Herman’s orientation function

While photographic techniques may allow one to obtain some average orientation values for a deformed crystalline polymer there is need for a quantitative measure of this distribution. In general, the distribution in orientation is determined for a single (hkl) plane—usually a (/lOO), (OkO) or (OOO plane if sufficient diffraction exists. The data are then presented either in a pole figure or may be used to determine the Herman s orientation function defined as... 

The degree of orientation of crystals is calculated using the Herman s orientation function." ... 

To understand the deformation of chains in a network it is useful to take into account the orientation of the network chains with respect to the stretching direction. The second order (Herman s) orientation function of the statistical segments with respect to the stretching direction is given by... 

Herman s orientation function/is a measure of chain orientation usually relative to the nanofiber axis its value varies fi-om zero for random orientation to unity for perfect alignment. It is calculated as follows /= 3(cos 0 - l)/2 where 6 is the angle between a polymer chain segment and the nanofiber axis. 

Thus, the polymer chains main orientation can be defined by the Herman s orientation function,/, given by ... 

Figure 7 Plan-view SEM micrographs of PS-b-PMMA deposited on top of corrugated substrates, t/h Is thickness ratio of the block copolymer to the depth of corrugation. Rc and 4nent are the maximal radius of curvature of the corrugated surface and Herman s orientation function, respectively.

The classical theory of IR dichroism " relates Herman s orientation function/to the dichroic ratio D by... 

Using the intensity as a function of angle, %, around each diffraction ring, Herman s orientation factor can be determined. Herman s orientation factor (P2) is defined in Equation 8.2 ... 

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),... 

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 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. 

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).

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 ... 

Pedicini and Farris (2003) used the dichroic ratio of the — NH stretching band (3320 cm ) in polyurethane nanofibers electrospun from 7% DMF to estimate the orientation function / for nanofibers. Herman s function, /, varies from/= 0 (isotropic) to/= 1 (perfect chain orientation). The value of/is related to the average angle 6 made by the oriented chain axis to that of the fiber (Fig. 9.13) ... 

Herman s function is a mathematical description of the degree of orientation of the axis of polymer chains within the fiber relative to some other axis of interest (in this case the fiber axis). 

Equation (1.4) gives the full expression, in which intensity varies with both polar angle 0 and azimuthal angle yt. K fiber symmetry is assumed, the intensity will be the same with given 0 and s and the integration of yields a constant In. Usually, a flat 2D detector is used to record the scattering pattern, which is a stereo-projection of 3D scattering intensity distribution. In this case, Fraser correction is needed to correct the distortion due to the stereo-projection [94]. The Hermans orientation function varies between -0.5 and 1. When /= -0.5, the normal and the reference axes are perpendicular to each other when /= 1, the normal is in parallel to the reference axis when / = 0, the system has random orientation. 

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