Dynamic Theory for Nematics

Continuum theory for nematic liquid crystals has its origins in the 1920s in the work of Oseen [1] and Zocher [2], who largely developed the static theory. The first to attempt the formulation of a dynamic theory was Anzelius [3], who was a student of Oseen, but an acceptable version had to await developments in non-linear continuum mechanics many years later, as well as further experimental studies by Zwetkoff [4] and Miesowicz [5]. A full account of the early development of dynamic theory for nematics can be found in a paper by Carlsson and Leslie [6]. 

The SD is a phase separation process usually occurring in systems consisting of more than two components such as in solutions or blends. However, in the present case the system employed is composed of one component of pure PET. In this case, what triggers such an SD type phase separation Doi et al. [24, 25] proposed a dynamic theory for the isotropic-nematic phase transition for liquid crystalline polymers in which they showed that the orientation process... 

This paper presents summaries of unique new static and dynamic theories for backbone liquid crystalline polymers (LCPs), side-chain LCPs, and combined LCPs [including the first super-strong (SS) LCPs] in multiple smectic-A (SA) LC phases, the nematic (N) phase, and the isotropic (I) liquid phase. These theories are used to predict and explain new results ... 

As discussed in Sec. 2.2.2.1, the foundations of the continuum model were laid by Oseen [61] and Zocher [107] some seventy years ago, and this model was reexamined by Frank [65], who introduced the concept of curvature elasticity to describe the equilibrium free energy. This theory is used, to this day, to determine splay, twist, and bend distortions in nematic materials. The dynamic models or how the director field behaves in changing from one equilibrium state to another have taken much longer to evolve. This is primarily due to the interdependency of the director it (r, t) and v (r, /) fields, which in the case of chiral nematics is made much more complex due to the long-range, spiraling structural correlations. The most widely used dynamic theory for chiral... 

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... 

Couette flow is considered in Section 5.7 and Poiseuille flow is discussed in Section 5.8. Some comments on a scaling analysis for Poiseuille flow of a nematic liquid crystal are contained in Section 5.8.3. These scaling results are important in establishing and validating the Ericksen-LesUe dynamic theory of nematics since they make predictions about flow which were confirmed experimentally. 

This form of threshold has been obtained and discussed by Pelzl, Schiller and De-mus [219], who also go on to discuss their theoretical and experimental results for some SmC materials. Although the dynamic theory for SmC liquid crystals has yet to be reached in Section 6.3, it seems appropriate to record here that these authors additionally considered switch-on and switch-off times (rise and decay times, respectively) and arrived at results analogous to those for a nematic liquid crystal given by equations (5.420) and (5.424) which were encountered when the dynamics of the Freedericksz transition for nematic liquid crystals was discussed in Section 5.9.1. These results for SmC are... 

The dynamic theory for SmC liquid crystals is based upon the same three conservation laws for mass, linear momentum and angular momentum that were employed in the derivation of the dynamic theory of nematics developed in Section 4.2.2. They are given by equations (4.29), (4.30) and (4.31) and, as before, the incompressibihty assumption applied to the mass conservation law leads to the familiar constraint that the velocity vector v satisfies... 

T. Carlsson and F.M. Leslie, The development of theory for flow and dynamic effects for nematic liquid crystals, Liq. Cryst, 26, 1267-1280 (1999). 

Lopatina and Selinger recently presented a theory for the statistical mechanics of ferroelectric nanoparticles in liquid crystals, which explicitly shows that the presence of such nanoparticles not only increases the sensitivity to applied electric fields in the isotropic liquid phase (maybe also a possible explanation for lower values for in the nematic phase) but also 7 N/Iso [327]. Another computational study also supported many of the experimentally observed effects. Using molecular dynamics simulations, Pereira et al. concluded that interactions between permanent dipoles of the ferroelectric nanoparticles and liquid crystals are not sufficient to produce the experimentally found shift in 7 N/ so and that additional long-range interactions between field-induced dipoles of nematic liquid crystal molecules are required for such stabilization of the nematic phase [328]. 

The theory for various molecular dynamics simulation algorithms for the calculation of transport coefficients of liquid crystals is presented. We show in particular how the thermal conductivity and the viscosity are obtained. The viscosity of a nematic liquid crystal has seven independent components because of the lower symmetry. We present numerical results for various phases of the Gay-Berne fluid even though the theory is completely general and applicable to more realistic model systems. 

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... 

Equations (10.38) and (10.39) give a nonlinear integro-differential equation for W, and its mathematical handling is not easy. A guidance of how to proceed is obtained from the phenomenological theory in nematics. De Gennes showed that the dynamics of nematics is essentially described by the Landau theory of phase transition and proposed a phenomenological nonlinear equation fof the order parameter tensor... 

The static continuum theory of elasticity for nematic liquid crystals has been developed by Oseen, Ericksen, Frank and others [4]. It was Oseen who introduced the concept of the vector field of the director into the physics of liquid crystals and found that a nematic is completely described by four moduli of elasticity Kn, K22, K33, and K24 [4,5] that will be discussed below. Ericksen was the first who understood the importance of asymmetry of the stress tensor for the hydrostatics of nematic liquid crystals [6] and developed the theoretical basis for the general continuum theory of liquid crystals based on conservation equations for mass, linear and angular momentum. Later the dynamic approach was further developed by Leslie (Chapter 9) and nowadays the continuum theory of liquid crystal is called Ericksen-Leslie theory. As to Frank, he presented a very clear description of the hydrostatic part of the problem and made a great contribution to the theory of defects. In this Chapter we shall discuss elastic properties of nematics based on the most popular version of Frank [7]. 

In addition to the commonly used kinematic and dynamic variables of continuum theories, the L-E theory contains a unit vector, n, called the "director", to describe the orientation of the liquid crystal. There is an elastic energy, W, associated with spatial variations of the director. For nematics, this is expressed as... 

The constitutive hydrodynamic equations for uniaxial nematic calamitic and nematic discotic liquid crystals are identical. In comparison to nematic phases the hydro-dynamic theory of smectic phases and its experimental verification is by far less elaborated. Martin et al. [17] have developed a hydrodynamic theory (MPP theory) covering all smectic phases but only for small deformations of the director and the smectic layers, respectively. The theories of Schiller [18] and Leslie et al. [19, 20] for SmC-phases are direct continuations of the theory of Leslie and Ericksen for nematic phases. The Leslie theory is still valid in the case of deformations of the smectic layers and the director alignment whereas the theory of Schiller assumes undeformed layers. The discussion of smectic phases will be restricted to some flow phenomena observed in SmA, SmC, and SmC phases. 

Magnetic resonance methods have been used extensively to probe the structure and dynamics of thermotropic nematic liquid crystals both in the bulk and in confined geometry. Soon after de Gennes [27] stressed the importance of long range collective director fluctuations in the nematic phase, a variable frequency proton spin-lattice relaxation Tx) study [32] showed that the usual BPP theory [33] developed for classical liquids does not work in the case of nematic liquid crystals. In contrast to liquids, the spectral density of the autocorrelation function is non-Lorentzian in nematics. As first predicted independently by Pincus [34] and Blinc et al. [35], collective, nematic type director fluctuations should lead to a characteristic square root type dependence of the spin-lattice relaxation rate rf(DF) on the Larmor frequency % ... 

Oseen [1] and Frank [2] far before the development of LCD technology. The dynamic continuum theory of nematics, which is frequently called the nematodynamics, was developed by Ericksen [3] and Leslie [4] (hereafter referred to as E-L theory) based on the classical mechanics just in time for the upsurge of LCD technology. In conjunction with the electrodynamics of continuous media, the static and dynamic continuum mechanics of Oseen-Erank and E-L theory provided theoretical tools to analyze quantitatively key phenomena, e.g., Freedericksz transition of various configurations and associated optical switching characteristics. For the details of E-L theory [5-7] and its development [9,10], please refer to the articles cited. 

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