Thermodynamic calculation of temperatures of possible and probable isoconcentration nucleation in binary liquid alloys

Scripta METALLURGICA

Vol. 17, pp. 711-716, 1983 Printed in the U.S.A.

Pergamon Press Ltd. All rights reserved

THERMODYNAMICCALCULATIONOF T~PERAIIJRES OF POSSIBLE AND PROBABLE ISOCONCENTRATION NUCLEATIONIN BINARY LIQUID ALLOYS. P. GRESSIN, N. EUSTATHOPOULOS,P. DESRE. Laboratoire de ll~ermodynamique et Physico-Chimie M~tallu~iques, E.N.S.E.E.G. - Domaine Universitaire, B.P. 75 - 38402 Saint Martin d'H~res (France).

(Received January 12, 1983) (Revised April 18, 1983) Introduction Over the last years many papers have dealt with the preparation of metastable crystalline or amorphous metellic phases by rapid quenching of the melt. In order to predict the a b i l i t y of a liquid alloy to form a supersaturated solid solution, the concept of the T temperature has been used in many cases.(1)(2)(3). T~ is defined (4) as the temperature bell~w which the free enthalpy AG of the crystellisation process Liquid

(XLA)÷ Solid (X~ =

X~)

(I)

corresponding to the formation of a solid solution with the same composition as the i n i t i a l binary a11oy, is negative, h i s "isoconcentration" transformation is of great interest because i t takes place without any fluctuation of concentration in the liquid phase : For a given supercooling this process would be more probable, the more rapid the cooling rate. Consequently transformation (1) is in strong competition with the glass transformation process. Moreover TO can be easily calculated from the thermodynamic properties of the bulk phases by writing :

Xs=xL,AG=xsmA+ (i-XSA)mB=O

(2)

where Xi , XAL are the molar fractions of component A in the solid and liquid phases and I A , m B , the differences in the chemical potential of components A and B in the solid and liquid alloys. Calculations of T_ have been performed either to show the change of this quantity with composition in particular°systems(i) (2) or to study the influence on To of the kind of A - B interactions in modelized binary alloys (3). H~ever, the supercooling ATo = TL - TO (where TL is the liquidus tempera~re in the phase diagram) represents only a minimum value below which "isoconcentration" crystallisation becomes possible. In this paper we present a calculation of the supercooling AT' = T, - T' for which this kind of transformation becomes probable and we discuss Me relation existing between ATo and AT'. Theory For an alloy A - B at constant temperature, pressure and composition of the liquid phase, the radius r and the reversible work of formation W of a critical nucleus, corresponding to a point of maximum resistance i~ homogeneous nucleation, are given by the expressions (4) (9) :

Gv(X Wc(r = r c, Xs) = .c(XS) =

1G~o3(xSA)

where o is the solid-liquid interracial tension , AGv = AG~M (VM : the molar volume of the solid) and r the radius of nucleus supposed to be spherical.

711 0036-9748/83/060711-06503.00/0

Copyright (c) 1983 Pergamon Press Ltd.

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NUCLEATION IN BINARY LIQUID ALLOYS

The~(value XAS~f X~ which minimizes W c corresponds to the = r e radius rc :

Vol. 17, No. 6

stable critical nucleus of

WcW= {Wc(XS)}min w = rc(XS = XS w)

rc

(5) (6)

In the regular solution approximation, the chemical potential ~iJ of component i (A or B) in the phase j(S or L) is written : " iJ =~

' +k j ( 1 - X~) • 2 RTln X~

(7)

so that : Aui =u)

xS i L AHf,i xS)2 L( L 2 -u i (Tf, i - T) +RTLn ,-iF--+kS(1 - i" - k I - Xi) (8) Tf,i Xi where Apf,i and Tf, i are the enthalpy and temperature of melting of component i and k s, k L the molar exchange energies of the solid and liquid regular solutions, respectively. In order to calculate the Wc(X~) curve one must know the composition dependence of the interfacial tension. While several models exist in this field (based on various approximations on the width of the compositional interface (5)(6)(7)I here we will use the more simple model of Becket (8) in which the compositional transition a't the interface is sharp ("zero-layer" model) Even though i t is a very rough approximation, this model has the advantage of being compatible with the capillary description used here for the nucleation process, which admits explicitly a sharp discontinui~ of both structure and composition at the nucleus-liquid boundary. Moreover i t is probable that the adsorption equilibrium at the nucleus-liquid interface is not reached during fluctuations in a rapidly cooled system. Using Becker's approximation the interfacial tension is written (g) :

°--°A O where oA, o~ are the solid-liquid interfacial tensions of A and B pure metals, m a structural parameter and ~ is the mean molar interfacial area. The ability of an alloy to form a supersaturated solution during rapid cooling will depend on the difference

calculated for given ~alues of XLAand supercooling AT = TL - T (Fig. 2). The limiting temperature TO is deduced from equation (2) which is simply written as (3) : TO =

AHf,AXA +AHf,BXB - (ks- kL)XAXB

(Ii)

XA ASf,A + XB ASf,B where X~ = X) = XI and ASf,i is the melting entropy of pure element i . Results and discussion As in a previous work (3), in orderi~8 sh~ clearly the influence of the heteratomic A - B interaction energies on the difference Wc - Wc, calculations have been performed using the same values of parameters Tf, i , Lf, i , VM, ~ et a~ (values which we have chosen arbitrariTy equal to those of t~e Ag(=mA) - Cu(= B) system, Table 1) but for different values of the molar exchange energies k ~ and L ' . In all cases we used m = 0.25 which corresponds to the more

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NUCLEATION

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713

densely-packed interfacial plane of an Fcc structure (the {111} plane). TABLE 1

TABLE 2 Metal

A Tf(K) I(KJ

Lf mo1-1)

o°_2

(mJ .cm ) 7 (m2.~w)l -I) (m 3.moIFm-1)

System

B

1234

i356

11,3

13,06

iR KJ.mol-

TE at

T = TE

(K) Xc~ B

~S

>L

2 5

15

1090 9,5.10-2

9.10-2

b

40

15

1050 B,5.10"3

7,5.1Q-3

c

40

0

803

< < 10-2

d

40

- 15

559

< < 10-2

a

~A'

(Ag- Cu)

155

200 '4,2 x 104

8,69

x 10-6

These calculations have been performed for four pairs of XS, ~L values (Table 2) all corresponding to sin~ole eutectic phase diagrams (Fig. 1). The latter have been computed by resolving for each temperature the system of equations ~ = AuD = 0. In table 2, we give for each pair LS, ~L the eutectic temperature (TE) and the v~ues a~c T = T~ of molar fractions of A and B components in the solid solutions B (ri ch in B) and ~(rich itl A), respectively. I t can be seen that the higher the difference X~ -~L, the deeper the eutectic and the lower the mutual solub i l i t y in the solid state. Figure 2 shows the Wc(Xs) curves calculated for three different compositions of systems a (figure 2a) and c (Figure 2b}'. Figure 3 shows the AT6/TI versus composition curves calculated from eqn11 for the four systems of Table 2. ( TL is the~temperature of liquidus). From these results the following comments can be made : a - At any ToXL values ~he molar fraction XSWof the more stable nucleus, corresponding to the Iowest~minimum'o~ the W(X°~) curves. Figure 2,his certainly different but yery close to the value of X~ which m.inimizes tRe 6G (X~) function (9). This signifies that XSA ~ is pratically insensitive to the composition dependence of o. This result is fundamentally B i f f ~ e n t from that obtained for nucleation in perfectly coherent miscibility gap systems where the X~ value is strongly modified when the compositional dependenceof ~ is taken into account{10~. This different behaviour is due to the existence here, of a structural contribution to the interfpcia~ tension (terms in ~o in equation 9) which leads to high values of a for all values of X~I, X~. For this reason r~^is here a monotonically decreasing function of supercooling. On the ~ontl~ary, in perfectly ~oherent miscibility gap systems a structural contribution to ~ does not exist and o can now reach zero when the composition Qf the nucleus tends towards the i n i t i a l composition of the system. Fo~ this reason computed r c (AT) curves (at constant XA) present a minimum at some intermediate T between the equilibrium and the spinodal temperatu~s where rW c tends to i n f i n i t y (10) (9). b - For a given system and composition of the liquid phase the Wr(XSA) curves flatten when the supercooling AT increases so that the difference AW (eqn 10 and FiguYe 2a) decreases rapidly with decreasing temperature. This effect is more pronounced for compositions lying near the pure metals (see curves ~or XA = 0.2 and X~ = 0.8 of Figure 2a) as compared to intermediate compositions (Figure 2a, X~ = 0.5). MoreoverAW decreases more rapidly wiht increasing values of AT for systems with a Tol~ value of }.S-xL than for systems for which this value is high (compare for a same value of X~ curves of Figure 2a and 2b). c - Results for the system a are given in Figure 4 in the form of an M versus AT/TL curve, where : I W Kiso wiSO w c M ~

KW

= exp

- Wc k T

(I2)

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T (K)

T(K)

.~.

1300

1200

1200

11oo

1100

9oo

/. ................-,,,./,ooo ://

o

if,. 900

~oo

,/

o~ o~ o~ o~ o~ ~ ~7 o,e o~

,

0 0,1 q2 0,3 0.4 0.5 0.6 0.7 0.8 09

XA

Fig. i : Lines of s o l i d - l i q u i d equilibrium of systems a and c (Table 2). b~/c cA~,a

~^1

CAS: C

i

o

5o~

~^/ o

CO

;

.i,, o.2- ~

Q2

0~4

~

dB

0j6

O~

x~

x~,

,

: Reversible work of formation W_ of c r i t i c a l nucleus as a function of the molar fracnucleus for different comp~it~ons of the liquid phase and different values of supercooling. Systems a and c. Wc in 10- J. •

M

"~'LT° 1[

7

:1/

L 0 10150 XA= .

0.5

0

C i o,1

10lOc

1050

~.~¢~. o.,o

,Fi~. 3 : Reduced supercooling of possible isoconcentration nucleation as a function of molar fraction for the four systems of Table 2.

Fi~. 4 : M = exp

o.~0 -

o.~,o as a function of

reduced supercoolingkforT system a.

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NUCLEATION

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7IS

Here, IX, Iis° are the nucl~ation (~quencies of the more stable and the "isoconcentration" critical nucleus respectively, K and K the corresponding pre-exponential factors and k Boltzmann's constant. On the same Figure 4 we have noted the ATn/T r values of the three alloyso~hich lie between 0.04 and 0.09. We can see that at these low su~r~ooloingsfs ~ is as high as I0 ~uu so that it is unlikely that purely kinetic factors would lead to I = I = I^ : In these ranges of supercoolings , if nucleation takes place (for example on a solid substrate) it leads to the n~)~ stable critical nucleus. However, as ~T/T m increases, M decreases, flrst~ rapidly, up to 14 = lO"Uand then slowly to unity in an as~pt~ical waJl~^As an example, for X) = 0.8 and AT/TI. =0.40 we have 14 = i0 such that, in this case, even if K '~" ~, the probablllt~) of a "isoconcentration" nucle.

.

.

.

=

"

.

.

iSO

W

.

ation is no longer negllglble. Whlle the a ~ of the pre-exponentlal factors ratlo K IK Is not known, i t is possible to perform a rough evaluation from nucleation experiments that have led to supersaturated solid solutions. So, in the Ag (= A) - Cu (= B) system (phase diagram close to that of system a,,~(gure 1) supersaturated.~u-rich solid solutipns (XB = 0.2) have been obtained @)~T/TL -- O_3tvl. At this vllue M -- 10zu (Figure 4, curv~nXA= 0.2) so that, following equation t~J~ we can conclude that KIS°/K~ is of the order of 10- .~STmilarly, Perepezko and Rasmussen~ i obtained supersaturated Sn - Bi solid solutions in the range XBi =0.15 to 0.30 during nucleation at AT/TL -- 0.30 to 0.35. Note that, as in the case of Ag-Cu, the Sn (= A) - Bi (=B) system has a simple~utectic phase~iagram with a rather important mutual solubility in the solid state (at T = TE, X~. = 0;12 and ~n -- 0.03). Taking KiS°/K X = 10"i~ we find from Figure 4, curve XL = 0.2, that the reduced supercooling AT'/T~ of "probable isoconcentration nucleation" is five iimes higher that the v@lue ATn/T= for "possibl@ isoconcentration nucleation". For the same system, AT'/ATo is -- 7 for X~ = O.8"an~l --9 for X~ = 0.5. However, i t is interesting to note that even thougB very different, the supercoolings AT'/T L change in a similar way with either composition (for a given system) or with the kind of system (for a given composition). Concl usi ons From the above discussion we can conclude that the thermodynamic tendancy of supersaturated s~lid Folutions to nucleate at high supercoolings is c r i t i c a l l y dependent on the difference Xo ~= of the molar exchange energies of solid and liquid alloys. In metallic systems this difference is generally a positive quantity nearly equal to the strain energy resulting from size differences between solvent and solute atoms in the solid solution. During rapid quenching of the melt the blockage of concentration fluctuations in the liquid made that nucleation of supersaturated solid solutions and amorphization would be in strong competition. For systems having low values of xS _ ~L, the tendency to form supersaturated solid solutions will be great and, as a consequence, amorphi~atioQ of these alloys by melt-quenching will be d i f f i c u l t . For systems with high values of Xo -X=, which preciFely correspond to those with strong attractive heteroatomic interactions in the liquid state (X" < < O) and with deep eutectic phase diagrams, the tendency to form supersaturated solutions will be either very weak (at the two extremes of the phase diagram where even i f AG < 0 the supercoolings leading to low values of M are quite unrealistic) or nul (in the intermediate range of compositions where AG > O, Figures 2 and 3). During slow cooling, which allows concentration fluctuations in the liquid to take place, these alloys would nucleate to the stable crystalline phase even at vew high supercoolings. During melt-quenching, the blockage of such fluctuations, required for both "isoconcentration" nucleation~and amorphization, would lead to an amorphous phase. The greater the differences Xb - >,L and X) - XL, the greater the tendancy to form a metallic glass. This general condition of glass-fo~ming Ability is compatible with the semi-empiriF~icriterion of a deep eutectic and of attractive heteroatomic interactions in the liquid alloys ~'=j. Finally, even i f in a given system the temperatures TO and T' of possible and probable "isoconcentration" nucleation ar~ ver~ different, we have shown that they vary in a similar way as a function of the quantity X~ -X= and of the composition. For this reason TO, which is readily calculable from the thermodynamic properties of bulk phases, can be used to predict easily amorphi zable systems and/or compositions. References (1) T.B. Massalski, Proc. of 4 th Int. Conf. on R Q M, Vol. 1, p. 203, Japan Inst. Met. (1982). (2) K.N. Ishihara and P.H. Shingu, Scriota Met., 16,837, 1982. (3) P. Gressin, N. Eustathopoulos, P. Desr~, J. Chim. Phys., 79, 545, 1982. (4) D.S. Kamenetskaya in "Growth of Crystals" , Ed. N.N. Sheftal, Consultant Bureau, Vol. 8. -

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NUCLEATION IN BINARY LIQUID ALLOYS

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p. 271, New York, 1969. N. Eustathopoulos, J.C. Joud et P. Desr~. J. Chim. Phys., 6__99,1599, 1972, et 71, 777, 1974. A. Nason, W.A. Tiller, Surface Sci., 40, 109, 1973. L. Coudurier, N. Eustathopoulos, P. De~r~, Fluid Phase Equilibria, 4, 71, 1980. R. Becker, Ann. Physik, 3_2, 128, 1938. P. Gressin, Report DEA, L.T.P.C.M., Grenoble, 1980 and Th~se Grenoble 1982. H. Reiss, M. Shugard, J.Chem. Phys. 28, 258, 1976. J. Perepezko and D. Rasmussen, 17 th--~erospace Sciences Meeting, New Orleans, 1979, Am. Inst. Aeronautics and Astronautics. (12) M.H, Cohen and D. Turnbull, Nature, 189, 131, 1961. (5) (6) 7) 8) (9) (10) (11)