Pair correlation function in liquids

Figure 4.32 Pair correlation function in liquid water. The first peak of 0-0 corresponds to...

A.K. Soper and R.N. Silver, Hydrogen-hydrogen pair correlation function in liquid water, Phys. Rev. Lett., 49 (1982) 471-474. 

This is almost the same as the solvaton-solvaton pair correlation function in liquids (see, for example, section 7.15). The only difference is the appearance of the condition P, i.e., the presence of the polymer. 

Pair correlation function in molecular theory of liquids (Part 6). 

The local structure in liquids can be measured by X-ray diffraction and described by either a radial distribution function or the pair correlation function. In particular, the oxygen-oxygen pair correlation function or reduced radial distribution function for water, goo(r) Fig. 11.74, can be obtained from... 

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. 

The above analysis demonstrates the importance of the pair correlation function in estimation of the thermodynamic properties of simple liquids. In the following section, the properties of the simplest fluid, namely, one based on non-interacting hard spheres, are developed on the basis of the relationships presented in this section. 

Fig. 2. Schematic illustration of the relationship between the pair correlation function in an ordered crystalline solid and the corresponding disordered liquid...

More modem approaches borrow ideas from the liquid state theory of small molecule fluids to develop a theory for polymers. The most popular of these is the polymer reference interaction site model (PRISM) theory " which is based on the RISM theory of Chandler and Andersen. More recent studies include the Kirkwood hierarchy, the Bom-Green-Yvon hierarchy, and the perturbation density functional theory of Kierlik and Rosinbeig. The latter is based on the thermodynamic perturbation theory of Wertheim " where the polymeric system is composed of very sticky spheres that assemble to form chains. For polymer melts all these liquid state approaches are in quantitative agreement with simulations for the pair correlation functions in short chain fluids. With the exception of the PRISM theory, these liquid state theories are in their infancy, and have not been applied to realistic models of polymers. 

Unlike the solid state, the liquid state cannot be characterized by a static description. In a liquid, bonds break and refomi continuously as a fiinction of time. The quantum states in the liquid are similar to those in amorphous solids in the sense that the system is also disordered. The liquid state can be quantified only by considering some ensemble averaging and using statistical measures. For example, consider an elemental liquid. Just as for amorphous solids, one can ask what is the distribution of atoms at a given distance from a reference atom on average, i.e. the radial distribution function or the pair correlation function can also be defined for a liquid. In scattering experiments on liquids, a structure factor is measured. The radial distribution fiinction, g r), is related to the stnicture factor, S q), by... 

Hwang L-P and Freed J H 1975 Dynamic effects of pair correlation functions on spin relaxation by translational diffusion in liquids J. Chem. Rhys. 63 4017-25... 

Figure. 3 (a) Partial pair correlation function.s gij(B.) in liquid K-Sb alloys, (b) Total, partial, and local electronic densities of states in liquid Ko.soSbo.so- Cf. text. 

A quantity of central importance in the study of uniform liquids is the pair correlation function, g r), which is the probability (relative to an ideal gas) of finding a particle at position r given that there is a particle at the origin. All other structural and thermodynamic properties can be obtained from a knowledge of g r). The calculation of g r) for various fluids is one of the long-standing problems in liquid state theory, and several accurate approaches exist. These theories can also be used to obtain the density profile of a fluid at a surface. 

Much more detailed information about the microscopic structure of water at interfaces is provided by the pair correlation function which gives the joint probability of finding an atom of type/r at a position ri, and an atom of type v at a position T2, relative to the probability one would expect from a uniform (ideal gas) distribution. In a bulk homogeneous liquid, gfn, is a function of the radial distance ri2 = Iri - T2I only, but at the interface one must also specify the location zi, zj of the two atoms relative to the surface. We expect the water pair correlation function to give us information about the water structure near the metal, as influenced both by the interaction potential and the surface corrugation, and to reduce to the bulk correlation Inunction when both zi and Z2 are far enough from the surface. 

The relaxation equations for the time correlation functions are derived formally by using the projection operator technique [12]. This relaxation equation has the same structure as a generalized Langevin equation. The mode coupling theory provides microscopic, albeit approximate, expressions for the wavevector- and frequency-dependent memory functions. One important aspect of the mode coupling theory is the intimate relation between the static microscopic structure of the liquid and the transport properties. In fact, even now, realistic calculations using MCT is often not possible because of the nonavailability of the static pair correlation functions for complex inter-molecular potential. 

It will be useful now to review some elementary facts regarding the structure of liquids at equilibrium. When a crystalline solid melts to form a liquid, the long range order of the crystal is destroyed. However, a residue of local order persists in the liquid state with a range of several molecular diameters. The local order characteristic of the liquid state is described in terms of a pair correlation function, g-i(R)> defined as the ratio of the average molecular density, p(R), at a distance R from an arbitrary molecule to the mean bulk density, p, of the liquid... 

The pair correlation function is a short range quantity in liquids, decaying to unity after a few molecular diameters, the correlation length However, in supercritical fluids g(r) has a much longer range and t, becomes considerably larger than the mean inter-molecular separation at the critical density. For instance, for carbon dioxide , = 5.5 nm at Tc compared to the mean intermolecular separation of 0.55 nm (Eckert, Knutson and Debenedetti 1996). 

The solvation structure around a molecule is commonly described by a pair correlation function (PCF) or radial distribution function g(r). This function represents the probability of finding a specific particle (atom) at a distance r from the atom being studied. Figure 4.32 shows the PCF of oxygen-oxygen and hydrogen-oxygen in liquid water. 

The liquid-like order present in the pair correlation function manifests itself as a peak in the static structure factor (S(q)). The scaling of the position qm of this maximum with the density has attracted much attention in the literature [40, 51-53]. Scaling arguments suggest [35, 42, 49, 51] that qm obeys the relation qm p1/3 for dilute solutions and qm pv/(3v 1) for semidilute solutions. Here v is the scaling exponent for the end-to-end distance, i.e., RE hT. The overlap threshold concentration is estimated as p N1 3v. As a conse-... 

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