Orientation probability distribution function

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. 

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... 

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

For a more comprehensive description of the molecular orientation, higher-order parameters are typically needed to add independent information for the reconstruction of a probability distribution function [14]. The ability to detect molecular orientation changes during ET events of surface-confined molecules is an extremely... 

In the absence of a field the orientation of the non-linear species in an isotropic polymer will be random. An applied field will tend to orient the non-linear species, but this is opposed by their thermal motion. The macroscopic non-linearity of poled films is determined by the orientation of the nonlinear species, which can be calculated if the ground state dipole and the principal component of the first hyperpolarisability are assumed to be parallel, a reasonable approximation for axially elongated molecules. The probability distribution function of the molecular orientation can be written as ... 

It is instinctive to first consider isotropic rotational diffusion of an atom, which is a simple but somewhat accurate description of a molecular hquid like liquid methane just below rmeiting, a Situation wherein the centers of mass of the molecules remain stationary, but the molecules rotate freely about their centers of mass. If we use the polar angle — 9, (p) to define the orientation of the vector d, the probability distribution function, G (see equation 12), which... 

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.

Watanabe and Klein have reported MD simulations of the hexagonal mesophase of sodium octanoate in water with hexagonal symmetry. The singlet (i.e., one atom) probability distribution functions of the carbon atoms on the hydrocarbon chains show close similarity to those in the micelle. The dynamics of water molecules close to the head groups shows lower mean square displacements, and their orientational correlation function decays more slowly than those of waters farther from the head groups, as was seen in a recent bilayer simulation.6 ... 

The free energy profiles across the interface are consistent with the amphiphilic nature of alcohols. More information about the behavior of amphiphilic solutes can be obtained by examining the statistical properties that charaeterize the positions and orientations of the alcohols at the interface. The probability distribution functions of finding the hydroxyl oxygen and terminal methyl carbon... 

To describe directly the orientation of the hydrocarbon chain with respect to the interface we define an end-to-end vector pointing from the methylene carbon atom adjacent to the hydroxyl group, to the methyl carbon atom. Then, we calculate the probability distribution function, P 9), of finding the angle 9, formed between this vector and the normal to the interface directed from hexane to water. P 9) is normalized to describe the probability in the same sohd angle for all values of 9. 

In addition to influencing the free energy profiles, the presence of the interface affects the orientational distributions of polar solutes. This is particularly clear in the case of isoflurane and desflurane, both bearing a considerable dipole moment. To describe this effect, we consider the probability distribution function, P 0), of finding the angle 0 between the molecular dipole moment of the solute and the normal to the interface, pointing from hexane to water. P 6) is defined as in Eq. (3). In the isotropic environment of... 

The direction data mainly come from the scan lines, which are divided into three groups, (SLOl-17, SL17-30 and SL31- 0) for the three discontinuity sets, respectively. The mean and variance of the discontinuity dip directions and dip angles are evaluated by the maximum likelihood estimation method (Table 1). Then, the frequency histograms of the orientation data are drawn out and combined with the x test the probability distribution functions of the... 

As an alternative to the continuum theory, the mesoscopic approach can be based in a dynamic field theory for the tensor order parameter, Q. This tensor can be viewed as a coarse-graining of the microscopic probability distribution function 0(u, r, t) for the molecular orientation u. In this sense, Q corresponds to the symmetric, traceless part of the tensor of second moments of ip at the point r and time t ... 

Another strategy for topological bias moves involves the use of an analytical probability distribution function (for ideal, non-self-avoiding random walks) to bias the choice of trial orientations [4,35]. Again, this algorithm is applicable to relatively flexible molecules. 

Then, the probability distribution function (pdf) for the end-to-end projection R is this of a sequence of N segments with orientational pdf Q, and its square average can be derived as ... 

In addition to the density profile and the diffusion constants, one-body orientational distribution functions were also measured. The orientation angles 9 and 4> for the dipole direction vector are defined in Figure 8. For the dipole unit vector, the probability distribution functions Pd( ) and / d() in regions A-H were calculated and are plotted in Figures 9 and 10, respectively. Each point represents an average over 11°. For... 

Using this basic idea, now let us consider a system of cross-linked network in a rectangular coordinate frame of reference OXi (i = 1, 2, 3). With the assumption that these individual units are basically similar and are decoupled from their surroundings. For any arbitrary unit we can expect to obtain the components of the stress tensor (ij = 1, 2, 3) in the vicinity of the central position of that unit. If the density of the probability distribution function of orientation of all units in the network system is known together with the number of units per unit volume, the expected stress tensor can be calculated. 

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. 

Without derivation, note that Doi and Edwards developed expressions for the probability distribution function for the chain as well as the relationship between the stress and the chain orientations to arrive at an evolution equation for the stress to the chain orientation process, which is a function of the macroscopic deformations. The resulting constitutive equation is... 

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)... 

Typical shapes of the orientation distribution function are shown in figure C2.2.10. In a liquid crystal phase, the more highly oriented the phase, the moreyp tends to be sharjDly peaked near p=0. However, in the isotropic phase, a molecule has an equal probability of taking on any orientation and then/P is constant. 

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