Hermans orientation parameter

Uniaxial orientation parameter (Hermans orientation function) Scattering data evaluation program by A. Hammersley (ESRF) Free Electron Laser Hamburg Full width at half-maximum... 

Figure 9.4. The orientation of structural entities (rods) in space with respect to the (vertical) principal axis and the values of for, the uniaxial orientation parameter (Hermans orientation function) for (a) fiber orientation, (b) isotropy, (c) film orientation...

Nematic phases are characterized by an unordered statistical distribution of the centers of gravity of molecules and the long range orientational order of the anisotropically shaped molecules. This orientational order can be described by the Hermans orientation function 44>, introduced for l.c. s as order parameter S by Maier and Saupe 12),... 

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) ... 

The parameter/ defined by (3.75) is called the Hermans orientation parameter or simply the orientation parameter. The value of / is equal to 1, 0, or —1/2 when the pole is parallel to Z, random, or perpendicular to Z, respectively. 

Figure 3.23 A point plotted on this diagram, giving the Hermans orientation parameters fa and fy for two orthogonal poles a and b in the crystal, represents the state of average orientation of crystallites in the sample.

The Hermans orientation function can be given a relatively simple interpretation. A sample with orientation / may be considered to consist of perfectly aligned molecules with mass fraction / and randomly oriented molecules with mass fraction 1 — /. PLCs are often characterized by their order parameter (denoted s), a concept coined by Tsvetkov [16]. This quantity is basically equivalent to the Hermans orientation function. For nematics and smectics the director represents the average mesogen direction and the order parameter can thus only take positive values. For cholesteric phases, on the other hand, the director is chosen perpendicular to the layers and in this case the order parameter takes negative values. 

The mathematics of spherical harmonics is an accepted tool in many scientific disciplines and is treated in several classic texts. In 1939 the method was applied to orientation in materials by Hermans and Platzek, who used just P2), which is often referred to as the Hermans orientation function. Spherical harmonics were applied to liquid crystals in the theoretical work of Maier and Saupe, who again emphasized only (Pj)- However, they called it the order parameter and designated it S, setting a nomenclature which is now standard in liquid crystal studies. 

The parameters average values. Note that fj is the Hermans orientation function. The full description of uniaxial orientation /(0) can thus not be given by a single measurement of birefringence. 

Infrared dichroism measurement provides information about the orientation of groups. If the angle between the transition moment vector and the chain axis is known, it is possible to determine a Hermans orientation parameter for the chain axis. The c-axis orientation (crystal-phase orientation) can be obtained from wide-angle X-ray scattering. The orientation of an amorphous polymer can be assessed by IR. 

Then the orientation parameter, S, was calculated from (6.8) by using the Herman orientation parameter. 

Einally, X-ray diffraction experiments in two dimensions help in obtaining more detailed information on the orientation of the structural features within the sample [48,50,70]. An advantage of such a two-dimensional approach is the possibility to quantitatively assess the orientation of the filler or of the polymer crystallites by calculating the Hermans orientation parameter [71]. 

We can obtain the orientation of the crystallites along the thread axis from the intensity integration as a function of azimuth angle at the radial position of equatorial (120) and (200) peaks. We use the Hermans orientation function / = (3 (cos (/)) — 1)/2, where cf> is the angle between the c axis and the fiber axis. The parameter / is 0 for random orientation in fibers and 1 if all crystals are perfectly aligned with respect to each other. For two reflections, (200) and (120), which are not orthogonal but have a known geometry in the equatorial plane, we have... 

For unidirectional molecular orientations, such as for uniaxially drawn polymers, these dichroic parameters can be related to the Herman orientation function, F. This quantity is equivalent to the second moment of the orientation distribution function for the molecular axis and is given by [19]... 

As noted before, MLCs and PLCs share essentially the same kinds of phases these are nematic, cholesteric, and a variety of smectic phases. These three names have been proposed by Friedel [1] in 1922 who imagined that such phases should exist—long before his concepts were confirmed by diffiractometric experiments. In all these phases the entire molecules (in MLCs) or the LC sequences in the chains (in PLCs) are oriented approximately— but not quite—perpendicularly to a preferred axis in space called director. The degree of alignment is characterized by the order parameter (also called the anisotropy factor) defined in 1946 by Hermans [36] as... 

In Eq. (11.1) (cosx) is the average cosine that the poles of a given family of planes make with the preferred (fibre axis) direction and (P2(cosx)) the second order Legendre polynomial of argument (cos ). Equation (11.1) is also known as Herman s orientation parameter, where P2(cosj ) = 1 corresponds to an ideal case of perfect alignment of the poles toward a preferential direction, P2(cosx) = 0 corresponds to isotropic case and P2(cosx) = 0.5 corresponds to an ideal... 

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