Microsegregation in solidification for ternary alloys
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Mathematical descriptions can thus be developed based on the geometrical models proposed above. In the early stage of solidification, when there is only one primary solid phase formed, mathematical equations were derived in our previous study [41 and given below. (1 - f ~ ' ) d x ~
(-7)
= ( x ) - * x ] ) d f '~ -
,o
\ 0a/a=x,
[1AI (1 - f ~ ) d x ~
= (x~ - *x~
'~ -
\OX/x_adO_ [1BI
The concentration gradient at the interface and the solute redistribution in the solid is obtained by solving Fick's second law,
Ox7
-
-
~
00
ot
Ox ~ O0
Ox7
Dj 0 •
2
[2A]
OEx ~ -
D~
- -
0A 2
[2BI
Solidification progresses with only one primary solid phase forming until the solidification path intersects the liquidus valley. Then, two phases, a + /3, will precipitate simultaneously. The geometry is shown in Figure 3(c). Equations [3A] and [3B] are obtained from the conservation of mass in the control unit volume: d(ft.#)
+ d ( f ' L f T ) + d(fO.f~) = 0
[3AI
d(fL.f~) + d(fa.f'~) + d ( f t J . f f ) = 0
[3B]
Since the liquid composition is uniform, the compositional change in the liquid phase can be written as d ( f L . f ~ ) = d ( f L x l f ) = f t dx~ + xjL dfL
[4A]
d ( f L ~ f ) = d ( f ~ x f ) = f t d x f + x~ d f t
[4B]
C(k)
A(i)
B(j)
Fig. 1 - - A solidification path of an alloy with the composition (x~ ~ superimposed on the liquidus projection of a ternary A-B-C system with three binary eutectics and one ternary eutectic. METALLURGICAL TRANSACTIONS A
The change of the average composition of the solid phase is due to two factors. One is the deposition of an increment of the solid phase, as given in the simple Scheil model. Another is diffusion at the solid/liquid interface. For simplicity, only one-dimensional diffusion, i . e . , diffusion along the A direction is considered. As given in the first term of the right-hand side of Eqs. [5A] through [5D], * x ~' d f '~ and * x ~ d f ~ represent the newly solidified solid at the interface. The second terms in Eqs. [5A] through [5D] account for the change due to diffusion, which is equal to the diffusional flux, j r (or J~), at the interface multiplied by the cross-sectional area, A ~ (or As), and the time interval, dO. d(f'~s
= *x'; d f '~ - [(JT)~=jA~=~, dO
[5AI
d(f'~s
= *x~ d f ~' - [(J~)~=~,]A~=~, dO
[5B]
d(f~s
= *x~ d f ~ - [(Jf).=.,]Af=~, dO
[6A]
d(f~.gf)
= *x~ d f ~ - [ ( J f ) . = . , ] A f = . , dO
[6B]
The values of the diffusional flux J" (or j o ) can be obtained from Fick's first law. By substitution of Eqs. [4A]. [5A]. and [6A] and [4B]. [5B], and [6B] into Eqs. [3A] and [3B]. respectively, two new equations can be obtained.
Fig. 3 - - A schematic diagram of the distribution of phases in a control volume during a ternary alloy's solidification process: (a) at the start of the solidification, the alloy is in molten state; (b) in the early stage of the solidification, there is a one primary solid phase with liquid phase; (c) two solid phases, a + /3, solidify simultaneously as the solidification path intersects the liquidus val
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