Order parameter nematic

Equation (39) shows that nematic degrees of freedom couple to simple shear, but not the smectic degrees of freedom the modulus of the nematic order parameter has a non-vanishing spatially homogeneous correction (see (39)), whereas the smectic order parameter stays unchanged. The reason for this difference lies in the fact that J3 and /3 include h and p, respectively, which coupled differently to the flow field (see (22) and (23)). Equation (38) gives a well defined relation between the shear rate y and the director tilt angle 9o, which we will use to eliminate y from our further calculations. To lowest order 0O depends linearly on y ... 

In contrast to the director tilt the lowest order correction to the nematic order parameter is quadratic in the shear rate (tilt angle) ... 

Fig. 11 Out of the material parameters connected with the order parameter, only /3 jn — has a measurable effect on the critical values. Some more parameters can influence the amplitudes of the order parameter undulation, namely L and Mo (the latter is only present in the case of the nematic order parameter). All amplitudes have been normalized such that = 1. Note that...

Following the lines proposed above will give a prediction of the pattern formed above onset. For a transition from undulating lamellae to reorientated lamellae or to multilamellar vesicles, defects have to be created for topological reasons. Since the order parameter varies spatially in the vicinity of the defect core, a description of such a process must include the full (tensorial) nematic order parameter as macroscopic dynamic variables. 

Lemaire, B J., Panine, P., Gabriel, J.C.P. and Davidson, P. (2002) The measurement by SAXS of the nematic order parameter of laponite gels. Europhysics Letters 59, 55-61... 

What does your best value for the exponent j8 indicate about the nature of the critical behavior underlying the N-I transition (mean-field second-order or tricritical) Note that /3 and P describe the temperature variation of the nematic order parameter S, which is a basic characteristic of the liquid crystal smdied. Thus, the same j8 and P values could have been obtained from measurements of several other physical properties, such as those mentioned in the methods section. 

Flexibihty of the moieties is another important parameter because it was recently shown that flexible objects require larger volume fractions to undergo nematic ordering. Flexibihty also reduces the nematic order parameter at the transition. Intuitively, very flexible mineral polymers should not show any orientational order at rest, but may display a strong flow birefringence. Thus, any soluble system where the structural unit in the solid state is anisotropic may not necessarily be a lyotropic Hquid crystal. For example, in solution a polymer is much less constrained than in the soHd state, and hence one must consider the elastic properties of the polymer chain and whether the anisotropic units still exist in solution. As shown recently for the case of the complex fluid with a min-... 

Figure 31. Coupling between the nematic order parameter S and the smectic order parameter 4/ for the calamitic system GB(3, 5, 2, 1) (TV = 256) at three state points along the isochor at density p = 0.32. At the nematic phase (T = 1.194 bottom), at the smectic phase (T = 0.502 top), and at the nematic-smectic transition region (T = 0.785 middle). The order parameters are for instantaneous configurations. (Reproduced from Ref. 161.)...

There are several ways of deriving the nematic order parameter S from the X-ray scattering pattern of an aligned sample of the nematic phase. We present... 

Therefore, a diffuse ring of scattering centered at the origin of reciprocal space is attached to each particle. Thus, the nematic order parameter which characterizes the distribution function of a single particle is derived from the interferences among a cluster of particles. This assumption is somewhat similar to a mean-field treatment and tends to overestimate S. 

The fairly good quality of the fits validates both Leadbetter s assumptions and the Maier-Saupe distribution function. However, the values of S obtained and even the quality of the fits obviously depend on the odd or even number of (CH2) groups in the flexible spacer. This odd-even effect is widespread and well known in the field of main-chain LCPs and will be discussed later in this article. The nematic order parameter of main-chain LCPs may reach values as high as 0.85 which demonstrates the very high orientation of the nematic phase of these polymers. Such a large orientation is undoubtedly responsible for the good mechanical properties of this type of materials. The treatment described above therefore provides a very easy way of characterizing the orientational order of a nematic phase. It has also been tested for thermotropic side-chain LCPs and found to be satisfactory as well [15]. Unfortunately, it has not been used yet in the case of lyotropic LCPs except for some aqueous suspensions of mineral ribbons (Sect. 5) which are not quite typical of this family of materials. 

Through the study of the different topics considered in this article, it was shown how X-ray scattering is a useful tool to characterize the most salient features of the mesophases of LCPs. For instance, a simple procedure can be used to measure the nematic order parameter and it is so far valid for all kinds of LCPs based on rod-like moieties. In the case of main-chain polymers, useful information about the conformation of the repeat unit can also be deduced from the diffuse scattering. In the case of side-chain polymers, not only the smectic period but also the amplitude and shape of the smectic modulation can be derived from the measurement of the smectic reflection intensities. Moreover, fluctuations and localized defects may be detected through their contribution to the diffuse scattering. The average distance between lyotropic LCPs can be measured as a function of concentration which tells us the kind of local packing of the particles. 

X=LILq is plotted as a function of the reduced temperature red at constant nominal stress CTn = 2.11xlO N mm . Here Lg is the loaded sample length at Tred l-OS. These results will also be used below to establish a close connection between the strain tensor and the nematic order parameter. It has also been shown that a quadratic stress-strain relation yields in the isotropic phase above the nematic-isotropic phase transition a good description of the data for ele-ongations up to at least 60% [4]. 

By making use of the form of the Landau energy proposed by de Gennes [17], all coefficients in this expression have been evaluated from experimental data [4]. Using these data to calculate U, one obtains a value that is considerably smaller than the one determined from the linearized analysis outlined above. As has been noted, however, a consistent analysis should not only include terms quadratic in the strains and bilinear coupling terms of the order parameter and the strain, but rather also nonlinear effects as well as nonlinear coupling terms between strain and the nematic order parameter [4]. 

Optical techniques to investigate the isotropic-nematic transition in liquid single crystal ela.stomers have been used [10, 11, 18]. The nematic order parameter (the modulus of the order parameter gy) has been determined by measuring the absorbances AII and Aj using the integrated intensities of the CN absorption band with the incident light polarized parallel and perpendicularly to the director of the monodomain [10, 18]. The result is plotted in Fig. 15 as a function of reduced temperature. It is similar to these obtained for polydomains or-... 

In Fig. 16 we show the nematic order parameter 5n obtained from the type of IR dichroism measurements just described as a function of the deformation (L/Lq (mon)) of the same elastomer, where Lo(mon) is the length of the LSCE at Tn i. As one can see... 

Figure 15. Nematic order parameter 5n vs. reduced temperature red for a LSCE (reproduced with permission from [18]).

Figure 16. Nematic order parameter S as function of the deformation (Z./Lo(mon)). Lo(mon), length of the LSCE at (reproduced with permission from [10]).

Figure 20. Nematic order parameter 5n (o) and director order parameter 5u (x) as function of internal mechanical stress

As the experimental investigations have focused predominantly on static properties of the nematic-isotropic transition, most of the theoretical papers have used a Ginzburg-Landau description involving an expansion in the nematic order parameter to describe static properties [4, 17, 22-25]. 

The most important coupling to deformations of the network is the one that is linear in both the strain of the network and the nematic order parameter. As has been discussed earlier in this section this leads to the consequence that the strain tensor can be used as an order parameter for the nematic-isotropic transition in nematic sidechain elastomers, just as the dielectric or the diamagnetic tensor are used as macroscopic order parameters to characterize this phase transition in low molecular weight materials. But it has also been stressed that nonlinear elastic effects as well as nonlinear coupling terms between the nematic order parameter and the strain tensor must be taken into account as soon as effects that are nonlinear in the nematic order parameter are studied [4, 25]. So far, no deviation from classical mean field behavior concerning the critical exponents has been detected in the static properties of this transition and correspondingly there are no reports as yet discussing static critical fluctuations. 

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