Radial distribution function and

Figure 4.1-13 Comparison of the experimental without (—) and with (—) triphenylphosphine at (solid line) and fitted (dashed line) (a) EXAFS 80 °C and in the presence of triphenylphosphine and (b) pseudo-radial distribution functions and reagents at 50 °C for 20 min (—). Repro-...

Examination of the EXAFS formulation in wave vector form reveals that it consists of a sum of sinusoids with phase and amplitude. Sayers et al32 were the first to recognize the fact that a Fourier transform of the EXAFS from wave vector space (k or direct space) to frequency space (r) yields a function that is qualitatively similar to a radial distribution function and is given by ... 

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. 

Figure 8. The structure of hydrated Na and CP ions at the water/Pt(IOO) interface (dotted lines) compared with the structure in bulk water (solid lines). In the two top panels are the oxygen ion radial distribution functions, and in the two bottom panels are the probability distribution functions for the angle between the water dipole and the oxygen-ion vector for water molecules in the first hydration shell. (Data adapted from Ref. 100.)...

The calculations of g(r) and C(t) are performed for a variety of temperatures ranging from the very low temperatures where the atoms oscillate around the ground state minimum to temperatures where the average energy is above the dissociation limit and the cluster fragments. In the course of these calculations the students explore both the distinctions between solid-like and liquid-like behavior. Typical radial distribution functions and velocity autocorrelation functions are plotted in Figure 6 for a van der Waals cluster at two different temperatures. Evaluation of the structure in the radial distribution functions allows for discussion of the transition from solid-like to liquid-like behavior. The velocity autocorrelation function leads to insight into diffusion processes and into atomic motion in different systems as a function of temperature. 

The interaction parameters for the water molecules were taken from nonempirical configuration interaction calculations for water dimers (41) that have been shown to give good agreement between experimental radial distribution functions and simulations at low sorbate densities. The potential terms for the water-ferrierite interaction consisted of repulsion, dispersion, and electrostatic terms. The first two of these terms are the components of the 6-12 Lennard-Jones function, and the electrostatic term accounts for long-range contributions and is evaluated by an Ewald summation. The... 

Despite the formation of clathrate-like clusters and complete 512 cages during these simulations, the increased ordering observed from the radial distribution functions and local phase assignments resulted in the authors concluding that their simulation results are consistent with a local order model of nucleation, and therefore do not support the labile cluster model. 

In order to use the above expressions for calculating the thermodynamic properties, appropriate expressions for the radial distribution function and for the equation of state for the hard-sphere reference system are required which are given in Appendix A. Fortunately, accurate information for the hard-sphere fluid as well as for the hard-sphere solid is available and this enables the determination of the properties of the coexisting dilute and concentrated phases of colloidal dispersions. 

Radial Distribution Functions and Reference Interaction Site Model... 

We should hasten to note that these fundamental difficulties do not mean that this theory does not often work. The most common application of IBC theory points to its particularly simple prediction for the dependence of relaxation rates on the thermodynamic state of the solvent with the Enskog estimate of collision rates, the ratio of vibrational relaxation rates at two different liquid densities p and p2 is just the ratio of the local solvent densities [pigi(R)//02g2(R)], where g(r) is the solute-solvent radial distribution function and R defines the solute-solvent distance at... 

Monte Carlo techniques were first applied to colloidal dispersions by van Megen and Snook (1975). Included in their analysis was Brownian motion as well as van der Waals and double-layer forces, although hydrodynamic interactions were not incorporated in this first study. Order-disorder transitions, arising from the existence of these forces, were calculated. Approximate methods, such as first-order perturbation theory for the disordered state and the so-called cell model for the ordered state, were used to calculate the latter transition, exhibiting relatively good agreement with the exact Monte Carlo computations. Other quantities of interest, such as the radial distribution function and the excess pressure, were also calculated. This type of approach appears attractive for future studies of suspension properties. 

A distribution function that describes the end-to-end distance regardless of direction can be obtained by fixing one end of the chain at the origin of a coordinate system and then finding the probability that the other end lies in an element of volume AytF dR (Figure 8-33). This is called the radial distribution function and is simply given by Equation 8-7 ... 

Structures of liquids in general are dominated by influences of intermolecular repulsions. Intermolecular attractions have a comparatively minor effect on the radial distribution function and, in the case of asymmetric molecules, on intermolecular correlations as well. At the high density and close packing prevailing in the liquid state, the spatial arrangement of the molecules of a liquid can be satisfactorily described, therefore, by representing the molecules as hard bodies of appropriate size and shape whose only interactions are the excessive repulsions that would be incurred if one of them should overlap another. Once the liquid structure has been characterized satisfactorily on this basis, one may take account of the intermolecular attractions by averaging them over the molecular distribution thus determined. Mean-field theories are useful in this connection. 

New MD simulations were performed using the set of parameters summarized in Table 13. The potential functions and parameters obtained as described in the previous sections were used to test the effect of different potential functions on radial distribution functions and translational times. Structural and dynamical results obtained are summarized in Table 14. We report values of R[o, the minimum after the first peak of the radial distribution... 

One of our objectives is to determine if the present model of copper can reproduce the variation of its properties with temperature. Since the simulations at 300 K and 1000 K that were discussed above were performed with different cutoffs, there is a potential element of inconsistency in comparing the two sets of results. We showed earlier that this is not significant for the radial distribution function and the mean square fluctuation, which are relatively insensitive to the cutoff being used. The coefficient of thermal expansion, on the other hand, was found to be much more dependent upon this parameter. Accordingly we repeated the 300 K simulation using the same cutoff, 5.40 A, as at 1000 K. 

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