Free Energy Calculations of Cu-Sn Interfaces
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287 Mat. Res. Soc. Symp. Proc. Vol. 398 01996 Materials Research Society
METHODOLOGY AND POTENTIALS The method employed in the evaluation of the Cu-Sn interfacial free energies is based on calculating the reversible work required to pull appart the surfaces which define the interface. This basic idea has been applied to LJ liquids [4] and to LJ solid-liquid interfaces [5], however in the present case we have eliminated the need of so called external cleaving potentials which are quite cumbersome to use in heterogeneous cases like the present one. The work required to separate the surfaces at the interface can be equated to the energy of adhesion which for our Cu-Sn system can be written as follows: WCuSn = Fs(Sn) + Fs(Cu) - FI(Cu-Sn),
(1)
where WCuSn is the energy of adhesion, Fs(Sn) and Fs(Cu) are the surface free energies of Sn and Cu respectively and FI(Cu-Sn) is the interfacial free energy. The work done in separating the surfaces is evaluated by employing adiabatic switching in a Molecular Dynamics (MD) framework [7,8]. Within this method, the free energy difference between two systems whose interactions are described by Hamiltonians H0 and H1 say, is given by AA=
f(H1 - Ho)dt
(2)
where AA is the free energy difference between system H0 and H1 .In the present system, H0 is a Cu-Sn slab with Cu-Sn interactions and system H1 is comprised of Cu and Sn slabs with 2 free surfaces each and with no Cu-Sn interactions. This is shown schematically in figure 1. The final system (HCu+HSn) is obtained by adiabatically switching off the Cu-Sn interactions.
HcuZn
cu+
H sn
FIG. 1. Schematic of Cu-Sn system used to calculate excess free energy. The work done in
separating the Cu-Sn surfaces at the interface is given by AA
=
(+)
-
-
The surface free energies of Cu and Sn at each sampled temperature were calculated employing the same formalism. The excess interfacial free energy FI(Cu-Sn) was evaluated employing equation (1). We used slabs of 2000 and 2500 atoms with almost equal number of Cu and Sn types. The Cu side of the interface was Cu(100) which was matched to both [3 and cubic Sn crystals at OK. The z-direction is defined normal to the interface. These configurations were annealed to 900K to create a solid Cu
-
liquid Sn system. This structure was then equilibrated at various temperatures in
288
the range 300-1100 K. Once thermalized, the MD switching runs were done at constant temperature and the excess interfacial free energy was computed. The switching was done over time scales of 5 to 10 psec (r in equation (2)).
Ec
re
a
A
p(0)
p(I)
p(2)
p(3)
t(I)
t(2)
t(3)
P0
Cu
3.62
2.50
5.11
1.07
3.63
2.2
6
2.2
3.14
2.49
2.95
1.0
Sn
3.14
2.81
5.09
1
5.0
20.0
10.0
5.0
2.41
5.93
-0.13
1.1
Cu 3 Sn
3.50
2.68
5.10
Table 1: Parameters for the MEAM. Values listed are the cohesive energy Ec (eV), the equilibrium
nearest neighbor distance re (A), the exponential decay factor for the universal energy function c, the scaling factor for the embedding energy A, the exponential
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