Ornstein-Zernicke equation

J. S. H0ye and G. Stell, Ornstein-Zernicke equation with a core condition and direct correlation function of Yukawa form. Mol. Phys. 32, 195-207 (1976). 

The integral equations themselves are obtained by inserting various approximations to the direct correlation function c, (r) into the Ornstein-Zernicke equation. It is common practice to omit the bridge graphs Bij in these approximations. 

Once the integral equation is solved together with the Ornstein-Zernicke equation for a given potential Uijir), equations of, the type of Eqs. (69) and (70) may be used to obtain thermodynamic properties from the resulting g,y(r) functions. At the MM-level, Eq. (69) yields the osmotic pressure H instead of the total pressure p of the solution. 

These conditions together with the Ornstein-Zernicke equation allow the calculation of c, (r) at short distances and gijir) at long distances. This approximation is called... 

The starting point of many modern theories of fluids is the Ornstein Zernicke equation (0Z)(10), which relates the direct correlation function c j(r) and the total correlation function h j(r)... 

Equation 12 can also be considered as an Ornstein-Zernicke equation describing the degree of thermal composition fluctuations of correlation length The correlation length is evaluated from V 2 S(0) and becomes infinite at the critical point as described by the scaling law The... 

A adsorption model for a templated porous material may be posed in terms of seven replica Ornstein-Zernicke (ROZ) integral equations [58-60] that relate the direct and total correlation functions, c, (r) and hij (r), respectively, of the matrix-adsorbate system... 

The mode coupling approximation for m (0 yields a set of equations that needs to be solved self-consistently. Hereby the only input to the theory is the static equilibrium structure factor 5, that enters the memory kernel directly and via the direct correlation function that is given by the Ornstein-Zernicke expression = (1 - l/5,)/p, with p being the average density. In MCT, the dynamics of a fluid close to the glass transition is therefore completely determined by equilibrium quantities plus one time scale, here given by the short-time diffusion coefficient. The theory can thus make rather strong predictions as the only input, namely, the equilibrium structure factor, can often be calculated from the particle interactions, or even more directly can be taken from the simulations of the system whose dynamics is studied. 

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