Journal of Crystal Growth 80 (1987) 383—392 NorthHolland, Amsterdam
383
DENDRITE SPACING IN UNIDIRECTIONALLY SOLIDIFIED AlCu ALLOY AN Geying and LIU Lixin Department of Metallu Materials and Technology, Harbin Institute of Technology, Harbin, People’s Rep. of China Received 6 February 1986, manuscript received in final form 6 November 1986
Directional solidification experiments have been carried out to study the variation in primary and secondary arm spacings with solidification parameters in the Al—Cu system. It is found that the primary arm spacing Z 5 in the high velocity regime or low temperature gradient ~egime, where a and1 b obeys are constants, the common the value correlation of K included the composition dependence, G is the temperature gradient in the liquid and V the growth rate; however, it does not obey = KG~V this correlation for the low velocity regime or high temperature gradient regime but goes through a maximum or a catastrophe as a function of V or G at V= or G = ~ where k is the equilibrium distribution coefficient, and ~ and ~ are the critical velocity and temperature gradient at the limit of constitutional undercooling respectively. The initial secondary arm spacings Z 54 (mm). The secondary arm spacing Z 20 are nearly independent of G and malnly depended on V, Z20 0.016 V° 2 tends to coarsen with time and thus is a function of coarsening time tf, Z2 = 0.016t?~~ (mm). Theoretical analyses of the primary arm spacing and the initial secondary arm spacing have been proposed, and the derived relationships agree reasonably well with the above experimental results.
1. Introduction Dendritic growth is the most common crystallization in cast materials, and dendritic structures are indeed always present in alloys which are solidified under normal casting conditions [1]. These structures form with a solute content significantly different from the average solute cornposition of the alloy. The difference in solute content between dendrites and the interdendritic region results in microsegregation. The solute segregation pattern is characterized mainly by the primary and secondary dendrite spacings; moreover, the degree of microsegregation which can be obtained in a casting alloy with commercially practical homogenization treatments depends on dendrite spacings. Consequently, there has been considerable interest in recent years in the mechanism controlling the dendrite spacing. The major morphological features of a dendrite are characterized by the primary arm spacing, the secondary dendrite spacing and the tip radius. Many experimental studies [2—9]have been carried out to measure primary and secondary dendrite spacings in binary alloys. Most directional solidifica
tion studies [2—4]have shown that the primary arm spacing decreases as the growth rate or ternperature gradient is increased. These results can be replotted to give the following relationship:
z1
=
KG~V”.
(1)
Experimental evidence available indicates that the secondary dendrite spacing is only influenced by the cooling rate R (or local solidification time t~) and alloy content C0, and is expressed as follows [8,9]: ~ K R_~ K tb (2) =
2
=
1
2 1’
where a and b are constants and the values of K1 and K2 depend on the composition. However, from detailed experimental work in the Pb—Sn system, Klaren et al. [6] found that Z1 does not continuously increase with a decrease in velocity but that it goes through a maximum. Only in the high velocity regime did they find the relationship given in eq. (1) to hold true. The same conclusion was obtained in a succinonitrile— acetone system by Somboonsuk et a!. [7]. Theoretical analysis of the primary arm spacing
00220248/87/$03.50 © Elsevier Science Publishers B.V. (NorthHolland Physics Publishing Division)
384
An Geving, Liu Lixin
/
Dendrite spacing in unidirectionally solidified Al — Cu alloy
was given at first by Brown and Adams [10] as follows: Z1=8D~Ct/(1—k)C0,
(3)
where D is the solute diffusion coefficient in the liquid, ~C the surfacetocenter content difference in the liquid region, t the local solidification time, k the equilibrium distribution coefficient, and C0 the initial solute concentration. Subsequently, Rohatgi and Adams [11] showed the spacing to be a function of solidification rate instead of solidification time in eq. (3). Another theoretical analysis was proposed by Okamoto and Kishitake [4]:
Fig. 1. Dendrite coarsening model: (a) Kattamis model [9]; (b) Chernov model [14]; (c) Kahlweit model [15].
Z1
where
=
2a( —mD(1 R
—
k)C0
(4)
)1/2
where a is a constant which is smaller than unity, m the slope of the liquidus, and R the cooling rate. However, in these two models the effects of solidification rate and temperature gradient were not isolated. Two detailed theoretical models have been proposed by Hunt [12] and Kurz and Fisher [13], respectively. The Hunt model [12] can be expressed as: 2Z~ _64DF[m(1 k)C VG 0 + kGDV’], (5) =
—
where F is the Gibbs—Thompson parameter (F a/AS), with a specific liquid—solid interface energy and ~S the melting entropy. According to the theory, the primary dendrite spacing decreases with an increase in velocity at V 2k~S for a constant G, where V~~=GD/L~T~ z~T0=—mC0(1—k)/k, 1, =
=
and the maximum of Z1 occurs at this velocity. Another detailed theoretical model to characterize the primary arm spacing was obtained by Kurz and Fisher [13] as follows: for the low velocity (V< V~~/k) regime [6~T’ ~D k ~T 0 \11/2 Zi~_~j_(~j~_ G ‘
(6)
for the high velocity (V> V~~/k) regime 4G1/2V1/4, =
4.3(DF/k)’/
(7)
J’~J1~ J1J1... __fl.__,~._fL. (a)
p
(b)
1 k, a~T’ (1 GD/VL~T0)~T0/p. The Hunt model and the Kurz—Fisher theory are different only by a constant in the high veloc=
—
=
—
ity regime. However, the low velocity regime, these two models giveforsignificantly different results. The Kurz—Fisher theory predicts a sharp increase in Z1 with a decrease in velocity at V fr~~/k and a maximum in Z1 to occur at V= 2J’~/(1+ k) for a constant G. The theoretical analysis of the secondary arm spacing was given at first by Kattamis et al. [9]. Three simple idealized isothermal coarsening models are sketched in fig. 1. Their analysis can be expressed in the following equations: For the Kattamis coarsening model [9], 3 [F(J~)]2[ln(1 —f) +Jj, mCL(1DF k)d =
—
(8) where CL is the liquid concentration in the interdendritic spaces, f re/a, F(f~) a/d, and d =
=
Z2/2.
=
For the Chernov model [14], 3 mCL(1 k)d~[F(J~)] DF [ln(1
—f) +f+ 1f2]
(9)
Because the secondary spacing tends to coarsen with or with distance away from dendrite tip. ittime, is necessary to investigate the the variation in
An Geying Liu Lixin
/
Dendrite spacing in unidirectionally solidified Al—Cu alloy
385
s
the initial and the final secondary arm spacing with solidification variables under directionally solidified conditions.
t~ ~‘ ___
___
~.
v
~
2. Experimental procedure and method
~

Alloys were prepared from 99.999% pure Al and 99.99% pure Cu in evacuated and sealed graphite crucibles. These alloys were quenched and swaged into 7 mm diameter rods. The apparatus used for unidirectionally solidified specimens is shown in fig. 2. A temperature gradient of between 5 and 30 K/mm in the melt2cm/s and growth were rates ranging from 5 x 10~to 5 X 10 obtained by this apparatus. Unidirectional growth was achieved by withdrawing the furnace and cooling systems upwards to minimize convection in the liquid. Experiments showed that the withdrawing rate was equal to the growth velocity of the interface over the entire range of growth 6
Fig. 3. Schematic representation of the measurement of seeondary dendrite spacings; Z
2 = I/n and Z20 = l’/n, where / and 1’ are volume lengths and n is the number of arms; S = distance and time z~ = S/V.
2
/
24 E
g’
2.3
.
X
2.2
~
2.1
1—
,O

2.0~ 0
I
10
20
30
I
40
I
50
V (pm/s) Fig. 4. Primary arm spacing Z 1 versus growth velocity V for 8
—
2
Al—2%Cu alloy; (0) cell, (X) dendrite; (1) G = 99.2 K/cm, (2) G = 68.4 K/cm. 1~1
I _
—
12
~2.5
13
2.2 Fig. 2. Schematic view of the apparatus used for unidirectional solidification: (1) to vacuum, (2) controlling thermocouple, (3) insulation, (4) water bath, (5) water in, (6) thermocouple, (7) specimen tube, (8) furnace tube, (9) heater, (10) withdrawing system, (11) specimen, (12) water out, (13) liquid metal coolant, (14) steel rod.
1 2. ~
I
10
I
I
20
30
I I
I
40 50 V ~m / s)
Fig. 5. Primary arm spacing Z1 versus growth velocity V for Al—4.5%Cu alloy: G = 80K/cm; (0) cell, (X) dendrite.
386
An Coving, Liu Lixin =~— —~~—
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~
Cu alloy
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Dendrite spacing in unidirectionally solidified Al—Cu alloy
~
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Dendrite spacing in unidirectionally solidified Al — Cu alloy
conditions. Temperature measurements in the specimen were made using a Pt/PtRh thermocouple placed 5 mm away from the interface, made from 0.1 mm diameter wire and insulated from the specimen using an alumina tube of 0.2 mm ID and 0.8 mm OD. The temperature gradient in the liquid at the growth front was determined from the slope of the temperature versus time record. The alloy was contained in an alumina crucible of 8 mm OD, 7 mm ID and 400 mm total length. The specimen was grown at the desired rate until about 50 mm had solidified. The remainder of the sample was then quenched, delineating the solid—liquid interface in the microstructure. The quenched unidirectionally grown specimen was removed from its tube, a longitudinal section which included the quenched interface was cut from it to observe the cellular or dendritic morphology and to measure the secondary arm spacing, and a transverse section 10 mm away from the quenched interface was prepared to measure the primary arm spacing. The primary arm spacing was measured at two to five different locations by a method described by Jacobi and Schwerdtfeger [2]. The secondary arm spacing was measured at different distances away from the dendrite tip and the initial secondary spacing was measured close to the dendrite tip for five different dendrites, as sketched in fig. 3.
240.
~
3. Results 3.1. Primary arm spacing The effect of V on Z1 was investigated at a constant temperature gradient. The results are shown in figs. 4 and 5. The curves shown in figs. 4 and 5 confirm that the primary arm spacing does
290
(a)
~
N
250
~ ‘.
2 210
• 
~“
“N..
SN
.
170 I I
80
120
I
40
I
“I
160 G(K/cm)
________________________________ b 250.
()
E 210
.\
90
~
120
.
5
50
I I
70
90
I
I
110 130
I
150 170
G (K/cm) Fig. 7. Effect of G on Z1 at constant V for Al—2%Cu alloy: (1) V = 30 ~sm/s, (2) V = 21 ~sm/s.
40
I
80
I
I
120
I
160
I
G(K/cm) Fig. 8. Effect of G on Z1 at constant V for Al—4.5%Cu alloy. Panel a: (1) V 51.6 1sm/s, (2) V = 29.3 ism/s, (•) dendrite. Panel b: V = 9.8 Om/s, (0) cell, (S) mix, (X) dendrite.
An Geying Liu Lixin
/
—
Dendrite spacing in unidirectionally solidified Al Cu alloy
Al—2%Cu alloy and Z1 ~ V°24 for Al—4.5%Cu alloy. Figs. 7 and 8 show the change in behaviour of Z 1 54 with for a given of p.m/s Z1 ~ for GAl—2%Cu, Z V. Relationships 6at V= 9.8 G° 1 ~ G°~ and 29.3 ftm/s, and Z 49 at V= 51.6 p.m/s 1 ~ temperature G° in Al—4.5%Cu for low gradient regime (G < kG~~)were obtained. Z 1 decreased sharply with an increase in G at G 120 K/cm for V= 9.8 p.m/s (G kG~~), and this catastrophe of Z1 is correlated with the dendritetocell transition, as shown in fig. 8b.
N
40
• 20
389
•____
. 2
=
0
=
80
120
160 G (K/cm)
Fig. 9. Variation of Z20 with G in Al—4.5%Cu alloy: (1) V = 20.4
~srn/s,
3.2. Initial secondary arm spacing
(2) V = 29.3 ~sm/s.
not continually increase with decreasing velocity but goes through a maximum. Below V 10.2 p.m/s at G 68.4 K/cm, V= 13.2 p.m/s at G 99.2 K/cm (as shown in fig. 4) and V= 20.4 p.m/s at G 80 K/cm (as shown in fig. 5), i.e. V (1—3)V~ 5/k, the primary spacing decreases sharply with decreasing velocity. The quenched interface changes from cellular to developing dendritic to developed dendritic structures, as shown in fig. 6. For the high velocity regime, least 28 squares for linear regression analyses give Z1 cx V° =
=
=
=
=
50
The variation of Z20 with G and V are shown in figs. 9 and 10, respectively. Fig. 9 shows that the temperature gradient G has nearly no effect on the initial spacing Z20. Z20 decreases as V increases 54(mm), according to thein relationship Z20 as shown fig. 10. 0.016V° =
3.3. Coarsening secondary arm spacing Results of the coarsening secondary spacing are shown intipfig. The distances away from dendrite are11. exchanged into coarsening timethe t~
—
80
Chernov model
40’\%\~~
:: I
10
L~ttamismode~~~
I
20
30 Fig. 11. Comparison of experimental results in Al—4.5%Cu alloy with the Kattamis coarsening theory [9]: G = 80 K/cm.
Fig. 10. Variation of Z 20 as a function of V for Al—4.5%Cu alloy: C 80 K/cm.
Values of V (in tem/s) are: (0) 9.8, ~<) 15.4, (~)20.4, 29.3, (0) 38.1.
(5)
390
An Geying, Liu Lixin
/
Dendrite spacing in unidirectional/v solidified Al—Cu alloy
by the relation tf S/V. A leastsquares linear regression of all the data in fig. 11 gives Z2 34(mm). 0.016t~ =
=
where C~is the tip composition. For a marginal stability condition, Trivedi tip composition is given by:[19] calculates that the c~=(~A+BL/P2),
4. Discussion
where P is the Peclet number (P and B are parameters,
4.1. Primary arm spacing
(11) =
VR/2D), A
A =2GD/Vk.~T 0, B= VF/2Dk~T0,
The Kurz—Fisher theory predicts that the primary arm spacing will increase sharply as the growth velocity is decreased at V= J~~/k.However, experimental results show just the opposite behaviour; the primary arm spacing decreases sharply near J’~~/k. The Hunt model predicts that the maximum occurs at V 2J’~~, which is much smaller than that observed experimentally. The Hunt model does not predict a catastrophe to occur at G kG~5, and the Kurz—Fisher theory =
=
shows the opposite catastrophe to appear at the critical G. For the high velocity or low temperature gradient regime, both theoretical analyses give 2V’~4for a given composition, which Z1 cz G’~ agrees with the experimental results qualitatively. We now propose a simple theoretical analysis for the effect of the temperature gradient, growth velocity and physical properties of the alloy on the primary arm spacing. Two major assumptions are made in this analysis as in the Hunt model [12]: (1) a dendrite or a cell tip may be approximated by a smooth steady shape even when dendrite sides have arms formed; (2) behind the tip the composition and temperature in the liquid are almost constant in the radial direction perpendicular to the growth direction. A shape for the primary dendrite was obtained by Hunt [12], based on these two assumptions and a general solute balance. However, it does not describe the shape near the tip because the liquid
and L 4(1 + 1)(l + 2) for the spherical approximation of the dendrite front, where 1 is the harmonic of perturbation (3 < 1< 6). Inserting eq. (11) into eq. (10) gives =
Z~ 4I~LF/GP =
GZ~ R=
4V~[mC~(1
—
k)
+
GD/V]
(10)
8V~DLF/GVR.
(12)
The relationship between the tip radius R and solidification variables may be obtained by assuming marginal stability of the tip as follows [13,16]: for the low velocity regime (V < J’~~/k) R 2D/Vp + 2mC0/G, (13) for the high velocity regime (V> fr~~/k) R 2~(Dr/Vk~T 1~2. (14) 0) Inserting eqs. (13) and (14) into eq. (12) respectively, we obtained the final results as: for V < =
=
Z 2(GD/p+ mVC 2, 1 V> 2.38(DLF)~’ 0)” for k~~/k
(15)
z,
(16)
=
=
i .34L1/2(DFk~Tc~)l/4G1/2V1/4.
Eqs. (15) and (16) show that the primary dendrite spacing goes through a maximum as a function of velocity at V V~~/k or decreases sharply with an increase in temperature gradient at G kG~ 5,where ~ ~TQ/D. The result given by eq. (16) is similar to that given by eqs. (5) and (7) but different by a constant, for the high velocity or low temperature gradient regime. A comparison of this analysis, the Hunt model and the Kurz—Fisher theory with experimental results in Al—2%Cu and Al—4.5%Cu alloys is shown in figs. 12 and 13, using the physical data of tables I and 2. It is found that the experimental results are just opposite to the result given by the Kurz—Fisher theory =
=
=
is not really homogeneous in the radial direction there. Hunt circumvented this problem by fitting part of a sphere to the derived shape at the growing front. He derived the relationship between the primary arm spacing and the tip radius R:
=
An Geying~Liu Lixin
/
Dendrite spacing in unidirectionally solidified Al—Cu alloy
Table 1 Values of the various parameters used, Al—4.5%Cu alloy [9] 3 m = —3.33 K/%Cu H = —250 cal/cm o=1.2X106cal/cm2 k=0.18 D=5X105cm2/s f=0.5 Tm = 635°C= 908 K F(f~)= 0.5 CL = 7.7%Cu experimental results and the Hunt model near
1000
E
3 N
— 400 600
.
300
.
Kuri~r
1_...~
at lower velocities or higher temperature gradients. A great difference also exists between the
I
200
391
V~ 5/k and kG~5. The correlation between this
100
Hunt
.
I
0.6
I I
1
2
analysis 4.2. Initial and secondary the abovearm results spacing is good.
(his model
I
I I
4 5
10
I
I I
I
15 20 30 50 100 V(ylm/s)
Fig. 12. Comparison of the three theories with experimental data in Al—2%Cu alloy: G= 68.4 K/cm; (0) cell, (x) dendrite,
Kurz
3.0

Many studies [16,17]have presented that Z20/R is a constant and is expressed as: Z20/R
=
K1,
(17)
where K1 2.1 ±0.03. When the velocity is so large as to form secondary arms, the tip radius R may be expressed as [18]: 2=K VR 2, (18) =
where K2 is a constant. By inserting eq. (18) into (17), we obtain 2, (19) Z20=KV_t/ where K K t~2.Eq. (19) predicts that the mitial secondary dendrite spacing is dependent on 1K2
Fisher
3 N
=
2.4
Kurz

Fisher
This model
2.2
temperature gradient, which agrees well with the the growth velocityresults but isshown independent above experimental in fig. 10.of the 4.3. Coarsening secondary arm spacing
2.0
theFig. coarsening secondary arm spacing results with the 11 compares the experimental of
~ Hunt
18
Table 2 Physical properties of Al—2%Cu alloy [13,20] 1.6 I
1.8 I
2.0 I
2.2
2:4 log G(K/cm)
Fig. 13. Comparison of the three theories with experimental results in Al—4.5%Cu: V = 9.8 ~sm/s, (0) cell, (•) mix, (X) dendrite.
= —2.6 K/%Cu k = 0.14 D 3 X i0~ cm2/s ~ = 2.4 ~ io—~K cm
ni
392
An Geying, Liu Lixin
/
Dendrite spacing in unidirectionally solidified Al — Cu alloy
coarsening theory proposed by Kattamis et a!. [9] (eqs. (8) and (9)) using the constants of table 1. A good relationship is obtained between the theory and the experimental results. Comparing these results with the data obtained by Kattamis et al. [9] and Bower et al. [8], it is found that coarsening occurs under common solidification and isothermal coarsening conditions at roughly the same rate as that observed under unidirectional solidification conditions. Consequently, it is suggested that the Kattamis coarsening theory is applicable to metallic solidification processing.
5. Conclusion (1) It has been found that the primary arm spacing does not continually decrease with an increase in the growth velocity and temperature gradient, but that it goes through a maximum as a function of velocity at V= I/~5/k or changes sharply at G kG~5 as a function of G. This maximum and catastrophe of Z1 is correlated with a transition of cell to developing dendrite to developed dendrite. Only for the high velocity or low temperature gradient 2V regime find the rela1/4 do for we a constant contionship Z1 KG ‘~ tent to hold true. The theoretical analysis proposed in this paper may explain the variation in Z 1 with G and V. (2) The initial secondary dendrite spacing is only dependent on the growth velocity for a given composition. (3) The secondary arm spacing tends to coarsen with time and thus is a function of the coarsening time. It is found that the coarsening occurs under common solidification and isothermal coar=
=
sening conditions at roughly the same rate as that observed under unidirectional solidification conditions.
References
[1] MC. Flemings, Solidification Processing (McGrawHill. New York. 1974). [2] H. Jacobi and K. Schwerdtfeger, Met. Trans. A, 7A (1976) 811. [3] D.C. McCartney and J.D. Hunt, Acta Met. 29 (1981) 1851. [4] T. Okamoto and K. Kishitake, J. Crystal Growth 29 (1975) 137. [5] J.T. Mason, J.D. Verhoeven and R. Trivedi, J. Crystal Growth 59 (1982) 516. [6] CM. Klaren, J.D. Verhoeven and R. Trivedi, Met. Trans. A. hA (1980) 1853. [7] K. Somboonsuk, J.T. Mason and R. Trivedi, Met. Trans. A. iSA (1984) 967. [8] T F Bower, H D Brody and MC. Flemings, Trans TMSAIME 71(1966) 624. [9] T.Z. Kattamis, J. Coughlin and MC. Flemings, Trans. Met. Soc. AIME 239 (1967) 1504. [10] P.E. and879. CM. Adams, J. Trans. Am. Foundrymen’s Soc. Brown 69 (1961) [11] P.K. Rohatgi and CM. Adams, Trans. Met. Soc. AIME 239 (1967) [12] J.D. Hunt. 879. Solidification and Casting of Metals (Metals Society, London, 1979) p. 3. [13] W. Kurz and D.J. Fisher, Acta 1Met. 29 (1981) [14] A.A. Chernov, Kristallografiya (1956) 583. 11. [15] M. Kahlweit, Scripta Met. 2 (1968) 251. [16] J.S. Langer and H. MhillerKrumbhaar, Acta Met. 26 (1978) 1681. [17] SC. Huang and ME. Glicksman, Acta Met. 29 (1981) 701. [18] J.S. Langer, R.F. Sekerka and T. Fujioka. J. Crystal Growth 44 (1978) 414. [19] R. Trivedi, J. Crystal Growth 49 (1980) 219. [20] M. Gunduz and J.D. Hunt, Ada Met. 33 (1985) 1651.