Nematodynamics

Leslie, F.M. Introduction to nematodynamics. In Dunmur, D., Fukuda, A., Luckhurst, G., INSPEC (eds.) Physical Properties of Liquid crystals Nematics, pp. 377-386, London (2001). Parodi, O. Stress tensor for nematic liquid crystals. J. Phys. (Paris) 31, 581-584 (1970) Miesowicz, M. The three coefficients of viscosity of anisotropic liquids. Nature 158, 27 (1946) Influence of the magnetic field on the viscosity of para-azoxyanisole. Nature 136, 261 (1936). 

To finalize the description of nematodynamics in the presence of an electric field we have to write the Maxwell equations for space charge Q... 

Figure 2.22 shows the temperature dependences of In ry (Fig. 2.21) on the inverse temperature 1/T, which are very close to linear functions [68]. As seen from Fig. 2.22 the isotropic viscosity rjs = a4/2 does not undergo considerable change near the phase transition region, while Vi and rj2 vanish in the isotropic phase. Several approaches have been proposed to describe the temperature dependences of the anisotropic liquid crystal viscosity coefiicients [75, 76]. In [75] phenomenological viscosity coefficients in the equations of nematodynamics were found in terms of the order parameter S — S T)... 

The surface flexoelectric energy, which is found from (4.2) and (4.3). Attaining the minimum of the nematic free energy, (4.5) or (4.6), it is possible to derive the equilibrium director distribution in a static case. To find the response times, we have to solve the equations of nematodynamics in the electric field. The corresponding analysis shows that the director reorientation is always accompanied by the macroscopic flow, the so-called backflow [5]. (The only exclusion is the pure twist rotation of the director [1].) Backflow considerably affects the characteristic times of the electrooptical effects in uniform structures, especially in the case of strong deformations of the initial director orientation [3, 5]. 

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]). 

Thus, in general, the electric and viscoelastic properties of liquid crystal polymers are field dependent and the response of the materials to an external field is essentially nonlinear. Unfortunately, in the major part of the electrooptical experiments this nonlinearity is not taken into account, and the results are interpreted in terms of the conventional nematodynamics with constant material parameters. 

The chevron appearance is described by means of a time-dependent system of linearized equations of nematodynamics in an electric field, which could be reduced as follows ... 

The first paper [87] which considered the role of flexoelectric polarization (3.19) in nematic liquid crystals predicted the possibility of an oscillatory instability, the wave vector of rolls being parallel to the initial director Lq. Later Madhusudana et al. [88] and Thom et al. [89] pointed to the important role of a flexoelectric contribution to the equations of nematodynamics in an electric field. 

The macroscopic nematodynamic equations describe the dynamics of the slowly relaxing variables, which usually are either connected with conservation laws or with the Goldstone modes of the spontaneously broken symmetries. To formulate them we wUl follow the traditional approach [65-67] rather than the one based more directly on the principles of hydrodynamics and irreversible thermodynamics [68]. In the nematic state isotropy is spontaneously broken and the averaged molecular alignment singles out an axis whose orientation defines the director n, i. e. an object that has the properties of a unit vector with n = -n. The static properties are conveniently expressed in terms of a free energy density whose orientational elastic part is given by [69]... 

Zakharov, A.V., Vakulenko, A.A., (2010). Orientational nematodynamics of a hybrid-oriented capillary. Physics of the Solid State Vol. 52, pp. 1542. 

Nematodynamics Les Houches-lectures Notes, Gordon and Breach, London and New York (1973). 

Liquid Crystalline Polymers Theories, Experiments, and Nematodynamic Simulations of Shearing Flows... 

As mentioned, a lot of theoretical and experimental studies have been performed to understand physics and rheological properties of lyotropic LCPs. The molecular Doi approach with many improvements and experimental tests is well presented in the literature (e.g., see Ref. [5]). But the thermotropic LCPs were poorly understood till recently, in spite of many attempts to develop either nematodynamic or molecular description of their flow properties. The beauty of continuum approach is that it can be applied to molecular nematics ofboth different types, as well as to the nonyielding suspensions with shaped particles. Yet, general nematodynamic theories are multi-parametric. For example, the general LEP continuum LC theory contains five constitutive parameters [2]. Similarly, de Gennes potential proposed for the monodomain description of general weakly elastic behavior of LCE has also five parameters [37]. Because viscoelasticity is a combination of elastic and viscous effects, it is expected that even in easy theoretical schemes, the continuum approach to viscoelastic polymer nematodynamics should involve at least 10 constitutive parameters. 

As mentioned, there exists neither molecular nor continuum theory for describing complicated properties of thermotropic LCPs, although many experimental data for this type of LCPs have been accumulated. One of the objectives of the new continuum theory of weakly nonlinear viscoelastic nematodynamics [22, 23] is to interpret and simulate experimental data, and create models of processing for LPCs. 

This effect has mostly been observed for lyotropic LCPs, sometimes also for thermotropic ones. The existence of region I in Figure 11.1 is explained by the formation of texture, a domain structure observed in many, mostly lyotropic LCPs. The texture occurs during relaxation when the stress levels are very low, that is, when approaching the rest state. Such a three-region flow curve was first observed in Ref. [50] and explained theoretically for lyotropic LCPs in Refs [51, 52] (see also Refs [4, 5, 53]). These theoretical descriptions are typically complementary to the more fundamental monodomain nematodynamic theories of both the molecular and the continuous types. 

The simulations in this and following sections are based on the continuum theory of weak viscoelastic nematodynamics [22, 23]. The closed set of constitutive equations... 

Our simulations of the above viscoelastic nematodynamics theory require the reliable and representative rheological data for LCPs, obtained for steady and transient shear flows and relaxation. We chose literature rheological data for two commercial LCPs, Titan and Zenith 6000 [49], as well as for two model polymers, a main-chain LCP, PSHQ9, and a side-chain LCP, PI-14-5CN [53]. 

It seems that the majority of thermotropic LCPs exhibit flow-aligning behavior. Thus, to describe the experimental observations for these polymers, the general viscoelastic nematodynamic theory [16,17] is used in our simulations with aligning assumption. 

On the Time-Temperature Superposition in Weakly Viscoelastic Nematodynamics... 

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