Orientational pair correlations

Figure 3. (a) Two-dimensional, bond orientational order parameter average values in the molecular fluid layers of LI ecu confined in a multi-walled carbon nanotube of diameter D=9norder parameter values for the contact, second, third and fourth layers, respectively. The dotted line represents the bulk solid-fluid transition temperature, (b) Positional and orientational pair correlation functions in the unwraiqred contact layer of U CCU confined in a multi-walled carbon nanotube of diameter D=9.1< (5 nm) showing liquid phase at 7=262 K and crystal phase at 7=252 K. 

From absolute total intensity light scattering (Malmberg and Lippincott, 1965) measurements the value of the static orientational pair-correlation term fN for neat chloroform is found to be 1.0 .01. Using this value of fN, the concentration-dependence of rt, and the single-particle reorientation time is in Eq. (12.3.29), the value of the dynamic correlation parameter gN is found to be 0.0 0.1 for neat chloroform. This value of gN is also consistent with the value of t t for the whole range of solutions studied. 

The integrated anisotropic spectrum, i.e. the total depolarized scattering intensity, has frequently been used by experimentalists to study orientational pair correlations in molecular liquids. This total intensity I is proportional to the correlation function... 

Here p is the particle number density and N the number of particles in the scattering volume. The quantity <1 orientational pair correlation factor... 

Here, Xj g, the light scattering correlation time is related to the single particle correlation time Tg = 1/(6Dj ), where Dj is the rotational txwnbling diffusion constant of the molecule, 92 the orientational pair correlation factor introduced above and j2... 

As pointed out in section 3 the total depolarized Rayleigh intensity provides information on the orientational pair correlation factor 92 defined by equation (20). Equation (19) indicates, however, that the total intensity is made up not only of an orientational but also of a collision induced contribution, which must be removed. As an example as to how 92 can be... 

Although the large value of g2 in CgPg points to orientational pair correlations in this liquid, the values close to unity in CS2 and CgHg should not be taken... 

The structure of a fluid is characterized by the spatial and orientational correlations between atoms and molecules detemiiued through x-ray and neutron diffraction experiments. Examples are the atomic pair correlation fiinctions (g, g. . ) in liquid water. An important feature of these correlation functions is that... 

The interfacial pair correlation functions are difficult to compute using statistical mechanical theories, and what is usually done is to assume that they are equal to the bulk correlation function times the singlet densities (the Kirkwood superposition approximation). This can be then used to determine the singlet densities (the density and the orientational profile). Molecular dynamics computer simulations can in... 

Spohr provides a detailed discussion of the water pair correlation function at the water/Pt(100) interface." His results are shown in Fig. 3 for the oxygen-oxygen, oxygen-hydrogen and hydrogen-hydrogen pair correlations when one of the reference atoms is in either the first or the second layer, but otherwise a complete averaging over the locations and orientations of the other atom has been performed. The pair correlations... 

The internal dynamics of a short sequence in a chain is studied according to the dynamic RIS scheme. Conformational transitions with dynamic pair correlations are considered. Resistance to dynamic rearrangements resulting from environmental effects and constraints operating at the ends of a sequence are incorporated into the calculation scheme. Calculations for a short sequence in a PE chain show that pair correlations do not significantly affect the orientational relaxation of a vector affixed to a bond in the sequence. Contributions from constraints, on the other hand, are dominant and slow down the orientational motions. 

In the study of structural properties of fluids of particles interacting through orientation-dependent potentials, it is convenient to decompose the full pair-correlation function A(l,2) = h ri2,Qi, Q2) ... 

B. M. Ladanyi and T. Keyes. The role of local fields and interparticle pair correlations in light scattering by dense fluids I. Depolarized intensities due to orientational fluctuations. 

In this book, we shall only be interested in homogeneous and isotropic fluids. In such a case, there is a redundancy in specifying the full configuration of the pair of particles by 12 coordinates (X, X"). It is clear that for any configuration of the pair X, X", the correlation g(X, X ) is invariant to translation and rotation of the pair as a unit, keeping the relative configuration of one particle toward the other fixed. Therefore, we can reduce to six the number of independent variables necessary for the full description of the pair correlation function. For instance, we may choose the location of one particle at the origin of the coordinate system, R = 0, and fix its orientation, say, at (j) = O = t// = 0. Hence, the pair correlation function is a function only of the six variables X" = R", S2". 

Similarly, the function g(Rl,R") is a function only of the scalar distance R= R" — R. For instance, R may be chosen at the origin R = 0, and because of the isotropy of the fluid, the relative orientation of the second particle is of no importance. Therefore, only the separation R is left as the independent variable. The function g(R), i.e., the pair correlation function expressed explicitly as a function of the distance R, is often referred to as the radial distribution function. This function plays a central role in the theory of fluids. 

The compressibility equation involves the radial distribution function even when the system consists of nonspherical particles. We recall that previously obtained relations between, say, the energy or the pressure, and the pair correlation function were dependent on the type of particle under consideration. The compressibility depends only on the spatial pair correlation function. If nonspherical particles are considered, it is understood that g(R) in (3.109) is the average over all orientations (3.105). In the following, we shall remove the bar over g(R). We shall assume that the angle average has been taken before using the compressibility equation. 

One of the most important applications of the theory of PS is to biomolecules. There have been numerous studies on the effect of various solutes (which may be viewed as constituting a part of a solvent mixture) on the stability of proteins, conformational changes, aggregation processes, etc., (Arakawa and Timasheff 1985 Timasheff 1998 Shulgin and Ruckenstein 2005 Shimizu 2004). In all of these, the central quantity that is affected is the Gibbs energy of solvation of the biomolecule s. Formally, equation (8.26) or equivalently (8.28), applies to a biomolecule s in dilute solution in the solvent mixture A and B. However, in contrast to the case of simple, spherical solutes, the pair correlation functions gAS and gBS depend in this case on both the location and the relative orientation of the two species involved (figure 8.5). Therefore, we write equation (8.26) in an equivalent form as ... 

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