JOURNAL OF APPLIED PHYSICS VOLUME 36, NUMBER 1 JANUARY 1965

A Stability Function for Explicit Evaluation of the Mullins-Sekerka Interface Stability Criterion

R. F. SEKERKA.

Westinghouse Research Laboratories, Pittsburgiz, Pennsylmnia

(Received 23 January 1964; in final form 3 August 1964)

.\ stability function S for explicit evaluation of the Mullins-Sekerka interface stability criterion is intro­duced and tabulated. Their criterion is then written in terms of S and compared with the constitutional supercooling criterion for interface stability. From these results, experimental data can be analyzed and a conclusive test of the stability theory can be made.

I N a previous paper by Mullins and Sekerka (called M-S),1 it was shown that during the unidirectional

solidification of a dilute binary alloy at constant veloc­ity, a planar solid-liquid interface is stable when

We thus confine attention to the case for which A < 1. Let us introduce a stability function S defined by the equation

(4)

(1) From the above considerations, it follows that

and unstable when the inequality is reversed. The nota­tion here is the same as used bv 1\1 ullins and Sekerka; it is summarized in Appendix i. We shall refer to (1) as the M-S stability criterion; h is given by the expression

h= Imaxf(w) I, (2) where

(5)

Then (1) becomes, for stability (note reversal of the in­equality caused by multiplication by -1)

(6)

Introducing the dimensionless variables

(3) y=(TMr/mGc)w2~O and tJ=.J-kU

Examination of Eqs. (2) and (3) shows that the de­termination of h by straightforward application of the calculus involves the solution of a cubic equation in w. But due to the cumbersome form of the analytic solu­tion of a cubic equation, it is difficult to assess the M-S criterion. It is the purpose of this paper to develop ex­plicit expressions for h and to define and tabulate a re­lated quantity S as a function of certain dimensionless parameters. In this way, the predictions of the M-S theory can be readily evaluated and compared with other theories and with experiment; the use of dimen­sionless parameters alleviates the need to specify con­stants characteristic of a given material (often poorly known) in order to give a complete theoretical analysis.

INTRODUCTION OF A STABILITY FUNCTION S

It was shown in M-S that for w~O

(1) f(w)~O,

(2) O~ Imaxf(w) I ~mGc (note mGc~O),

(3) few) has one and only one maximum,

(4) for A ~ 1, where A is the dimensionless parameter,

k2 rv TilT rv TM V ---- -- ----= k-----, (1-k) D (-mk" D mGcD

h = mG c and instability is impossible. -----

1 W. W. Mullins and R. F. Sekerka, J. Appl. Phys. 35, 444 (1964).

and defining a function £(y) by the equation

few) 2k A:(y) = ---= y+------- (7)

mC;, (1 +tJy)!-l +2k'

it follows that "c(y) has a minimum which obeys the inequality

o~ [min£(y)J~ 1 (8) and that

s= l-[min£(y)]. (9)

To find the value y= y", where "c(y) has its minimum value we set

8£(y)/ --- =0

8y y~ljm

This condition yields

{}k (1+tJYm)l=~---_----. (10)

[(1+tJYm)I-1 + 2kJ2

Since (1+tJYm)L1+2k>O (for k>O) one can take the square root of both sides of (10) without ambiguity of sign. Introducing the substitution r= (1 +tJYm)t leads readily to the following cubic equation for r:

r~+ (2k-l)r- (2k/.ll) = O. (11)

Writing (11) in the form

r3= (2k/ A 1)+(1- 2k)r, (12) 264

]\[ F L L I 1'\ S - S EKE R K A I)J" T E !{ F ACE S L<\ B r LIT Y C R I T E RIO N 265

one can plot the left and right sides of Eq. (12) as we have done for three representative cases in Fig. 1. Xote that the straight line (2k.'.1 i)+(1-2k)r always has a positive intercept on the ordinate for positive A and k while its slope may be positive or negative and depends only on k. Hence, corresponding to k ~ 0, there exists one and only one root r>O of Eq. (11); it is this positive root which corresponds to positive w. Henceforth in this paper, we shall refer to r as the real positive root of (11).

Obviously, from (11), r is a function of two dimen­sionless variables, A and 1<, and since

y",=(rL ll(A/4k), (13)

it follows from (9) that S is also a function of only "1 and k. Using (7), (9), (10), and (11) we readily find

.1 3Al .4(1-2k) S(A,k)= 1+- - ---r - ---~r2. (14)

4k 2 4k

EVALUATION OF ,~(A,k)

Before treating the case of general A and k, we shall present three approximate analytic expressions for SeA ,k) corresponding to: (a) the limiting case A --> 0, (b) the limiting case k --> 0, and (c) the special case k=!. We consider these cases in turn.

Case (a). For verv small A (such that 2k/ A 1»1) it is clear from Oi) that r»1 and, in particular, r3»(2k-1)r. A useful approximation amounts to ne­glecting the linear term in (11) to get

and using (14)

(2k)1/3 r~--­

A-+ll A 1 If, (15)

(1-2k) A ,\)(.1,k) ~ 1-t(2k.1)!----(2kA)i+- (16)

A ... O 4k 41, or

SeA ,k) ~ I-H2kA)t, • 4 ... 0

(17)

where (17) is the leading term of (16). Case (b). In the mathematical limit 1?=O, we have

r=l; accordingly for k«l we set r=l+~, where ~«1 and proceed to do a simple perturbation calculation to first order in small quantities. Thus (11) becomes approxima tel y

(18)

from which

~~k(~-l)+k~(~-~-!), (19) k ... o A I A! 2.1

where we have selected the root of (18) which makes

FIG. 1. Plot of Efj. (11) as a func­tion of r for (a) case 1-··1 positin root, 2 conjugate imaginary roots, (b) case II-l positive root, 2 equal negative roots, and Ic) case III-l posi· tive root, 2 unequal negative roots. -- ~+(1-2k)r A'12

• Roots

t«1. Using (1-1-) and (19) then gives finally

S(A,k) ~ (1+A-2.1t)(I-k). (20) k .... O

Case (c). For k=!, the linear term in (11) is zero so r(A,!)=A-1I6. Then (14) readily yields

S(A,!)= 1-!A 1+..1/2. (21)

Note that (17) agrees with the leading term of (21) for 1?=!, while (16) reduces exactly to (21) in that case.

For general A and k it is expedient to program the calculations for computation by an IBM 7090 com­puter. Since only the positive root of (11) is required for substitution into (14) and since (11) does not contain a quadratic tern1, it is easier to make use of a straight­forward application of the cubic formula than to use a standard program for the three roots of a general cubic and then select the positive root. Three cases arise, depending on the relative sizes of A and k; the resulting fornlUlas for r are given in Appendix II for reference . The program was run for 16 values of k and for A ranging from 10-10 to 1. Using these results, Figs. 2 and 3 were constructed; in Fig. 2, S is plotted as a function of A. for various k, while in Fig. 3 S is plotted against log A. Together the plots cover the range of .1 where S is substantially different from ° or 1.

THE M-S STABILITY CRITERION IN TERMS OF S (A,k)

According to Eq. (6), stability obtains when

(g'+g)/2mG c> SeA ,k).

Using the formulas 1

DkGc K V --(g'-g)

c",(k-l) L

(6)

(22)

266 R. F. SEKERKA

0.9

0.8

0.7

0.2 0.3

/Sfable

Unstoble

0.6 0.7 0.8 0.9

FIG. 2. Plot of the stability function S versus the dimensionless stability parameter A for several values of the solute partition coefficient k. A smooth interface is stable if the test function 'I' fans above the appropriate curve.

and the definitions of ~J' and g, (22) can be written in terms of experimentally measurable quantities as

2KL k [L GJ D ~--- --+-- --->S(A,k). Ks+Kdl-k) 2KL V (-mk"

(23)

It is interesting to divide the parameters of this problem into two classes. The first class consists of the three variables of operation, G, V, CoO' which can be varied independently of the base material and kind (but not amount) of solute, while the second class consists

0.9

O.B

0.7

0.6

S 0.5

0.4

Unstoble 0.3

0.2

0.1

910 -9 -8 -7 -6 -5 -4 -3 Log,oA

FIG. 3. Plot of the ~tability function S versus logA for several values of k. A smooth interface is stable if the test function 'I' falls above the appropriate curve.

of material constants which are specified by a choice of base material and solute and contains all other param­eters except the above three variables of operation. Within the approximation of dilute solutions, the ma­terial constant.s are independent of the kind and amount of solute. "'ith the preceding classification of param­eters, let us consider all of the material constants to be fixed and suppress their explicit appearance in the stability uiterion. Accordingly, we define a test func­tion ,(G/V,c,,) by the equation

Proceeding t.o suppress the explicit appearance of the material constants, we call

s( VI coo) = SeA ,k), (25)

where we continue to use the symbol S without am­biguity because the left side of (25) is a function of only one variable, the ratio (Vic",). Then for stability, we need the test function to be greater than the sta­bilit:-" function, i.e.,

,(G/ V,c",) > s( V / c",,). (26)

For extremely small values of A where S -+ lone obtains

'l(G/V,c",) > 1, (27) A~()

which is the modified constitutional supercooling cri­terion of lVI-So It is very important to note that Eq. (26) contains three independent variables of operation (con­structed from the three fundamental variables, G, V, CoO,

while Eq. (27), as well as the constitutional supercooling criterion of Tiller et al./· 3 depends on only two inde­pendent variables of operation. Indeed, to get the con­stitutional supercooling criterion for stability, one sets KL=K,~ and L=O in (24) and uses (27). Thus, the in­clusion of capillarity (S= 1 without capillarity) breaks the ratio G/V, so to speak, and makes G, V, c'" in­dividually important. It is for this reason that one can have stability despite the presence of constitutional supercooling, as pointed out by lVI-So

Let us cast Eq. (23) in still another form where it can be easily compared with the constitutional supercooling criterion. Using the notation4 "gradient of constitu­tional supercooling," dS/dx=mGc-G, along with Eq. (22) and reversing the sense of the inequality in (23) we find that the :M-S theory predicts instability when

dS 2KL [LV J1 ->--- --+G --G lix K 8+KL 2KL S

(28)

2 W. A. Tiller, K. A. Jackson, J. W. Rutter, and B. Chalmers, Acta Met. 1, 428 (1953).

3 J. W. Rutter and B. Chalmers, Can. J. Phys. 31, 15 (1953). 4 W. A. Tiller, J. App!. Phys. 33,3106 (1962).

1\1 U L LIN S- S EKE R K A 1 NT E R FA C EST A B I LIT Y eRr T E RIO N 267

or

(29)

whereas the constitutional supercooling criterion pre­dicts instability when

dS/dx>O. (30)

Equations (29) and (30) show that for instability to occur, not only must constitutional supercooling exist but a definite gradient of constitutional supercooling must exist. Indeed, as $ ----> 0, the amount of constitutional supercooling needed for instability increases, finally becoming infinite at $=0 which is the onset of the absolute stability considered by 1\1-S.

CONCLUSIONS

We have explicitly evaluated the M-S stability cri­terion in terms of a stability function S and have com­pared the M-S theory with the theory of constitutional supercooling. We have seen that the theories are some­what similar for the case S~l but differ appreciably for S substantially different from 1. We shall not attempt at this time to make a detailed comparison with experi­ment. However, let us note that most experiments to date have apparently been conducted in the range where A«1, and consequently [see Eq. (17)J S-1. The usual procedure is then to plot G/V vs Coo and, according to (24) and (27), get a straight line whose slope is related to the diffusion coefficient D. Hence, since diffusion co­efficients of liquid metals are often poorly known, it may be useful to begin experiments in the region of low ve­locities and high solute content so that A --j- 0 and from the slope of a straight line plot of G/V and c"" determine D. Then one can proceed to higher velocities and lower concentrations and use the determined value of D to evaluate A.

ACKNOWLEDGMENTS

This work was partially supported by the U. S. Air Force Office of Scientific Research under Contract AF 49(638)-1029. The author would like to thank Dr. W. W. Mullins, Department of Metallurgical Engineer­ing, Carnegie Institute of Technology, and Dr. W. A. Tiller, Crystallogenics Section, Westinghouse Research Laboratories, for their many helpful suggestions and for their help in preparing the manuscript.

APPENDIX I

v = average" velocity of freezing (em/sec) G=average5 temperature gradient III the liquid

(deg/cml Gs =average5 temperature gradient In the solid

(deg/cm)

TjJ = melting point of the pure solvent (OK) L=latenl heat of fusion per unit volume of the pure

sol ven t (erg/ cm 3) K L= thermal conductivity of the liquid (erg/cm­

sec-deg) K 8= thermal conductivity of the solid (ergs/ cm-sec-deg) K=(K L +Ks)/2=average thermal conductivity of

the system (ergs/cm-sec-deg) g=G(KL/K)=generalized gradient in the liquid

(deg/cml g'=Gs(KsiK)=generalized gradient in the solid

(degcm) Coo = concentration of solute in the liquid far from the

interface (wt.%) 1<= partition coefficient of solute in solven t (dimen-

sionless) m= slope of liquidus line on phase diagram (deg/wt.%) D= diffusion coefficient in the liquid (cm2/sec) Gc =average5 solute gradient in the liquicl (wt.%/cm) 'Y=solid-liquid surface free energy/unit area (erg/

cm2) 1'= '}'/L = capillarity constant (cm) w= 27r per wavelength of sinusoidal perturbation of

interface (em-I)

k2 rv TM rv 1'11'1 V A=---·-----=k·-·--

(1-k) D (-m)c", D mGeD

= stability parameter (dimensionless),

APPENDIX II

A straightforward application of the cuhic formula6

leads to the following expression for r, the positive root of (11):

r=a+l1, (Al)

where a and f3 are given below for the three cases which occur. Define

Then Case (1)

k2 (2k-l)3 F=-+---.

A 27

If F?;O and (F)!~k/A~,

a= [ kl + (F)!]t 4' ."

(A2)

(A3a)

(A3b)

" Means an li\·erage O\·er a cross section of the sample at the interface.

6 Handbook 0/ Chemistry and Physics (Chemical Rubber Company, Cleyeland, Ohio, 1961), 43rd ed., p. 318.