Editing Reference hyper-netted chain
Jump to navigation
Jump to search
The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.
Latest revision | Your text | ||
Line 1: | Line 1: | ||
(Note: the reference-HNC (RHNC) is somtimes referred to as the modified-HNC (MHNC) Ref.1). | |||
Rosenfeld and Ashcroft (1979) (Ref. 1) proposed the `anzatz of universality': | Rosenfeld and Ashcroft (1979) (Ref. 1) proposed the `anzatz of universality': | ||
The basis of the method is to solve the modified | {\it ``...the bridge functions.. ..constitute the same family of curves, irrespective of the assumed pair potential".} | ||
(with inclusion of the one-parameter | |||
appropriate to | The basis of the method is to solve the modified HNC equation | ||
(related to the hard-sphere diameter) by requiring | (with inclusion of the one-parameter bridge functions | ||
Fred Lado | appropriate to hard spheres), and determine the only free parameter <math>\eta</math> | ||
noticed that the | (related to the hard-sphere diameter) by requiring thermodynamic consistency. | ||
Fred Lado \cite{PRA_1973_08_002548} and Rosenfeld and Ashcroft (1979) \cite{PRA_1979_20_001208} | |||
noticed that the OZ equation can always be written in the form | |||
:<math>\gamma_{12} = \rho \int_V (h_{13} - \gamma_{13}) h_{23} ~{\rm d}(3)</math> | :<math>\gamma_{12} = \rho \int_V (h_{13} - \gamma_{13}) h_{23} ~{\rm d}(3)</math> | ||
In view of this a hybrid solution between the | In view of this a hybrid solution between the HNC | ||
approximation and the Malijevsky-Labik formula for hard spheres | approximation and the Malijevsky-Labik formula for hard spheres \cite{MP_1987_60_0663,MP_1989_67_0431} | ||
was proposed. The philosophy of this method is that the bridge functional of the | 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. | liquid is fairly similar to that of the hard sphere fluid. | ||
See also \cite{PLA_1982_89_0196,PRA_1983_28_002374}.\\ | |||
The RHNC closure is given by (Eq. 17 | The RHNC closure is given by (Eq. 17 \cite{PRA_1983_28_002374}) | ||
:<math> c\left(r\right) = h(r) - \ln [g(r) e^{\beta \ | :<math> c\left(r\right) = h(r) - \ln [g(r) e^{\beta \phi(r)}] + B_0(r) </math> | ||
along with the constraint (Eq. 18 | along with the constraint (Eq. 18 \cite{PRA_1983_28_002374}) | ||
:<math>\rho \int [g(r) - g_0(r)] \delta B_0(r) dr_3 = 0 </math> | :<math>\rho \int [g(r) - g_0(r)] \delta B_0(r) dr_3 = 0 </math> | ||
Incorporating a reference potential $\phi_0(r)= \phi_0(r;\sigma,\epsilon)$ | |||
Incorporating a reference potential | this equation becomes (Eqs. 19a and 19b \cite{PRA_1983_28_002374}) | ||
this equation becomes (Eqs. 19a and 19b | |||
:<math>\rho \int [g(r) - g_0(r)] \sigma \frac{\partial B_0(r)}{\partial \sigma}dr_3 = 0 </math> | :<math>\rho \int [g(r) - g_0(r)] \sigma \frac{\partial B_0(r)}{\partial \sigma}dr_3 = 0 </math> | ||
and | and | ||
Line 26: | Line 27: | ||
These are the conditions that will determine the optimum values of <math>\sigma</math> and <math>\epsilon</math> | These are the conditions that will determine the optimum values of <math>\sigma</math> and <math>\epsilon</math> | ||
that minimise the free energy. | that minimise the free energy. | ||
The RHNC satisfies the 'Hiroike' termodynamic relation test | The RHNC satisfies the 'Hiroike' termodynamic relation test \cite{PRA_1983_28_002374,JPSJ_1957_12_00326}, i.e. | ||
:<math>\left.\frac{\partial U}{\partial V}\right\vert_{T} = T \left.\frac{\partial p}{\partial T}\right\vert_{V} -p</math> | :<math>\left.\frac{\partial U}{\partial V}\right\vert_{T} = T \left.\frac{\partial p}{\partial T}\right\vert_{V} -p</math> | ||
For highly asymmetric mixtures see | For highly asymmetric mixtures see \cite{JCP_2005_123_174508}. | ||
==References== | ==References== | ||
#[ | #[PRA_1979_20_001208] | ||