Orientation density function

The specimen intensity transform X is a type of convolution product of the particle intensity transform Ip and the particle orientation density function ( 1,2). The procedure that we have used to simulate Ip involves firstly the calculation of the intensity transform for an infinite particle, with appropriate allowances for random fluctuations in atomic positions and for matrix scattering. A mapping of Xp is then carried out which includes the effects of finite particle dimensions and of intraparticle lattice disorder, if this is present. A mapping of Is is then obtained from Tp by incorporating the effects of imperfect particle orientation. 

It has been shown ( 1, 2) that for an orientation density function of the type... 

Figure 20. (a) Orientational correlation time t in the logarithmic scale as function of the inverse of the scaled temperature, with the scaling being done by the isotropic to nematic transition temperature with Ti-N. For the insets, the horizontal and the vertical axis labels read the same as that of the main frame and are thus omitted for clarity. Along each isochor, the solid line is the Arrhenius fit to the subset of the high-temperature data and the dotted line corresponds to the fit to the data near the isotropic-nematic phase boundary with the VFT form, (b) Fragility index m as a function of density for different aspect ratios of model calamitic systems. The systems considered are GB(3, 5, 2, 1), GB(3.4, 5, 2, 1), and GB(3.8, 5, 2, 1). In each case, N = 500. (Reproduced from Ref. 136.)... 

To proceed further, we need to know the dependence of the function on density, which can be obtained by resorting to the virial expansion. Now, the molecules having different orientations may be regarded as belonging to different species, so that in effect we have a multicomponent system. The virial expansion of the free energy of a gaseous mixture in ascending powers of the density is well known for our present purpose we use the form... 

The Orientation Density Function (ODF) is defined and the calculations for the general knowledge about texture analysis in poly crystalline materials. The difference between local and global textures allows to determine the ODF more precisely and to describe the grains orientation relationships. 

In a volume-oriented density function such as that used by ROCS, Gaussian functions are atom-centered. In the surface-oriented formulation, the M. of Equation 2.2.1 are Gaussians with peaks at the atomic surface (set by the atomic radii, denoted fi). By itself, the sum over the M produces internal molecular surfaces in addition to external ones. The E[ of Equation 2.2.2 defines Gaussians on local radial co-ordinates around each observer point from set P, with peaks at the molecular surface (set by the minimum distances from the observers to the molecule, denoted d,). When yis chosen carefully, the integral of the product of two molecules surface-density functions R (defined in Eq. 2.2.3) is very closely approximated by the morphological similarity function used by Surflex-Sim [31]. 

The far infra-red and Rayleigh spectra are related to the correlation function of the collective orientation density fluctuations... 

For this study, the dimensions of the fiber are known and fixed. Therefore, in order to determine the fluid property that can hold the fibers is suspension, Eqns. (13) and (15) must be rewritten to solve for critical shear stress as a function of density difference and fiber diameter for a fiber oriented horizontally and vertically, respectively. For a fiber oriented horizontally, the critical yield stress is ... 

Fig. 4.20 Covalent and noncovalent contribution to the stress tu — /22 in a randomly oriented melt subject to a uniaxial extension of A = 2 as a function of density. (From Refs 182, 239).

Jeon B S (2001), Theoretical orientation density function of spunbonded nonwoven fabric , Textile Research Journal, 71,509-513. 

The vector q is the difference between vectorial wave numbers of incident and scattered rays such that q = q, as defined above. The nonsubscript i has the usual significance of (-1) and F(ry) is the probability distribution function (probability density) for the vector r,y. Since the molecules in solution exhibit no preferential orientation in space, F(rji) is spherically symmetric, and P(q) assumes the form (7)... 

There is, of course, a mass of rather direct evidence on orientation at the liquid-vapor interface, much of which is at least implicit in this chapter and in Chapter IV. The methods of statistical mechanics are applicable to the calculation of surface orientation of assymmetric molecules, usually by introducing an angular dependence to the inter-molecular potential function (see Refs. 67, 68, 77 as examples). Widom has applied a mean-held approximation to a lattice model to predict the tendency of AB molecules to adsorb and orient perpendicular to the interface between phases of AA and BB [78]. In the case of water, a molecular dynamics calculation concluded that the surface dipole density corresponded to a tendency for surface-OH groups to point toward the vapor phase [79]. 

LS. In the LS phase the molecules are oriented normal to the surface in a hexagonal unit cell. It is identified with the hexatic smectic BH phase. Chains can rotate and have axial symmetry due to their lack of tilt. Cai and Rice developed a density functional model for the tilting transition between the L2 and LS phases [202]. Calculations with this model show that amphiphile-surface interactions play an important role in determining the tilt their conclusions support the lack of tilt found in fluorinated amphiphiles [203]. 

S. Chains in the S phase are also oriented normal to the surface, yet the unit cell is rectangular possibly because of restricted rotation. This structure is characterized as the smectic E or herringbone phase. Schofield and Rice [204] applied a lattice density functional theory to describe the second-order rotator (LS)-heiTingbone (S) phase transition. 

Noncrystalline domains in fibers are not stmctureless, but the stmctural organization of the polymer chains or chain segments is difficult to evaluate, just as it is difficult to evaluate the stmcture of Hquids. No direct methods are available, but various combinations of physicochemical methods such as x-ray diffraction, birefringence, density, mechanical response, and thermal behavior, have been used to deduce physical quantities that can be used to describe the stmcture of the noncrystalline domains. Among these quantities are the amorphous orientation function and the amorphous density, which can be related to some of the important physical properties of fibers. 

Positional Distribution Function and Order Parameter. In addition to orientational order, some Hquid crystals possess positional order in that a snapshot at any time reveals that there are parallel planes which possess a higher density of molecular centers than the spaces between these planes. If the normal to these planes is defined as the -axis, then a positional distribution function, can be defined, where is proportional to the... 

Here the functions g(0) and /(0) are defined in a suitable way to produce the desired phase behavior (see Chapter 14). The amphiphile concentration does not appear expHcitly in this model, but it influences the form of g(0)— in particular, its sign. Other models work with two order parameters, one for the difference between oil and water density and one for the amphiphile density. In addition, a vector order-parameter field sometimes accounts for the orientional degrees of freedom of the amphiphiles [1]. 

The aromatic portion of the molecules discussed in this chapter is frequently, if not always, an essential contributor to the intensity of their pharmacological action. It is, however, usually the aliphatic portion that determines the nature of that action. Thus it is a common observation in the practice Ilf medicinal chemistry that optimization of potency in these drug classes requires careful attention to the correct spatial orientation of the functional groups, their overall electronic densities, and the contribution that they make to the molecule s solubility in biological fluids. These factors are most conveniently adjusted by altering the substituents on the aromatic ring. 

Draw ratio Density of the amorphous material da) (g/cm-" ) Amorphous orientation function fa) Crystallite length Oc) (nm) Long period (L) (nm) Degree of crystallinity (X=>) Substructure parameter (A) Axial elastic modulus ... 

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