On the non-steady-state growth of lamellar eutectics

Volume

10, number

MATERIALS

11,12

On the non-steady-state Junmin

March

LETTERS

1991

growth of lamellar eutectics

Liu

Laboratory of Solid-State Microstructures, Nanjing University, Npnjing 210008, China Received

19 March

1990; in final form 29 November

1990

The non-steady-state growth of the typical anomalous Al-5 eutectic is investigated experimentally by means of two methods of non-steady-state growth: abrupt change of growth rate and constant-acceleration (deceleration) growth. The uniqueness of steady-state growth of the eutectic is verified, and the details of the dependence on non-steady-state growth of the response features, such as silicon interflake spacing, to varying growth rate, are shown. A model for non-steady-state growth of a lamellar eutectic is proposed to describe the response dynamics of a eutectic: a new concept.

Three primary obstacles met by the theory of eutectic growth are: ( 1) whether steady-state growth is unique or not; (2) whether the eutectic structures are dependent on details of non-steady-state growth; ( 3 ) what is the response feature of the lamellar spacing during non-steady-state growth. In order to solve these problems, we must analyze non-steady-state growth of a eutectic (i.e. transient eutectic growth). Mollard and Flemings [ 1,2] studied transient growth of Pb-Sn eutectic alloys, theoretically and experimentally, but they paid attention to those conditions that favor plane front solidification of binary alloys containing a eutectic, and the dependence of solute redistribution in the liquid in front of the growing interface on variation of growth rate. At least, their work cannot directly answer all the above obstacles. On the other hand, they declared that for eutectic growth, the initial transient is of the order of the characteristic diffusion length. However, we find that the transient distance between two steady states of growth is up to ten times the length, and depends on response laws of the eutectic to variations in growth rate. Therefore, it is necessary that the transient growth of eutectics be investigated once more. Here, the typical anomalous Al-% eutectic was selected for experimental investigation (not the regular Pb-Sn eutectic, as done by Mollard and Flemings [ 1,2 1, because we think some new results of non-steady-state growth of a eutectic can be obtained by investigating an anomalous eutectic) by 0167-577x/9

I/$ 03.50 0 199 1 - Elsevier Science Publishers

means of two techniques of non-steady-state growth. One is the abrupt change of growth rate during growth (mechanism A) and the other is the constant-acceleration (and deceleration) growth (mechanism B). We also tried to develop a theory of non-steady-state growth of a eutectic. The experiments were done on a unidirectional solidification apparatus [ 3 1, and a detailed description of these experiments is omitted here. We will pay attention to response laws and adjusting mechanisms of the inter-Bake spacing, 2, of A14 eutectic for varying growth rates V. We define an abrupt change factor, p= VJV,, for mechanism A ( VI and V, are the growth rates before and after the abrupt change, respectively), and denote the acceleration of growth by the symbol a for mechanism B. For mechanism A, the response curves of I to the abrupt change of growth rate, from the measured data,areshownintig. laasp>l andfig. lbasp 10). For p> 1, the retardation of the response is more remarkable. We denote by S, the retarded distance. This distance is defined as the growth length from the position where the abrupt change happens to the position where A= d, after this

B.V. ( North-Holland

)

521

Volume 10, number Ii,12

12

I

21 0

MATERIAL LETTERS

04

‘P

20

30

40

50

60

S (mm)

Fig. 1. The interflake spacing J., as a function of the growth distance, S. GL=89 K/cm. The dotted lines correspond to the positions where the abrupt change of growth rate happens. (a) p> 1, (b)

PC

1.

abrupt change. We have concluded from a number of measurements that, with I’, held unchanged, the larger the term ]p- 11, the longer the distance S,; while p is held unchanged, the larger (asp> 1) or the smaller (as p< 1) the growth rate, V,, the longer the distance S,. We first operate the abrupt change of growth rate with p> 1, and then operate the abrupt change of growth rate with p < 1, on the same sample. Even if the growth parameters satisfy Vi lP,,= VZIP<,, the response curve for p> 1 differs from the one for p < 1. Therefore, we conclude that the response law of I for varying growth rate is dependent on the details of non-steady-state growth. For mechanism B, as a > 0, the response of R to the varying growth rate is also retarded and wavy, as shown in fig. 3 (see below), where .rZ, is the measured response curve. The larger the parameter a, the stronger the retardation. From the general tendency, the response function of 1 with respect to growth rate V, ( I’,= I’,+&, V, the initial growth rate, t is time), is exponential When a < 0, within the range inves522

March 1991

tigated, no response of L to the varying growth rate has been observed, except when the acceleration a approaches zero. In other words, while {al is held unchanged, the retardation of response is much more remarkable as a< 0 than when a> 0. Similarly, we first operate the growth with a > 0, then operate the growth with a < 0, on the same sample, by making the growth satisfy the condition: the final growth rate equals the initial one. We have observed that the spacing R cannot return to the value 1, ] V=VO.So we conclude, for mechanism B, the response of the spacing is irreversible, and depends also on detailed growth. For mechanism A, as ps 1, the response mechanism of the spacing to the abrupt change of growth rate is immediate branching of silicon flakes, as shown in fig. 2a. In practice, growth with p% 1 is uncommon. Commonly, adjustment of the spacing for p> 1 results from the so-called “cluster-branching” of silicon flakes, as shown in fig. 2b. The latter is dynamically superior to immediate branching, and is more effective than the so-called “termination mechanism” [4]. For p< 1, continuous clusterbranching of silicon flakes has also been observed, accompanied by growth-stopping of both single silicon flakes and clusters of silicon flakes. For mechanism B, similar response mechanisms to those for mechanism A have been observed. We can be convinced that, during non-steady-state growth, silicon flakes have a strong tendency for branching. It is necessary, by these mechanisms, that continuous and wavy response of I to varying growth rate take piace. Because of the strong tendency for branching of silicon flakes, the retardation of the response is more remarkable for p< 1 than for p> 1. For mechanism B, the result of a strong tendency for branching is that, for a < 0, the spacing nearly has no response to varying growth rates. Furthermore, we investigated the transition to the steady state from the non-steady states of mechanisms A and B for times long enough for each of the transitions. The measured results show that the spacing i always tends to 1,; the selection of A corresponding to steady-state growth. This clearly demonstrates that the steady-state selection of eutectic growth is unique, and independent of the detailed process. So far, we have resolved the three problems

Volume 10, number 11, I2

March 1991

MATERIALS LETTERS

a+2 a*c v, ac ~+~+~~=~at’

I

ac

(1)

where X, Z are the coordinates parallel and perpendicular to the growing interface, D the solute diffusion coefficient in the liquid, t the time and V,= V2 for mechanism A, V,= V, for mechanism B. We assume the solution of eq. ( 1) has the form C( X, Z, t) =A (X)B( Z) Y(t). Satisfying suitable initial and boundary conditions, the solution can be written as follows. For mechanism A:

;

xexp(Dyt-_2Z) for mechanism C(X,Z,

t)=CE+

xexp(Dyt--M,Z)

B: cD &VoCo ___ cos (2nnXIA) n=, (nz)2D

c

,

(3)

in which 1,=A,\ ,,=“,, ;iO=jl,\ v=vO. CO is the difference between the two maximum solubilities for the a-Al and Si phases, respectively, CE the concentration of the eutectic point, M, = V,/2D+ Fig. 2. Photomicrographs showing the response mechanism of the silicon interflake spacing, 1, to the abrupt change of growth rate; G,=89 K/cm, magnification 200x. (a) The abrupt branching of silicon flakes for V, = 1 pm/s, pz 70. (b) The cluster-branching of the silicon flakes, for V, =47 pm/s, vz=69 pm/s.

out above, with the aid of our experimental investigations. The non-steady-state growth of a eutectic can be treated theoretically in the light of the correlation of changing the solute distribution in the liquid with the detailed process of growth. This distribution is determined by the diffusion of solute in the liquid, as discussed by Mollard and Flemings [ 11, and by the ability for branching and growth-stopping of the silicon phase while the growth rate changes. The solute concentration in the liquid near the growing interface, C, satisfies pointed

[ ( V,/2D)2+y+

(2nn/L)*]

“2,

i=2, t , andy=Y’(t)/Y(t).Fromeqs.(2)and(3)weknow that the factor y is a parameter that relates the control of the response dynamics of the eutectic phases on the solute diffusion field. The response dynamics describes branching and growth-stopping of the eutectic phases during non-steady-state growth. As y tends to zero, it is very difficult for the eutectic phase to branch or stop growing while the growth rate changes; the solute distribution in the liquid changes very slowly with time. In other words, the smaller the term I ~1, the more slowly the distribution changes. Therefore, the eutectic phases can hold no branching and no stopping of growth for a larger interval of time. This explains reasonably the retarded response of the spacing to the varying growth rates. We can also derive from eqs. (2) and (3) the following properties of the parameter y: 523

Volume 10, number 11,12

y>O

forp> 1 ora>O,

y=O

forp= 1 or a=0 ,

y
forp< 1 or a<0 .

MATERIALS LETTERS

With the aid of the response function for interface supercooling [ 51 and the steady-state selection of eutectic growth [ 41, we can easily derive the relation between il and time (i.e. the response law of 1 to the varying growth rate). For mechanism .A: L2V2=kaL/Q,L ,

~u+e2c

QL=

s

ED

x $r

0) 0

I

10

I6

Onv,

(A)’sin’ (E)

cxp(Dyt);

20

30

40

50

60

70

V, (pm 1s.)

(4a)

nlvl -

March 1991

M

I

!

(4b)

for mechanism B:

01

A=hLtDIQ:,

(5a)

’

70

60

50

40

30

20

10

0

V&m/s)

Q; = ( 1+E)~C’~;~~ V, Fig. 3. Response curves of the spacing as a function of the growth rate V,, for mechanism B. GL=89 K/cm. (a) V,=l urn/s, ~~0.072 um2/s. (b) If,,=70 pm/s, a= -0.078 um*/s.

Here k is the revised factor, which can be obtained from experiments; uL and e are constants related to the physical parameters of the eutectic. The parameter y can be easily obtained from experiments. For example, for mechanism A, y=O.O05/D for p> 1, y= -0.001/D forp< 1; for mechanism B, y=O.OOl/ D for a>O, y= -0.0001/D for ac0. In order to test the model described above, for mechanism A, we list Table 1 Retarded distance, S,, from calculation, and S,, from measurements, of response of 1 to the abrupt change of growth rate under various conditions for mechanism A. CL, the temperature gradient in front of the growing interface in the liquid, is equal to 89 K/cm Y

sr (mm)

sm (mm)

P

VI @m/s)

V2 (w/s)

0.005/D

3.53 2.17 1.72

6.89 24.33 52.91

24.33 52.91 91.99

3.75 5.02 6.66

4 7 8

-0.001/D

0.14 0.67

68.65 66.97

9.85 44.95

23.74 10.97

22 13

(pm-*)

524

in table 1 the measured retarded distance, S,,,,, and the calculated one, S,, for various growth conditions. For mechanism B, the response curves from measurementslf, II,, and from calculations, &, of the spacing with respect to the varying growth rate, are shown in fig. 3. The consistency of the calculated results with the measured ones indicates that the above model is effective.

References [ 1 ] F.R. Mollard and M.C. Flemings, Trans. Metall. AIME 239 (1967) 1526. [2] F.R. Mollard and M.C. Flemings, Trans. Metall. AIME 239 (1967) 1534. [ 31 J.-M. Liu, Y.-H. Zhou and B.-L. Shang, Acta Metall. Sinica 25B (1989) 386. [4] K.A. Jackson and J.D. Hunt, Trans. Metall. AIME 236 (1966) 1129. [ 5 ] J.F. Barker and J.W. Cahn, Solidification (ASM, Metals Park, 1971) p. 23.