Herman orientation function

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),... 

In favorable situations, segmental orientation in polymer blends can be followed using the technique of IR dichroism. The results can be expressed in terms of the Hermans orientation function shown in Equation 1. 

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) ... 

Figure 1.52, Electron diffraclion patterns of (a) PANI-DBSA fibril (Hermans orientation function calculated from the arc of the (002) reflection /=0.84) (b) emeraldine base fibril obtained after de-doping of the PANI-DBSA fibril with NH4OH. (Photograph by courtesy of the authors of ref 330. Copyright (1994) Elsevier Science Ltd, Kidlington 0X5 1GB, UK.)...

Hermans Orientation Function [301,302] in Equation 2.79 gives a quantitative expression for specifying the degree of axial orientation in crystalline fibers ... 

PLC and PES were found to be incompatible and formed a self-reinforced system. The chain orientation of the PLC component was constant in the melt flow direction (Hermans orientation function, defined by equation 8.6, s = 0.39 + 0.03) in these blends. The aL parallel to the melt flow direction was always lower than that in the lateral direction. Values of ay of PES + PLC blends decreased markedly in an almost linear manner with increasing PLC content. There is moderate increase in aj. A comparison with the Takayanaga model [101] gives the best fit value for b — 0.50. A slight deviation between predicted and experimental values was obtained for the blends with 60 to 80% PLC, showing a higher b value due to the greater reinforcement. 

The Hermans orientation function is probably the quantity most frequently used to characterize orientation. This orientation function was introduced by P.H. Hermans in 1946 and is part of an equation which relates optical birefringence (An) to chain (segmental) orientation ... 

Figure 10.6 Values of the Hermans orientation function for three simple cases. The directors are marked with arrows.

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. 

It is possible to assess the Hermans orientation function by birefringence measurements. X-ray diffraction and infrared spectroscopy. The full description of uniaxial orientation fief) cannot thus be attained by a single birefringence measurement. 

Absorption bands with ij/ = 5, 76 show no dichroism (R = 1). Absorption bands associated with a perpendicular transition moment vector have R = 0 and the Hermans orientation function is given by... 

Glass Fiber (%) Thickness (mm) Tensile Strength (MPa) Young s Modulus (MPa) Break Elongation (%) Hermans Orientation Function... 

Figure S..25 Hermans orientation function of Thermx LNOOl fibers spun at diffa-ent take-up speeds (from Seo [18], reproduced with permission from Society of Plastics Engineers).

From the WAXS pattern, the ciystalline orientation can be estimated using the Hermans orientation function defined as follows... 

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. 

PTT waxd crystal orientation function/ o can be measured from the azimuthal scan of 010 reflection. The Herman orientation function equation using this reflection has been given by (38), and is based on Wilchinsky s treatment of uniaxial orientation ... 

A very elaborate theory exists for the analysis of fiber diffraction data, although no recent monographs on the subject exist the book Diffraction of x-rays by Chain Molecules (70) contains still the most comprehensive treatment. A large part of the mathematical framework was developed for the famous studies that were done to unravel the structure of DNA (Fig. 13) (71). This subject has remained popular to this day (72). Unfortimately, although the theory exists, there are no standard data analysis packages. In the recent literature on SR-based S5mthetic polymer research, there are few examples to be found in which the authors extract more information than Hermans orientation functions or spacings of diffraction maxima from their data. 

For an axially symmetric system, only even series of Legendre polynomials are retained. A full description of the segmental orientation function requires that the individual component be defined. The coefficients (7s, characteristic of the distribution fimction, represent the contribution from each Legendre polynomials. The second-order coefficient is the well-known Herman orientation function (201) and is generally the only one used for structural analysis in pol5uners. 

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