Finite-Temperature Molecular Dynamics Study for Atomic Structures of Grain Boundary in Transition Metals Fe and Ni
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equations of Gaussian constraint method [7,12] is applied as dqidt- in'(1) pi dt
mi_'
dpi
0(2)
,
"(, -L..,6
(3)
where 0 is potential energy; pi is generalized momentum; q, and 4i are generalized coordinate and velocity respectively; Wis a Lagrangian undetermined multiplier. The numerical algorithm [14] for Gaussian thermostat method is established by us and is employed to study the atomic structure of a-Fe E5 [1001 GB. In order to adapt the theory to various practical experimental conditions, the equations of motion on the basis of Hamiltonian formulation [8,9] is employed as follows: dq =__(4) pi dt
dp, dpi
mi
84
_
aqi
p
- gkT)/Q, d
i
(5)
(6)
m,
where W is a variable which is associated with the mass parameter Q of heat bath [15]. The expressions (4), (5) and (6) constitute the equations of motion for a closed system and a canonical distribution in extended phase space (p, q, W) can be produced. In the present paper, a numerical algorithm based on Nosd-Hoover thermostat formulation is set up and used for investigating the atomic structure of Ni Ell [1101 GB. Calculation mode In the present paper, the reference atomic configuration of GB is given as an initial state for the constant temperature MD simulation. In order to reveal the characteristic of the crystallography of GB and the related atomic disorder phenomenon, the average fluctuation of the atomic period of GB with the temperature is introduced as follows:
ATz(T)
A~zTZ,(T) Tz,(T)
-
(7)
n( , ATxj(T)
ATx(T
Tx,()
(8)
where the Tzi and the Tx, are the atomic periods of GB along the Z and X directions respectively; ATz, and ATx, are the corresponding fluctuations with the temperature; nz (nx) is the number of the Tz (Tx) involving in the simulation calculation. In addition, the average distortion of the structure unit Adj(T) in the relaxation region of GB and the relative maximum displacement of the atom (AR(T)/R(T)) are calculated at various temperatures. The corresponding definitions are AdT(T) =
,,,(TA Ld,(T) n Sdi (T) ,
578
(9)
and
AR(T) R(T) ) .,
-
- X0)2 + (Y'- yo), + W- Zo)_ /(X - X0)2 + (y - yo), + (Z- Zo)2 / max
(P,
(10)
where the (xo,yo,zo) is reference origin, (x',y',z') and (x,y,z) are the coordinates with and without displacement respectively. di is the length of side of the periodic structure unit, Adi(T) is the fluctuation of the side with the temperature, nd is the number of the sides involving in the simulation. The time step is taken as At = 2 - 5 x 10-15 sec and the range of force for the interaction of atoms is taken as R = 1.3a 0 (ao is lattice parameter). For the a-Fe E5 [100] GB, the temperature range is from 10 to 1132K (in consideration of the phase transformation temperature), the total number of atoms in the calculating system is taken as N = 2377. For the Ni EllI [110] GB, the temperature range is from 10 to 1700K. The volume effect with the temperature is considered in the MD simulation and the modification of the lattice parameter is carried out by using the coefficient of cubical expansion of
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