Ornstein-Zemike equation

D. Henderson, G. Stell, and E. Waisman, Ornstein-Zemike equation for the direct correlation function with a Yukawa tail, /. Chem. Phys. 62,4247-4259 (1975). 

Fluctuations in thermodynamics automatically imply the existence of an underlying structure that has created them. We know that such structure is comprised of molecules, and that their large number allows statistical studies, which, in turn, allow one to relate various statistical moments to macroscopic thermodynamic quantities. One of the purposes of the statistical theory of liquids (STL) is to provide such relations for liquids (Frisch and Lebowitz 1964 Gray and Gubbins 1984 Hansen and McDonald 2006). In such theories, many macroscopic quantities appear as limits at zero wave number of the Fourier transforms of statistical correlation functions. For example, the Kirkwood-Buff theory allows one to relate integrals of the pair density correlation functions to various thermo-physical properties such as the isothermal compressibility, the partial molar volumes, and the density derivatives of the chemical potentials (Kirkwood and Buff 1951). If one wants a connection between detailed correlations and integrated moments, one may ask about the nature of the wave-number dependence of these quantities. It turns out that the statistical theory of liquids allows an answer to such a question very precisely, which leads to new types of questions. The Ornstein-Zemike equation (Hansen and McDonald 2006), which is an exact equation of the STL, introduces the concept of correlation length which relates to the spatial extension of the density and/or concentration (the latter in the case of mixtures) fluctuations. This quantity cannot be accessed from pure... 

Physically pg(r) is the density of particles at distance r from a given particle. Most thermodynamic properties of interest can be computed from a knowledge of g(r) and the interparticle pair potential v(r). The starting point in calculating the radial distribution hmction is the well known Ornstein-Zemike equation [5] ... 

Moreover the components of vector A change to become Aa = aA/vA, etc. An Ornstein-Zemike (OZ) approach (referred to as the integral equation theory) describing multicomponent compressible polymer blend mixtures has been extensively investigated [35]. The multicomponent OZ equation relates the direct correlations matrix C and the total (i.e., direct and indirect) correlations matrix H as ... 

The results summarised here are for pure water at the temperature 25°C and the density 1.000 g cm , and are obtained by solving numerictdly the Ornstein-Zemike (OZ) equation for the pair correlation functions, using a closure that supplements the hypernet-ted chain (HNC) approximation with a bridge function. The bridge function is determined from computer simulations as described below. The numerical method for solving the OZ equation is described by Ichiye and Haymet and by Duh and Haymet. ... 

Equation (2.2.11) is the Ornstein-Zemike like equation for site-site correlation functions first derived by Chandler and Andersen.It is more familiar when written as an integral equation. 

The polymer RISM theory relates the set of total correlation functions (r) = g(r)-l to the set of site-site direct correlation functions, (c(r), and the set of intramolecular distribution functions, w(r), via the nonlinear site-site Ornstein-Zemike-like (SSOZ) integral matrix equation [41]... 

To calculate the properties of the RPM, one approach is to solve the relation between the direct correlation function and the pair correlation function given by the Ornstein-Zemike (OZ) integral equation. The solution of the RPM in the MSA yields a simple analytical expression for the Helmholtz energy, given by, ... 

Figure 10.14 Comparison between radial distribution functions for microemulsions obtained by means of the Ornstein-Zemike integral equation and Brownian dynamics simulations. The parameters are as in Figure lO.JOa (see text). The volume fractions are 0.16 (lower curves), 0.26 (middle curves) and 0.42 (upper curves)...

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