Pattern selection during directional solidification

Journal of Crystal Growth 82 (1987) 747-756 North-Holland, Amsterdam

747

PATTERN SELECTION DURING DIRECTIONAL SOLIDIFICAnON B. BILLIA, H. JAMGOTCHIAN and L. CAPELLA Laboratoire de Physique Cristalline, UA 797, Faculte des Sciences de St. Jerome, Case 151, Rue H. Poincare, F-13397Marseille Cedex 13, France Received 15 September 1986; manuscript received in final form 1 November 1986

A model has been constructed for pattern selection during the directional solidification of binary alloys, in terms of characteristic shape parameters. A step by step procedure allows a direct comparison with experiments at each level of the analysis. The introduction of nondimensional periodicity and radius of curvature makes a unified approach possible, which depends only on a parameter Ie and on the solute partition coefficient. It is found that the Pelce-Pumir relation for the tip supersaturation agrees well with available data and that marginal stability at the tip should be a limiting case of a more general capillarity condition whose formulation is still unknown. It is finally shown that the quantitative evaluation of the periodicity of the structure is very sensitive to the precision of the model, which is not the case for the tip radius of curvature and solute concentration.

1. Introduction During directional growth of a binary alloy, the solidification front is either planar, cellular or dendritic. Phenomenological models have been developed to describe non-planar growth (1-7}. In order to uniquely define the structure a criterion of either minimum undercoating or marginal stability at the tip is used. The models based on the marginal stability conduction are rather successful in predicting the parameters for dendrites. Two major difficulties are encountered with such models when they are applied to cellular growth. Indeed ad-hoc shapes have to be assumed so as to determine the periodicity and supersaturation at the tip is approximated by the expression for a sphere or an isolated dendrite, i.e. without taking into account the interaction between neighbouring cells. By making an analogy with the equations describing fingering in a Hele-Shaw cell [8], Peke and Pumir recently derived an exact solution for the shape of a dendrite growing in a isothermal capillary [9]. For unidirectional solidification in a temperature gradient these authors further showed that the Saffman-Taylor shape is still an exact solution near the cap. It should be mentioned that

these results were obtained in the small Peclet number limit, the Peelet number being in this case the ratio of half the wavelength to the thickness of the solute boundary layer at a planar solid-liquid interface, and neglecting surface tension. In this paper a 2D model for pattern selection during directional solidification, in terms of characteristic shape parameters, is developed. We shall proceed in three steps: for tip supersaturation the Pelce-Pumir relation will be used, the capillarity condition will next be considered for marginal stability and in the light of the results for fingering in a porous medium [IO}, and the solute conservation during stationary growth.

2. Two-dimensional model for pattern selection During directional solidification the solidliquid interface is a free boundary that has to be taken as an unknown of the problem. For instance, cellular morphologies with a given wavelength are computed by Brown and coworkers [11]. For our purpose it will be sufficient to characterize a cell by three shape parameters: the tip radius of curvature R o' periodicity A of the structure and tip displacement CPo from the plane front

0022-0248/87/$03.50 e Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

748

B. Bitlia et al.

I

Pattern selection during directional solidification

z

m~eI stricktly ap.plies to an isolated parabola which should be right for dendrites but may b questionable for cells. e The Peclet number P being defined as )(

P = )..,/2 Is,

(3)

Pelce and Pumir [9] have obtained; without capil_ larity and in the smaIl Peclet number limit n ' a · for 01' re Ianon : Fig. 1. Shape parameters of a solidification front and a Saffman-Taylor cap fitting the actual pattern

!l =

~ Coo

IT keo

+.!l'(l _ Is IT

Coo)

ic; .

(4)

with position as defined by the solidification isotherm (fig. 1). As we have done previously [4], it will be assumed that liquid and solid have equal thermal conductivities and that latent heat is negligible.

(5) and IT

= mCoo(k -l)/Gk,

(6)

2.1. Supersaturation at the tip

The tip supersaturation will be defined as

n=

(Co - Coo)/Co(l - k),

(1)

where Co and Coo are respectively the solute concentration in the melt at the tip and far from the interface, and k is the partition coefficient of the solute. Neglecting capillarity, various expressions are usually used, which are obtained for the isothermal growth of a cylinder with a hemispheral tip or a paraboloid of revolution [6],i.e. only !l and R o are considered together with one characteristic length for directional solidification, the solutal one: Is = DjV,

(2)

where D is the diffusion coefficient of solute in the melt and V the growth velocity. Directional solidification actually involves three characteristic lengths: Is, a thermal length IT and a capillarity length Ic • Including capillarity, Trivedi studied the growth of a parabola during directional solidification of a binary alloy (2]. The supersaturation D has then a complex form which would be difficult to handle in a phenomenological approach. Besides the

where m is the liquidus slope and G the tempera_ ture gradient. The great merit of eq. (4) is that it establishes a dependence of the tip characteristics, R o and U (or Co), on the periodicity ~ of the structure Relation (4) will be used henceforth, i.e, we shaIi use for !l the expression tor supersaturation at a Saftman-Taylor cap growing at a velocity V with the same wavelength, radius of curvature and tip shift than the real cell or dendrite (fig. 1). Uie equation for the cap [9] is thus z ST

= q, + ~(1 -.!l') In 1 + cos(2'ITx/.!l')..,) 0

21T

2 ' (7)

The geometrical meaning of 2 is shown in fig 1. .!l' gives an upper limit. in fractions of ~, of th~ length over which the Saffman-Taylor cap fits the actual cell. For 2 different from unity, i.e, fot non-zero )..,jR o• one gets from the definition of .!l': -

)..,

s, =

'IT -(1 -.!l'). .!l'2

(8)

It is worth noticing that ~/Ro neither depends on material properties nor on the degree of insta_ bility but on .!l' only, which can be expressed from

B. Billia et al. I Patternselection duringdirectionalsolidification

749

eq. (4) as a function of Co:

It'= 1 + _k .,....I_-_C--:oo:::::/:...k_C....::o~_ 1 - k 1 - US/IT) ( C.c,/kCo) .

(9)

At the onset of morphological instability, Co equals Cao/k so that It' is unity from eq. (9) and AIR o is zero from eq. (8). Far from the threshold, Co is close to Cao and Is/IT 4:: 1, .!t' is thus nearly zero and A/ R o tends toward infinity. The ratio A/R o can thus be plotted versus .!t' (fig. 2) together with experimental points from the literature [12-14]. For cells (roughly .!t'~ 0.5) the agreement between experiment and theory is satisfactory, despite the fact that the experiments on metallic alloys are three-dimensional and often with convection in the liquid phase. When P is no longer small and dendrites are observed, the data points are more scattered, but on the average there is still a fairly good fit. This suggests that the range of validity of eq. (4) might extend somewhat beyond the limit of small Peclet numbers. In the following we shall consider R o' A and It' as the three characteristic parameters of a cell rather than R o' A and ~o' This is possible due to the link between ~o, R o and .!t' through the condition of thermodynamic equilibrium at the solid-liquid interface. By defining a new variable U which is of first degree in Co:

U=

l-kCo/Cao , (1- k)(l-/ s/ I T )

(lOa)

and related to .!t'

I -.!t' U= l-(l-k)It"

(lOb)

we can alternately write eq. (8)

A 1-(l-k)U -='1TkU . {1-U)2 Ro

(11)

Like .!t', U is between 0 and 1 but the ratio

A/R o is now a function of both U and k, Different curves giving A/ R o versus U have thus to be drawn for various values of k to make a comparison with several available experimental results (fig. 3). Using .!t' gives a more unifying result than

o

.1

.2

.3

.4

.5

.6

.7

.8

.9

1

C Fig. 2. Variation of AIR o with :1'. Experimental points for: (*) k - 0.1 from Esaka and Kurz [13) (Co is obtained from the

tip temperature by assuming equilibrium solidification) ; (.) k - 0.14 from Miyata, Suzuki and Uno [14); (0) k - 0.68 from lin and Purdy [121.

using U, but this advantage is of no use for a practical application to a specific alloy with a definite k and, moreover, it will be suppressed when the capillarity effect is included. The choice between .!t' and U should rather depend on what is under study, the former stretching the dendritic region and the latter the cellular one. At this point we can thus consider that a pattern may be parametrized by .!t' and A/R o, or by U, AIR o and k, and must fall on the appropriate curve in fig. 2 or fig. 3. The open question is where? 2.2. Pattern selection through capillarity For fingering in a Hele-Shaw cell, MacLean and Saffman [10] have shown that the inclusion of

750

B. Billia et al. / Pal/em selection during directional solidification

where (J is the solid-liquid surface energy, L the latent heat of fusion per unit volume and TM the melting point of the solvent. Eq. (12) can be put in a more familiar form: G- mVCo(k-I)

D

+

TM(J

=0

417'2

L>..2 (I-!l')2"

.

(13)

This relation expresses a balance between supercooling which is responsible of the growth of a perturbation and the stabilizing effect of surface tension. Besides it is similar to the one deduced from the criterion of marginal stability at the tip . Both relations can be written (14) where F is a function which is specific to the criterion which is used. Let us define dimensionless the X and Ro by

X= x ~

~

~

A

~

~

J

~

~

-1) = 2PV IT -Is (I5a) GL (IT _ 1) = 2P R VIT- Is Is

GL (IT Is

TM (1

1

U

Ie

o A

TMG

'

Ie'

Fig. 3. Variation of >"IR o with U for different values of k . Experimental points as for fig. 2.

(15b) where

surface tension effects remove the shape degeneracy. Indeed capillarity induces a relation between the parameters " and !l' (see fig. 4 in ref. [10)). Although the Saffman-Taylor shape is not exact solution far from the tip for a non-zero temperature gradient, what should be crucial because the selection involves transcendental effects [IS], one can make the same correspondence as done by Pelce and Pumir for the case of solidification in a thermal gradient (section 4 in ref. (9)). This gives for" " =

'17'2 ~ T M(1 (I-!l')2 >..2 mL

le=VTMG/DGL

(16)

is a capillarity length which involves the tempera, ture difference over !he solutal length. The dimensionless periodicity ;\ is thus simply linked to the Peelet number P and the ratio between superCOOling (destabilizing contribution characterized by IT -I.) and capillarity (stabilizing contribution characterized by Ie). The interesting point is then that after direct transformation of eq. (14) we get ~ and R as 0 functions of !l' (or U) and k only

~2 = [1- (1- k)!l'] F= k

x{c.(k-l)~[l- mV~~-l)r,

R~= Ri [1- (1- k)!l'] F= Ri ;\2

(12)

k

(17a)

F

I-(I-k)U' ;\2

F 1 - (1 - k) U'

(17b)

B. Billia et at. / Pattern selection during directional solidification

where >../R o is given by eq, (8) or eq. (11). For any k it is thus possible to draw >../R o, X and Ro versus Ie (or U). without further reference to a specific alloy. For the sake of clarity and because they can be confronted with numerous experimental results on succinonitrile-acetone alloys [13,16], curves for k = 0.1 only will be pre-

751

104rT---

~

• D • A

•

•5 1.3 3.ll 4

w'' '

Ac

o 11.2

sented here. Marginal stability at the tip is first considered as the selecting mechanism, i.e. •

(18) The corresponding X and Ro are shown in fig. >../ R o. The great difference in variation between Ro which is nearly flat and X which is asymptotic to Ie= 1 and Ie= 0 on this logarithmic scale. thus reflecting the behaviour of

4, together with

o •

o 10'

a

k ••1

10000 10

AIR,

0

.1

.2

.3

.5

.4

.1

.1

.1I

l:

1

X':\.5

R,.R,..s

1000

1000

R,

100

..

. •...• , t·

11

o

10

k ••1

o

o

.1

~

Fig. 4. Variation of

~

A

X. Ro and

~

~

3

~

S

.1

b .2

.3

.4

.5

.Il

.7

.8

.9

t

1

1

C

A/R o with 3' for k - 0.1.

Fig. 5. Variation of (a) X, (b) Ro for k - 0.1. Experimental points for succinonitrile-acetone alloys (from refs. {13,16\).

752

AIR o'

B. Billia et at. / Pattern selection during directional salidification

has to be noticed. We shall further come back to this point in the next section. The experimental points (fig. 5) have been placed according to the following procedure. The ratio AI R o is first calculated from the published values for A and R o; the corresponding !I! is then deduced from eq. (5) or fig. 2. After evaluation ~ and li o are plotted for the same abscissa !I! and the agreement with the theoretical curves is checked. When the solute concentration or temperature at the tip is known, it is otherwise possible to obtain an experimental value for !I! from eq. (9) and thus to plot directly the experiments in fig. 5, but this procedure suffers from a lack of precision for ::R amplifies any error in Co' It clearly stems that there is a convergence towards the theoretical predictions from marginal stability considerations as the structures become more and more dendritic (i.e. as !I! decreases), the agreement being ultimately excellent for acetone concentrations ranging from 1.3 to 9.2 wt%. For 0.5 wt% acetone the curves are similar but shifted by a factor of about 2 at low!l!. Conversely there is an important misfit when cells are observed. Besides the experimental lio are a decreasing function of ::R, whereas the theoretical li o are an increasing function of !I!. This comparison rather comforts marginal stability at the tip as the selecting mechanism for dendrites, but only as a limiting case (!I!--+ 0) for which the interaction between neighbours is probably weak so that a dendrite can legitimately be considered as isolated. Moreover, this comparison suggests to seek for another selecting mechanism (if not others) for high and intermediate !I!, i.e . to seek for one adequate function (or more) for F to be put in eqs, (17). This suggestion may have some additional support from the work of Trivedi and Somboonsuk who have shown that there exist two different mechanisms of pattern formation for cells and dendrites [16]. This should be a difficult task, out of the scope of the present paper. Nevertheless some insight can be gained from a new formulation of the extensive experimental results of Esaka and Kurz [13] and of Somboonsuk, Mason and

Trivedi {16]. From eq. (17a) we get F=k~2/[1-(1-k)!I!].

(l9)

so that it is possible to plot F as a function of experimental A/ R o as it is shown in fig. 6a Equivalently we can also draw the correspondin~ .!I! versus F - 1 (fig. 6b). It results from both figures that there are two distinct limiting regimes for selection through capillarity, with a cross-over between them. For dendrites, experiments fit well with F =: 40 A2/ R~ , i.e. with the expression widely used for marginal stability [2], for concentrated alloys. for suceinonitrile-O.5% acetone, F == 250 2 A / R~, ~hi:h roughly ~orresponds to 1= 16 in eq, (18). This difference still needs to be clarified, one possibility being a dependence on concentration that quickly saturates. Moreover, the exponent is such that there is no longer a dependence on the periodicity in the capillarity condition, as it should be the case for non-interacting dendrites. For cells there is a need for more points so that we can only tentatively propose F =: 3600{A/R o • It is worth noticing that fig. 6b is the equivalent for ~irect~ona~ solid ification of fig. 4 in ref. [10] for fingering In a Hele-Shaw cell. The main difference is now that !I! can be less than 0.5 for directional solidification. Let us mention that all the dendritic points, for suceinonitrile-acetone as well as for other metallic alloys like Al-Cu [14] or Pb-Tl (17), correspond to !t'< 0.5, so that it may be a question ~hether or n.ot !t'= 0.5 (or AIR o = 2'IT) has something to do With the transition to the dendritic regime. Finally, as an illustrative but highly speculative attempt a function Ie qualitatively analogous to the one for fingering and that may be related to the cellular regime has otherwise been used:

,,= (.!I!- 0 .5)4/(1 -.!I!t

(20)

The corresponding '5. o.s and Ro.o.s are shown in fig. 4. Any comparison with experiment is impossible due to a lack of data far enough from the cross-over. So as to completely predict a pattern, supersaturation and the right capillarity condition been picked out, one further needs a condition to select !.e or V.

B. Billia et al. / Patt ern selection during directional solidification

2.3. Solute conservation during stat ionary solidif ication

753

formed is equal to the one Coo in the homogeneou s liquid far from the interface (fig. 7). From the assumption of equilibrium solidification one has :

For growth to be stationary we should consider that the average solute content ( Cs ) in the solid

(21a)

lO'..---.--~r-r-TT'1!'TT"--.---r-r-T"TTTr'--'-T""T1TTnr---r---r--r,.,."TT'tl • •5

w t 'I At

t3 • J.6 L> 4 9.2

F

6

10

~

.'

.

.

•

• I

. ••• •

• • ".

S lll'-

':

I',

,

10

L>

a 10

r

10 0

.\/R.

10 0 0

.9 .8 .7 .6

...

.5 .4

•3 .2

.,

•

:~

~

.1

b

o OL1'-:O.~-----_---I-=--------_-:-"-";i'"--------::~.--------=-:-;;.---=::-----l 2.Hi'

3.10'

Fig. 6. Succinonitrile-acetone alloys (from refs. [13,16» : (a) variati on of F with X/ R o; (b) varia tion of !i' with F -

I.

B. BII/ia et al. / Pal/ern selection during directional solidification

754

z

z

x

x Fig. 7. Stationarity condition used for the evaluation of 2: (Cs ) - Coo'

where C.,. is the solute concentration at the interface q,. The stationarity condition thus reads:

I

I

Fig. 8. Sa£fman-Taylor approximation fOJ a whole cell or dendrite . The link with the Saffman-Taylor shape at the cap is also shown.

From eq. (2Ib) we get, after integration:

q, _ o

~ 1-!t'lX+~(1-~1)2TMO =0 7T.q>1

X.q>l

GL

'

(24a)

(2Ib) with where :£ is the curvature at a point on the interface and the angular brackets mean the "average value over a period". For any quantity A, ./2 (A)=X 0 Adx.

~=

00

~

n-l

22n - 1 1 00 1 --...,...----..,.- ~2 2n n(2n + 1) m-l m 2n '

(24b)

and

(2Ic)

Coo k - 1 ( Is) 1 -!t' TMO cf>o=m G - k- I-IT l-(l-k)!l'- RoGL'

To explicitate eq. (21b), a relation for q, is needed. Rather than an ellipse or a parabola, a Saffman-Taylor shape is assumed (fig. 8):

(24c)

Xl (

)

ep = q,o + 217' 1 -.q>1 In

1 + COS(27TX/.q>1 Xl)

2

. (22)

cf> is different from ZST which describes the cap

but is linked with:

R 1 =R o'

(23a)

~lXl

(23b)

=X,

so that (23c) This choice has the advantage that cf> and zST coincide at the onset of morphological instability for the planar front (!t'-+ 1).

so that .q> can now be obtained. It should be noticed that material properties explicitly appear in relations (24), so that a unified presentation, involving k and IT/Is only, is not possible at this step. The variation of !t' with IT/Is is shown in fig. 9 for a succinonitrile - 4 wt% acetone alloy growing at a temperature gradient of 67 K/cm. Marginal stability at the tip is assumed to be the capillarity condition. The parameters h and R o are shown in fig. 10. Concerning R o, a convergence between experiments and theory is again observed for dendritic growth, whereas for h there is a functional convergence but a discrepancy by a factor of 6 between theory and experiment. For IT/Is = 1000, the selected and experimental 2 are indicated by open and full triangles in fig. Sa, and one can see

B. Billia et al. / Pattern selection during directional solidification

•• •

••

•

•

.1

Fig. 9. Variation of .It' with IT/Is. Succinonitrile-4 wt% acetone. G ~ 67 K/cm. Experimental points from ref. [16].

that the experimental !i' is greater than the theoretical !i'; this shift corresponds to a 10% difference only in Co. Other approximations for ep have been tested (ellipse, parabola, sine) and the

.l,R. (em)

.1

•01

.001

755

best fit was obtained for the Saffman-Taylor shape. Solute conservation selects the right g> with a precision which is sufficient to get R o, but poor for A when the front is dendritic, i.e. when X is getting asymptotic to Ie= O. On a general ground, we think that this should be the reason for which several models for dendrites give good quantitative estimates for R o and a functional agreement only for A. At high and intermediate .P there is a need for more experimental data and a weII established capiIIarity condition before being able to enlarge this discussion. 3. Conclusion A model has been constructed for non-planar growth during directional solidification of binary alloys. Three steps have been evidenced and treated so as to allow a direct confrontation with experiments at any level of the analysis. With this procedure it should be possible to immediately detect if an assumption is founded or not and determine its range of validity. Moreover, a unified approach has been feasible, A/ R o depending only on Ie and the nondimensional periodicity X and the radius of curvature Ro on g> (or U) and on k . It results that, for instance, all the experimental A/R o, 1 or Ro values for succinonitrile-acetone alloys [13,16] must fallon a single curve in figs. 2 to 5. Marginal stability at the tip appears to be a limiting case, valid for dendrites of a more general capiIIarity condition, whose formulation for cellular growth is stiII needed. On a diagram giving A/R o, Xor Ro versus 2, dendritic growth concerns only a limited region, so that we suggest that experiments be carried out at high and intermediate !i' in order to give a firm support to the construction of the model for cells, particularly for determining the capillarity condition.

.0001 L---J.--L..J-Iu.J..UJ._..L.....L....L.L.U-LLL.-_I..-L....L...L..L.u.u 100 100e 10 1

Acknowledgements Fig . 10. Variation of A and R o with IT/Is. Succinonitrile-4 wt% acetone . G - 67 K/cm. Experimental points from ref. [16].

Fruitful discussions with P. Pelee are gratefully acknowledged. The authors are also greatly inde-

756

B. Bil/ia et al. / Pattern selection during directional solidtficauan

bted to Professor W. Kurz for communicating results on succinonitrile-acetone alloys prior to their publication, thus allowing a much more detailed analysis of the theoretical model to be made. References [IJ J.D. Hunt, in : Solidification and Casting of Metals (Metals Society, London, 1979) p. 3. [2J R. Trivedi, 1. Crystal Growth 49 (1980) 219. (3) W . Kurz and DJ. Fisher, Acta Met. 29 (1981) 11. [4J B. Billia, H. Jamgotchian and L. Capella, J. Crystal Growth 66 (1984) 596. {5J V. Laxmanan, Acta Met. 6 (1985) 1037. [61 W. Kurz and DJ. Fisher. Fundamentals of Solidification (Trans. Tech., Aedermannsdorf, Switzerland, 1984). (7) W. Kurz, B. Giovanola and R. Trivedi, Acta Met. 34 (1986) 823.

[8J P.G. Saffman and GJ. Taylor. Proc, Roy . Soc. (London) A245 (1958) 312. [9J P. Pelce and A. Pumir, J. Crystal Growth 73 (1985) 337. POJ J.W. Mclean and P.G. Saffman, J. Fluid Mech. 102 (1981) 455. [llJ L.H. Ungar and R.A. Brown, Phys. Rev. B31 (1985) 5931. (12) I. Jin and G.R. Purdy, J. Crystal Growth 23 (1974) 37. [13J H. Esaka and W. Kurz, J. Crystal Growth 72 (1985) 578' H. Esaka, PhD Thesis, Ecole Polytechnique Federale d~ Lau sanne (1986). [14J Y. Miyata, T. Suzuki and J.I Uno, Met Trans. A16 (1985) 1799. 115) T. Dombre, V. Hakim and Y. Pomeau, Compt. Rend. (Paris) (Ser, II) 302 (1986) 803. /161 K. Somboonsuk, J.T. Mason and R. Trivedi, Mel Trans. A15 (1984) 967; R. Trivedi and K. Somboonsuk, Acta Met. 33 (1985) 1061. [17J H. Jamgotchian, B. Billia and L. Capella, J. Crystal Growth 64 (1983) 338.