Orientation probability function

The orientation can be viewed in more general terms in terms of an orientation probability function /(), where the angle (j> is the angle between the director and the molecular segment. It is implicit in this treatment that the orientation in the plane perpendicular to the director is random, i.e. that the orientation is uniaxial (Figure 10.7). The in-plane orientation is the same in the zx and zy planes. It is possible to represent /(( ) by a series of spherical harmonic functions. 

In nmr spectroscopy the solute orientation is often described by an orientational probability function P(0,0) [45]. TO,0) is a measure for the probability (per unit solid angle) that the applied magnetic field (or the optic axis of the liquid crystal) assumes the spherical coordinates (dy 0) in the molecule-fixed coordinate system (x, y, z). P(0, 0) can be expressed in terms of spherical harmonics of second order as follows [45] ... 

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. 

Fig. 2. Schematic representation of the orientational distribution function f 6) for three classes of condensed media that are composed of elongated molecules A, soHd phase, where /(0) is highly peaked about an angle (here, 0 = 0°) which is restricted by the lattice B, isotropic fluid, where aU. orientations are equally probable and C, Hquid crystal, where orientational order of the soHd has not melted completely.

A small step rotational diffusion model has been used to describe molecular rotations (MR) of rigid molecules in the presence of a potential of mean torque.118 120,151 t0 calculate the orientation correlation functions, the rotational diffusion equation must be solved to give the conditional probability for the molecule in a certain orientation at time t given that it has a different orientation at t = 0, and the equilibrium probability for finding the molecule with a certain orientation. These orientation correlation functions were found as a sum of decaying exponentials.120 In the notation of Tarroni and Zannoni,123 the spectral denisities (m = 0, 1, 2) for a deuteron fixed on a reorienting symmetric top molecule are ... 

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. 

The state of particle orientation at a point can be fully described by an orientation distribution function, x, y). The distribution is defined such that the probability of a particle, located at x and y at time t, being oriented between angles i and 4>2, is given by... 

Essentially different situation is encountered in the case of a high potential barrier or, equivalently, at low temperatures both these conditions are expressed by the relation ct 1. Applying the Boltzmann law (4.24) to the two-well potential (4.33), we arrive at the conclusion that the orientational probability is almost totally localized in exponentially small vicinities of the directions If = 0 and vf = re. It is also obvious that in a system with the energy function (4.33) at full equilibrium, the populations of both wells are equal. 

The distributions of both segment and bond densities allow us to calculate any structural property of the mono-layer. The bond orientational probabilities (forward, lateral, and backward) as well as the bond order parameter of a chain as a function of bond number along the chain are given below. 

FIG. 5. Bond orientational probability as a function of bond number accounted from the attached end. 

Bond orientational probabilities as a function of bond number to the attached end are plotted in Fig. 5 for a bmsh with 400 bonds. The first bond has a large forward probability. The orientational probability of a bond to be lateral increases rapidly while that of a bond to be forward decreases rapidly with increasing bond number. After a small critical layer number, the orientational probability for a bond to be lateral increases slowly while that to be forward decreases... 

Nematic phases are characterised by a uniaxial symmetry of the molecular orientation distribution function f(6), describing the probability density of finding a rod with its orientation between 6 and 6 + d0 around a preferred direction, called the director n (see Fig. 15.49). An important characteristic of the nematic phase is the order parameter (P2), also called the Hermans orientation function (see also the discussion of oriented fibres in Sect. 13.6) ... 

Conformer sequence probabilities Radial distribution functions Scattering functions Orientation correlation functions Mechanical properties Distribution of free volume... 

In quantum mechanics, in characteristic contrast to classical mechanics, a freely movable polyatomic molecule has a centrally symmetric and, particularly in its lowest state, a spherically symmetric structure, i.e. a spherically symmetric probability function. That means that on the average, even in its lowest state, a free molecule does not prefer any direction, it changes its orientation permanently owing to its zero-point motion. If another molecule tries to orientate the molecule in question a compromise between the zero-point motion and the directing power will be made, but only for... 

Probably the most critical question one needs to address in understanding the structure in a molecular liquid is where, in the space defined by the local frame of the central molecule, are we likely to find a neighboring particle. Only after having localized this neighboring particle can we begin to worry about its orientation. The function that provides a direct answer to this question is what we have termed the spatial distribution function (SDF)... 

Let y(t) be some function of time, such as the orientation of the intemuclear vector and some other function f (e.g. the dipolar interaction) that depends on y. It is possible to define a probability function p(y,t) which is the probability that at a time t, the inter-nuclear vector has some orientation y. Then... 

There are a number of models for polarization of heterogeneous systems, many of which are reviewed by van Beek (23). Brown has derived an exact, though unwieldly, series solution using point probability functions (24). For comparison to spectra for the thermoplastic elastomers of interest here, the most useful model seems to be the one derived by Sillars (25) and, in a slightly different form, by Fricke (26). The model assumes a distribution of geometrically similar ellipsoids with major radii, r-p and rj which are randomly oriented and randomly distributed in a dissimilar matrix phase. Only non-specific interactions between neighboring ellipsoids are included in the model. This model includes no contribution from the polarization of mobile charge carriers trapped on the interfacial surfaces. 

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.

A complete description of the texture (or preferred orientation) is formulated as a probability for finding a particular crystallite orientation within the sample this is the orientation distribution function (ODF). For an ideally random powder the ODF is the same everywhere (ODF=l) while for a textured sample the ODF will have positive values both less and greater than unity. This ODF can be used to formulate a correction to the Bragg intensities via a fourdimensional surface general axis equation) that depends on both the direction in reciprocal space and the direction in sample coordinates ... 

Molecular order is descnhedhy the orientational distribution function P(0) [Mcbl]. This is the probability density of finding a preferential direction n in the sample under an angle 0 in a molecule-fixed coordinate frame (Fig. 3.2.2(a)). For simplicity, macroscopically uniaxial samples with cylindrically symmetric molecules are considered. Then, one angle is sufficient to characterize the orientational distribution function. In practice, not the angle 0 itself but its cosine is used as the variable and for weak order the distribution function is expanded into Legendre polynomials P/(cos 0),... 

Fig. 2. Increased acyl chain unsaturation and increased temperature produce dissimilar increases in acyl chain disorder. (A) Difference order-parameter profiles, S(n), of deuterium NMR measurements on perdeuterated 18 0 in the sn-1 position as a function of changes in temperature and unsaturation at the 5n-2 position. A,SYm) between 18 0, 22 6 PC and 18 0, 18 1 PC O A5fn) for 18 0, 18 1 PC between 27°C and47°C. Carbon atoms are numbered beginning at the glycerol backbone. (Data from Gawrisch and Holte, 1996 used by permission of K. Gawrisch). (B) Difference in orientation probability for the fluorescent membrane probe DPH. Orientation distribution of DPH in 16 0,18 1 PC at 40°C minus that in 20°C (—), and the distribution of di22 6 PC minus that of 16 0,18 1 PC at 20°C (—).

AJV = Wj - JVjj is the net anisotropic anchoring constant. NPs tend to orient parallel to n for AW>0, and recent experiments suggest this condition. Furthermore, taking into account Eq. (16) the distribution probability function of the NPs within a homogeneously aligned nematic LC phase... 

Once the probabilities are known, other physical quantities, which are function of the occupation probabilities, can be calculated from (A) — J2yPy y- or order parameters for order-disorder phase transitions. Different examples will appear in the following. For instance, the orientational contribution to the absolute polarization of the ferroelectric compound pyridinium tetrafluoroborate was estimated from 2H NMR temperature-dependent measurements on the perdeuterated pyridinium cations.116 The pyridinium cation evolves around a pseudo C6 axis, and the occupation probabilities of the different potential wells were deduced from the study of 2H NMR powder spectra at different temperatures. The same orientational probabilities can be used to estimate the thermodynamical properties, which depend on the orientational order of the cation. Using a generalized van t Hoff relationship, the orientational enthalpy changes were calculated and compared with differential scanning calorimetry (DSC) measurements.116... 

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