Computer Investigations on Phase Decomposition in Real Alloy Systems Based on a Discrete Type Phase Field Method
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THEORETICAL BASIS The total free energy of microstructure Gy~tcm is given by a sum of the chemical free energy Go, the interfacial energy E.,f and the elastic strain energy E.,. Gsys1em
+E:_ +EfG+)dv
(1)
In the present work the phase decomposition of Fe-Al-Co ternary ordering alloy is performed. The chemical free energy is evaluated on the basis of Bragg-Williams-Gorsky approximation, where the pairwise interactions up to the 2nd nearest neighbor and also the magnetic interchange energies up to the 2 ndorder are taken into account. Therefore, we concretely define following 15 parameters to represent various states of Fe-Al-Co alloy, i.e. the atomic compositions ci (i=Fe, Al and Co), the I'" nearest neighbour long range order parameters X i, the 2nd neighbour long range order parameters yj and zi, and the spin order parameters si, where 4 subordinate parameters are included. The concrete expression of the chemical free energy G., is given by equation 2, according to our previous paper[ 10, 11] 201 Mat. Res. Soc. Symp. Proc. Vol. 481 ©1998 Materials Research Society
G(r) =U.+ ,•CJ,(,
",>- IE [,I(W') +M,()+3,> + -)) u,)] (U-[+MU'))-3(WV(2+Mt(,2)]j+'YjC~j(+Ic~c~xx•[(W,(' +3J +RTEC,["c`1Xq-N4(c"¶'()
x )J•+ 3(w( 2 M"
((--
(2)
2)
-)+ 1.R .. 4 .. C,(I-X,)[(I-Z,)In(I- Z,)+(I+ Z,)In(I+Z,)]
-+(
,where
=
1
-J"5')q+J5 (q 22J~")q~q
) M(7
+V
-2v'i ")+
The interfacial energy E,,.f is defined by equation 3 on the basis of so-called Cahn's interfacial energy[ 12]. 2 2 2+ E.,f =,C.(VC.) + . (vCo)', + K (VCy)2 +K..IVX. ,Iv l+X,1 +•l X +• +1 ,.,+ (3) 2 2 uIVS,I "+ F -uIV ScI ,cIV K " + S ,I K + z I I "+ K,1v z + K, cIVZl +I + '•,•, The elastic strain energy for the cubic lattice crystal is given by equation 4 based on Landau elasticity theory[ 13,14,15], where, Cg, 1 is the elastic stiffhess, e,ý is the constrained strain and q7 is the lattice mismatch between the pure metals. It should be noted that the composition dependence of E,. is taken into consideration, as is clearly known from the {c-ce) term in eq.4. 2
(3/2)(C',, + 2C•°m) 7 {c(r) 2
+ (I/ 2)C ... (,
(r) + e2,r
2
2
c,}2
-rl(C,*,
+ 2C,*,,){c(r) -c,} {e,(r) + e4 (r) + eL(r))
,2
+ e ,(r) + C,,, {e( (r)eý (r) + eý (r)e, (r) + e, (r)
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