Morphological stability of a planar interface under rapid solidification conditions

Acta metall. Vol. 34, No. 8, pp. 1663-1670, 1986 OOOI-6160/86 $3.00 +o.oo Printed in Great Britain Pergamon Journals Ltd

MORPHOLOGICAL STABILITY OF A PLANAR INTERFACE UNDER RAPID SOLIDIFICATION CONDITIONS

R. TRIVEDIT and W. KURZ Department of Materials Engineering, Swiss Federal Institute of Technology, Lausanne, Switzerland

(Received 10 July 1985)

Abstract-The linear perturbation theory of Mullins and Sekerka for the stability of a planar interface is extended to the case of large thermal peclet numbers. It is shown that an absolute stability criterion for a planar interface exists for undercooled melts also. In light of these results, the conventional constitutional supercooling criterion is reexamined and a more general planar interface stability criterion is proposed which is valid for low as well as high growth rate conditions. The results of this stability analysis are applied to dendritic growth from pure undercooled melt.

RCsum&Nous avons etendu la theorie lineaire de la perturbation de Mullins et Sekerka pour la stabilitt d’une interface plane au cas des grands nombres de P&let. Nous montrons qu’il existe un critere de stabilite absolue pour une interface plane dans un matiriau surfondu. A la lumitre de ces resultats, nous reexaminons le critere de surfusion constitutionnelle classique et nous proposons un critire plus general pour la stabilitt dune interface plane, critire valable aussi bien pour les faibles que pour les fortes vitesses de croissance. Nous appliquons les r&hats de cette analyse de la stabilitt a la croissance dendritique dans un corps pur surfondu.

Zusammenfassung-Die lineare Storungstheorie von Mullins und Sekerka fur die Stabilitlt einer ebenen Grenzflache wird auf den Fall groBer thermischer Pecletzahlen erweitert. Es wird gezeigt, daB ein absolutes Stabilitatskriterium fur eine ebene Grenzfllche such fur unterkiihlte Schmelzen besteht. Im Hinblick auf diese Ergebnisse wird das herkiimmliche Kriterium der konstitutionellen Unterkiihlung neu betrachtet. Es wird ein allgemeineres Kriterium fiir die Stabilitlt einer ebenen Grenzfllche vorgeschlagen, welches fur Wachstumsbedingungen mit kleiner und grol3er Geschwindigkeit gilt. Die Ergebnisse dieser Stabilitats- analyse werden auf das dendritische Wachstum aus einer reinen unterkiihlten Schmelze angewendet.

INTRODUCTION

The linear stability analysis of a planar interface, first carried out by Mullins and Sekerka [l], has become the key to our understanding of morphological features of solid-liquid interfaces [2-lo]. One of the important results of the theory is the prediction of the absolute stability of a planar interface at high velocities under constrained growth conditions. The treatment by Mullins and Sekerka [l] was limited to the case in which the thermal diffusion length was much greater than the wavelength of perturbations. Under this assumption, their results predict that a planar interface will be unstable at all velocities in undercooled pure or alloy melts. Although their assumption is reasonable under most normal solidification conditions, it will not be valid for the rapid solidification of the undercooled melt where the thermal diffusion length can be quite small. In fact, experimental observations on rapidly quenched alloys show segregation free regions which indicate the existence of the absolute stability of a planar interface in undercooled melts. The main purpose of this paper is to extend the linear stability analysis

tPresent address: Ames Laboratory, USDOE and Depart- ment of Materials Science and Engineering, Iowa State University, Ames, IA 50011, U.S.A.

of Mullins and Sekerka for the rapid solidification case. Our results show that the absolute stability of a planar interface exists not only for the constrained growth of alloys but also for undercooled pure and alloy melts. We shall derive conditions for the absolute stability of pure and alloy undercooled melts. In order to understand these absolute stability criteria,

we propose that the traditional constitutional supercooling criterion for the stability of a planar interface be reexamined. The latter is shown to be valid only

under specific conditions.

THEORETICAL MODEL

Consider an initially planar interface which is moving with a constant velocity Vin the Z-direction. Let the planar interface coincide with the plane Z = 0. If the coordinate system is attached to the interface, the steady-state solute and thermal fields are governed by the following equations

v2c + (v/D)(ac/dz) = 0 (1)

V2T,_ + ( I’/aL)(t?T,/aZ) = 0 (2)

and

V2Ts + ( V/as)(3Ts/dZ) = 0 (3)

where C is the concentration of solute in the liquid,

1663

1664 TRIVEDI and KURZ: STABILITY OF A PLANAR INTERFACE

and r, and T, are the temperature fields in the liquid and solid, respectively. D is the solute diffusion coefficient in liquid and aL and a, are the thermal diffusivities in liquid and solid, respectively. In this treatment we shall assume the diffusion in solid to be negligible. The effect of the solute diffusion in the solid is quite small and its effect on the stability is discussed in Appendix I.

We shall examine the stability of the planar interface by imposing an infinitesimal perturbation in the shape of the interface such that the interface is described by the equation

determined from the boundary conditions. Substitut- ing equations (6) and (7) into equation (5) we obtain

a=mb-Tw’. (14)

The value of the constant b is determined from the requirement that the velocities on any element of the interface calculated from heat flow and diffusion considerations must coincide, i.e.

o(X) = kH [&(~Tsl=), - K,(8T,/dZ),]

Z = 4(X, t) = s(t) sinwX (4) = ]Dl{C& - 1)IlW/W, (15) where 6 is the amplitude, o the wave number (=2x/n), and I the wavelength. The perturbed inter-

where k is the equilibrium partition coefficient of

face must satisfy the capillarity conditions at the solute, and KL and KS are thermal conductivities of

interface liquid and solid, respectively. AH is the latent heat of fusion per unit volume. We now substitute the tem-

Th = TM + mC, - To26 sinwX (5) perature and concentration gradients at the interface,

where TM is the melting point of pure solvent, m the Z = 6 sinwX, from equation (8) to (10) into equation

slope of liquidus, T+ the temperature, C, the concen- (15). Using the relationship given by equation (14),

tration (liquid side) at the interface and r = y/AS in and equating the like Fourier coefficients, we obtain

which y is the interfacial energy and AS the entropy to the first order in 6, the value of b as

of fusion per unit volume. The far field boundary conditions must also agree with the fields due to the b+ (16) unperturbed interface. 2

The solutions of equation (l-3) can now be where

obtained such that b, = GChx2( Ksos + K,wJ

T4 = To + ad sinwX (6) + Gc] &%Gs + &o,G,I

C, = C,, + b6 sinwX (7) - I’GcKK,G,Ia,) - (&G/41

where To and C,, are the values for a flat interface and the constants a and b determine the corrections due to the infinitesimal change in the shape of the interface. As shown by Mullins and Sekerka [l], the solutions of equations (l-3) which satisfy equations (6) and (7) on the interface Z = 6 sinoX are

C - C, = (GcD/V)[l - e-(y’D)z]

+ 6(b - G,) sinwX e-“c’

T, - To = (G,a,/ V) [ 1 - e -( ““~)z]

(8)

+ GcVAH [oc -(V/D)] (1W

b2 = mG,( Ksws + K,.q)

+ (KsG, - KLGL)~ - (VID)(l -k)}

(16b)

The stability condition, b/S, can now be obtained from equation (15) by substitution v(X) = V + 6 sinoX, and by evaluating the temperature gradients from equation (9) and (10). Equating the like Fourier coefficients gives

+ 6(a - GJ sinwX e-“‘Lz (9) V = (K,G, - K,G,)/AH (17)

and

T, - To = (G,a,/V)[l - e-(“‘*)a

+ 6 (a - Gs) sin OX e”g

i/S = (l/AH) [ Ksos(a - G,) + K,o,(a - GJ

- ( VKsGJas) + ( ~fkG,Ia,)l. (18) (10) _.

where Gc, GL and G, are the concentration gradient, Substituting the value of a from equation (14) and

the temperature gradient in liquid and the tempera- using equation (16), we get

ture gradient in solid at an unperturbed flat interface, s and s =(V/b,)[-h2(Ksw,+ KLwL)

wc = (V/2D) + [( V/2D)2 + 07”’ (11) x {WC - (VID)(l - k)I

q = ( V/2a,) + [( V/2a,)2 + cd]“2 (12) - { KsG,w, + K,GLm,_ - ( VKLGLI~L)

0s = - ( V/2as) + [( V/2as)2 + wy. (13) + ( VKsGs/a,)> {WC - ( V/D)0 -k))

The values of the constants a and b can now be + mGc(@s + &wL){wc - ( V/D)1 (1%

TRIVEDI and KURZ: STABILITY OF A PLANAR INTERFACE 166.5

STABILITY ANALYSIS

The stability of a planar interface depends on the sjgn of 6s. The planar interface is stable when S/S c 0, and it is unstable when B/S > 0. Thus, the marginal stability condition can be obtained by ex- amining the condition S/S = 0. Since ( V]b,)(&,w, +- KLot) {we - ( VjD)(l - k)) is always positive, we may write the marginal stabiiity criterion as

+ mGc WC - (V/D)

UC - (V/-D>U -k) 1 =o (20)

where

&, = Kr.i(Ks + Kr.) and Ks = Ks/(Ks + Kd

substituting

tc = 1% - (VID)I/I% -(V/D)@ - k))

the marginal stability criterion, equation (20), becomes

- Twz - [&,Gr<, + &G&] + mG,& = 0. (21)

The function cc is exactly the same as that obtained by Mullins and Sekerka [I]. Equation (20) reduces to the result of Mullins and Sekerka when &. = rs = 1. This occurs when w $ V/a, or V/as, which gives cc,_ c?z os N w and rL = ts = 1. Thus, for large velocities, the simplified equation of Mullins and Sekerka considers the stability of the interface only with respect to very small wavelength perturbations. For example, for V = 10 cm/s and a,_ = 10e3 cm2/s, the wavelength of perturbation must be much less than 1 pm for the assumption w 9 aL to be valid.

In order to gain an insight into the nature of function CL and &, we shall consider a special case in which Ks = KL and aL = us. For example, these assumptions are valid for the solidification of succinonitrile. For this case, we obtain

V/2aL tL = I - [( V/2a,)* + w2]“*

VI% ” = ’ + [( V/2as)’ -I- w2]‘/*’

(23)

Thus, when w % ( V/2aL), these functions approach unity, so that the equation (21) corresponds to the Mu&m+-Sekerka stability criterion. However, when

and ?js in the limit V/Za,o + 1 can be represented as

& N 2CD2at/v2 (24)

and

2w2a2 g,N2-7T

The variations in the functions &, ts and SC with peclet numbers are shown in Fig. 1, where we define the peclet number pL = V;i/2a,, ps = Vl/2as, and p = VA/;(!2D in which 1 = 2x/o.

DISCUSSION

(a) Stability criterion

We shall first consider the case of the solidification of a pure material. The stability condition for this case can be obtained from equation (18) by substituting Gc = 0 (b = 0). This gives

For G, and G, > 0, all the terms on the right hand side of equation (26) are negative. Thus a planar interface will always be stable under these conditions during the, solidification of pure materials. We also note that 6 /S will be negative and the planar interface stable, if Gs > 0 and G, = 0, a condition which is commonly encountered during the solidification of castings.

Let us examine now the case of an unstable front which leads to dendrites. A planar interface stability criterion has been used by Huang and Glicksman [4] to predict the stable dendrite tip radius in an undercooled pure melt. They have shown that the experimental dendrite tip radius, R, is approximately equal to I,. In order to examine the value of L, for large

(V/&) & o the function tt. approaches zero, and the Fig. 1. The variations in the functions &, Ts and SC with function & approaches 2. The behavior of function & appropriate peclet numbers p,_. ps and pc, respectively.

AM M!8-M


velocities, we substitute in equation (26) thermal conditions which are present during the dendrite growth at the tip, i.e. G, x 0 and G, = - VAH/K,.

This gives

B - = f’(K,o, + KLwL) fi

X -(I-02/VAH) + ;u-y;; . (27) s s L L 1

The stability of the planar interface thus depends on the sign of the large bracket in equation (27). This stability behavior can be readily visualized by consid- ering the special case as considered before in which KS = KL and a, = aL, which gives, using equation (22)

B - =fV(w,+w,) 6

Substituting v = TC,V/2a,AH and Y = 2a,w/V, and separating the w dependent terms, we obtain

B - =fV(w,+w,)[l -f(o)] 6

(29)

where

f(w)=vY’+[l + Y2]_“2. (30)

Note that a planar interface will be stable when f(o) > 1. It can be readily shown that the function vY2 +[1 + Y2]-‘/2 2 1, for all values of Y when v > l/2, where the equality sign is valid when Y = 0 and v is finite. Thus, a planar interface will always be stable with respect to all wavelengths if v > l/2. Thus, v > l/2 represents the absolute stability condition for a planar interface growing from a pure undercooled melt. This is a new behavior which was not shown by the Mullins-Serkerka analysis since their treatment assumed the last term in the large bracket of equation (28) to be negligible.

The physical reason for the absolute stability condition can be seen by rewriting the condition, v > l/2, as

l-C P>a, AH V

(31)

which shows that a planar interface is stable when the capillary length is larger than the thermal diffusion length. We denote this thermal absolute velocity as (V.&, which is given by

( V& = y (32)

where ATh = AH/C,, which corresponds to the hypercooling limit.

We shall now consider the solute case for which the thermal gradient terms in equation (19) are negligible.

Such a condition is valid for constrained growth under large growth rates. For this case, the planar interface will be stable when

Tw2 > mG&. (33)

For large values of V, & -+ w’D~/V’~. Substituting this limiting value of &-, and the value of mG, from the solute flux balance at the interface, we get the absolute velocity, ( Va&, above which a planar interface in the presence of solute is always stable, i.e.

( I’& = D AT,lrk. (34)

This absolute stability criterion is the same as that obtained by Mullins and Sekerka.

We shall now examine the stability criterion for an undercooled alloy melt in which G, = 0. Substituting the large peclet number expansion of the functions tL and lc in equation (20), we obtain for KS = KL

Vabs =x ~ DAT, + aLAT,,

r or

Vabs = (Vabs)C + ( V&s)T. (35)

Thus, above this velocity a planar interface will be stable for undercooled alloy melts. Note that the Mullins-Sekerka treatment did not predict this absolute stability since they assumed small thermal peclet numbers in which case (I’&r was infinity.

(b) Stability us constitutional supercooling

The constitutional supercooling criterion, proposed by Tiller et al. [ 111 has been extensively used in the literature to predict the conditions under which a planar interface will be unstable. A modified version of this principle was subsequently given by Mullins and Sekerka [1] and Tiller [12] in which the temperature gradient in liquid, G,, was replaced by G, the thermal conductivity weighted average temperature gradient in solid and liquid at the interface. This modified criterion was found to be nearly identical to the stability criterion at low velocities except for a very dilute solution [13]. The modified supercooling criterion, however, cannot explain the existence of the absolute stability condition of Mullins and Sekerka for the planar interface stability under constrained growth conditions [13]. Furthermore, the constitutional supercooling criterion will always predict the interface to be unstable for undercooled melts. Con- sequently, the constitutional supercooling criterion is not a general principle to characterize the stability of a planar interface, and a better criterion is desired which is also valid under high growth rate conditions.

In order to gain an insight into the stability criterion, we shall first take a simple approach, as shown in Fig. 2, in which we compare the velocities of the highest and the lowest points of the interface. If v.,, and va are the velocities of the perturbed interface at points A and B, respectively, then for a planar interface to be stable, Au = vA - vg must be

TRIVEDI and KURZ: STABILITY OF A PLANAR INTERFACE 1667

Fig. 2. A perturbed interface with wavelength 1. A and B are the highest and the lowest points on the interface.

negative. We first consider the case of a pure undercooled melt with Gs = 0. If AC, is the difference in thermal gradients in liquid as points A and B, then, from equations (5), (8), and (9), we obtain

AC, = 2w,6[Tw2 + G,{ 1 - (V/a,o,)}]. (36)

Using the flux balance equation (15) gives

Au = (2w,6KLlAH)

x [ - To2 - CL{ 1 - (v/a,wL)}]. (37)

The first term in the large bracket of equation (37) is the surface energy effect and this term will always be negative so that it will tend to stabilize a planar interface. The second term, for an undercooled melt, will be positive (since G, is negative), and it will favor the instability of a planar interface. We may rewrite the second term by defining an effective gradient,

(G&r> which is

(G&r = G,{l - ( V&w,)). (38)

Thus, the stability equation (37) becomes

Au = (2w,6KL/AH)[ -Tw2 - (G&l. (39)

The constitutional supercooling criterion ignores the surface energy effect. Furthermore, it assumes

(G&r = G, so that at large undercooling values, ]GJ, increases which would predict that a planar interface will always be unstable. The key point to note here is that the assumption (G,),, = CL is no longer valid at very large undercooling values since V becomes large. Consequently, as the undercooling is increased, G, increases but (G&r decreases. Thus, at some undercooling values, (G&e becomes sufficiently small that it has the same magnitude as the surface energy effect. This condition then defines the absolute stability criterion.

Similar arguments will show that it is not the mG, - G function that controls the stability of a planar interface, but it is m(G,),, - (G),, function which must be considered. At large rates, this function can become sufficiently small that capillarity effects can no longer be ignored. From the simple arguments presented here, we may now examine the general stability criterion given by equation (21) and rewrite it in the form that is convenient for comparison with the constitutional supercooling criterion. A general stability criterion given by equation (21) can now be written which shows the condition for planar interface instability as

where

m(G,),,- (G),, > Tw’ (40)

(G&t = G&c and (G),, = R,G,Sr. + F&G&.

Since the functions tc and <r approach zero at large velocities, Fig. 1, the term m (G,),, - (G),, differ significantly from mG, - G. Thus, constitutional supercooling criterion cannot be used to predict the general condition for a planar interface stability.

Equation (40) which represents a general condition for a planar interface instability, predicts both the low and the high velocity limits of stability, For low velocities, (G&N G, and (G),,- G so that if To2 is negligible then equation (40) is equivalent to the modified constitutional supercooling criterion. For rapid solidification, equation (40) also correctly predicts the absolute stability criteria for growth from an undercooled alloy melt and that for constrained growth.

(c) Dendritic growth in pure undercooled melt

During dendritic growth, the stability of the dendrite tip plays key role since it selects a specific solution from an infinite number of possible steady- state solutions [2,3]. Experimental studies in pure undercooled dendrite [4] and in constrained alloy growth [5-81 have conclusively shown that the experimentally observed dendrite tip radius matches with its theoretical value calculated at the limit of its stability. This marginal stability criterion for dendrite tip radius was first proposed by Langer and Miiller- Krumbharr [2], and their theoretical analysis was limited to small undercooling conditions where the solute and thermal peclet numbers were small. Under these conditions, they predict the relationship I/R2 =


constant to hold true, where V and R are dendrite velocity and tip radius, respectively. Recent experimental studies on constrained growth have shown that this relationship holds true for a range of velocities, except near the dendrite to cell transition, where VR2 increases as the velocity is decreased. Similarly, one would expect large peclet numbers at very high velocities, where the criterion of Langer and Miiller-Krumbhaar would also have to be modified.

Langer and Miiller-Krumbhaar showed that the dendrite tip radius, R, is related to the marginally stable wavelength, 1,, predicted by the linear stability analysis of a planar interface, i.e.

R = 1, = 271/c+. (41)

This relationship has been employed earlier by Kurz and Fisher [lo], and experimental studies on dendrite growth have confirmed the validity of this criterion [8]. Thus, substituting equation (41) in the general marginal stability criterion, equation (20), and solv- ing transport equation, one obtains a relationship between R, V, and undercooling, AT, for all peclet number conditions.

We shall consider the case of a pure undercooled melt for which the thermal diffusion field around a dendritic front give the following relationship

AT = g Iv@,) + F + l (42) LP A PO

where Iv(p,) is the Ivantsov function and h is the interface kinetic coefficient. The simultaneous solution of equation (42) and the stability criterion, equation (20) will give the relationship between V or R and AT. We shall first consider the case in which the thermal field in equation (42) is characterized by the Ivantsov solution only, neglecting capillarity and interface kinetics effects.

Figure 3 shows the relationship between the growth rate and the undercooling for pure succinonitrile. The dotted line in the figure is the result obtained by using the Langer and Miiller-Krumbhaar analysis which is valid for small undercooling conditions [2]. A significant difference is obtained at large undercooling values. At unit dimensionless undercooling (AT = 23.8 K), the dotted line shows the velocity to be infinite, whereas the present model predicts a finite velocity. This maximum velocity corresponds to the absolute stability value so that a planar interface would be stable for velocities larger

than ( V&r. Figure 4 shows the variation in the dendrite tip

radius with undercooling. The dotted line, which is the Langer and Miiller-Krumbhaar stability result, shows the radius decreasing continuously with undercooling. The result of the present model shows that the dendrite tip radius goes through a minimum and approaches infinity at a unit dimensionless undercooling where a planar interface becomes stable.

One of the predictions of the Langer and Miiller- Krumbhaar theory was that VR2 = constant for small

v)

;

>

AT, K

Fig. 3. The variation in the dendrite growth rate with undercooling for pure succinonitrile. The dotted line in the figure is the result obtained by using the Langer and

Miiller-Krumbhaar [2] analysis.

undercooling values. Figure 5 shows the variation in VR2 with undercooling. The present model shows that the Langer and Miiller-Krumbhaar model is valid for small and medium undercooling values. However, at large undercooling values, VR2 is found to increase rapidly and approach infinity at the unit dimensionless undercooling value.

The transition from nonplanar to planar interface occurs at the hypercooling limit (or unit dimensionless undercooling) when the Ivantsov equation characterizes the thermal field. However, just below the hypercooling temperature the velocity becomes sufficiently large so that interface kinetics effects become important. Consequently, both the capillary effect and the kinetics effect will enable the dendritic structures to exist beyond the hypercooled limit. This was indeed shown experimentally by Glicksman and Schaefer [14] who were able to grow phosphorous dendrites from a melt which was hypercooled. Their results showed that the dendrite velocity did not increase sharply near the hypercooling limit. Instead, a gradual increase in velocity was observed for

Id41 ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ 3 i ’ ’ ’ 0 4 8 12 16 20 24 26

AT, K

Fig. 4. The variation in the dendrite tip radius with undercooling for pure succinonitrile. The dotted line uses the

stability result of Langer and Miiller-Krumbhaar [2].

TRIVEDI and KURZ: STABILITY OF A PLANAR INTERFACE 1669

IO”-

c IO’-

E i

N- 106- (r >

IO 4 _---_---__--4

IO4 I ’ . ’ 5 ’ . ’ .S’ ’ ’ 0 4 a 12 16 20 24 28

AT, K

The general stability criterion, presented in this paper, can also be used to evaluate alloy dendrite growth characteristics. However, for large undercooling values, the changes in the thermal and solute diffusion coefficient with undercooling should be taken into account. Furthermore, the phase diagram properties, such as the liquidus slope and the solute distribution coefficient may vary appreciably with undercooling. Also, the solute trapping effect becomes important and it should be incorporated into the model. These aspects are discussed in subsequent papers [16, 171.

Fig. 5. The variation in VRz with undercooling for pure succinonitrile. CONCLUSIONS

values of dimensionless supercooling of about two. These results are consistent with the model presented here, and we shall now quantitatively evaluate the growth rate as a function of undercooling for phosphorous dendrites.

Figure 6 shows the effect of capillarity and interface kinetics on the growth of phosphorous dendrites. The values of the physical constants were taken from Kotler and Tarshis 1151, and the surface energy value of 5 mJ/m* was used in these calculations. Glicksman and Schaefer [14] observed faceted growth for undercooling up to 9 K. Above 9 K linear kinetics were shown to exist. Consequently, we have used linear attachment kinetics and compared our results with the experimental data above 9 K undercooling values. Figure 6 shows that the undercooling at which nonplanar to planar transition occurs increases when surface energy and interface kinetics are taken into account. Also, this maximum undercooling for dendrite growth increasing as the kinetic effects increase

A planar interface stability analysis is carried out using a first order perturbation theory. A general stability criterion is developed which predicts absolute stability of a planar interface for free and constrained growth. It is shown that the modified constitutional supercooling criterion is valid only in the limit of small velocity and negligible capillarity effect. Application of the theory to dendrite growth shows a significant deviation in velocity or radius of the dendrite tip from the earlier models when the undercooling or velocity becomes very large.

~c~~Q~~e~ge~e~~~-The authors are grateful to W. J. Boettinger and M. E. Glicksman for valuable discussions. and to B. Giovanola and Ph. Trevoz for numerical calculations. The authors would also like to acknowledge financial support from “Kommision zur Fiirderung Wissenschaft- lither Forschung” and Ames Laboratory, which is operated for USDOE by Iowa State University, under contract No. W-7405-Eng-82, supported by the Director of Energy Research, Office of Basic Energy Sciences.

(or k decrease). The theoretical model agrees quite well with the experimental data for ir, N 0.17 m/s K.

0.00 10.00 20.00 30.00 40.00 50.00

UNDERCOOLING, K

Fig. 6. The effects of capillarity and interface kinetics on the velocity of pure phosphorous dendrite. The experimental data points are the results obtained by Glicksman and Schaefer [14].


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REFERENCES

W. W. Mullins and R. F. Sekerka, J. appl. Phys. 35,444 (1964). J. S. Langer and H. Miiller-Krumbhaar, Acta metall. 26, 1681 (1978). J. S. Langer, Rev. Mod. Phys. 52, 1 (1980). S. C. Huang and M. E. Glicksman, Acta metall. 29, 701 (1981). K. Somboonsuk, J. T. Mason and R. Trivedi, Metall. Trans. AH, 967 (1984). R. Trivedi and K. Somboonsuk, J. Mater. Sci. Engng 65, 65 (1984). K. Somboonsuk and R. Trivedi, Acta metall. 33. 1051 (1985). H. Esaka and W. Kurz, J. tryst. Growth. 72, 578 (1985). R. Trivedi, J. crvst. Growth. 73. 289 11985). W. Kurz and D: J. Fisher, Act; meta)/. 2d, 11 (1981). W. A. Tiller, K. A. Jackson, J. W. Rutter and B. Chalmers, Acta metall. 1, 428 (1953). W. A. Tiller, Art and Science of Growing Crystals. Wiley, New York (1963). R. F. Sekerka, Proc. of the Flat-Plate Solar Array Project Research Forum on the High Speed Growth anh Characterization of Crystals for Solar Cells, p. 93. Jet Propulsion Lab., Pasadena, Calif. (1984). M. E. Glicksman and R. J. Schaefer, J.’ tryst. Growth 1, 297 (1968). G. R. Kotler and L. A. Tarshis, J. tryst. Growth 2, 222 (1968). W. Kurz, B. Giovanola and R. Trivedi, Acta metall. In press. J. Lipton, W. Kurz and R. Trivedi, Acta metall. 34,823 (1986).

APPENDIX I

The effect of solute dt&sion in solid on the stability of a planar interface

When the diffusion in the solid phase is significant, the solute flux balance equation (15) should be modified as

a(x) = [f/{C,(k - Wl[WClW, - WWz),l 641)

where Ds is the diffusion coefficient of solute in solid and Cs is the solute concentration field in solid. The solute field, C,, for a perturbed interface is governed by the relationship

C, - kc, = (G,,D,/V)[I - e-(v’Ds)z]

+ 6 (bk - G,,) sin&’ eocsZ (A2)

where Cc, is the solute gradient at the unperturbed interface and

ocs = -( V/2D,) + [( V/2D,)‘+ ~~1”~. (A3)

The concentration gradient in the solid at the purturbed interface is

@C&Q), = G,, + &s(bk - G,,)

- ( VG,,/D,)] 6 sin ox. (A4)

Substituting this result in equation (Al) and following the procedure described in the main text, the marginal stability criterion, S/S = 0, is obtained as

--2-~~G~[~~~ivK~~~-R,Gs[~~~~~~~~

+ mG, WC-(V/D)

WC - ( V/D ) (I- K) + 1 k~,s(DslD) =O. (A5)

Comparison of this resuit with the stability equation (20) shows that the effect of solute diffusion in solid modifies the last term of equation (20) with an additional factor of ko,,(D,/D) in the denominator. Alternatively, we may rewrite equation (A5) as

- To2 - [ KLGLSL + KsG&] + mG,<& = 0

where the function <c will be

(‘46)

5;= bc - ( V/D )I

[o+ - (V/D)0 - k)l + @WCs (A7)

where

a = D,k/D.

The effect of solute diffusion in solid is to reduce the magnitude of the solute term, and thus to stabilize the planar interface.

Documents

Morphological stability of a planar interface under rapid solidification conditions