Nematic liquid crystals order parameter

While the equilibrium statistical mechanics of the nematic liquid crystal order parameter and related physical properties near are now well imderstood, the dynamical responses of the order parameter remain relatively imejqrlored. This is probably due to the fact that most studies of the order parameter near the phase transition point, such as the critical erqronent, changes in molar volume, and other physical parameters, are directed at imderstanding the phase transition processes themselves and are usually performed with temperature changes occurring at very slow rates. 

An aligned monodomain of a nematic liquid crystal is characterized by a single director n. However, in imperfectly aligned or unaligned samples the director varies tlirough space. The appropriate tensor order parameter to describe the director field is then... 

Nematic. In a nematic liquid crystal, the long axes of the molecules remain substantially parallel, but the positions of the centers of mass are randomly distributed. Therefore, there is orientational order and a nonzero orientatinnul order parameter, but there is no positional order. 

The application of an electric field between the electrodes results in a realignment of the nematic liquid crystal mixture and the dichroic dye molecules parallel to the electric field resulting in a lower optical density (absorption) and, theoretically, the disappearance of colour assuming an ideal order parameter (S = 1) of the nematic liquid crystal director and the dye molecules. A residual absorption in this state gives rise to a display with a strongly coloured background and weakly coloured information. 

Unlike ferromagnets, where G((p) = G(-(f), the free energy of nematic liquid crystals exhibits an asymmetry G(S) G(-5), as discussed previously and shown in Fig. 3b. To reproduce this behavior, a term that is an odd power of the order parameter S is needed in the Landau expression for the free energy of a nematic liquid crystal. A term linear in 5 is not allowed since the equilibrium condition dQdS = 0 could not then be satisfied in the disordered isotropic phase where 5=0. The presence of a cubic term will lead to the desired asymmetry in G as a function of 5 and to the emergence on cooling of a second minimum in G at a finite 5 value. 

The order parameter in this sample as determined from the NMR spectra varied from 0.88 to 0.72 over the nematic temperature range. This value is higher than the usual values for low molecular weight nematic liquid crystal compounds. 

A. V. Komolkin, Aatto Laaksonen and A. Maliniak, Molecular Dynamics Simulation of a Nematic Liquid Crystal, J. Chem. Phys. 101 (1994) 4103 M. R. Wilson, Determination of Order Parameters in Realistic Atom-Based Models of Liquid Crystal Systems, J. Mol. Liq. 68 (1996) 23 ... 

A molecular dynamics simulation has been performed on 4-n-pentyl-4(-cyanobiphenyl (5CB) in the nematic phase. Order parameters and dipolar couplings have been calculated and used to test theoretical models. Theoretical models have also been developed to explain the shielding of a noble-gas atom in an anisotropic environment and applied to explain the medium-induced shielding of the noble gases Xe and Ne in the nematic liquid crystal 4(-ethoxybenzylidene-4-n-butylaniline (EBBA). ... 

As mentioned earlier, the proton NMR spectra of small molecules oriented in nematic liquid crystals become rapidly complex with the increase of number of interacting nuclei. In order to obtain information on orientation and structure, it is therefore essential to develop strategies to interpret such spectra with the help of special techniques. Techniques such as Automatic Analysis and the use of Multiple Quantum Spectroscopy have been employed for these purposes during the period under report. A method based on the Automatic Analysis of Multiple Quantum proton spectra of flexible molecules dissolved in liquid crystalline phases has been suggested for extracting starting parameters for the... 

If the liquid crystalline molecules are not cylindrically symmetrical, S must be expressed by a tensor (traceless tensor). The nematic liquid crystal state (a uniaxial symmetrical system) is the main concern of the book. For such a phase, the order parameter can be denoted by the scalar in Equation 1.11. 

By means of the Maxwell equal area rule, the dashed line in the figure is replaced by an equal pressure solid line. The regime of low pressure and low density is the isotropic phase while the regime of high pressure and high density corresponds to the anisotropic phase — nematic liquid crystal phase. The order parameter is determined by... 

Landau used a very simple form of free energy to describe systems such as the para-magnetic system where the magnetic polarization M is the order parameter. The idea has been extended by de Gennes (1973) to deal with nematic liquid crystals. The free energy is given in the extended Landau-de Gennes form as... 

The Maier-Saupe theory of nematic liquid crystals is founded on a mean field treatment of long-range contributions to the intermolecular potential and ignores the short-range forces [88, 89]. With the assumption of a cylindrically symmetrical distribution function for the description of orientation of the molecules and a nonpolar preferred axis of orientation, an appropriate order parameter for a system of cylindrically symmetrical molecules is... 

Fig. 2.3.6. Orientational order parameter ( P icosOy) in three nematic liquid crystals from Raman measurements a) 4 -n-pentyl-4-cyanobiophenyl (5CB) b) 4 -n-heptyl-4-cyanobiphenyl (7CB) (c) 4 -n-octyloxy-4-cyanobiphenyl (80CB). (After reference 82.)...

In principle, a free energy expansion of this type should be valid for nematic liquid crystals, with s denoting the usual orientational order parameter defined by (2.3.1). The term of order is not precluded by symmetry, for the states s and —s represent two entirely different kinds of molecular arrangement which are not symmetry related and do not have equal free energies. In the former case, the molecules are more nearly parallel to the unique axis, while in the latter they are more nearly perpendicular to it. However, in the nematic phase s is usually quite large (greater than about 0.4) so that very many more terms have to be included in the expansion in order to draw any valid conclusions. Consequently, the... 

In equations (5)-(8), i is the molecule s moment of Inertia, v the flow velocity, K is the appropriate elastic constant, e the dielectric anisotropy, 8 is the angle between the optical field and the nematic liquid crystal director axis y the viscosity coefficient, the tensorial order parameter (for isotropic phase), the optical electric field, T the nematic-isotropic phase transition temperature, S the order parameter (for liquid-crystal phase), the thermal conductivity, a the absorption constant, pj the density, C the specific heat, B the bulk modulus, v, the velocity of sound, y the electrostrictive coefficient. Table 1 summarizes these optical nonlinearities, their magnitudes and typical relaxation time constants. Also included in Table 1 is the extraordinary large optical nonlinearity we recently observed in excited dye-molecules doped liquid... 

When the liquid crystal exhibits a direct isotropic to smectic-A phase transition, the capillary condensation of the smectic phase in a gap between the surfaces is possible, since the isotropic to smectic-A phase transition is a first order phase transition due to the coupling between the nematic and smectic order parameters [26]. 

As discussed abfeve, ellipsometry is directly sensitive only to the interfacial variations of the nematic order parameter, which is connected to the optical refraction indices. The interfacial smectic order, which has no direct influence on the optical properties, can only be observed due to its coupling to the nematic (orientational) order. The same experimental setup as described in Sect. 4.1.3 has been used to study the interface between smectic liquid crystal dodecylcyanobiphenyl (12CB) in the isotropic phase and the silanated glass. Although only orientational order is observed, the temperature dependence of pb is in this case quite different from the case with the nematic liquid crystal, as evident from Fig. 4.4. 

By considering a simple spatial dependence of the order parameter, i.e. S z) = exp(—2/ ), the integral in (4.3) can be calculated and the ellip-ticity coefficient for the surface-induced layer of a pre-nematic liquid crystal on an isotropic substrate is... 

The Landau theory assumes that the order parameter is small in the vicinity of the transition, so that only the lowest terms required by symmetry and preventing the free energy from diverging are kept in the expansion. In the case of nematic liquid crystals, the order parameter is a tensor and its scalar invariant is its trace. Thus, the Landau free energy reads... 

In bulk nematic liquid crystals, the average equilibrium order is uniaxial with the order parameter Qo = aoTo and oq = S. Thus, the correlation lengths of the 5 independent degrees of freedom of the nematic order read... 

M.A. Osipov, The order parameter dependence of the flexoelectric coefficients in nematic liquid crystals, J. Physique Lett. 45(16), 823-826, (1984). 

The NMR technique has provided useful information regarding the orientational characteristics of nematic liquid crystals [96]. In many examples, the order parameters of the mesogenic core comprising a linear array of aromatic nuclei have been accurately determined from the observed dipolar (Dhd) and quadrupolar (Av) splittings by using deuterium-substituted samples. An attempt was made to elucidate the nematic conformation of the polymethylene-type spacer involved in dimer compounds, o, ct)-bis(4-cyanobiphenyl-4 -yloxy)alkanes (CBA- , with = 9,10), by the combined use of NMR and RIS analysis. The chemical structures and the phase transition data obtained by differential scanning calorimetry (DSC) are shown in... 

In experiment on nematic liquid crystals, both positive and negative anisotropy <, is observed, the sign depending on chemical structure. The magnitude of e is often proportional to orientational order parameter S. In the isotropic phase the anisotropy disappears. Typical temperature dependencies of n and , are shown in Fig. 7.5. These observations can be accounted for by the Maier-Meier theory [5]. The latter is based on the following seven assumptions ... 

For nematic liquid crystals, the synunetry is reduced and we need additional variables. The nematic is degenerate in the sense that all equilibrium orientations of the director are equivalent. According to the Goldstone theorem the parameter of degeneracy is also a hydrodynamic variable for a long distance process 0 and the relaxation time should diverge, x—>oo. In nematics, this parameter is the director n(r), the orientational part of the order parameter tensor. For a finite distortion of the director over a large distance (L—>oo), the distortion wavevector 0 and the... 

A qualitative picture, Fig. 10.4, shows the distance dependencies of the orientational order parameter for homeotropically aligned nematic liquid crystal at the solid substrate. The problem is to explain such dependencies [6]. The influence of the surface on the orientational order parameter may be discussed in terms of the modified Landau-de Gennes phase transition theory. Consider a semi-infinite nematic of area A being in contact with a substrate at z = 0 and uniform in the x and y directions. When writing the free energy density a surface term -W8(z)S must be added to the standard expansion of the bulk free energy density ... 

Some solid surfaces induce disorder in nematic liquid crystals. It means that the order parameter at the interface is lower than the bulk value. For instance, evaporated SiO layers of a certain thickness due to their roughness decrease the order parameter of MBBA from the bulk value Sb 0.6 down to So 0.1-0.2. In some cases, the surface order parameter may be equal to zero (surface melting). 

Fig. 10.12 Ordoelectric surface polarization. In sketch (a) the order parameter is iarger at the surface and smaller in the bulk in sketch (b) the order parameter is smaiier at the surface than in the bulk. Corresponding gradient curves S z) for a nematic liquid crystal are qualitatively pictured in the bottom sketches. Vectors and m show the directions of the surface polarization...

The Maier-Saupe theory is very useful in considering liquid crystal systems consisting of more than one type of molecules, such as mixtures of nematic liquid crystals and dichroic dyes. The interactions between different molecules are different and the eonstituent molecules have different order parameters. 

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