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Initial transient solute redistribution during directional solidification with liquid flow
ELSEVIER
Journal of Crystal Growth 182 (1997) 212-218
Initial transient solute redistribution during directional solidification with liquid flow Weidong
Huang *, 1, Yuko Inatomi,
The Institute of Space and Astronautical
Kazuhiko
Kuribayashi
Science, 3-1-l Yoshinodai, Sagamihara, Kanagawa 229, Japan
Received 5 September 1996; accepted 22 May 1997
Abstract The initial transient solute redistribution during directional solidification with liquid flow is analyzed and approximate solutions for both finite and infinite sample length are given, based on two assumptions: (1) there exists a solute transport boundary layer within which diffusion is the only solute transport mechanism, while beyond which convective mixing makes the solute distribution completely uniform; (2) the concentration profile inside the diffusion boundary layer is of exponential form and the diffusion length is a function of time during the initial transient process. The solution for infinite sample length comes back to Warren and Langer’s approximate solution for pure diffusion when thickness of the boundary layer approaches infinity, and back to Burton et al’s accurate solution for steady state when time approaches infinity. The calculation results according to the models for initial transient solute redistribution during directional
solidification with liquid flow fit well with Inatomi et al.? experimental PACS;
results.
02.30.Jr; 66.10.Cb; 81.1O.Aj; 81.30.Fb; 47.90. + a
Keywords;
Solute redistribution;
Directional solidification; Liquid flow
1. Introduction Solute transport is one of the most important factors in determining solidification microstructures and quality of grown crystals. Because solute diffusion is much slower than heat diffusion and atomic attachment, the time required for the interface to reach the steady state depends on the time
* Corresponding author. ‘Present address: State Key Laboratory for Solidification Processing, Northwestern Polytechnical University, Xian 710072, People’s Republic of China. Fax: + 86 29 525 0199. 0022-0248/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PII SOO22-0248(97)00332-l
required for the solute distribution to reach the steady state. Therefore, the knowledge of time-dependent development of the solute distribution can help us to understand the time-dependent evolution process and the final steady state selection of the interface morphology. For example, the knowledge of morphological instability of a growing solid-liquid interface is an important basis to understand the complicated solidification microstructures. The classical theories [l-3] on morphological instability are based on an assumption that the solute distribution in front of the growing interface has already reached the steady state. In
W Huang et al. f Journal of Crystal Growth 182 (1997) 212-218
fact, any practically observed interface breakdown in experiments within finite time must have occurred before the solute distribution reaches the steady state. The big difference between theoretical prediction and experimental observation of initial wavelength of interface breakdown remains one of the unresolved important problems for the steadystate theories. This problem has been well resolved by Warren and Langer in a recent theoretical analysis of dendritic growth during directional solidification based on an approximate solution of the non-steady-state solute distribution [4]. Huang et al. gave the experimental evidence of time-dependent interface instability [S] and history-dependent steady-state selection of dendrite morphology [6]. These results showed that without detail knowledge of non-steady-state developing processes of crystal growth, it is difficult to understand the final solidification microstructures completely. The initial transient solute redistribution during directional solidification is a kind of Stefan problem and exact analytic solution has not been found yet. Tiller et al. [l] gave the first approximate solution for the problem, based on an assumption that to a first approximation, there exists a quasisteady-state solute distribution in front of the interface. Smith et al. [7] gave a more detailed and mathematically more accurate treatment of this problem. Both Tiller et al’s and Smith et al.‘s solutions are based on an assumption that solidification velocity is equal to the sample pulling velocity and keeps constant from the beginning to the end of the solidification process. In fact, the increase of interface concentration will slow down the interface velocity compared with the sample pulling velocity during the initial transient process. Huang et al. [S] gave a solution based on the quasi-steady-state approximation, which included the effects of the change of interface concentration on the interface velocity during the transient process. Recently, Warren and Langer [4] gave a new approximate solution for the problem, in which both the change of interface velocity and the change of diffusion length are taken into account. The above work considered diffusion as the only solute transport mechanism. However, liquid flow caused by natural convection or forced stirring is almost unavoidable during solidification on the
213
ground, which will make important contribution to the solute transport. The relation among liquid flow, solute and heat transport, and interface morphology is a subject of many interests and studies [9]. However, we have not found researches on the effects of liquid flow on the transient solute redistribution, and we think this is really an important problem needed to be resolved. For most type of liquid flow, there exists a hydrodynamic boundary layer in which the flow is parallel to the solid/liquid interface and the flow velocity tends to zero with approach to the interface, except for the volume-change convection during the solid/liquid transformation, which is usually a very weak liquid flow. Diffusion dominates the solute transport away from the interface inside the boundary layer, especially in the region very close to the interface, while liquid convection dominates the solute transport in regions further from the interface. According to this concept, Burton et al. [lo], and later Hurle [l l] and Delves [12] assumed that there exists a solute transport boundary layer with a thickness of 6. Within the distance 6 from the interface diffusion is the only solute transport mechanism, while beyond 6 the flow keeps liquid concentration uniformly equal to C6 because of sufficient mixing of liquid. Based on the assumption, they gave a solution for steady state solute distribution in front of the solid/liquid interface during directional solidification with liquid flow. In this paper, we check which of the above approximate solutions are better by comparing them with numerical calculation of the differential equation for pure diffusion in the liquid. We find that the Warren-Langer solution is the best of them. Then we incorporate the WarrenLanger approximation with Burton et al’s solute transport boundary layer approximation to give a solution for the initial transient solute redistribution during directional solidification with liquid flow, which is then compared with the experimental results by Inatomi et al. [13].
2. Check of the approximate
solutions
All of the approximate solutions for the initial transient solute redistribution mentioned above are
214
W. Huang et al. 1 Journal of Crystal Growth 182 (1997) 212-218
compared with numerical calculation of the differential equation for pure diffusion in the liquid during directional solidification. Basic assumptions for the numerical calculation are as follows: (1) A semi-infinite rod with uniform cross-section solidifies at constant temperature gradient G and at constant sample pulling velocity I/, with respect to the isotherms. The solid/liquid interface is plane and perpendicular to the axis of the specimen. The co-ordinate system is fixed to the moving interface. (2) Before solidification the specimen is of uniform concentration Co. (3) Diffusion coefficient is constant in the liquid and is negligible in the solid. (4) The interface is in local-equilibrium state, the equilibrium partition coefficient k. is constant and less than unity. The liquid concentration satisfying the above conditions is then determined by the following equations:
C=Da2c+j25 at
az2
(1)
CW) =
co
(4
’az’
1
C(~J)=G,
(3)
Dac = - I/iCi(l - k,) ) az z=o
(4)
v,
= L
I/
+ P
mdc(oJ) G
dt
’
where D is the solute diffusion coefficient in the liquid, m the liquidus slope, kO the solute partition
Table 1 Material properties
of succinonitrile-acetone
Property Melting point Diffusion coefficient of acetone Liquidus slope Equilibrium partition ratio
in succinonitrile
coefficient, I/i the interface velocity with respect to the solidifying material, and Ci = C(O,t) is the interface concentration in the liquid phase. The Eqs. (lH5) are numerically calculated with the one-dimensional finite differential method. In numerical calculation, Eq. (3) is changed to C(lOD/I/,,
t) = Co.
(6)
The calculation is carried out for SCN-acetone alloys of three different initial concentrations, 0.1, 1 and 10 mol% acetone, directionally solidified at the same temperature gradient of 5 K/mm and at the same sample pulling velocity of 1 pm/s. The material properties used in the calculation are shown in Table 1. The selection of the alloy concentrations and the experimental parameters cover three typical situations for directional solidification: ATTo/G is small for Co = 0.1 mol%, middle for Co = 1 mol% and large for Co = 10 mol%. Where AT, is the liquidus-solidus range at Co. The calculated results are shown in Figs. 1 and 2. One can find that Smith et al’s solution fits the numerical calculation very well at small value of AT,/G, poor at middle value of AT,/G, and very bad at large value of AT,/G. Tiller et al’s solution is not far from the numerical calculation only at the middle value of AT,/G. Huang et al’s solution is not far from the numerical calculation only at the large value of ATo/G. Warren and Langer’s solution is the one which fits the numerical calculation well for all of the three situations especially at large value of AT,/G. The above results indicate that the influence of the difference between Vi and VP on the initial transient solute redistribution becomes more important with increase the value of AT,/G, and that the constant diffusion length during the initial transient process is not a very good approximation.
alloy Symbol
Value
Ref.
Till D
331.24 K 1.27 x 1O-9 m’/s - 2.22 K/mol% 0.10
1141 Cl51 1141 Cl41
215
W. Huang et al. 1 Journal of Crystal Growth 182 (1997) 212-218
h? ---
0.24
- - - Tiller -.-_
0.22
s 0.16 B E = 0
Smith _____.Hwng
0.22
-Warren
0.20
(a)
Tiller
*
-.--smith
.-.--.Huang
-warren
0.16
0.16
Nunwid
o
0.16 014
(a) 0.10 0.00
0.60
1.00
1.50
2M
250
3.00
&mm
2.4
(b)
---Tiller - - - Tiller -. - - Smith .____.l+ang
2.2
-.-. ‘\
-Warren o Numerical
/
-----.Huang -Warren
'\ 1.80
,/
Smith
o
'\
Numerical
i
1.6
04
Cc)
---Tiller - .-. Smith .----.Huang -Warren
0.00 0
2w
400
600
800
1WO
12iUl
fS 1. Interface concentration in the liquid as a function of time during directional solidification of SCN-acetone alloys VP = 1 pm/s, (a) Co = 0.1 mol%, and at G = SK/mm (b) C, = 1 mol%, and (c) Co = 10 mol%.
050
o
Numetical
1.00
1.60
2w
250
3.00
&mm
Fig. 2. Liquid concentration profile in front of the interface during directional solidification of SCN-acetone alloys at G = 5 K/mm and VP = 1 pm/s, 1200 s from the beginning of solidification, (a) Co = 0.1 mol%, (b) Co = 1 mol%, and (c) C, = lOmol%.
216
W. Huang et al. J Journal
of CrystalGrowth
3. Mathematical model for initial transient solute redistribution with liquid flow
Here, we consider first a sample with finite length of liquid phase for directional solidification with liquid flow. There exists a solute transport boundary layer 6 in front of the interface, within 6 diffusion is the only solute transport mechanism and beyond 6 the liquid concentration is of a uniform concentration Cd. The other assumptions are the same as those for pure diffusion in the above section. Then Eq. (1) is only valid within the range 0 < z < 6, and Eq. (3) becomes C(z > &t) = Cd.
side of Eq. (11) is
(13) Then we obtain i =
Cie-6if + (Ci - Cd)e-“’ , Odzd6, 1 _ ,-w
Vi[Ca - koCi - Ci(l - ko)eP”I’]
-
’ -!!]C*-[iPl~eY~j~]Ci) [ (1 - e-‘/l)
~{(Ci-C~)[1-I’(:ISel:,~)2]}-”
(7)
We assume an approximate solution for Eqs. (1) and (2), (4), (5) and (7) with the following form Ca -
and the left-hand ~
3.1. Finite sample length
C(z,t) =
I82 (1997) 212-218
(14)
To determine Cg, consider the conservation the solute in the whole sample
of
LCo=j:koCiVidt+~~Cdz
(8) where 1 is the time-dependent diffusion length, and 1 = 0 at t = 0. Our task is to determine Ci, Cd and 1 as functions of time. Eq. (4) combined with Eq. (8) gives Wci-
Vi =
D(ci
m
‘C dz = Vi(Ca - koCi) - C a( L - S s0
-
cc5)
_
v
1
P
fs
in which
(10)
Vi dt = V,t + ~ (Ci - Co).
Compare the right-hand (16), one can find that
c, =
I
Dac az
&.=*
-
side of Eq. (11) is
sides of Eqs. (11) and
V,C,(l - ko)eP6” L - 6 - T/,8?- z (Ci - CO)
I
X + Vi(CJ - Ci) DazzEo
= Vi[C6 - k 01C. - C.(l 1 - k,)e-6’f]
(17)
0
of Eq. (1) from 0 to 6 yields
The right-hand
J:Vidt)
(16)
with Eq. (9) gives
/C,(l - k,)(l - ema”)
Integration
where L is the initial length of liquid phase. Differentiation of Eq. (15) with respect to time yields &
1Ci(l - k,)(l - e-“I’)’
2
(15)
cd)
Eq. (5) combined Ci
+ C,(,-,-~~Vidt),
(12)
Then, Ci, Cd, Vi and 1as functions of time can be easily determined from Eqs. (9) and (lo), (14) and (18) by numerical method.
W. Huang et al. JJournal
211
of Crystal Growth 182 (1997) 21Zp218
i = Vi(CO - klJCi) - ICi
3.2. Infinite sample length
(27)
’
Ci - CO When C(Z,t)
L~S+Vpt+~(Ci-CO)
(19)
the change of C6 with time can be neglected, i.e. c d z C,. Then, l/i, Ci and I can be determined from the following equations:
= Co + (Ci
Wci - cO) 1Ci(l - k,)(l - ee8”) ’
(20)
Co)ep”!‘,
Z 3
0.
(28)
These are just the results obtained by Warren and Langer [4] for the case of pure diffusion (please notice the difference of a factor of 2 in 1 because of different definition). 4.2. Comparison
vi =
-
with the solution for steady state
For infinite sample length, steady state can be reached when t approaches infinity, i.e., Vi = VP, ci = 0, and i = 0, and Eqs. (20) and (21) give I( ‘X ) = ;_
(29)
P
Ci( X) =
ci
2
L
Wci
-
co)
_
m ICi(l - k,)(l - ems”)
v
1
(24
’
,-W’
1 _
,
O
C(z,t) = co,
z 3 6.
(24)
4. Discussion 4.1. Comparison
with the Warren-Lunger
When li approaches come respectively, v,
=
’ c,
=
infinity,
Wci cO) ICi(1 - ko)’
solution
Eqs. (20 j(23)
be-
-
‘;
Wci
-
cO) _
’ m [ 1Ci(l - ko)
(25)
v
1
’
3
C(z,
‘cc
)=
co
k, + (1 - k,)e- “pziD k. + (1 - ko)e-‘pd’D ’
Odz<<. (31)
is given
by CO - Cie-*‘I + (Ci - C,)e-“’
Coke k.
and Eq. (23) becomes
The solute profile in front of the interface
C(z,t) =
co k. + (1 - ko)eP”n’“‘D=
(26)
These are just the solutions obtained by Burton et al. [lo-121 for steady state solute distribution during directional solidification with liquid flow. 4.3. Comparison
with experimental
results
Inatomi et al. [13] gave experimental results of the interface velocity Vi and concentration gradient Gc as functions of time during directional solidification of SCN-10 mol% acetone and SCN6.5 mol% acetone. The experiments were carried out in sample cells with finite initial length of about 20 mm for the liquid phase. The diffusion boundary layer and convection mixing region were observed in front of the interface, and the boundary layer thickness 6 was measured. The comparison of Vi and G, between the experimental results and calculations according to several models are shown in Figs. 3 and 4. The models considering the liquid flow, especially the one for finite sample length, fit the experimental results well.
218
W. Humg
of Crystal
et al. /Journal
Growth 182 (1997) 212-218
0.30
/’
___._.~~e” 0.25
- - - L=infinity -
f
L
.
-L=2Cfnm
0.15
i 0.10
0.05
(4
(a)
0.00
o.25_: -----.Warren 1.40
0.20.:
-
L=2Omm
1.20
E 3
1.00 0.80
3
o
0.60 0.40
(b) 0
200
400
m0
em
1000
1Mo
ts
Fig. 3. Interface velocity as a function of time during directional solidification of (a) SCN-10 mol% acetone, G = 2.37 K/mm, T/, = 1.05 urn/s and (b) SCN-6.5 mol% acetone, G = 2.37 K/ mm, T/, = 0.70 pm/s.
References [I] [2] [3] [4] [S] [6] [7]
W.A. Tiller, K.A. Jackson, J.W. Rutter, B. Chalmers, Acta Metallurgica 1 (1953) 428. W.W. Mullins, R.F. Sekerka, J. Appl. Phys. 35 (2) (1964) 444. D. Wollkind, L. Segel, Phil. Trans. R. Sot. 268 (1970) 351. J.A. Warran, J.S. Langer, Phys. Rev. E 47 (1993) 2702. Weidong Huang, Proc. In Space ‘94. Tokyo, Japan, 1994. p. 249. Weidong Huang, Xingguo Geng, Yaohe Zhou, J. Crystal Growth 134 (1993) 105. V.G. Smith, W.A. Tiller, J.W. Rutter, Can. J. Phys. 33 (1955) 723.
0.20
aw~~“‘:~‘.‘:~~‘~:‘.‘.:~‘~‘:~‘.‘I
0
200
400
600
arm
lCC0
1203
ts Fig. 4. Concentration gradient as a function of time during directional solidification of (a) SCN-10 mol% acetone, G = 2.37 K,‘mm, I/, = 1.05 urn/s and (b) SCN-6.5 mol% acetone, G = 2.37 K,!mm, I/, = 0.70 urn/s.
PI Weidong Huang, Qiuming Wei, Yaohe Zhou, J. Crystal Growth 100 (1990) 26. of Crystal c91 S.H. Davis, in: D.T.J. Hurle (Ed.), Handbook Growth, vol. lb, Elsevier, Amsterdam, 1993, p. 859. Cl01 J.A. Burton, R.C. Prim, W.P. Slichter, J. Chem. Phys. 21 (1953) 1987. Cl11 D.T.J. Hurle, J. Crystal Growth 5 (1969) 162. 1121 R.T. Delves, J. Crystal Growth 314 (1968) 562. [I31 Yuko Inatomi, Hirofumi Miyashita, Eiichi Sato, Kazuhiko Kuribayashi, Kazuhisa Itonaga, Tetsuichi Motegi, J. Crystal Growth 130 (1993) 85. Cl41 SC. Huang, M.E. Glicksman, Acta Metall. 29 (1981) 701. N.B. Singh, Met. Trans. Cl51 M.A. Chopra, M.E. Glicksman, A 19 (1988) 3087.
Journal of Crystal Growth 182 (1997) 212-218
Initial transient solute redistribution during directional solidification with liquid flow Weidong
Huang *, 1, Yuko Inatomi,
The Institute of Space and Astronautical
Kazuhiko
Kuribayashi
Science, 3-1-l Yoshinodai, Sagamihara, Kanagawa 229, Japan
Received 5 September 1996; accepted 22 May 1997
Abstract The initial transient solute redistribution during directional solidification with liquid flow is analyzed and approximate solutions for both finite and infinite sample length are given, based on two assumptions: (1) there exists a solute transport boundary layer within which diffusion is the only solute transport mechanism, while beyond which convective mixing makes the solute distribution completely uniform; (2) the concentration profile inside the diffusion boundary layer is of exponential form and the diffusion length is a function of time during the initial transient process. The solution for infinite sample length comes back to Warren and Langer’s approximate solution for pure diffusion when thickness of the boundary layer approaches infinity, and back to Burton et al’s accurate solution for steady state when time approaches infinity. The calculation results according to the models for initial transient solute redistribution during directional
solidification with liquid flow fit well with Inatomi et al.? experimental PACS;
results.
02.30.Jr; 66.10.Cb; 81.1O.Aj; 81.30.Fb; 47.90. + a
Keywords;
Solute redistribution;
Directional solidification; Liquid flow
1. Introduction Solute transport is one of the most important factors in determining solidification microstructures and quality of grown crystals. Because solute diffusion is much slower than heat diffusion and atomic attachment, the time required for the interface to reach the steady state depends on the time
* Corresponding author. ‘Present address: State Key Laboratory for Solidification Processing, Northwestern Polytechnical University, Xian 710072, People’s Republic of China. Fax: + 86 29 525 0199. 0022-0248/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PII SOO22-0248(97)00332-l
required for the solute distribution to reach the steady state. Therefore, the knowledge of time-dependent development of the solute distribution can help us to understand the time-dependent evolution process and the final steady state selection of the interface morphology. For example, the knowledge of morphological instability of a growing solid-liquid interface is an important basis to understand the complicated solidification microstructures. The classical theories [l-3] on morphological instability are based on an assumption that the solute distribution in front of the growing interface has already reached the steady state. In
W Huang et al. f Journal of Crystal Growth 182 (1997) 212-218
fact, any practically observed interface breakdown in experiments within finite time must have occurred before the solute distribution reaches the steady state. The big difference between theoretical prediction and experimental observation of initial wavelength of interface breakdown remains one of the unresolved important problems for the steadystate theories. This problem has been well resolved by Warren and Langer in a recent theoretical analysis of dendritic growth during directional solidification based on an approximate solution of the non-steady-state solute distribution [4]. Huang et al. gave the experimental evidence of time-dependent interface instability [S] and history-dependent steady-state selection of dendrite morphology [6]. These results showed that without detail knowledge of non-steady-state developing processes of crystal growth, it is difficult to understand the final solidification microstructures completely. The initial transient solute redistribution during directional solidification is a kind of Stefan problem and exact analytic solution has not been found yet. Tiller et al. [l] gave the first approximate solution for the problem, based on an assumption that to a first approximation, there exists a quasisteady-state solute distribution in front of the interface. Smith et al. [7] gave a more detailed and mathematically more accurate treatment of this problem. Both Tiller et al’s and Smith et al.‘s solutions are based on an assumption that solidification velocity is equal to the sample pulling velocity and keeps constant from the beginning to the end of the solidification process. In fact, the increase of interface concentration will slow down the interface velocity compared with the sample pulling velocity during the initial transient process. Huang et al. [S] gave a solution based on the quasi-steady-state approximation, which included the effects of the change of interface concentration on the interface velocity during the transient process. Recently, Warren and Langer [4] gave a new approximate solution for the problem, in which both the change of interface velocity and the change of diffusion length are taken into account. The above work considered diffusion as the only solute transport mechanism. However, liquid flow caused by natural convection or forced stirring is almost unavoidable during solidification on the
213
ground, which will make important contribution to the solute transport. The relation among liquid flow, solute and heat transport, and interface morphology is a subject of many interests and studies [9]. However, we have not found researches on the effects of liquid flow on the transient solute redistribution, and we think this is really an important problem needed to be resolved. For most type of liquid flow, there exists a hydrodynamic boundary layer in which the flow is parallel to the solid/liquid interface and the flow velocity tends to zero with approach to the interface, except for the volume-change convection during the solid/liquid transformation, which is usually a very weak liquid flow. Diffusion dominates the solute transport away from the interface inside the boundary layer, especially in the region very close to the interface, while liquid convection dominates the solute transport in regions further from the interface. According to this concept, Burton et al. [lo], and later Hurle [l l] and Delves [12] assumed that there exists a solute transport boundary layer with a thickness of 6. Within the distance 6 from the interface diffusion is the only solute transport mechanism, while beyond 6 the flow keeps liquid concentration uniformly equal to C6 because of sufficient mixing of liquid. Based on the assumption, they gave a solution for steady state solute distribution in front of the solid/liquid interface during directional solidification with liquid flow. In this paper, we check which of the above approximate solutions are better by comparing them with numerical calculation of the differential equation for pure diffusion in the liquid. We find that the Warren-Langer solution is the best of them. Then we incorporate the WarrenLanger approximation with Burton et al’s solute transport boundary layer approximation to give a solution for the initial transient solute redistribution during directional solidification with liquid flow, which is then compared with the experimental results by Inatomi et al. [13].
2. Check of the approximate
solutions
All of the approximate solutions for the initial transient solute redistribution mentioned above are
214
W. Huang et al. 1 Journal of Crystal Growth 182 (1997) 212-218
compared with numerical calculation of the differential equation for pure diffusion in the liquid during directional solidification. Basic assumptions for the numerical calculation are as follows: (1) A semi-infinite rod with uniform cross-section solidifies at constant temperature gradient G and at constant sample pulling velocity I/, with respect to the isotherms. The solid/liquid interface is plane and perpendicular to the axis of the specimen. The co-ordinate system is fixed to the moving interface. (2) Before solidification the specimen is of uniform concentration Co. (3) Diffusion coefficient is constant in the liquid and is negligible in the solid. (4) The interface is in local-equilibrium state, the equilibrium partition coefficient k. is constant and less than unity. The liquid concentration satisfying the above conditions is then determined by the following equations:
C=Da2c+j25 at
az2
(1)
CW) =
co
(4
’az’
1
C(~J)=G,
(3)
Dac = - I/iCi(l - k,) ) az z=o
(4)
v,
= L
I/
+ P
mdc(oJ) G
dt
’
where D is the solute diffusion coefficient in the liquid, m the liquidus slope, kO the solute partition
Table 1 Material properties
of succinonitrile-acetone
Property Melting point Diffusion coefficient of acetone Liquidus slope Equilibrium partition ratio
in succinonitrile
coefficient, I/i the interface velocity with respect to the solidifying material, and Ci = C(O,t) is the interface concentration in the liquid phase. The Eqs. (lH5) are numerically calculated with the one-dimensional finite differential method. In numerical calculation, Eq. (3) is changed to C(lOD/I/,,
t) = Co.
(6)
The calculation is carried out for SCN-acetone alloys of three different initial concentrations, 0.1, 1 and 10 mol% acetone, directionally solidified at the same temperature gradient of 5 K/mm and at the same sample pulling velocity of 1 pm/s. The material properties used in the calculation are shown in Table 1. The selection of the alloy concentrations and the experimental parameters cover three typical situations for directional solidification: ATTo/G is small for Co = 0.1 mol%, middle for Co = 1 mol% and large for Co = 10 mol%. Where AT, is the liquidus-solidus range at Co. The calculated results are shown in Figs. 1 and 2. One can find that Smith et al’s solution fits the numerical calculation very well at small value of AT,/G, poor at middle value of AT,/G, and very bad at large value of AT,/G. Tiller et al’s solution is not far from the numerical calculation only at the middle value of AT,/G. Huang et al’s solution is not far from the numerical calculation only at the large value of ATo/G. Warren and Langer’s solution is the one which fits the numerical calculation well for all of the three situations especially at large value of AT,/G. The above results indicate that the influence of the difference between Vi and VP on the initial transient solute redistribution becomes more important with increase the value of AT,/G, and that the constant diffusion length during the initial transient process is not a very good approximation.
alloy Symbol
Value
Ref.
Till D
331.24 K 1.27 x 1O-9 m’/s - 2.22 K/mol% 0.10
1141 Cl51 1141 Cl41
215
W. Huang et al. 1 Journal of Crystal Growth 182 (1997) 212-218
h? ---
0.24
- - - Tiller -.-_
0.22
s 0.16 B E = 0
Smith _____.Hwng
0.22
-Warren
0.20
(a)
Tiller
*
-.--smith
.-.--.Huang
-warren
0.16
0.16
Nunwid
o
0.16 014
(a) 0.10 0.00
0.60
1.00
1.50
2M
250
3.00
&mm
2.4
(b)
---Tiller - - - Tiller -. - - Smith .____.l+ang
2.2
-.-. ‘\
-Warren o Numerical
/
-----.Huang -Warren
'\ 1.80
,/
Smith
o
'\
Numerical
i
1.6
04
Cc)
---Tiller - .-. Smith .----.Huang -Warren
0.00 0
2w
400
600
800
1WO
12iUl
fS 1. Interface concentration in the liquid as a function of time during directional solidification of SCN-acetone alloys VP = 1 pm/s, (a) Co = 0.1 mol%, and at G = SK/mm (b) C, = 1 mol%, and (c) Co = 10 mol%.
050
o
Numetical
1.00
1.60
2w
250
3.00
&mm
Fig. 2. Liquid concentration profile in front of the interface during directional solidification of SCN-acetone alloys at G = 5 K/mm and VP = 1 pm/s, 1200 s from the beginning of solidification, (a) Co = 0.1 mol%, (b) Co = 1 mol%, and (c) C, = lOmol%.
216
W. Huang et al. J Journal
of CrystalGrowth
3. Mathematical model for initial transient solute redistribution with liquid flow
Here, we consider first a sample with finite length of liquid phase for directional solidification with liquid flow. There exists a solute transport boundary layer 6 in front of the interface, within 6 diffusion is the only solute transport mechanism and beyond 6 the liquid concentration is of a uniform concentration Cd. The other assumptions are the same as those for pure diffusion in the above section. Then Eq. (1) is only valid within the range 0 < z < 6, and Eq. (3) becomes C(z > &t) = Cd.
side of Eq. (11) is
(13) Then we obtain i =
Cie-6if + (Ci - Cd)e-“’ , Odzd6, 1 _ ,-w
Vi[Ca - koCi - Ci(l - ko)eP”I’]
-
’ -!!]C*-[iPl~eY~j~]Ci) [ (1 - e-‘/l)
~{(Ci-C~)[1-I’(:ISel:,~)2]}-”
(7)
We assume an approximate solution for Eqs. (1) and (2), (4), (5) and (7) with the following form Ca -
and the left-hand ~
3.1. Finite sample length
C(z,t) =
I82 (1997) 212-218
(14)
To determine Cg, consider the conservation the solute in the whole sample
of
LCo=j:koCiVidt+~~Cdz
(8) where 1 is the time-dependent diffusion length, and 1 = 0 at t = 0. Our task is to determine Ci, Cd and 1 as functions of time. Eq. (4) combined with Eq. (8) gives Wci-
Vi =
D(ci
m
‘C dz = Vi(Ca - koCi) - C a( L - S s0
-
cc5)
_
v
1
P
fs
in which
(10)
Vi dt = V,t + ~ (Ci - Co).
Compare the right-hand (16), one can find that
c, =
I
Dac az
&.=*
-
side of Eq. (11) is
sides of Eqs. (11) and
V,C,(l - ko)eP6” L - 6 - T/,8?- z (Ci - CO)
I
X + Vi(CJ - Ci) DazzEo
= Vi[C6 - k 01C. - C.(l 1 - k,)e-6’f]
(17)
0
of Eq. (1) from 0 to 6 yields
The right-hand
J:Vidt)
(16)
with Eq. (9) gives
/C,(l - k,)(l - ema”)
Integration
where L is the initial length of liquid phase. Differentiation of Eq. (15) with respect to time yields &
1Ci(l - k,)(l - e-“I’)’
2
(15)
cd)
Eq. (5) combined Ci
+ C,(,-,-~~Vidt),
(12)
Then, Ci, Cd, Vi and 1as functions of time can be easily determined from Eqs. (9) and (lo), (14) and (18) by numerical method.
W. Huang et al. JJournal
211
of Crystal Growth 182 (1997) 21Zp218
i = Vi(CO - klJCi) - ICi
3.2. Infinite sample length
(27)
’
Ci - CO When C(Z,t)
L~S+Vpt+~(Ci-CO)
(19)
the change of C6 with time can be neglected, i.e. c d z C,. Then, l/i, Ci and I can be determined from the following equations:
= Co + (Ci
Wci - cO) 1Ci(l - k,)(l - ee8”) ’
(20)
Co)ep”!‘,
Z 3
0.
(28)
These are just the results obtained by Warren and Langer [4] for the case of pure diffusion (please notice the difference of a factor of 2 in 1 because of different definition). 4.2. Comparison
vi =
-
with the solution for steady state
For infinite sample length, steady state can be reached when t approaches infinity, i.e., Vi = VP, ci = 0, and i = 0, and Eqs. (20) and (21) give I( ‘X ) = ;_
(29)
P
Ci( X) =
ci
2
L
Wci
-
co)
_
m ICi(l - k,)(l - ems”)
v
1
(24
’
,-W’
1 _
,
O
C(z,t) = co,
z 3 6.
(24)
4. Discussion 4.1. Comparison
with the Warren-Lunger
When li approaches come respectively, v,
=
’ c,
=
infinity,
Wci cO) ICi(1 - ko)’
solution
Eqs. (20 j(23)
be-
-
‘;
Wci
-
cO) _
’ m [ 1Ci(l - ko)
(25)
v
1
’
3
C(z,
‘cc
)=
co
k, + (1 - k,)e- “pziD k. + (1 - ko)e-‘pd’D ’
Odz<<. (31)
is given
by CO - Cie-*‘I + (Ci - C,)e-“’
Coke k.
and Eq. (23) becomes
The solute profile in front of the interface
C(z,t) =
co k. + (1 - ko)eP”n’“‘D=
(26)
These are just the solutions obtained by Burton et al. [lo-121 for steady state solute distribution during directional solidification with liquid flow. 4.3. Comparison
with experimental
results
Inatomi et al. [13] gave experimental results of the interface velocity Vi and concentration gradient Gc as functions of time during directional solidification of SCN-10 mol% acetone and SCN6.5 mol% acetone. The experiments were carried out in sample cells with finite initial length of about 20 mm for the liquid phase. The diffusion boundary layer and convection mixing region were observed in front of the interface, and the boundary layer thickness 6 was measured. The comparison of Vi and G, between the experimental results and calculations according to several models are shown in Figs. 3 and 4. The models considering the liquid flow, especially the one for finite sample length, fit the experimental results well.
218
W. Humg
of Crystal
et al. /Journal
Growth 182 (1997) 212-218
0.30
/’
___._.~~e” 0.25
- - - L=infinity -
f
L
.
-L=2Cfnm
0.15
i 0.10
0.05
(4
(a)
0.00
o.25_: -----.Warren 1.40
0.20.:
-
L=2Omm
1.20
E 3
1.00 0.80
3
o
0.60 0.40
(b) 0
200
400
m0
em
1000
1Mo
ts
Fig. 3. Interface velocity as a function of time during directional solidification of (a) SCN-10 mol% acetone, G = 2.37 K/mm, T/, = 1.05 urn/s and (b) SCN-6.5 mol% acetone, G = 2.37 K/ mm, T/, = 0.70 pm/s.
References [I] [2] [3] [4] [S] [6] [7]
W.A. Tiller, K.A. Jackson, J.W. Rutter, B. Chalmers, Acta Metallurgica 1 (1953) 428. W.W. Mullins, R.F. Sekerka, J. Appl. Phys. 35 (2) (1964) 444. D. Wollkind, L. Segel, Phil. Trans. R. Sot. 268 (1970) 351. J.A. Warran, J.S. Langer, Phys. Rev. E 47 (1993) 2702. Weidong Huang, Proc. In Space ‘94. Tokyo, Japan, 1994. p. 249. Weidong Huang, Xingguo Geng, Yaohe Zhou, J. Crystal Growth 134 (1993) 105. V.G. Smith, W.A. Tiller, J.W. Rutter, Can. J. Phys. 33 (1955) 723.
0.20
aw~~“‘:~‘.‘:~~‘~:‘.‘.:~‘~‘:~‘.‘I
0
200
400
600
arm
lCC0
1203
ts Fig. 4. Concentration gradient as a function of time during directional solidification of (a) SCN-10 mol% acetone, G = 2.37 K,‘mm, I/, = 1.05 urn/s and (b) SCN-6.5 mol% acetone, G = 2.37 K,!mm, I/, = 0.70 urn/s.
PI Weidong Huang, Qiuming Wei, Yaohe Zhou, J. Crystal Growth 100 (1990) 26. of Crystal c91 S.H. Davis, in: D.T.J. Hurle (Ed.), Handbook Growth, vol. lb, Elsevier, Amsterdam, 1993, p. 859. Cl01 J.A. Burton, R.C. Prim, W.P. Slichter, J. Chem. Phys. 21 (1953) 1987. Cl11 D.T.J. Hurle, J. Crystal Growth 5 (1969) 162. 1121 R.T. Delves, J. Crystal Growth 314 (1968) 562. [I31 Yuko Inatomi, Hirofumi Miyashita, Eiichi Sato, Kazuhiko Kuribayashi, Kazuhisa Itonaga, Tetsuichi Motegi, J. Crystal Growth 130 (1993) 85. Cl41 SC. Huang, M.E. Glicksman, Acta Metall. 29 (1981) 701. N.B. Singh, Met. Trans. Cl51 M.A. Chopra, M.E. Glicksman, A 19 (1988) 3087.