Spatial pair correlation function

Fluid microstructure may be characterized in terms of molecular distribution functions. The local number of molecules of type a at a distance between r and r-l-dr from a molecule of type P is Pa T 9afi(r)dr where Pa/j(r) is the spatial pair correlation function. In principle, flr (r) may be determined experimentally by scattering experiments however, results to date are limited to either pure fluids of small molecules or binary mixtures of monatomic species, and no mixture studies have been conducted near a CP. The molecular distribution functions may also be obtained, for molecules interacting by idealized potentials, from molecular simulations and from integral equation theories. 

Figure 1 shows the predicted spatial pair correlation functions, yg. reduced separation distance, r = for a typical, dilute supercritical solution using the LJ parameters given... 

As in a one-component system, the functions gap(R, R") depend only on the scalar distance R= R" — R7. Hence, for the spatial pair correlation function, we have... 

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. 

Here, pa is the average number density of molecules of species a, i.e., pa = Na)/V, with V the volume of the system. We also recall the definition of the spatial pair correlation function... 

Equation (4.12) is a connection between the cross fluctuations in the number of particles of various species, and integrals involving only the spatial pair correlation functions for the corresponding pairs of species a and p. 

First, the theory is valid for any kind of particles, not necessarily spherical particles. Only the spatial pair correlation function features in Gaig, even when the particles are not spherical. Second, no assumption on pairwise additivity of the total potential energy is invoked in the theory. Finally, we note that in this book, we discuss only classical systems the Kirkwood-Buff results, however, hold for quantum systems as well. 

Fig. 2. 39 The spatial pair correlation function for water-like particles in two dimensions. (The hydrogen bond energies she are indicated near each curve.)...

Fig. 2.42 The spatial pair correlation function for a system of water-like particles, with parameters given in (2.6.26). The HB energies shb/ bT are indicated next to each curve. The locations of the various maxima are indicated on the abscissa.

Finally, we derive some general relations between the spatial pair correlation functions of the various quasi-components. We begin with the simplest case of a two-structure model. We denote Fy gafi R) the pair correlation function for the pair of species a and p. Then, Paga iR) is the local density of an a molecule at a... 

N y/V. We also recall the definition of the spatial pair correlation function... 

Perhaps some of the most important information on the mode of molecular packing of water in the liquid state is contained in the radial distribution function, which, in principle, can be obtained by processing X-ray or neutron scattering data. There are, however, several difficulties in extracting the proper information from the experimental data. First, it should be kept in mind that the full orientation-dependent pair correlation function cannot be obtained from such an experiment. Instead, only information on the spatial pair correlation function is accessible. We recall the definition of this function,... 

Relation (7.113) is very general. First, it applies to any two-component system at chemical equilibrium, and to any classification procedure we have chosen to identify the two quasicomponents. Second, because of the application of the Kirkwood-Buff theory of solutions, we do not have to restrict ourselves to any assumptions on the total potential energy of the system. Furthermore, the quantities appearing here depend on the spatial pair correlation functions go, R), even though we may be dealing with nonspherical particles. 

Fig. 8.23. The spatial pair correlation functions for a two-component system in two dimensions. The computations were carried out by solving the four Percus-Yevick equations (8.150) with the parameters listed in (8.153). [For more details see appendix 9-E.]...

Only the spatial pair correlation functions appear in the relations, even when the particles are not spherical. 

G(r, t) - space- and time-dependent pair correlation function = Debye-Waller temperature factor Ge(r) = equilibrium spatial pair cor relation function for atoms Go(r) == instantaneous spatial pair correlation function... 

The information obtained from SANS experiments reported in the preceding section enables the prediction of the spatial pair correlation function for both dry networks and networks swollen at equilibrium in a good solvent. 

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