PyOZ

A python implementation of a solver to the Ornstein–Zernike equation

In the integral theory of fluid ¹, the Ornstein–Zernike equation ² connects the radial distribution function with the direct correlation function through that

$$ h(\boldsymbol{r}) = c(\boldsymbol{r}) + \rho \int d\boldsymbol{s}\ c(\boldsymbol{r}-\boldsymbol{s}) h(\boldsymbol{s})$$

where $h(r) = g(r)-1$ is the total correlation function, $g(r)$ is the radial distribution function, and $c(r)$ is the direct correlation function.

To solve this equation we need a closure relation. We can define a auxiliary quantity $\gamma(r)=h(r)-c(r)$, such that

$$c(r) = e^{-\beta u(r)+\gamma(r)+b(r)}-\gamma(r)-1$$

and $b(r)$ is the bridge-function. The following closure relations gives different bridge functions:

• Percus-Yevick (PY) - $b(r) = \log(1+\gamma(r))-\gamma(r)$
• HiperNetted Chain (HNC) - $b(r) = 0$
• Mean-Spherical Approximation (MSA) -

Examples

On the folder 'examples' you can find different applications of the OZ solver.

Hard-Sphere Fluids

Fig.1 - The radial distribution function of a pure hard-sphere fluid for three different densities. The symbols represent MC data.	Fig.2 - The contact value of the radial distribution function of a pure hard-sphere fluid as a function of the bulk density. The symbols represent MC data.

References

Footnotes

1. Hansen, Jean-Pierre, and Ian Ranald McDonald. Theory of simple liquids: with applications to soft matter. Academic press, 2013. ↩

2. Ornstein, L.S.; Zernike, F. Accidental deviations of density and opalescence at the critical point of a single substance. Proceedings of the Royal Netherlands Academy of Arts and Sciences. 17 (1914): 793. ↩