Orientation function diagrams

Consider Fig. 17 which shows an equilateral triangle having a point, P, within its perimeter. If we allow the line that bisects each apex angle and which extends to the opposing side to have a length of unity, then the distance along or parallel to each of these bisectors can be related to 

In some crystalline systems where there are no reflecting planes normal to the desired crystallographic direction, one can still obtain the orientation for this direction providing the crystalline structure is defined. This method has been developed by Wilchinsky, where he showed that by measuring the orientation functions associated with various known hkl) planes, one can calculate the value of the desired orientation function. This method should not be confused with the application of the direction cosines relationship as discussed earlier. A specific example is given for the case of polypropylene in eqn. (22) where the value of cos ooi, can be found (and therefore the orientation of the chain axis) by measuring the diffraction from the (110) and (OkO) planes.  

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Fig. 17. Diagrams a-c illustrate different experimental methods for measuring the dichroism needed to calculate the orientation function - see text for discussion of (a) two spectra technique, (b) differential technique and (c) waveplate technique...

Fig. 50. Orientation function triangle diagram for the b and c-axes for isotactic polypropylene (O) hot drawn samples (Il(fC) ( ) melt spm fibres (—) cold...

Technical Documentation Technical documentation necessary for the installation, operation, and maintenance of the machine shall be supplied. The information must be provided by the machine supplier to the buyer before delivery of the equipment. Circuit documentation (diagrams, tables, descriptions) serve to explain the function of circuits, power connections, and process-oriented functions (see lEC 204-2,617-1,848, 1082). 

In addition to the orientation function described earlier, block flow diagrams are used to sketch out and screen potential process alternatives. Thus, they are used to convey information necessary to make early comparisons and eliminate competing alternatives without having to make detailed and costly comparisons. 

Figure 28 Rheo-optical elongation recovery experiments of PDMS/PC (50/50% (w/w) and 80/20% (w/w)) block copolymers at 22 °C. (a, b) Stress-strain diagrams and (c, d) orientation function/strain diagrams of the PDMS and PC segments.

Fig. 4.6. Piezoelectric pulse diagrams can be used to obtain explicit representations of the time dependent electric fields in piezoelectric substances. The magnitudes and orientations of these electric fields are critical to development of shock-induced conduction. As an example, the diagram on the left shows the polarization and displacement relations for a location at the input electrode. The same functions for a location within the crystal is shown on the right (after Davison and Graham [79D01]).

Our treatment has been oriented towards using tableaux to represent functions rather than Rumer diagrams, and it will be convenient to continue. Thus, corresponding to the five canonical diagrams for a ring of six orbital symbols one can write... 

A primary focus of our work has been to understand the ferroelectric phase transition in thin epitaxial films of PbTiOs. It is expected that epitaxial strain effects are important in such films because of the large, anisotropic strain associated with the phase transition. Figure 8.3 shows the phase diagram for PbTiOs as a function of epitaxial strain and temperature calculated using Landau-Ginzburg-Devonshire (lgd) theory [9], Here epitaxial strain is defined as the in-plane strain imposed by the substrate, experienced by the cubic (paraelectric) phase of PbTiOs. The dashed line shows that a coherent PbTiOs film on a SrTiOs substrate experiences somewhat more than 1 % compressive epitaxial strain. Such compressive strain favors the ferroelectric PbTiOs phase having the c domain orientation, i.e. with the c (polar) axis normal to the film. From Figure 8.3 one can see that the paraelectric-ferroelectric transition temperature Tc for coherently-strained PbTiOs films on SrTiOs is predicted to be elevated by 260°C above that of... 

The next step in the calculation of the rotational diffusivity is to evaluate the radius, ac. The subscript c on this parameter is meant to remind us that it is a function of concentration. Clearly, as the concentration is increased, this radius must decrease. In addition, one can anticipate that ac will depend on the orientation of the rods. As the rods become increasingly aligned parallel to one another, the hindcrance to rotation will drop off. To obtain ac, one must determine the number of rods, A (ac), which can cut through the imaginary cylinder constraining the test rod. To determine the probability of such an intersection, consider the diagram below. 

This function can be plotted on a polar diagram and used to predict the shape of the surface energy plot cusps in the Wulff constmction. The results are semi-quantitative but useful for finding the relative anisotropic surface energy, in that for cubic crystals, minima are found at low-index (111), (110), and (100) orientations. The interested reader is referred to Venables (2000) and Howe (1997) for details. 

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