen theory has been the most widely used LC flow theory to date. 

Predictions of the Leslie-Ericksen Theory for Shear Flows... 

The first attempt at a theory of negative which we are aware of (we do not dignify our simple picture presented in [17] as a theory ) was a letter from P.K. Currie communicated to us by K.F. Wissbrun (December 14,1979). Currie s analysis was based on the Leslie-Ericksen theory for MLC nematics and is of debatable relevance to polymeric liquid crystals. Nevertheless, he does conclude that a negative is possible and would occur for a narrow range of boundary orientations. This analysis is available to interested parties from G. Kiss. In MLC nematics, may change from negative to positive as the shear rate increases [75]. 

Figure 4. Apparent viscosity in Poiseuille flow of p-azoxyanisole with perpendicular wall orientation. Points experimental, from data in Fig 2. Lines calculated from leslie-Ericksen theory. Note that abscissa is wall shear rate times square of radius (after Tseng, Silver, and...

The Leslie-Ericksen theory for flow of nematics is a continuum theory which considers the coupling between velocity field and director field. Details about this important theory are presented in Vertogen and de Jeu (1988). 

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]. 

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