Test of Kinetic Models for Interface Velocity, Temperature, and Solute Trapping in Rapid Solidification
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The solute trapping measurements, however, only partially test the models, leaving an ambiguity by not evaluating the predictions for the second interface response function: the interface velocity-undercooling relation. Prior to this work, the question of whether or not the solid-liquid interface is slowed down by a solute drag effect during alloy solidification has remained unresolved [2]. The recent development of an accurate and reproducible time-resolved temperature measurement technique for PLM experiments [8] has allowed us to address this question [9] and produce a complete test for theories of interface kinetics during alloy solidification. During solidification of a pure element, the interface velocity, v, is given by [10] v = vo(Ti) [I - exp(AGeff/RTi)] ,
(1)
where Ti is the interface temperature, vo(Ti) a kinetic rate constant, R the gas constant, and, for pure elements, AGeff is the molar Gibbs free energy change upon solidification. This relation has been extended to alloys in different ways. In the CGM "without solute drag" Eq. (1) applies with AGeff = AGDF - Xs A9L1B + (I - Xs) AItA.
119 Mat. Res. Soc. Symp. Proc. Vol. 398 0 1996 Materials Research Society
(2)
where AGDF is the "driving free energy", Xs the solute mole fraction of the solid at the interface, and A91B and A.LA the changes in chemical potential on solidification for solute and solvent respectively. To include solute drag, it is assumed that part, AGd, of this free energy is dissipated by the solute-solvent redistribution, and not available to drive the interface motion. Eq. (1) then applies with AGeff = AGDF - AGd, where, for sharp interface models, AGd is determined using the thermodynamics of irreversible processes. In the CGM "with solute drag", AGd
(3)
= (XL - Xs) (AA - A91tB) ,
where XL is the solute mole fraction in the liquid at the interface. The partition coefficient, k, in the CGM is given by X
k(v) =where the diffusive speed, distance and
Vd,
(V/Vd)+±1E
-
, -
XL
/______ (v/Vd)+ 1--(-KE)XL
,V
(4)
is the ratio of solute diffusivity at the interface to an inter-atomic
KE(XL,Xs,T) =_(Xs / XL) exp[-(AB -Aga)
/ RT]
.
(5)
The two interface response functions relating XS, XL, v, and Ti, are obtained in the CGM by solving equations (1) and (4). In another sharp interface model due to Agren [5], AGd is given by half the right hand side of Eq. (3) and the partition coefficient is obtained from a linearized diffusive flux-driving force relation. Continuum models assume that the solid-liquid interface has a finite width, 6. The concentration profile across the interface is determined by solving the steady state diffusion equation given by [3] v(X(y)-Xs)+D(y) X
RTi
(y)) d .tB-l dy
A)=
0,
(6)
with y the coordinate normal to the interface and D(y) and X(y) the local diffusivity and solute mole fraction. In this case, AGd is given by [2] Ad
fj(X~y A 8 ( - Xs) ddB
dy gddy)
(7)
0
In the Hillert-Sundman model [3], the diffusivity is assumed to vary exponentially across the interface, matching the solid and liq
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