An atomistic study of solid/liquid interfaces and phase equilibrium in binary systems
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INTRODUCTION
i ii (r) ⫽ LJ (r) ⫺
THE
Lennard–Jones potential[1] has been used extensively over the years to model a wide range of materials. However, the necessity for including many-body interactions led Daw and Baskes to the development of the embedded atom method (EAM).[2,3] Holian et al.[4] proposed an extension of the Lennard–Jones potential that allows for manybody interactions. That was further developed by Baskes.[5] We present here a generalization of the method for a binary system. The Lenard–Jones bonding energy between two atoms of type i is
冋
i LJ (r) ⫽ i
册
1 2 12 ⫺ 6 r r
[1]
where i is the well depth and r is the atomic separation in 6 units of r 0i ⫽ 冪 2 i, with being the diameter of the atom, when approximated by a sphere. The following relationships summarize the description of the energy of a mixture of atoms. The energy is given by the usual EAM form:
冋
E ⫽ 兺 Fi (i) ⫹ i
1 ij (rij) 2兺 j⫽i
册
[2]
The pair potential ij denotes the interactions between atoms i and j separated by a distance rij . Note that for binary systems, three pair potentials are required, two between like atoms and a third between unlike atoms. The embedded function is chosen to be Ai Z0i Fi ( ) ⫽ [ln ( ) ⫺ 1] 2
i ⫽
1 (rij ) Z0 兺 j⫽i
[5]
where the atomic densities have the form
(r) ⫽ exp (⫺i (r ⫺ 1))
[6]
All summations were done over the j neighbors of an i atom. For the results presented subsequently, the interactions were smoothly truncated after the second neighbors. The background density was calculated by the linear superposition of the atomic densities . Two dimensionless parameters are required to describe the system: A, which quantifies the extent of the many-body bonding; and , which describes the decay in the electron density with distance. In this study, fixed values of A ⫽ 0.5 and  ⫽ 6 have been used for both atom types. The choice is consistent with selecting the fcc solid solution as the ground state. For convenience, A was chosen to be 1 eV, but all results presented subsequently scale with this energy. To obtain the unlike pair potential AB , and L10 reference structure with c/a ⫽ 1 was used. The energy per atom of this structure is given by ELI0 ⫽
1 [F ( ) ⫹ FB(B)] ⫹ AA ⫹ 4AB ⫹ BB 2 A A
[7]
1 1 [2B ⫹ A] and B ⫽ [2A ⫹ B] 3 3
[8]
where
[3]
By taking the energy of this system as
冢r
And the like atom pair potential is
METALLURGICAL AND MATERIALS TRANSACTIONS A
[4]
The background electron density is given by
A ⫽
M.I. BASKES and M. STAN, Technical Staff Members, are with the Los Alamos National Laboratory, Los Alamos, NM 87545. Contact e-mail: [email protected] This article is based on a presentation given in the symposium “Fundamentals of Solidification” which occurred at the TMS Fall meeting in Indianapolis, Indiana, November 4–8, 2001, under the auspices of the TMS Solidification Committee.
2 Fi (i (r)) Z0
AB ⫽ 6AB ELI0 ⫽ 6 LJ
with r in units of r0AB, we obtain
AB ⫽
冋
1 12
⫺
冣
1 r6
[9]
册
1 1 AB 6 LJ ⫺ [FA(A) ⫹ FB(B)] ⫺ AA
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