Reference hyper-netted chain

(Note: the reference-HNC (RHNC) is sometimes referred to as the modified-HNC (MHNC) Ref.1).

Rosenfeld and Ashcroft (1979) (Ref. 1) proposed the `anzatz of universality':

{\it ``...the bridge functions.. ..constitute the same family of curves, irrespective of the assumed pair potential".}

The basis of the method is to solve the modified HNC equation (with inclusion of the one-parameter bridge functions appropriate to hard spheres), and determine the only free parameter $\eta$ (related to the hard-sphere diameter) by requiring thermodynamic consistency. Fred Lado (Ref. 2) and Rosenfeld and Ashcroft (1979) (Ref. 3) noticed that the OZ equation can always be written in the form

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 \gamma_{12} = \rho \int_V (h_{13} - \gamma_{13}) h_{23} ~{\rm d}(3)}

In view of this a hybrid solution between the HNC approximation and the Malijevsky-Labik formula for hard spheres (Ref. 4 and 5) was proposed. The philosophy of this method is that the bridge functional of the liquid is fairly similar to that of the hard sphere fluid. (See also Ref.s 6 and 7) The RHNC closure is given by (Eq. 17 Ref. 7)

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 c\left(r\right) = h(r) - \ln [g(r) e^{\beta \phi(r)}] + B_0(r) }

along with the constraint (Eq. 18 Ref. 7)

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 \rho \int [g(r) - g_0(r)] \delta B_0(r) dr_3 = 0 }

Incorporating a reference potential 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_0(r)= \phi_0(r;\sigma,\epsilon)} this equation becomes (Eqs. 19a and 19b in Ref. 7)

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 \rho \int [g(r) - g_0(r)] \sigma \frac{\partial B_0(r)}{\partial \sigma}dr_3 = 0 }

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 \rho \int [g(r) - g_0(r)] \epsilon \frac{\partial B_0(r)}{\partial \epsilon} dr_3 = 0 }

These are the conditions that will determine the optimum values 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 \sigma} 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 \epsilon} that minimise the free energy. The RHNC satisfies the 'Hiroike' termodynamic relation test Ref. 7 and 8, i.e.

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 \left.\frac{\partial U}{\partial V}\right\vert_{T} = T \left.\frac{\partial p}{\partial T}\right\vert_{V} -p}

For highly asymmetric mixtures see Ref. 9.

References

1. [PRA_1979_20_001208]
2. [PRA_1973_08_002548]
3. [PRA_1979_20_001208]
4. [MP_1987_60_0663]
5. [MP_1989_67_0431]
6. [PLA_1982_89_0196]
7. [PRA_1983_28_002374]
8. [JPSJ_1957_12_00326]
9. [JCP_2005_123_174508]