Martynov Sarkisov

Martynov and Sarkisov proposed an expansion of the Bridge function in terms of basis functions:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(\rho, T, r)= - \sum_{i=1}^\infty A_i (\rho,T) \phi^i (\rho, T, r)}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} is the chosen basis function and $A_{i}$ are the coefficients determined from thermodynamic consistency conditions. The Martynov-Sarkisov closure is based on the expansion of the Bridge function in powers of the thermal potential.

(1983 Eq.16 Ref. 1) closure in terms of the bridge function, for hard spheres, is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B[\omega(r)]= - A_2 \omega(r_{12})^2 = \sqrt{(1+2\gamma(r))}-\gamma(r) -1}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(r)} is the thermal potential and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_2=1/2} . (This closure formed the basis for the Ballone-Pastore-Galli-Gazillo closure for hard sphere mixtures). Charpentier and Jaske (Ref. 2) have observed that the value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_2} differs drastically from 0.5 for temperatures greater than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^*\approx 2.74} , thus the MS closure is deficient in the supercritical domain.

References

1. [MP_1983_49_1495]
2. [JCP_2001_114_02284]
3. [JCP_1993_99_03926]
4. [JCP_1999_110_03961]
5. [JCP_2001_114_09496]