With this notebook you can calculate:

(1)  the Laplace transform G_{i,j}(s) of  r g_{i,j}(r) (given by the 
Mathematica function G[i_,j_,eeta_,con_,ratio_,alfa_,yy_]), g_{i,j}(r) being 
the radial distribution function, and 
(2)  the  cavity function at distance r  (yr[i_,j_,r_,alfa_,yy_])

 by means of  the RFA method  [A. Santos, S. B. Yuste and M. López de Haro, (
J. Chem. Phys. vol 109, num. 16, 22 October 1998)] for a BINARY mixture of 
STICKY Hard Spheres (this "mixture" can be one-component) and for 
-> a given value (alfa) of the \[Alpha]  parameter, and 
-> given values (yy) of the cavity function at contact.

The parameter alpha can obtained by means of the notebook alphashs.nb. In 
the notebook ycon_chi.nb we show how to estimate the contact values of the 
cavity function and the isothermal susceptibility of a SHS mixture by means of
 formulas (3.1) and (3.2) of the paper referred above.

The mixture is defined by:
-> nco=number of species
-> eeta=volume packing fraction
-> con=list with the molar fractions of each species,e.g, {x_1,x_2}, 
where  x_1 is the molar fraction of the species with the largest diameter
-> ratio=list with the diameter ratios, e.g., {1,ratio_2} 
 where ratio_2=sigma_2/sigma_1=sigma_2
-> taus=list with the tau parameters characterizing the adhesiveness of the 
intermolecular potential , e.g., {{tau_11,tau_12},{tau_21,tau_22}}.

Here you have to introduce the values of nco, eeta, con , ratio and taus  
corresponding to your sticky-hard-sphere mixture

These are the values for our Example  1.

In this example we will use the value  alpha= 0.020153002097351269`, which
 corresponds to the isothermal compresibility  1/14.8617 and to the cavity 
functions at contact  y[1,1,nco]=y[2,2,nco]=2.88962; y[1,2,nco]=y[2,1,nco]=
2.80328  (see the notebook alphashs.nb)

nco=2;   
eeta=0.4;
con={1/2,1/2};
ratio={1,1};
taus={{1,10},{10,1}};

These are the values for our Example  2.

In this example we will use the value  alpha= 0.00667728271322821`, 
which corresponds to the isothermal compresibility  1/11.682802 and to the 
cavity functions at contact  y[1,1,nco]=2.603812; y[1,2,nco]=2.4605960; y[2,1,
nco]=2.4605960; y[2,2,nco]=2.4910791;  (see the notebook alphashs.nb)

nco=2;
 con={2/3,1/3};
ratio={1,1/2};
eeta=0.4;
taus={{1,10},{10,1}};

numterm is a parameter used in the inversion of the Laplace transform G_ij(s).
  A good value is numterm=100, but if you prefer faster calculations  and are 
not very concerned with the precision of the results, try a smaller number . 
For example, numterm=50 leads also to very good results and  halves the  
computation times.

numterm=100;

Now  we evaluate the Laplace transform, G_ij(s), of   r*g_ij(r)  as prescribed
 by the RFA method
        You must evaluate this cell again if you change the mixture

Examples

Example 1.    A binary mixture with
nco=2;   
eeta=0.4;
con={1/2,1/2};
ratio={1,1};
taus={{1,10},{10,1}};

alpha=0.0201530020973512691;
y[1,1,nco]=y[2,2,nco]=2.88962; y[1,2,nco]=y[2,1,nco]=2.80328;

In this example we have used the value  alpha= 0.0201530020973512691, 
which corresponds to the isothermal susceptibility  1/14.8617 and to the 
cavity functions at contact given above.

alpha1=0.0201530020973512691;
ycontact1={{2.88962,2.80328},{2.80328,2.88962}};

Here we have used numterm=100

{yr[1,1,1.05,alpha1,ycontact1],
yr[1,2,1.05,alpha1,ycontact1]}

Example 2.    A binary mixture with
nco=2;
 con={2/3,1/3};
ratio={1,1/2};
eeta=0.4;
taus={{1,10},{10,1}};

alpha=0.00667728271322821065;
y[1,1,nco]=2.60381230280949838; y[1,2,nco]=2.46059600372653219; 
	y[2,1,nco]=2.46059600372653219; y[2,2,nco]=2.4910791716714335;

In this example we have used the value  alpha=0.00667728, which corresponds
 to the isothermal susceptibility chivalue=1/11.6828023268096048; and to the 
cavity functions at contact given above.

alpha2=0.00667728271322821065;
ycontact2=
{{2.60381230280949838,2.46059600372653219},
{2.46059600372653219,2.49107917167143355}};

Here we have used numterm=100

{yr[1,1,1.05,alpha2,ycontact2],
yr[1,2,0.76,alpha2,ycontact2],
  yr[2,2,0.51,alpha2,ycontact2]}