Orientational order orientation distribution function

This can be inserted in equation (02.2.3) to give tlie orientational distribution function, and tlius into equation (02.2.6) to deteniiine the orientational order parameters. These are deteniiined self-consistently by variation of tlie interaction strength iin equation (c2.2.7). As pointed out by de Gemies and Frost [20] it is possible to obtain tlie Maier-Saupe potential from a simple variational, maximum entropy metliod based on tlie lowest-order anisotropic distribution function consistent witli a nematic phase. 

Fig. 2. Schematic representation of the orientational distribution function f 6) for three classes of condensed media that are composed of elongated molecules A, soHd phase, where /(0) is highly peaked about an angle (here, 0 = 0°) which is restricted by the lattice B, isotropic fluid, where aU. orientations are equally probable and C, Hquid crystal, where orientational order of the soHd has not melted completely.

Pulsed deuteron NMR is described, which has recently been developed to become a powerftd tool for studying molectdar order and dynamics in solid polymers. In drawn fibres the complete orientational distribution function for the polymer chains can be determined from the analysis of deuteron NMR line shapes. By analyzing the line shapes of 2H absorption spectra and spectra obtained via solid echo and spin alignment, respectively, both type and timescale of rotational motions can be determined over an extraordinary wide range of characteristic frequencies, approximately 10 MHz to 1 Hz. In addition, motional heterogeneities can be detected and the resulting distribution of correlation times can directly be determined. 

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]. 

History. Wilke [129] considers the case that different orders of a reflection are observed and that the orientation distribution can be analytically described by a Gaussian on the orientation sphere. He shows how the apparent increase of the integral breadth with the order of the reflection can be used to separate misorientation effects from size effects. Ruland [30-34] generalizes this concept. He considers various analytical orientation distribution functions [9,84,124] and deduces that the method can be used if only a single reflection is sufficiently extended in radial direction, as is frequently the case with the streak-shaped reflections of the anisotropic... 

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Note 5 Even for molecules with cylindrical symmetry, does not provide a complete description of the orientational order. Such a description requires the singlet orientational distribution function which can be represented as an expansion in a basis of Legendre polynomials with L an even integer. The expansion coefficients are proportional to... 

The order parameter S is the orientational average of the second-order Legendre polynomial P2(a n) (n = the director), and if the orientational distribution function is approximated by the Onsager trial function, it can be related to the degree of orientation parameter ot by... 

With increasing flow rate, the orientational state in the nematic solution should change. Larson [154] solved numerically Eqs. (39) and (40b) with Vscf(a) given by Eq. (41) for a homogeneous system (T[f ] = 0) in the simple shear flow to obtain the time-dependent orientational distribution function f(a t) as a function of k. The non-steady orientational state in the nematic solution can be described in terms of the time-dependent (dynamic) scalar order parameter S[Pg.149]

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. 

In elastomer samples with macroscopic segmental orientation, the residual dipolar couplings are oriented as well, so that also the transverse relaxation decay depends on orientation. Therefore, the relaxation rate 1/T2 of a strained rubber band exhibits an orientation dependence, which is characteristic of the orientational distribution function of the residual dipolar interactions in the network. For perfect order the orientation dependence is determined by the square of the second Legendre polynomial [14]. Nearly perfect molecular order has been observed in porcine tendon by the orientation dependence of 1/T2 [77]. It can be concluded, that the NMR-MOUSE appears suitable to discriminate effects of macroscopic molecular order from effects of temperature and cross-link density by the orientation dependence of T2. 

Nematic phases are characterised by a uniaxial symmetry of the molecular orientation distribution function f(6), describing the probability density of finding a rod with its orientation between 6 and 6 + d0 around a preferred direction, called the director n (see Fig. 15.49). An important characteristic of the nematic phase is the order parameter (P2), also called the Hermans orientation function (see also the discussion of oriented fibres in Sect. 13.6) ... 

From the potential Vnem(vt). one can obtain the rod orientation distribution function 1/ (u), and hence the order parameter S, by a self-consistent calculation. At equilibrium, the distribution function V (u) is related to the potential Fnemfu) by Boltzmann s equation ... 

An axially symmetric molecule is characterized by its linear polarizability in the principal axes a x and a y = a" and a" = af/. It is a good approximation to assume that its second- and third-order polarizability tensors each have only one component and respectively, which is parallel to the z principal axis of the molecule. For linear and nonlinear optical processes, the macroscopic polarization is defined as the dipole moment per unit volume, and it is obtained by the linear sum of the molecular poiarizabilities averaged over the statistical orientational distribution function G(Q). This is done by projecting the optical fields on the molecular axis the obtained dipole is projected on the laboratory axes and orientational averaging is performed. The components of the linear and nonlinear macroscopic polarizabilies are then given by ... 

Polar alignment can be described physically through a knowledge of the orientational distribution function f ) (32). A related quantity that can be expressed as a single number is the order parameter. The order parameter describing the induced polar alignment G is given by... 

From the knowledge of the order parameter from the measurement of the variation of the optical absorption spectrum due to poling, one can estimate the x parameter (Eq. (43)) intervening in the above developments. For poled polymers one can use also another alternative description of linear and nonlinear optical susceptibilities by expanding the orientation distribution function in the series of Legendre polynomials, where the expansion coefficients are order parameters ... 

If the orientation dependence of the resonance frequency of a spin 5 is determined by just one interaction, it can be exploited for use as a protractor to measure angles of molecular orientation. In powders and materials with partial molecular orientation, the orientation angles and, therefore, the resonance frequencies are distributed over a range of values. This leads to the so-called wideline spectra. From the lineshape, the orientational distribution function of the molecules can be obtained. These lineshapes need to be discriminated from temperature-dependent changes of the lineshape which result from slow molecular reorientation on the timescale of the inverse width of the wideline spectrum. The lineshapes of wideline spectra, therefore, provide information about molecular order as well as about the type and the timescale of slow molecular motion in solids [Sch9, Spil]. 

Molecular order is descnhedhy the orientational distribution function P(0) [Mcbl]. This is the probability density of finding a preferential direction n in the sample under an angle 0 in a molecule-fixed coordinate frame (Fig. 3.2.2(a)). For simplicity, macroscopically uniaxial samples with cylindrically symmetric molecules are considered. Then, one angle is sufficient to characterize the orientational distribution function. In practice, not the angle 0 itself but its cosine is used as the variable and for weak order the distribution function is expanded into Legendre polynomials P/(cos 0),... 

Fig. 3.2.3 Relationships between coordinate systems for the description of molecular order. The orientation of the laboratory coordinate system in the principal axes system of the coupling tensor determines the angular dependence of the resonance frequency. The orientations of the preferential sample direction in a molecule-fixed coordinate frame determines the orientational distribution function.

A 2D spectrum is obtained by 2D Fourier transformation over the spinner phase and the acquisition time t2. It exhibits spinning sideband signals in both dimensions. Those in the additional dimension o)i are characteristic of the molecular order. No sidebands appear in this direction if the sample is isotropic. The 2D sideband signals can be analysed to obtain the orientational distribution function. 

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