Monodomain theory

Even worse, the complicated problem of polydomain behavior of model LCPs, PSHQ9 and PI-14-5CN, used in Ref. [54], carmot make reliable our simulations. Simply speaking, the monodomain theory we used is generally not suitable for the... 

The molecular theory of Doi [63,166] has been successfully applied to the description of many nonlinear rheological phenomena in PLCs. This theory assumes an un-textured monodomain and describes the molecular scale orientation of rigid rod molecules subject to the combined influence of hydrodynamic and Brownian torques, along with a potential of interaction (a Maier-Saupe potential is used) to account for the tendency for nematic alignment of the molecules. This theory is able to predict shear thinning viscosity, as well as predictions of the Leslie viscosity coefficients used in the LE theory. The original calculations by Doi for this model employed a preaveraging approximation that was later... 

Comparing these results, we find that the observed shear-induced disruption in orientation, relative to that of a monodomain, is roughly twice as large as predicted by the mesoscopic theory. 

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 Doi theory captures the molecular viscoelasticity of LCP, i.e., the relaxation of the orientation distribution under flow. But it completely ignores distortional elasticity and is limited to monodomains. The assumption of spatial uniformity underlies all its key elements the nematic potential, the kinetic equation, and the elastic stress tensor. Therefore, its successes are restricted to situations where distortional elasticity is insignificant. 

Fig. 4.1.5. Reflexion spectrum from a monodomain cholesteric film at normal incidence. Full curve experimental spectrum for a mixture of cholesteryl nonanoate, cholesteryl chloride and cholesteryl acetate in weight ratios 21 15 6 at 24 °C (intensity in arbitrary units). Broken curve spectrum computed from the exact theory for a film thickness of 21.0 ftm and pitch 0.4273 foa. (After Dreher...

The theory of propagation inclined to the optic axis is, of course, very much more complicated, and analytical solutions have not so far been found. The first attempt at solving the problem numerically was by Taupin, but the most complete calculations are those of Berreman and Scheffer who also carried out a precise experimental study of reflexion from monodomain samples at oblique incidence. Fig. 4.1.15 presents their observed reflexion spectra for two polarizations. 

A direct confirmation of the existence of these two branches has been found by Liao, Clark and Pershan from their Brillouin scattering experiments on a monodomain sample of jff-methyl butyl /K(p-methoxy-benzylidene)amino) cinnamate. This compound shows the nematic, smectic A and smectic B phases. Choosing both the incident and the scattered light to be polarized either as ordinary or extraordinary waves, they observed two peaks corresponding to the two modes, the angular dependence of which is in excellent agreement with the theory (fig. 5.3.11). 

As a test of the revised theory, further experiments were conducted [Zhao et al., 2005] on nematic solutions of a SCLCP. ER measurements indicated, via application of the Brochard hydrodynamic model, a slightly prolate conformation, R /R = 1.17 0.02, consistent with small-angle neutron scattering measurements, which indicated, that= 1.12 0.06. Observations of the shear stress transient response of a homeotropic monodomain indicated that at a concentration between 0.01 and 0.02 g/mL, the solution exhibited a transition from director-aligning to director-tumbling behavior. The latter result is inconsistent with the original Brochard model [see Eq. (1.94)], which predicts such a transition (i.e., Sas > 0) only for a polymer with an oblate shape but agrees with the modified theory [Eq. (1.96)]. 

Doi [23] had already noted that his theory was restricted to a monodomain or textureless sample. The extension by Marucci and Maffettone [68] retains that restriction. This issue was addressed by Larson and Doi [72], who proposed a model for the rheology of textured lyotropic solutions in the tumbling regime. In the linear Larson-Doi polydomain model the response of the material is expressed in terms of a variable / proportional to the defect density. The defect density is proportional to the shear rate, so that texture refinement is a feature of this model. The steady state predictions for the order parameter S are independent of shear rate. 

The theory of the propagation of light along the optic axis was considered in [11], [13]-[17j. The kinematical approach could explain many experimental results. However, for quantitative explanation of the experiments, a more detailed consideration is necessary. For this purpose, the dynamical theory should be used [15]. Precise measurements of the reflection spectrum from a monodomain CLC show good agreement with theory [18]. A solution... 

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. 

New mathematical techniques [22] revealed the structure of the theory and were helpful in several derivations to present the theory in a simple form. The assumption of small transient (elastic) strains and transient relative rotations, employed in the theory, seems to be appropriate for most LCPs, which usually display a small macromolecular flexibility. This assumption has been used in Ref [23] to simplify the theory to symmetric type of anisotropic fluid mechanical constitutive equations for describing the molecular elasticity effects in flows of LCPs. Along with viscoelastic and nematic kinematics, the theory nontrivially combines the de Gennes general form of weakly elastic thermodynamic potential and LEP dissipative type of constitutive equations for viscous nematic liquids, while ignoring inertia effects and the Frank elasticity in liquid crystalline polymers. It should be mentioned that this theory is suitable only for monodomain molecular nematics. Nevertheless, effects of Frank (orientation) elasticity could also be included in the viscoelastic nematody-namic theory to describe the multidomain effects in flows of LCPs near equilibrium. 

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. 

Figures 11.11 and 11.12 describe the evolution of normalized shear stress o (y,t)/a and first normal stress difference Nj (y, t)/Ni with strain yt for PSHQ9 in start-up shear flow. The flow temperature was 130 °C and shear rate was y = 1 s . The experimental data are shown by dots and simulated curves by dashed line. The simulated overshoots for shear and normal stresses are the same as those for experimental data, but the overshoots in calculated curves occur at a relatively low yt and are very narrow. One can attribute such large values of o (Y,t)/a ratios, characteristic for liquid crystalline polymers, to the existence of a lot of polydomains in the nematic state when the start-up flow initiated. Recall that the theory used for simulation utilizes monodomain approach, whereas PSHQ9 exhibits polydomains in nematic state in start-up flow. Thus, deviation of simulated results from experimental data seems reasonable.

Later on, semi-soft elasticity concept has been extended to the dynamical case. The theory, which is based on the separation of time scales between the director and the network, describes the mechanical properties of monodomain NEs in the linear response regime, when the sample is subjected to a sinusoidal shear of small amplitude (Terentjev and Warner 2001). When the shear is applied in a plane containing the director, the theory predicts the existence of a low frequency semi-soft elastic plateau, in addition to the usual rubbery plateau observed at higher frequencies. 

Theory for Rigid Rodlike Macromolecules with Monodomains... 

Here, we present the molecular theory of Doi and coworkers (Doi 1981, 1983 Doi and Edwards 1978a, 1978b Kuzuu and Doi 1980, 1983, 1984) for predicting the rheological behavior of concentrated solutions of monodomains consisting of rigid rodlike macromolecules. 

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