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					THERMODYNAMICS OF BINARY LIQUID METAL ALLOYS IN THE FRAMEWORK OF THE ORPA AND PSEUDOPOTENTIAL THEORY
N.E.Dubinin, T.V.Trefilova, N.A.Vatolin Institute of Metallurgy, Ekaterinburg, Russia Abstract We represent the application of the optimized random phase approximation (ORPA) to the thermodynamic study of binary liquid metal alloys. The ORPA is more complicated method of the thermodynamic perturbation theory (TPT) which describes some effects of the perturbation on the structure. Pair interactions are estimated in the framework of the local Animalu-Heine model pseudopotential with the VashishtaSingwi approximation for the exchange-correlation function. Earlier, we used this pseudopotential model for the same purpose in conjunction with other TPT methods (Weeks-Chandler-Andersen, and variational). It is shown that the ORPA results are more accurate than those obtained by using the above mentioned TPT methods. INTRODUCTION The optimized random phase approximation (ORPA) was suggested by Andersen et.al. in series of works [1-4] as an extension of the random phase approximation (RPA) developed also in these papers. Both of these methods are related to TPT (thermodynamic perturbation theory) approaches going beyond the high-temperature approximation (HTA) for the calculation of effects of the attractive forces on the liquid structure. Initially, ORPA was used for study of models with relatively uncomplicated pair interactions such as the ionic solutions [2,3,5], the Lennard-Jones fluid [4,5], the square-well fluid [6,7]. Since ORPA successfully describes systems named above this method was applied by a number of authors to metallic fluids [8-11]. Such a transition from the model systems to metals is not trivial because in the last case the pseudopotential theory must be taken into account. Generalization of the ORPA together with the pseudopotential method to binary mixtures was made by Kahl and Hafner [12-14]. All of the named works on pure liquid metals [8-11] and binary metal alloys [12-14] are concerned with the description of fluid structure only. The ORPA formalism for calculation of thermodynamic properties for binary metal alloys was presented in our previous study [15]. Earlier, the analogous one, WCA (WeeksChandler-Andersen [16]), was developed by us in [17], where we have demonstrated an advantage of the WCA approach in comparison with the variational method [18] of the TPT for a quantitative description of metal alloys thermodynamics. Note, the ORPA is closely related to the WCA because the WCA separation of the pair interatomic potential of the system under consideration is the starting point for the ORPA procedure. The present work is devoted to calculations of thermodynamic properties (internal energy of mixing, and success entropy of mixing) of six equiatomic alkalialkali liquid alloys by means of the ORPA and local Animalu-Heine (AH) model

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pseudopotential [19,20] near a melting temperature. The ORPA-AH results obtained here are compared with the WCA-AH once obtained earlier in ref.[17]. THEORY The thermodynamic perturbation theory is based on the assumption that the potential energy of an actual fluid, U, can be represented as the sum of two terms the first of which is responsible for the repulsive forces in the system and the second - for the attractive ones [21]:
U  U 0  U1

(1)

where U0 is the potential energy of a reference system and U1 is a perturbation of the potential energy. Hereafter, a reference system and perturbation characteristics will be marked by “0” and “1”, respectively. By analogy with formula (1), for binary mixtures, we have φij(r)= φ0ij(r)+ φ1ij(r) gij(r)= g0ij(r)+ g1ij(r) cij(r)= c0ij(r)+ c1ij(r) (2) (3) (4)

where ij(r) the interatomic pair potentials; gij(r) the pair correlation functions; cij(r) the direct correlation functions; i,j=1,2. The RPA implies the asymptotic form of the direct correlation function and the hard sphere (HS) system as a reference one:
HS cij (r ), r   ij  cij (r )   1   ij (r ), r   ij 

(5)

where β=1/kT; k the Boltzmann constant; T - temperature; ij - HS diameters. The RPA gives unphysical behavior of the pair correlation functions inside the HS cores. In the framework of the ORPA the following correction of the RPA is introduced: g1ij(r)=0, r≤ij This condition leads to the next expression: (6)

  ij (r ), r   ij  1 cij (r )   1  ij (r ), r   ij 
Here,  ij (r ) are so-called optimized potentials:

(7)

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[ [ [  ij (r )  ij (ij )  aij1]  aij2] (r /  ij  1)  (r /  ij  1) 2  aijn3] Pn (2r /  ij  1) (8)



3



where

[m aij ]

n 0   - parameters; Pn (x ) the n-th Legendre polynomial; ij the positions of the

first minimum in ij(r). [m Values of aij ] are defined by minimizing the random-phase contribution to the free energy,  [φ1ij(r)], with respect to φ1ij(r): [φ1ij(r)]/ φ1ij(r) = 0, where
  1 HS 1 HS   ij (r )  1 / 16 3  d 3 q Tr     k  ik (q) S kj (q)  ln det  ij     k  ik (q) S kj (q)  (10)   
k 1

r≤ij

(9)

 

2

 

 

2

  

k 1

Here, i the partial number density of the i-th component; ij(q) the Fourier transform of the ij(r); SHSij(q) the partial HS Ashcroft-Langreth [22] structure factors. The term ij(ij) in eq.(8) arises due to the fact that the ORPA separation of the ij(r) is based on the WCA separation:
  ij (r )   ij (ij ), r  ij 0  ijW CA(r )   r  ij 0,    ij (ij ), r  ij 1  ijW CA(r )    ij (r ), r  ij 

(11)

(12)

 ij (r )   ij (r ), r   ij  0  ijORPA(r )   0 r   ij  ijW CA(r ),  r   ij  ij (r ),  1  ijORPA(r )   1  ijW CA(r ) , r   ij 
Thus

(13)

(14)

 ij (r )   ij (r ), r   ij  0  ijORPA(r )   ij (r )   ij (ij ),  ij  r  ij  r  ij 0,

(15)

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 ij (r ), r   ij  1  ijORPA(r )   ij (ij ),  ij  r  ij  r  ij  ij (r ) ,

(16)

Moreover, the ORPA does not imply any procedure for the defining of the HS diameters and the WCA approach is used for this aim:

i , j 1

 c c  g
2 i j 0



0 ij ( r ) 

HS g ij (r ) r 2 dr  0



(17)

where ci the concentration of the i-th component. Conditions (9) and (17) lead to the self-consistent calculation procedure for the free energy per atom, F, which in the ORPA scheme has the form:

FORPA  FW CA  2

1ci ci   ij (r )  ij (r )cijHS (r ) exp  ij (r )   ij (r )r 2 dr i, j 
0

2

 ij

(18)

where  is the number density,

FW CA  FHS

ij     HS 2 HS 0 2  2 ci c j  g ij (r ) ij (r )r dr  g ij (r ) ij (r )r dr    i , j 1  ij  ij 



2





(19)

RESULTS AND DISCUSSION We choose for consideration equiatomic Na-K, Na-Rb, Na-Cs, K-Rb, K-Cs, and Rb-Cs liquid alloys which are adequately described on the basis of the nearly-free electron model. The effective pair potentials, ij(r), are estimated by using the local version [20] of the Animalu-Heine model pseudopotential [19]. An exchangecorrelation correction to the Hartree dielectric function is approximated by the Vashishta-Singwi [23] function. The Nozieres-Paines interpretation formula [24] is used to estimate the electron gas correlation energy. The values of the AH parameters are taken from ref. [20] for pure metals at T=0 K. The results obtained for the internal energy of mixing, E , and the excess entropy of mixing, S ex , at T=373K are summarized in Tables 1 and 2 in comparison with the WCA results [17] and experimental data [25,26]. Unfortunately, the experimental data for S ex are available only for Na-K and Na-Cs systems. Therefore, the calculations for all six alloys do not listed in Table 2. It is shown that the best results in all cases are achieved by using the ORPA in comparison with the WCA. However, even the ORPA doesn‟t allow to achieve a well agreement of the results obtained with experiments for Cs-based alloys. In this case, the significant improvment of the pseudopotential approach is needed.

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Table 1. Internal energy of mixing, E (eV), obtained by using different TPT methods in comparison with experimental data [25] for equiatomic alkali-alkali liquid alloys at T=373K. WCA 0.0068 0.015 0.0292 0.0014 0.0053 0.0013 ORPA 0.0071 0.014 0.0255 0.0013 0.0041 0.0005 Experiment [25] 0.0076 0.013 0.0095 0.0013 0.0012 -0.0014

Na-K Na-Rb Na-Cs K-Rb K-Cs Rb-Cs

Table 2. Excess entropy of mixing, S ex / k , obtained by using different TPT methods in comparison with experimental data [25,26] for equiatomic Na-K, K-Rb, and Na-Cs liquid alloys at T=373K. WCA -0.041 -0.005 0.003 ORPA -0.029 -0.02 0.005 Experiment -0.012 [25] 0.018 [26] -

Na-K Na-Cs K-Rb CONCLUSIONS

The present work shows that a kind of the TPT approach plays an important role in thermodynamic calculations for binary liquid metal alloys. Even most perfect of the methods of the HTA, the WCA, is less accurate than the ORPA for quantity description of metal alloys. However, the last method needs sufficiently large computations. Therefore, in a number of cases, it is possible to use the WCA instead of the ORPA.

ACKNOWLEDGEMENTS The authors are grateful to the Russian Foundation for Basic Research (Grant N02-03-32313a) and the Committee of the Russian Academy of Science by work with youth (Grant N 188 of VI competition-expert 1999) for financial support. REFERENCES 1. Andersen H.C., Chandler D.: „Mode expansion in equilibrium statistical mechanics. I. General theory and application to the classical electron gas‟. J.Chem.Phys. 1970 53 (2) 547-56. 2. Chandler D., Andersen H.C.: „Mode expansion in equilibrium statistical mechanics. II. A rapidly convergent theory of ionic solutions‟. J.Chem.Phys. 1971 54 (1) 26-33. 3. Andersen H.C., Chandler D.: „Mode expansion in equilibrium statistical mechanics. III. Optimized convergence and application to ionic solution theory‟. J.Chem.Phys. 1971 55 (4) 1497-504.

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