Ericksen theory

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... 

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... 

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... 

Figure 10.14 Measured shear viscosity n against strain y for 8CB at 36.6°C (3.2°C below Tni) and a shear rate of y — 16 seer, fit by Eqs. (10-29) and (10-31) of the Ericksen theory with the director confined to the deforraation plane. (From Gu and Jamieson, reprinted with permission from J. Rheol. 38 555, Copyright 1994, American Institute of Physics.)...

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... 

As the shear rate increases, the numerical solutions of the Smoluchowski equation (11-3) begin to show deviations from the predictions of the simple Ericksen theory. Tn particular, the scalar order parameter S begins to oscillate during the tumbling motion of the director (for a discussion of tumbling, see Sections 11.4.4 and 10.2.6). The maxima in the order parameter occur when the director is in the first and third quadrants of the deformation plane i.e., 0 < 9 — nn < it j2, where n is an integer. Minima of S occur in the second and fourth quadrants. The amplitude of the oscillations in S increases as y increases, until S is reduced to only 0.25 or so over part of the tumbling cycle. 

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. 

Leslie recognized from early experiments that the anisotropy of the materials calls for multiple viscosity coefficients corresponding to different orientation of the LC relative to the flow. Combining this idea with the Ericksen theory leads to the Leslie-Ericksen (LE) theory, which comprises two elements one describing the evolution of n(r) in a flow field, and the other prescribing an extra stress tensor due to the evolving (r) field. 

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...

Doi developed a constitutive theory for liquid crystalline polymers that takes into account the rotational diffusion of the large rodUke molecules and reduces to the LesUe-Ericksen theory for slow deformation rates, and this has been generalized by Marrucci and Greco and others to incorporate a nematic potential that accounts for gradients in the orientation tensor. The theory has a structure that is... 

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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