Preferred orientation function

In Eq. 2.78 the multiplier 7 is calculated as a sum over all N symmetrically equivalent reciprocal lattice points and t is the preferred orientation parameter refined against experimental data. The magnitude of the preferred orientation parameter is defined as t = TJT, where Tx is the factor for Bragg peaks with reciprocal lattice vectors perpendicular, and T is the same for those which are parallel to the preferred orientation axis, respectively. In the case of the ellipsoidal preferred orientation function this parameter is equal to z for the needles (in-plane preferred orientation) and 1/t for platelets (axial preferred orientation). 

Figure 2.49. Preferred orientation functions for needles represented by the ellipsoidal (a) and March-Dollase (b) functions with the parameter t = Ti/T = 2.5, and the two functions overlapped when Ti/T = 1.5 (c). The two notations, T and Ti, refer to preferred orientation corrections in the directions parallel and perpendicular to the preferred orientation (PO) axis, respectively.

Here the preferred orientation parameter, represents needles and its inverse represents plates. An example of the March-Dollase preferred orientation function for needles with magnitude = 2.5 is shown in... 

The platelets and needles discussed above are the two limiting but still the simplest possible cases. Particles may (and often do) have shapes of ribbons. These particles will pack the same way needles do - parallel to the sample surface but the ribbons will not be randomly oriented around their longest axes - they will tend to align their flat sides parallel to the sample surface. This case should be treated using two different preferred orientation functions simultaneously one along the needle and one perpendicular to its... 

Figure 2.50. The illustration of the complex distribution of reciprocal lattice vectors modeled using a spherical harmonic preferred orientation function for the (100) reflection.

Lorentz and polarization factor at step i = exp(-P(tt(j) ), preferred orientation function = preferred orientation parameter = acute angle between the preferred orientation direction and the reciprocal lattice vector for H... 

Usually, phase scale factors and global parameters are refined first, while phase-dependent lattice parameters and peak shape parameters are refined in later stages. Intensity corrections, such as preferred orientation functions, are optimised last. Depending on the program, this parameter variation sequence can be preprogrammed or needs to be carried out manually. 

A summary of physical and chemical constants for beryUium is compUed ia Table 1 (3—7). One of the more important characteristics of beryUium is its pronounced anisotropy resulting from the close-packed hexagonal crystal stmcture. This factor must be considered for any property that is known or suspected to be stmcture sensitive. As an example, the thermal expansion coefficient at 273 K of siagle-crystal beryUium was measured (8) as 10.6 x 10 paraUel to the i -axis and 7.7 x 10 paraUel to the i -axis. The actual expansion of polycrystalline metal then becomes a function of the degree of preferred orientation present and the direction of measurement ia wrought beryUium. 

Fig. 2. Young s modulus corrected for porosity as a function of preferred orientation curve is based on theoretical model where = rayon-based fibers Q — PAN-based fibers and A = pitch-based fibers (2). To convert GPa to psi, multiply by 145,000.

For a removal attempt a molecule is selected irrespective of its orientation. To enhance the efficiency of addition attempts in cases where the system possesses a high degree of orientational order, the orientation of the molecule to be added is selected in a biased way from a distribution function. For a system of linear molecules this distribution, say, g u n ), depends on the unit vector u parallel to the molecule s symmetry axis (the so-called microscopic director [70,71]) and on the macroscopic director h which is a measure of the average orientation in the entire sample [72]. The distribution g can be chosen in various ways, depending on the physical nature of the fluid (see below). However, g u n ) must be normalized to one [73,74]. In other words, an addition is attempted with a preferred orientation of the molecule determined by the macroscopic director n of the entire simulation cell. The position of the center of mass of the molecule is again chosen randomly. According to the principle of detailed balance the probability for a realization of an addition attempt is given by [73]... 

The above results show that the structure of the system with the molecules self-assembled into the internal films is determined by their correlation functions. In contrast to simple fluids, the four-point correlation functions are as important as the two-point correlation functions for the description of the structure in this case. The oil or water domain size is related to the period of oscillations A of the two-point functions. The connectivity of the oil and water domains, related to the sign of K, is determined by the way four moleeules at distanees eomparable to their sizes are eorrelated. For > 0 surfactant molecules are correlated in such a way that preferred orientations... 

However, case (ii) above, where there is biaxial symmetry of the distribution function, but no preferred orientation of the structural units about their Ox3 axes is a feasible proposition. Kashiwagi et al.10) and later Cunningham et al. n) have given expressions for the second moment... 

The diffraction lines due to the crystalline phases in the samples are modeled using the unit cell symmetry and size, in order to determine the Bragg peak positions 0q. Peak intensities (peak areas) are calculated according to the structure factors Fo (which depend on the unit cell composition, the atomic positions and the thermal factors). Peak shapes are described by some profile functions 0(2fi—2fio) (usually pseudo-Voigt and Pearson VII). Effects due to instrumental aberrations, uniform strain and preferred orientations and anisotropic broadening can be taken into account. 

The used S5mbols are K, scale factor n, number of Bragg peaks A, correction factor for absorption P, polarization factor Jk, multiplicity factor Lk, Lorentz factor Ok, preferred orientation correction Fk squared structure factor for the kth reflection, including the Debye-Waller factor profile function describing the profile of the k h reflection. 

Data for the bulk fluid, line A, indicate that vz varies as a function of z but maintains a value near 0.75 of maximum velocity. The periodicity of vx and vy is clearly evident in the graph of line A and a 1800 out of phase coupling of the components is seen with one positive when the other is negative. This indicates a preferred orientation to the plane of the oscillatory flow and this feature was seen in all the biofilms grown throughout this study. The secondary flow components are 0.1-0.2 of the maximum axial velocity and are spatially oscillatory. The significant non-axial velocities indicate non-axial mass transport has gone from diffusion dominated, Pe = 0, in the clean capillary, to advection dominated, Pe 2 x 103, due to the impact of the biofilm. For comparison, the axial Peclet number is Pe L 2x 10s. Line B intersects areas covered by biomass and areas of only bulk... 

These models are semiempirical and are based on the concept that intermolecular forces will cause nonrandom arrangement of molecules in the mixture. The models account for the arrangement of molecules of different sizes and the preferred orientation of molecules. In each case, the models are fitted to experimental binary vapor-liquid equilibrium data. This gives binary interaction parameters that can be used to predict multicomponent vapor-liquid equilibrium. In the case of the UNIQUAC equation, if experimentally determined vapor-liquid equilibrium data are not available, the Universal Quasi-chemical Functional Group Activity Coefficients (UNIFAC) method can be used to estimate UNIQUAC parameters from the molecular structures of the components in the mixture3. 

If excited molecules can rotate during the excited-state lifetime, the emitted fluorescence is partially (or totally) depolarized (Figure 5.9). The preferred orientation of emitting molecules resulting from photoselection at time zero is indeed gradually affected as a function of time by the rotational Brownian motions. From the extent of fluorescence depolarization, we can obtain information on the molecular motions, which depend on the size and the shape of molecules, and on the fluidity of their microenvironment. 

More recently, electrostatic theory has been revived due to the concept of molecular electrostatic potentials. The potential of the solute molecule or ion was used successfully to discuss preferred orientations of solvent molecules or solvation sites 50—54). Electrostatic potentials can be calculated without further difficulty provided the nuclear geometry (Rk) and the electron density function q(R) or the molecular wave function W rxc, [Pg.14]

The result of this crystallization is a harder, less ductile film. The extent or the degree of crystalinity, either before or after heat treatment, is a compound function of phosphorous content, metalizing solution (bath) pH, temperature to which the sample was heated, time of exposure to the highest temperature, and a number of additional factors. By degree of crystalinity we mean here component crystalline sizes, possible preferred orientation and alike. 

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