Orientational probability distribution

The structure of the adsorbed ion coordination shell is determined by the competition between the water-ion and the metal-ion interactions, and by the constraints imposed on the water by the metal surface. This structure can be characterized by water-ion radial distribution functions and water-ion orientational probability distribution functions. Much is known about this structure from X-ray and neutron scattering measurements performed in bulk solutions, and these are generally in agreement with computer simulations. The goal of molecular dynamics simulations of ions at the metal/water interface has been to examine to what degree the structure of the ion solvation shell is modified at the interface. 

Fig. 2. Orientational probability distributions of the molecular axes in (a) a-nitrogen and (b) y-nitrogen. Contours of constant probability for the molecule in the origin, calculated in the mean field model, are plotted as functions of the polar angles (0, ) with respect to the crystal axes (Fig. 1). The angle 0 increases linearly with the radius of the plots from 0 (in the center) to tt72 (at the boundary) d> is the phase angle.

Fig. 5. Orientational probability distribution of the molecular axes for the delocalized (a) and localized (b) mean field states in /3-nitrogen, (a) applies to both molecules in the unit cell, (b) is drawn for one molecule in the unit cell the other molecule in the cell is rotated over = 180°. The distribution does not change qualitatively up to (at least) T = 70 K it just becomes slightly wider with increasing temperature. Reading of the contour plot as in Fig. 2.

This is known as Jeffery s solution.21 It may be noted that the last term in (2-101) is added to ensure that p = 1. When Brownian motion is present, the orientation of a particle must be given statistically by means of an orientation probability distribution function... 

For lamellar phases there is often an ideal swelling behavior where water molecules in the vicinity of the surfaces have their orientation probability distribution almost constant on increasing the water content and where the addition of water only increases the amount of free water. There is no appreciable long-range ordering effect on the water molecules. 

We consider a collinear magnetic system magnetized with reference to a single direction n at a temperature T. (A non-collinear generalization can be made by making the notation more complicated.) The orientational probability distribution is denoted by p( ( e ), and its average for a lattice site i,... 

Fig. 6. Orientational probability distribution of the N2 molecules in the disordered j5-phase of solid nitrogen, from Mean Field calculations [50]. This result, which would correspond to a nearly free precession of the molecules about the z-axis with a large spread in the precession angle, is thermodynamically unstable...

Fig. 7. Orientational probability distribution of the molecules in j9-nitrogen, calculated in Refs. [50, 52]. This result, which representssix orientational pockets, has a considerably lower free energy than the result in Fig. 6...

Fig. 4. Orientation probability distribution (left) and ordering of molecule on the surface (right).

Much more detailed information can be obtained from molecular dynamics and Monte Carlo simulations. This includes the solute orientational profile, which can be expressed using the orientational probability distribution P cp z). If d is a vector fixed in the molecular frame of a solute molecule, the probability distribution of the angle (p between this vector and the normal to the interface is calculated easily using computer simulations as a function of the solute location z. For relatively large dye molecules with a slow reorientation time, convergence can be slow, so it is important to verify that the computed F(0 z) is uniform for z values in the bulk region. Only a few molecular dynamics studies have been reported, with results that generally show an orientational preference with broad distributions. 

By varying the moments of these spectra about the isotropic chemical shift, an approximate orientational probability distribution of the chain axes about the direction of stress can be determined. These results are shown in Fig. 8.15 for samples with different draw ratios [58]. 

In order to find an explicit form for the orientational probability distribution function P( l,f 2o). the function is routinely expanded in... 

The character of DR spectra in liquid crystals refiects both the symmetry of these phases and the dynamics of molecules under the influence of the intermolecular ordering potential. In order to explain the observed DR spectra in liquid crystals, a number of different theories based on the use of a general expansion of the orientational probability distribution... 

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