Liquid crystals orientation parameter

Description of the threshold values of the domain occurrence as a function of the liquid crystal physical parameters (dielectric and conductivity anisotropy, viscoelastic properties, etc.), and experimental conditions (cell thickness, initial orientation, form, amplitude, frequency of the exciting field, etc.). A semiquantitative theoretical explanation of the threshold phenomena within the framework of the theory of the linear nematody-namics in an electric field is provided. The optical features of the arising domain patterns are also discussed. 

Thennotropic liquid crystal phases are fonned by rodlike or disclike molecules. However, in the following we consider orientational ordering of rodlike molecules for definiteness, although the same parameters can be used for discotics. In a liquid crystal phase, the anisotropic molecules tend to point along the same direction. This is known as the director, which is a unit vector denoted n. 

An orientational order parameter can be defined in tenns of an ensemble average of a suitable orthogonal polynomial. In liquid crystal phases with a mirror plane of symmetry nonnal to the director, orientational ordering is specified. 

McMillan s model [71] for transitions to and from tlie SmA phase (section C2.2.3.2) has been extended to columnar liquid crystal phases fonned by discotic molecules [36, 103]. An order parameter tliat couples translational order to orientational order is again added into a modified Maier-Saupe tlieory, tliat provides tlie orientational order parameter. The coupling order parameter allows for tlie two-dimensional symmetry of tlie columnar phase. This tlieory is able to account for stable isotropic, discotic nematic and hexagonal columnar phases. 

The orientational order parameter for a liquid crystal can be measured by first calculating the ordering tensor... 

The anisotropy of the liquid crystal phases also means that the orientational distribution function for the intermolecular vector is of value in characterising the structure of the phase [22]. The distribution is clearly a function of both the angle, made by the intermolecular vector with the director and the separation, r, between the two molecules [23]. However, a simpler way in which to investigate the distribution of the intermolecular vector is via the distance dependent order parameters Pl+(J") defined as the averages of the even Legendre polynomials, PL(cosj r)- As with the molecular orientational order parameters those of low rank namely Pj(r) and P (r), prove to be the most useful for investigating the phase structure [22]. 

On a molecular level the director is not rigorously defined, but the molecular director is typically considered to be the average long axis of the molecules, oriented along the macroscopic director with some order parameter less than one. This type of anisotropic order is often called long-range orientational order and, combined with the nonresonant optical properties of the molecules, provides the combination of crystal-like optical properties with liquidlike flow behavior characteristic of liquid crystals. 

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The values of these autocorrelation functions at times t = 0 and t = 00 are related to the two order parameters orientational averages of the second- and fourth-rank Legendre polynomial P2(cos/ ) and P4 (cos p). respectively, relative to the orientation p of the probe axis with respect to the normal to the local bilayer surface or with respect to the liquid crystal direction. The order parameters are defined as... 

This article reviews the following solution properties of liquid-crystalline stiff-chain polymers (1) osmotic pressure and osmotic compressibility, (2) phase behavior involving liquid crystal phasefs), (3) orientational order parameter, (4) translational and rotational diffusion coefficients, (5) zero-shear viscosity, and (6) rheological behavior in the liquid crystal state. Among the related theories, the scaled particle theory is chosen to compare with experimental results for properties (1H3), the fuzzy cylinder model theory for properties (4) and (5), and Doi s theory for property (6). In most cases the agreement between experiment and theory is satisfactory, enabling one to predict solution properties from basic molecular parameters. Procedures for data analysis are described in detail. 

The orientation dependent parameter p defined by Eq. (11) becomes unity in the isotropic state, and decreases as the polymers are uniaxially oriented. Therefore, it follows from Eqs. (9) and (10) that the wormlike hard spherocylinder system has a smaller translational entropy loss from the ideal solution in the liquid crystal state than in the isotropic state. This difference drives the system to form a liquid crystal phase. However, in order to determine the equilibrium orientation of the system, the orientation dependence of Sor has to be formulated, and this is done in Sect. 2.3. 

Norden et al. have examined the CD spectral changes of dyes oriented in a liquid crystal matrix, on variing the order parameter q, which is estimated by NMR observation 262). Kuball et al. presented a theoretical description of the optical activity of oriented molecules, and they found that the C D of the transition A of a given oriented molecule Ae (v) is described by ... 

Orientational Distribution Function and Order Parameter. In a liquid crystal a snapshot of the molecules at any one lime reveals that they arc not randomly oriented. There is a preferred direction for alignment of the long molecular axes. This preferred direction is called the director, and it cun be used to define- an orienlalional distribution function, f W). where flH win Vilb is proportional to the fraction of molecules with their long axes within the solid angle sinbdw. 

Positional Distribution Function and Order Parameter. In addition to orientational order, some liquid crystals possess positional order in that a snapshot at any time reveals that there arc parallel planes which possess a higher density of molecular centers than the spaces between these planes II the normal to these- planes is defined as the -axis, then a positional distribution function. g( ). can be defined, where gOd is proportional to the fraction of molecular centers between r and + Since yO is periodic, it can he represented as a Fourier scries (a sum uf a sinusoidal function with a periodicity equal to the distance between ihe planes and its harmonics). To represent the amount ol positional order, the coefficient in front of the fundamental term is used as the order parameter. The more Ihe molecules lend to form layers, the greater the coefficient in front of ihe fundaiucnlal sinusoidal lerm and [he greaicr the order parameter for positional order,... 

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 discovered dependence of kinetic parameters of orientation processes on the degree of polymerization 44) is a consequence of the duplex nature of LC polymers — that is the presence of the main chain and of mesogenic side groups. This is why a correct juxtaposition of the kinetic characteristics of orientational processes of low-molecular and polymeric liquid crystals requires an explicit knowledge of the degree of polymerization of a corresponding polymer. 

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