Leslie-Ericksen equations

Problem 10.3(a) (Worked Example) From the Leslie-Ericksen equations at high shear rate, derive Eq. (10-29) for the director angle 6 in a shearing flow of a tumbling nematic. 

Again, we express n in terms of the angle 9 using Eq. (AlO-22). We need the xy component of the stress tensor given by the Leslie-Ericksen equation, Eq. (10-10). We consider, in turn, the jry component of each term of this equation. We use Eq. (A 10-26) to evaluate the first term of Eq. (10-10) ... 

When De 1 and molecular elasticity is negligible, the flow properties of polymeric nematics can, in principle, be described by the Leslie-Ericksen equations (see Section 10.2.3). However, at moderate and high De, the Leslie-Ericksen continuum theory fails, and a molecular theory is required to describe the effect of flow on the distribution of molecular orientations. 

Inspired by a mesoscopic theory for mixtures of immiscible liquids (Doi and Ohta 1991 see also Section 9.3.3), Larson and Doi (1991) have derived mesoscopic equations for polydomain nematics by multiplying the Leslie-Ericksen equation (10-13) by the director... 

Some general predictions can be made with the aid of the scaling properties [19] of the Leslie-Ericksen equations. Neglecting the molecular inertia, the substitution... 

If the applied magnetic field greatly exceeds the critical strength, the relaxation will be non-exponential. The hyperbola rotation must then be calculated by means of the Leslie-Ericksen equations, and the rotational viscosity Yi is determined by a fit of the calculated hyperbola rotation to the observed one. A disadvantage of this method is that it is difficult to follow the rotation of the interference figure by eye or by means of automatic equipment. 

The six Leslie coefficients i to Og are the material constants in the stress tensor of the Leslie-Ericksen equations (see Chap. VII, Sec. 8.1 of Vol. 1 of this Handbook). The coefficients must be known for any calculation of flow phenomena and director rotations by means of the Leslie-Ericksen equations such as, for example, for the prediction of the transmission curve during the switching of a liquid crystal display. Because of the Parodi equation [90]... 

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

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). 

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... 

From slow-shear-rate solutions of the Smoluchowski equation, Eq. (11-3), with the Onsager potential, Semenov (1987) and Kuzuu and Doi (1983, 1984) computed the theoretical Leslie-Ericksen viscosities. They predicted that ai/a2 < 0 (i.e., tumbling behavior) for all concentrations in the nematic state. The ratio jai is directly related to the tumbling parameter X by X = (1 -h a3/a2)/(l — aj/aa). Note the tumbling parameter X is not to be confused with the persistence length Xp.) Thus, X < I whenever ai/a2 < 0. As discussed in Section 10.2.4.1, an approximate solution of Eq. (11-3) predicts that for long, thin, stiff molecules, X is related to the second and fourth moments Sa and S4 of the molecular orientational distribution function (Stepanov 1983 Kroger and Sellers 1995 Archer and Larson 1995) ... 

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. 

For low-molecular-weight nematics, it has been known that the constitutive equation of nematics is entirely different from that for isotropic liquids. A phenomenological theory for the hydrodynamics of nematics (of low molecular weight) has been constructed by Ericksen and Leslie. Their equation reads... 

Nematic polymers have much in common with conventional, low-mass nematics. Their behavior in an electric field is usually described in the firame-work of the same Leslie-Ericksen approach though, strictly speaking, some corrections must be done in the set of nematodynamic equations to take into account the coupling between the motion of mesogenic units and the backbone of a polymer [228]. The field behavior of thermotropic nematic polymers differs considerably from that of lyotropic solutions of long rodlike molecules (like poly-7-benzyl-glutamate) and the two systems will be discussed separately (see also a recent review article [279]). 

Much research in the last few decades focused on the simulation of LCPs for various processes. Suitable rheological constitutive equations are required for this simulation. Leslie-Ericksen (LE) theory describes the flow behaviour and molecular orientation of many LCPs. LE model is limited to low shear rates and weak molecular distortions. But at high shear rate, the rate of molecular distortions is too fast. Doi and Edwards developed their model to describe the complex dynamics of macromolecules at high shear rate (Doi and Edwards 1978). Doi theory is applicable for lyotropic LCPs of small and moderate concentrations. Due to the complex nature of Doi theory, it is also challenging for simulation. Leonov s continuum theory of weak viscoelastic nematodynamics, developed on the basis of thermodynamics and constitutive relations, consider the nematic viscoelasticity, deformation of molecules as well as evolution of director. 

The Ericksen-Leslie Dynamic Equations and note that this implies... 

It is convenient at this point to summarise the Ericksen-Leslie dynamic equations for nematics in the incompressible isothermal theory when the director inertial term is neglected. These are the most frequently used forms of the equations and we state them in the notation introduced in the previous Sections. They consist of the constraints... 

The linearised version of the Ericksen-Leslie dynamic equations coincides with that discussed by Martin, Parodi and Pershan [192], as mentioned by Leslie [168], with suitable reorganisation of the terms in the equations. 

The constraints in equation (4.118) are clearly satisfied. The governing Ericksen-Leslie dynamic equations (4.119) and (4.120) become, respectively,... 

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