Separation of scales for growth of an alloy needle crystal
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T ⫽ T⬁ ⫹ (TI ⫺ T⬁) and c ⫽ c⬁ ⫹ (cI ⫺ c⬁)
Here, P and Pc are thermal and solutal Peclet numbers defined by P⫽
V V , Pc ⫽ ⫽P 2 2D D
E1(u) ⫽
兰
u
exp (⫺) d
Here, L is the latent heat per unit volume and c is the specific heat of the liquid, both assumed to be constants. The latter equation can be solved for the interface concentration:
[3]
with thermal diffusivity and solute diffusivity D, both assumed to be constants. The exponential integral function is ⬁
Fig. 1—Dimensionless supercooling SL relative to the liquidus temperature according to Eq. [12] as a function of the thermal Peclet number P. The dashed curve is I(P), and the dot-dashed curve is the second term in Eq. [12]. The computation corresponds to an alloy of Al-0.15 mass fraction Cu, with D ⫽ 2.1(10⫺5) cm2/s, ⫽ 0.66 cm2/s, L /c ⫽ 374.0 K, m ⫽ ⫺800 K/mass fraction, and k ⫽ 0.37. In this case, Pc ⫽ 3.1(104)P, and there is an obvious separation of scales of thermal and solutal effects.
cI ⫽
I(P) ⫺ [4]
[5]
[6]
[7]
R.F. SEKERKA, University Professor, is with the Departments of Physics and Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA. S.R. CORIELL, Chemist, and G.B. McFADDEN, Mathematician, are with the National Institute of Standards and Technology, Galthersburg, MD. Manuscript submitted April 27, 2001.
METALLURGICAL AND MATERIALS TRANSACTIONS A
[9]
S⫽
TM ⫺ T⬁ L /c
[10]
is the dimensionless supercooling relative to the melting point of the pure material. The dimensionless supercooling relative to the liquidus temperature is SL ⫽
TM ⫹ mc⬁ ⫺ T⬁ L /c
[11]
which leads to
and (cI ⫺ c⬁) ⫽ Pc exp (Pc)E1(Pc) ⫽ I(Pc) cI(1 ⫺ k)
(mc⬁ /[L /c]) ⫽S [1 ⫺ (1 ⫺ k)I(Pc)]
where
where m is the liquidus slope, which is negative in the case of the distribution coefficient k being less than one. Both m and k are assumed to be constants. Conservation of energy and solute at ␣ ⫽ 1 gives (TI ⫺ T⬁) ⫽ P exp (P)E1(P) ⬅ I(P) L /c
[8]
Substituting Eqs. [8] and [5] into Eq. [6], we obtain
The interface temperature TI and liquid concentration cI are connected by the relation TI ⫽ TM ⫹ mcI
c⬁ [1 ⫺ (1 ⫺ k)I(Pc)]
I(P) ⫹
[m(k ⫺ 1)c⬁I(Pc)]/[L /c] ⫽ SL [1 ⫺ (1 ⫺ k)I(Pc)]
[12]
The coupled problem of thermal and solutal diffusion has been studied by a number of authors, including Ivantsov,[2] Bolling and Tiller,[3] Temkin,[4] Langer,[5] and Lipton et al.,[6,7] and has been discussed by Glicksman and Marsh (Ref. 8, p. 1114). The form of Eq. [12] can be understood as follows: the function I(P), which would be the only contribution to the supercooling for growth from a pure melt, has a sigmoidal shape and rises from zero to one, as shown by the lower dashed curve in Figure 1. The second term in Eq. [12] also has a sigmoidal shape, as shown by the dashed-dot curve,
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VOLUME 32A, OCTOBER 2001—2669
and rises from zero to m[(k ⫺ 1)/k]c⬁ /[L /c ]. The result of adding these terms can be understood very simply by taking advantage of the separation of scales of the Peclet
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