Heterogeneous nucleation in solidifying alloys

HETEROGENEOUS

NUCLEATION ALLOYS

IN SOLIDIFYING

B. CANTOR aod R. D. DOHERTY School of Engineering and Applied Sciences. Universtty of Sussex. Brighton BNI 901. England

variety of solution models are used to determine driving forces for nucleation in sohdifyrng alloys The resulting data are then used to analyse heterogeneous nucleation experiments tn which liqutd alloy droplets solidify within a surrounding solid matrtx. By comparing these experiments with homogeneous nucleation experiments on pure metals. the analysis makes it possible to determine solidliqutd- matrix contact angles and therefore quantify the catalysmg effect of the matrix on the nucleatton process. This approach is shown to be more useful than a simple comparison of undercoohng data. In almost all cases. the liquid alloy droplets are found to solidif! by heterogeneous nucleation with contact angles less than IX0 A careful analysis of the results supports previous suggestions that Turnbull’s classic experiments on solidification of pure metals did not show true homogeneous nucleation.

Abstract-A

R&me--Des modeles divers de solutions sont utilisk pour determiner les forces matrices de la germination dans les alhages en tours de solidification. On utilise les donnees qui en rtsultent pour analyser les experiences de germination hetirogene dans lesquelles des gouttelettes d’alliage hquide se solidifient dand une matric solide. En comparant ces experiences avec des experiences de germination homogene dans les mitaux purs. on peut determiner les angles de contact solide--liquide-matrice et quantifier ainst I’effet catalytique de la matrice sur le processus de germination. On montre que cette approche est beaucoup plus utile que la sample comparaison des donntes de surfusion. Dans presque tous les cas. les gouttelwettes d’alliage liquide se solidifient par germination hlterogene. avec des angles de contact inferieurs a 180’. L‘analyse precise des r&hats confirme des suggestions anttrieures selon lesquelles ies experiences classiques de Tumbull sur ia solidification des mttaux purs ne montraient pas de veritable germination homogtne. Zusammeafassuag-Eine Anzahl von L&ungsmodellen wird zur Bestimmung der treibenden Krlfte fiir die Keimbildung in erstarrenden Legierungen herangezogen. Die erhaltenen Ergebnisse werden danach benutzt. urn Experimente zur heterogenen Keimbildung. bei der Tropfchen der fliissigen Legierung innerhalb einer umpebenden festen Matrix erstarren. zu analysieren. Der Vergleich dieser Expertmente mit denjenigen zur homogenen Keimbildung an reinen Metallen und die Analyse erlauben es. die Kontaktwinkel zur Matris zwischen dem festen und fliissigen Beretch zu bestimmen und damrt den katalytischen EmfluB der Matrix auf den Keimbildungsprozess zu quantifizieren. Es wird gezeigt. daR dieses Vorgehen brauchbarer ist als ein einfacher Vergleich der Unterkiihlungsdaten. In fast allen Fallen erstarren dte fliissigen Legierungstriipfchen durch heterogene Keimbildung mit Montakwmkeln kleiner als 180 Eine sorgfalttge Auswertung der Ergebnisse stiitzt friihere Andeutungen. dab die klassischen Experimente von Tumbull zur Erstarrung reiner Metalle keine reine homogene Keimbildung zeigten.

I. INTRODUCIIOK

In investigations of the nucleation behaviour of solidifying liquids. the main experimental dificulty is the control of impurities which may act as nucleation catalysts. The classic experiments by Tumbull and coworkers [ 1.23 on homogeneous nucleation overcame this problem by dividing the liquid mass into many small droplets; only a few droplets contained impurities and the majority were not subjected to any catalytic effect. Most investigations of heterogeneous nucleation. i.e. the catalytic effect itself. have not been so successful because of the added difficulty of obtaining an impurity-free catalytic surface. Heterogeneous nucleation has been investigated by inserting nucleating particles [?-6) or a single nucleation surface [73 into the liquid. by observing liquid droplets on an

inert [g] or nucleating substrate [93, and by studying the nucleating effect of a primary solid phase in alloy droplets on an inert substrate [IO. 111. The results which have been obtained so far have sometimes been questioned because of a lack of control over extraneous impurities [IZ. 131. and no definitive atomic model has yet been produced to explain the catalytic heterogeneous nucleation process. In some recent experiments on heterogeneous nucleation. Southin and Chadwick [14] obtained impressively reproducible results by using a simple but elegant technique which is described in the next section. The objective of the present paper is to provide a detailed analysis of Southin and Chadwick’s results. and to shovv how they may be used to derive contact angles between catalytic nucleant surfaces and the solidifying melt.

34

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2. SOUTHIN AND CHADWICK’S EXPERIMENTS

Southin and Chadwick employed a technique described previously by Wang and Smith [ 151 which used alloys in a binary eutectic or monotectic system, with a composition c, such that the equilibrium structure contained only a small fraction of one of the phases (see Fig. 1). The alloys were chill-cast, then annealed at a temperature T4 between the liquidus and solidus to form small eutectic liquid droplets in a primary solid matrix. On cooling from the annealing temperature, differential therma analysis was used to determine the temperature at which the liquid droplets were nucleated by the surrounding solid. This technique eliminated problems of contamination, because the liquid droplets and the catalytic nucleating surfaces were formed internally during the heat treatment. Southin and Chadwick analysed their results as follows. Consider an A-B alloy of composition c, which at the annealing temperature consists of liquid droplets in an Q solid matrix (see Fig. 1). If the temperature is changed, the composition of the liquid can adjust to maintain equilibrium because there is no nucleation barrier to precipitation or dissolution of A atoms at the liquid-matrix interface. On cooling below the eutectic temperature, the /I phase is heterogeneously nucleated on the a matrix at a temperature 7& measured by differential thermal analysis. At this temperature. the liquid composition cL is given by the metastable equilibrium line obtained by extrapolating the liquidus (see Fig. 1). From this construction, it is possible to determine the liquidus temperature of the liquid droplets r,.

TMe

NUCLEATION

IN

SOLIDIFYING

ALLOYS

For each of the alloys investigated. Southin and Chadwick compared the undercooling parameter u hr, = (TL - TIs)/TL with a similar undercooling parameter for homogeneous nucleation of B in the absence of the r phase. This homogeneous nucleation parameter was obtained from Turnbull’s data [l] and is given by LJ’,,>,= (7$a - TYT)~7qBwhere T$e is the melting point of pure B and TV, is the homogeneous nucleation temperature of pure B as measured by Turnbull. When O,,, was smaller than Llhom,Southin and Chadwick took this to indicate that the 1 phase had a catalytic effect on the nucleation of p. Table I shows the nucleation temperatures and undercooling parameters in those alloys for which both sets of data are available. (In Table I the alloys are grouped according to criteria which will be explained in the following section.) In some alloys. Lihc, is very small and Southin and Chadwick suggest that 3 is an effective nucleation catalyst; in other alloys with U,,, 5 u h,Vn* Southin and Chadwick suggest that there is no catalytic effect and fi nucleates homogeneously in the liquid droplets. Nucleation takes place when the driving force for solidification becomes sufficiently large to overcome the surface energy barrier. In pure materials, this driving force is, to a very good approximation, directly proportional to the undercooling in the liquid. Thus, when nucleation occurs at a small undercooling, this indicates a strong catalytic heterogeneous nucleation effect. In alloys, however, the driving force for solidification is a function of both temperature and composition and the effect of composition is excluded from the above analysis. In Southin and Chadwick’s experiments, the B phase of composition cp is nucleated from liquid of composition cL in the presence of a catalytic surface of a; this is not the same as nucleation of pure solid /3 from pure liquid 5 on the catalytic surface. In fact, the driving force for solidification at the nucleation temperature depends upon the undercooling in the liquid (r, - 7,s) and the compositions of liquid and solid, cL and c,,. To estimate the driving force for solidification it is necessary to use a solution model for both liquid and solid. and this calculation is carried out in the next section. 3. THE DRIVING FORCE FOR SOLIDIFICATION

Figure 2 shows free the A-B alloy of Fig. the melting point. In bull’s on solidification force for solidification expressed by the usual latent heat of melting. I ’ %

co mole

fraction

A:

5. y3 of

A& = GLA- GSA

E

Fig. I. Typical eutectic and monotectic phase diagrams alloys studied by Southin and Chadwick [14].

energy-composition curves for I at any temperature T below an experiment such as Turnof a pure metal. the driving per mole of solid AG7 can be approximation in terms of the Thus for solidification of pure

for

(1)

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HETEROGENEOUS

NUCLEATION

IN SOLIDIFYING

35

ALLOYS

Table I. Nucleation temperatures and undercooling parameters for alloys studies in both Southin and Chadwick’s and Turnbull’s experiments. For a system A-B. I&., is the nucleation temperature and C’h.lll the undercooling parameter in Turnbull’s experiment on pure B: &s is the nucleation temperature and U,,, the undercooling parameter in Southin and Chadwick’s experiment on B-rich liquid droplets in an A-rich solid matrix r,r

System A-B

(K)

t%

c,<,,,,

C’,,,.,

AI-Pb

520

570

0.133

0.050

AI-Sn Cu-Pb

400 520

415 597

0.208 0.133

0. I 80 0.003

Zn-Bi Zn-Pb Zn-Sn

454 520 400

406 564 389

0.166 0.133 0.208

0.269 0.060 0.230

Ag-Cu

Cu-Ag

1120 1007

995 943

0.174 0.184

0.112 0.20’

NiAI,-AI Ag-Pb Ph-Ag AI-Ge Cd-Bi

803 520 1007 1004 454

907 575 550 582 3.58

0.140 0.133 0.184 0.184 0.166

0.006 0.003 0.177 0.333 0.185

CuAI,-AI Ge-AI Si-AI Bi-Sn Sn-Bi Pb-Sb I(AMgI-AI Cd-Pb Sb-Pb

803 803 803 400 454 768 803 520 520

799 638 803 361 375 500 816 483 501.5

0.140 0.140 0.140 0.208 0.166 0.150 0.140 0.133 0.133

0.048 0.121 0.086 0.245 0.148 0.100 0.040 0.130 0.076

Pb-Sn Sn-Pb

400 520

426 433.5

0.208 0.133

0.090 0.140

where GLA. GSAare molal free energies of pure liquid and solid A, L, is the latent heat of melting for pure A. and TMAis the melting point. Similarly, for solidification of pure B: AGr = GLB- GSB = +cr,, MB

- 7).

energies of A and B in the liquid and solid phases (see Fig. 2). The differences in free energy between liquid and solid phases of the pure components can again be expressed in terms of the latent heats of melting of the pure components LA. LB. as given by equations (1) and (2).

(2)

_r

where GLB, Gsa are molal free energies of pure liquid and solid E, LB is the latent heat of melting for pure E, and TMBis the melting point of pure E. In Southin and Chadwick’s experiments, the equilibrium solid at temperature T has composition c,, (see Fig. 2) and the driving force for solidification per mole of solid AGr is given by [16]

AGLB

1

AGS = C@(G,B- Gss) + (1 - C$(GL” - &I = cII(GL~- Gsa) + c&AC0 - A&)

+ (I - cB)(%,, - G,,) + (I - c,)(AC,, - A&,).

(3)

where cp is the mole fraction of B in the solid; CL,,, GSA, GLB. GssBare partial modal free energies of A and B in the liquid and solid phases: and ALLA. AGSA, AcLB, Ai?sB are the relative partial molal free

1

: :

:

CU CO CL mole fraction of B

cp

Fig. 2. Free energy composition curves for z. fl and liquid phases at any temperature in Southin and Chadwick’s heterogeneous nucleation experiments [ 141.

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Thus the driving force for solidification in an alloy can be obtained in terms of known parameters by assuming a solution model for the liquid and solid in order to estimate the relative partial molal free energies. For ideal solutions, the relative partial molal free energies are given as follows [17] AG$”

= R T In( 1 - cL) : AG$” = RT In( I - c,,)

AG$”

= RT In cL

: AG$’

= RT

In c,,,

(4)

where R is the gas constant. The effect of non-ideality can be estimated in several ways. Firstly, published values of relative partial molal free energies [18] can be extrapolated to the temperature T. Unfortunately. tabulated data are available for only a few alloy systems [IX]; and even for these alloys, the data have been determined at temperatures well above the nucleation temperatures measured by Southin and Chadwick. The published data are in the form of relative partial modal enthalpies and entropies measured above the melting point for the liquid, and at the eutectic temperature for the solid. If these enthalpies and entropies are assumed to be independent of temperature. then the relative partial molal free energies at T are given by ACLA = ARLA - TA$,, AC,

= Anu

: A&, = A&

- TA.S,

- TASUl : A&a = AAsB - I-A&j,

(5)

where ARti. AnsA. AH‘s, Agse are the published relative partial molal enthalpies measured at temperatures direrent from 7: and ALLA, ASsA. AS,, AS,, are the published relative partial molal entropies also measured at temperatures different from 1: For many systems, however, no data are available for the solid and the relative partial molal free energies can only be estimated with the additional assumption that the solid is ideal so equations (5) are modified to

NUCLEATION

A&, = Ar$;j.”

: Ai;,, = A@;”

AGLB= A@‘;” + RT In ;‘ : AGss = At?$“.

(8)

The value of ‘J can again be obtained by putting the driving force for solidification equal to zero at the eutectic temperature. In summary. the driving force for solidification in homogeneous nucleation experiments on pure B can be estimated at any temperature from known parameters by using equation (2). At the observed nucleation temperature TV,, the driving force AGr = AGyr and is given by equation (2) with T = T,,. In Southin and Chadwick’s experiments, the driving force for solidification at any temperature can be estimated from equation (3) after substituting for (GLA- GSA) and (G,, - Gsa) from equations (I) and (2). and using one of equations (4)-(8) for the relative partial molal free energies. At the observed nucleation temperature TVs, the driving force AGs = AGvs and is given by equation (3) with T = r,,. The various approximations required to estimate the relative partial molal free energies are improved and the above expressions simplified when either c,, c I or cL 1 cu 1: 1. The first case. c,, 2 1. arises in an alloy system with A atoms insoluble in solid B. The second case. cL = c,, = 1, arises when in addition the temperature is below a critical temperature at which the extrapolated liquidus intersects the righthand side of the phase diagram; then, all the A atoms have solidified as the 2 phase and the remaining liquid is pure B. It is interesting to notice that the driving force for solidification in an alloy is proportional to undercooling only if cL 1 c,, 2 I so that AGs = AGr = +(L Yil MB

The value of w can be estimated by substituting in equation (3) and letting the driving force for solidification vanish at the eutectic temperature. In the third method, the solid and one of the liquid components is taken as ideal, and the other liquid component is assumed to have an activity coefficient y independent of temperature and composition. Then

ALLOYS

the relative partial molal free energies are given by

AG,vs = +tl-

In the second method of estimating the effect of non-ideality, the solid is again assumed to be ideal and the liquid is assumed to be a regular solution with an interaction parameter w. Then the relative partial molal free energies are given by [I71

IN SOLIDIFYING

- T) TNS)= LJhc,

(9)

Thus, only for alloys with cL - c,, : I can undercoolinys be compared directly with Turnbull’s data [I, 23 (as in Southin and Chadwick’s analysis) because A atoms are insoluble in both liquid and solid B at the nucleation temperature. In the present work, the above equations have been used to calculate AGys and AGVT for all the alloys in Table I, using Turnbull’s [ I, 23 and Southin and Chadwick’s [ 143 data, and published phase diagrams [ 19-211 and thermodynamic data I:IS]. The results are shown in Tables 2-5. In all cases the driving force for homogeneous nucleation AG,vr was calculated from equation (2), and the alloys are grouped into different tables according to the methods used to calculate the driving force for heterogeneous nucleatton AG,vs. The alloys in Table 2 are those for which cL 2 c,, = 1 at the heterogeneous nucleation

CANTOR

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NUCLEATION

IX

SOLIDIFYING

37

ALLOYS

2. Nucleation data for alloys studied In both Southm and ChadwIck’s and Turnbull’s experiments. and for which (‘L- cl. 1 1. For a system .4-B. AG .,r is the drivmg force for solidification at the nucleation temperature III Turnhull‘s experiment on pure B: ACiys IS the driving force for solidification at the nucleation temperature

Table

in Southln

and Chadwick’s

experiment

on B-rich

liquid

droplets

in an .4-rich

solid matrix.

(‘L and c’,, are the

concentratrons of E m the liquid and nucleating solid. fJ the contact angle. A& the critlcal drlvmg force for the onset of homogeneous nucleation. l,5, the homogeneous nucleation frequent! at the nucleation temperature. 0 the sol&liquid surface energy. and ~,~-c,~ the difference in surface energies of liquid and nucleating \olld in contact with the 4-rich matrix. Alloys m nucleation type .4 nucleate heterogeneously: alloy In nucleallon type B nucleate homogeneousI! with a contact angle of 150 : alloys III nucleation type C also nucleate homopeneousl! but the contact angle IS not necessaril! 180

System

4-B

Al- Ph Al-Sn Cu-Pb Zn-Bi Zn-mPh Zn-Sn

AG\T (kJmol_‘~ 0.635 1.469 0.635 I.819 0.63C 1.469

AGVS (kJmol_ 0.X 1.359 0.024 2.789 0.286 I.623

(I

It

(de@

A& (LJmolK’l 1.36 _I ‘5

55 95 19 I80 61 I26

;:4; _ 1.32 I.lb

IS-’

IV,, rns31

IO_‘JJ IO’ I,). I‘IZh

(i (mJm_‘1

Kucleation type

32.4 56.0 32.4 56.3 32.4 56.0

IO” 10-R” 10’:

.4 4 .4 B .J c

~zL-.~i,.5 (mJm_ ‘1 166 -40 30.6 < - 56.3 15.’ - 32.9

Table 3. Nucleation data for alloys studies in both Southin and ChadwIck’s and Turnbull’s experiments: and for which both solid and liquid relative partial molal free energies are available For explanation of column headings see text or caption to Table 2

System .4-B Ag-Cu Cu-Ag

AGVT (kJmol_‘) 2.262 2.191

AGVS (kJmol_‘) 1.624 2.150

0 (de@

A’& (kJmol-‘)

78 IO4

temperature. For these alloys. the driving force for heterogeneous nucleation was calculated direct]! from equations (9). The alloys in Table 3 are those for which there are available solution data for horh liquid and solid phases. and the driving force for heterogeneous nucleation was calculated from equation (3) with the extrapolation of equations (5) for the relative partial molal free energies. For the remaining alloys it was necessary to assume that the solid /I phase was ideal. The driving force for heterogeneous nucleation was calculated from equation (3) in three ways: firstly. using equations (8) for an ideal A-component in the liquid and a non-idea1 B-component with activity coefficient fiAG$s): secondly. using equations (7) for a regular liquid solution with interaction parameter w (AGE,); and thirdly. where possible. using equations (6) to extrapolate thermodynamic data for the liquid (AGES). These remaining alloys are split into two further groups according to whether c,; 2 0.99. For the alloys in Table 4 with c,: > 0.99. non-idealit? of the solid is not expected to have a significant effect on the driving force. and the above approximations are expected to be reasonable. The alloys in Table 5 have c,, < 0.99; the assumption of an ideal solid is unavoidable but is likelq to be a poor approximation. For the alloys in Tables 2 and 3. the value of the driving force is expected to be a good approximation to the exact value. For many of the remaining alloys. the different methods of estimating the driving force lead to almost the same result. and this again indicates that the approximations are reasonable. Ho&-

(d’

l,Si me31

Nucleation (mJi_‘)

tYPe ,-l A

4.61 3.06

G,L?,S (mJm_‘) 3’.b - 32.7

ever. for those alloys indicated with an asterisk. the different methods give varied results, presumably because of non-ideality in the solid b phase. Driving forces for solidification of NiAIJ-AI. CuAI,-Al. and [(AlAg)-Al have been calculated in the same waq as the other alloys. although the intermetallic compounds must produce free energy-composition curves more complicated than shown in Fig. 2. The assumptions used to calculate the driving force should still be reasonable for NiAl,-Al with L’,:> 0.99: data for CuAl,-Al and C(AIAg)-Al should however be treated with more caution.

4. THE

NUCLEATION

FREQUENCY

Consider first a homogeneous nucleation experiment such as Turnbull’s in which a mass of liquid droplets of pure f3 IS progressively undercooled belou the melting point. At any temperature. 7. the homogeneous nucleation frequency per unit volume of liquid I, is given by [I] I, = A exp( - h’&f:AGtkT)

(10)

where fi is a shape factor equal to 16~ 3 for spherical nuclei. RS is the molar volume of solid B. 0 is the solid-liquid surface energy. and k is Boltzmann’s constant. The frequency factor +I is equal to [I] (II&T /I) exp( - AC,, k77 where II, = ‘V,,.‘R, is the number of atoms per unit volume of liquid. ,2‘, is Avogadro’s number. RL is the liquid molar volume. II is Plank’s constant. and AG, IS the activation free energ! for

-Al

Ag Pb Pb-Ag* AI-Ge* Cd-Bi

NiAl,

System A--B

AGj,s (LJmoJ ‘)

0.071 0.017 4. I93 7.201 2.180

AGw (kJmol-‘)

I.497 0.635 2.191 6.493 I.819

0.07 I 0.017 3.614 11.715 2.136

AGL (LJmol-‘)

2.263

AG& (kJmol_ ‘) I8 I4 180 II8 II7

I8 I4 180 93 I21

0” (dcg)

(degt

0,’

-130

Meg)

0s

8.43 I.41

3.34 1.44 -

(kJzi_

I)

lNSl m-j)

10ZO IO” 101s

JO- ,097, lo-‘s&31

(s-’

99.0. 32.4 I35 231 56.3

(mJi_‘)

C C

A B

A

type

Nucleation

<-I35 - I08 -36.2

94.2 31.4

e.L-0.s (mJm_‘)

of column Table 4. Nucleation data for alloys studied in both Southin and Chadwick’s and Turnbull’s experiments, and for which cL < I and cII > 0.99. For explanation headings see text or caption IO Table 2. The driving force Tar heterogeneous nucleation has been calculated in 3 ways: from the activity coefficient (AGis); for a regular solution (AG;s); and by extrapolation of liquid relative partial molal free energies (AC;,). Each driving force leads to a corresponding value of contacl angle (0;. 0”. Us). Alloys marked with an asterisk have a considerable spread in calculated driving forces

CANTOR

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HETEROGENEOUS

n

f -5

N 5

*

t

0 0 “0 b _--------__ 0

!

I

:

NUCLEATlOn:

:

:

3

_

0

-i

2

“0 b 0

::

c

IN SOLIDIFYING

ALLOYS

39

40

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transporting an atom across the solid-liquid interface. Following Turnbull [I], this activation energy is approximately equal to the activation energy for viscous flow in the liquid, so that exp( -AG,.ri’k7J k IO-’ at the point of most metals and meiting .41 IO" s-'m-‘. As the liquid temperature falls below the melting point. the nucleation frequency remains negligible until a critical temperature range is reached and the nucleation frequency rises sharply, by many orders of magnitude. Let 1: be the smallest nucleation frequency per unit volume which can be detected in the nucleation experiment; then the observed nucleation temperature Tvr is that temperature at which I, reaches rt /f = .4 exp( - Ka-‘R&AG,~~rk~,r).

(11)

Consider next Southin and Chadwick’s experiment in which liquid alloy droplets are cooled within a solid matrix. As the temperature falls, the liquid composition cL varies and is given by the extrapolated liquidus in Fig. 1: the composition c,, of the equilibrium solid which wants to form also varies and is given by the /I nhase solvus line as shown in Fig. I. Solidification can take place in two ways: firstly. by homogeneous nucleation within the bulk of the liquid droplets. and secondly. by heterogeneous nucleation in contact with the r matrix. The homogeneous nucleation frequency per unit volume of liquid fst at any temperature T is given by [I] IsI = A exp( - Ka3Ri,‘AG$T),

(12)

where it has been assumed that the frequency factor. shape factor, solid-liquid surface energy, and molar volumes are independent of composition. i.e. the same as those in homogeneous nucleation experiments on pure B. For heterogeneous nucleation of a solid cap on a flat catalytic surface. the nucleation frequency per unit volume of liquid Is, at any temperature 7 is given by [II Is2 = B exp( - KdRff(B)/AG;kT-),

(13)

wherei(0) = $(Z - 3 cos 0 + cos3 0) and 0 is the contact angle at the solid-liquid-catalyst triple junction. Figure 3(a) shows the case of a spherical solid cap on a flat catalytic surface. Figure 3(b) shows the expected situation in Southin and Chadwick’s experiments with catalytic z surfaces which are not flat; the effect of curved z surfaces will be considered towards the end of the next section. Equation (13) again assumes that the shape factor, solid-liquid surface energy. and molar volumes are independent of composition and equal to those in homogeneous nucleation. The frequency factor B is given by [I] (n,ki? /II expt -AG,/kT) where ns is the number of atoms at the liquid-catalyst surface per unit volume of liquid. The value of )I~ clearly depends upon the size of liquid droplets and is approximately equal to .Y’~‘(N&)2’ 3 where x is the number of liquid drop lets per unit volume of liquid. For an undivided Liquid

NUCLEATION IN SOLIDIFYING ALLOYS volume 1.0me3. similar Southin .Y= _. 74 1035s-lm-3

of l.Om’. the droplet diameter is I.lm. x = and the frequency factor B 2 1030s-’ me3 to the value given by Turnbull [ 11: for and Chadwick’s typical droplet size of 20 pm. x 10’” m-‘. and the frequency factor B I

In Southin and Chadwick’s experimental arrangement, let uL be the volume fraction of liquid droplets (typically -0.1). and let I,+ be the smallest nucleation frequency with can be detected per unit volume of specimerr. Then homogeneous nucleation should become detectable at a temperature T&r given by I,* = u,A exp( - Ka’R&‘AG<.s,kT,s,).

(14)

where Gvs, is the driving force for solidification at TVs,. Similarly, heterogeneous nucleation should become detectable at a temperature TVs2given by f; = u,E expt - Ka3n~f’(B)/AG.~,,kT,,,).

(I 5)

where GNS2 is the driving force for solidification at T,vs2. Whether the observed nucleation is homogeneous or heterogeneous is determined by which of the two frequencies ls, or Is2 first exceeds Lf as the temperature is reduced. Thus if &s, > Tysz nucleation is homogeneous and the observed nucleation temperature TVs = TVs,; alternatively, if r,,, c TVs2 nucleation is heterogeneous at an observed temperature r,,, = &s2. Another way of considering this is to use equations (12) and (13) with T = TV, to obtain expressions for the homogeneous nucleation frequency IVs, and heterogeneous nucleation frequency Ivs2 at the observed nucleation point. Whether nucleation is homogeneous or heterogeneous is determined by whether Iss, is greater than or less than Iys2. If Iss, > iVS2. nucleation IS homogeneous wtth INsi = 1:: if INSi < INSZ+ nucleation is heterogeneous with INsr = 15. Equations (12) and ( 13) can be used to write a condition that the homogeneous nucleation frequency at any temperature is greater than the heterogeneous nucleation frequency at the same temperature; Is, > Is2 when AC: > Ka’R:: I - /‘(0):,~kTln(AiB).

(16)

When .4= B this inequality cannot be fulfilled and homogeneous nucleation is never preferred; this cor-

a)

-iI2dL oc

54

QOCL

Fig. 3. Two possible models for spherical cap heterogeneous nucleation: (a) flat catalytic surface; and (b) curved catalytic surface.

CANTOR

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41

strains the heterogeneous nucleus to be other than a spherical cap. However. this is unlikely to have a large effect. In addition, equation (17) assumes that the solid-liquid surface energy and molar volume are independent of composition. The factor 2 was obtained as follows. Turnbull has given IF 2 IO- ’ s- ’ per 50 pm particle. i.e. - 10” s-’ rne3. He measured nucleation temperatures for a just-detectable nucleation frequency. whereas Southin and Chadwick’s nucleation temperatures are for the marirnu~~ frequency. so 12 may be one or two orders of magnitude greater than If. It is worth noting in passing that both Turnbull and Southin and Chadwick observed that the temperature difference between just-detectable and maximum nucleation frequencies was no more than a few degrees. I* = 1013 s-I m-3 1: = ]O’? s-i mm3, With ,,j = ]040s-’ m-3, ,j = ]035s-’ ms-3 and rL = 0.1. the ratio In(l5/c,B)/ln(l$/A) = 3. There is some uncertainty in the values of I$. 13, A and B; however. ,f(f?) is very insensitive to these parameters and any reasonable values lead to a value for ln(lz:cLB) ln(l*,,A) in the range 0.70-0.85. Using the driving forces derived in Section 3. contact angles have been calculated from equation (17) 5. RESULTS for each of alloys in Tables 2-5. For the alloys in Tables 4 and 5. each of the three driving forces AGbs, Southin and Chadwick’s data can be used to calcuAC:, and AGis were used to calculate the contact late contact angles for heterogeneous nucleation. angle. producing three values of contact angle. 0:. 0”’ Assume initially that none of their alloys was underand BE. cooled sufficiently IO fulfi] the inequality in equation For many of the alloys, the driving forces calcut 16) and promote homogeneous nucleation within the lated in Section 3 are expected to be good approximations; for those marked with an asterisk. however. bulk of the liquid droplets. The validity of this different methods of estimation lead to rather differassumption will be investigated later. Then Southin ent results. This uncertainty in the driving force is and Chadwick’s experimental nucleation temperatures are given by equation (15). with AGSS2 = AG,Vs not a serious problem when calculating contact angles. because in equation (17) the contact angle is and TVs2= TVs. In principle. f(0) and hence 0 can rather insensitive to the value of driving force For be obtained by substituting for the other parameters many asterisked alloys therefore. the contact angle. in equation (15). However, the most suitable values is virtually independent of the method of estimating for solid-liquid surface energies are those from homodriving force. The three exceptions to this. Bi-Sn. geneous nucleation experiments. In fact it is not Cd-Pb and Sb-Pb, are each marked with two asternecessary to know the solid-liquid surface energies isks in Table 5; for these three alloys, the present and the contact angle can be determined directly from methods of determining the contact angle are clearly a comparison of Southin and Chadwick’s and Turnnot reliable. bull’s experimental data as follows. In any particular Excluding Bi-Sn, Cd-Pb and Sb-Pb, the calculated A-B alloy. the nucleation temperature measured by contact angles can only be good approximations if Southin and Chadwick is given by equation (IS): for the assumptions of spherical homogeneous nucleation the corresponding pure B liquid. the nucleation temand spherical cap heterogeneous nucleation are valid. perature measured by Tumbull is given by equation In general. the anisotropy of solid-liquid surface (111.Dividing these two equations energy is small [23] and is therefore unlikely to affect the contact angles significantly. It might be argued that the correct model for heterogeneous nucleation in a liquid droplet should allow for curvature of the nucleation surface as shown in Fig. 3(b). When the Thus. i(6) and 0 can be obtained by a comparison standard method is used to estimate the work of nucof the results from two experiments-homogeneous and heterogeneous nucleation. Equation (17) is only leation for a spherical cap on a flat nucleating surface. valid with the assumption that the shape factor K Fig. 3(a), the result is [22] is the same in both experiments and this will not lV(flat) = Ka3Rff(0)/AGg. (18) be the case if. for instance. the nucleation surface con-

responds to the improbable situation when each liquid droplet contains only one atom. In any real experiment A > B and there are two possibilities. If the contact angle is 180.. there is no interaction between the catalyst and nucleating solid: as expected I(()) = 1 and equation (16) predicts that homogeneous nucleation is faster at all temperatures. For any other value of 0. .f(N) < 1 and homogeneous nucleation is faster only when the driving force is sufficiently large. i.e. only at a sufficiently high undercooling. At low undercoolings heterogeneous nucleation is favoured because of the catalytic surfaces: at high undercoolings homogeneous nucleation becomes dominant because of the greater number of potential homogeneous nucleation sites, i.e. because A > B. As the sample temperature falls. if the heterogeneous nucleation frequency reaches IJ before equation (16) is fulfilled, the observed nucleation will be heterogeneous. Otherwise. the observed nucleation will be homogeneous. With typical values of .4 1 1040s-’ rnm3, B 1 103’s-’ m-“. K = 1671 3. K = 1.4 x 10-2’JK-‘. 0 : O.ZJm-‘. 7 z 10°K. Q, z IO-sm3. equation (16) becomes AGs ; 1011 -,f(O):’ ’ kJmol-‘.

42

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Fig. 4. Correction factor /‘(0.& due to curvature of catalytic nucleating surface. When the curvature of the nucleating surface is small compared to the curvature of the spherical cap, the work of nucleation can be obtained by a similar method to first order in 4 (see Fig. 3b) W(curved) = Ku3Q~(f‘(f7) +f’(& $))/AGi

(19)

/vt?, 4) 1: &(8 sin 0 cos2 0 - 3 sin3 0).

(20)

As shown in Fig. 4, 3412 > f’(e, 4) > -3412 so that f’(0.4) can be neglected compared to f(0) when t$ is small. For alloys such as Cu-Pb, NiA13-Al and Ag-Pb, the contact angle t3 is small and the radius of curvature of the spherical cap is large (because the driving force is small). Thusf’@, 4) may be significant, and will tend to aid the nucleation process because the nucleus in Fig 3(b) has a greater volume to solidliquid surface area than the nucleus in Fig. 3(a)[22]. For all other alloys, f“(& I$) can be neglected. To calculate contact angles it was assumed initially that all of Southin and Chadwick’s alloys were nucleated heterogeneously by the CLmatrix. In other words it was assumed that none of the alloys were sufficiently undercooled for homogeneous nucleation within the bulk of the liquid droplets to become the faster process. Two related methods were used to in* Both AGc and l,rsl are very sensitive to variations in solid-liquid surface energy and driving force for solidificain Tables 2-5 were obtained by using slightly digerent values for the frequency factor A

tion. The contact angles

and the latent heats of melting LB compared with the values used by Tumbull [l]. (In the present work. latent heats were taken from Ref. 27.) Therefore, it was also necessary for consistency to use slightly different values of solid-liquid surface energy from those quoted by Tumbull [I]. These modified solid-liquid surface energies are included in Tables 2-5. For Ge, the present solid-liquid surface energy is 231 mJ me2 whereas that given by Tumbull is 181mJ m-‘; in all other cases, the difference is only a few per cent. Using Turnbull’s values of latent heats, frequency factor and solid-liquid surface energies would have almost no effect on the calculated contact angles, but would have a large effect on AGe and Iv,,. For all the alloys. equations (10) and (13) together with the contact angles and modified surface energies in Tables 2-5 give the homogeneous nucleation frequency in Turnbull’s experiments as IO“ s-’ m-‘, and the heterogeneous nucleation frequency in Southin and Chadwick’s experiments as 10” s _ ’ m-j. Thus the analysis is self-consistent because these were the values assumed originally when calculating the contact angles.

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vestigate this assumption. Firstly, equation (16) was used to calculate the critical driving force AGc at which the liquid droplets should begin to be nucleated homogeneously. Secondly, equation (12) was used to calculate the homogeneous nucleation frequency I,Vsr within the bulk of the liquid droplets at the nucleation temperature TVs observed by Southin and Chadwick. For these calculations, A and 5 were taken as 104’ and lO35s-’ me3 respectively and j’(0) was determined from the contact angles in Tables 2-5. For alloys in Tables 4 and 5, AGc was calculated only from BE,or, when this was not possible, from 0”. The results are included in Tables 2-j.* The alloys fall into three groups as indicated in the last column of each of Tables 2-5. Alloys in group A have f‘(0) c I and also AGgs c AGc. confirming that the liquid droplets are nucleated heterogeneously by the I matrix. For most of these alloys f,vvs,is very small, and for all of them it is less than lOI3 s-’ rne3, the expected detection frequency in Southin and Chadwick’s experiments. Alloys in group B have j‘(0) > 1 so that no value of contact angle will satisfy equation ( 17). This,means that the contact angle is effectively 180” and the liquid droplets are nucleated homogeneously because the x matrix is not wetted by the nucleating solid (see Fig. 3a). These alloys reach a high undercooling so that AGNS > AGc. However, the homogeneous nucleation frequencies I,vs, are in the range I020_1025 s- 1me3. considerably higher than any reasonable value for the detection frequency in Southin and Chadwick’s experiments (the assumed value of lOI s-’ mV3 is unlikely to be incorrect by more than say two or three orders of magnitude). For two of the alloys in group B, Bi-Sn and Cd-Pb, it has already been pointed out that the present data are unreliable. The high values of f,Ysl for the other group B alloys are discussed further in the next section. Alloys in group C havef(f3) < I like group .4, suggesting that the r matrix is wetted by the nucleating solid. However. these alloys have AGNs > AGc and I .vs1 in the range 10’3-10’5 s-’ m-3 so the assumption of heterogeneous nucleation is inconsistent and the calculated contact angles are not necessarily correct. Because the values of INS, are quite close to the assumed value of lOI s-’ m-‘, it is reasonable to conclude that the liquid droplets are nucleated homogeneously at the observed temperature TVs.The calculated contact angle is then only a lower limit, because a smaller contact angle would give too large a value for the heterogeneous nucleation frequency I #r2, If the true contact angle is in fact less than 180”. then the liquid droplets nucleate homogeneously, in spite of the solid wetting the z matrix. because of the greater number of homogeneous nucleation sites. Alternatively, if the true contact angle is equal to 180’. there is no wetting, and the liquid droplets nucleate homogeneously in the same way as alloys in group B.

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Except for Sb-Pb. Cd-Pb and Bi-Sn. Southin and Chadwick’s experiments together with the present analysis can be regarded as an indirect method for measuring contact angles. In alloys with small contact angles, the I phase is an effective catalyst for nucleation of B; in alloys with large contact angles, the LY phase is not an effective catalyst. With this interpretation, the present analysis sometimes leads to results which differ from those obtained by Southin and Chadwick using the undercooling parameter L,,%.,. For instance in Cu-Ag. U,,., > U,,,,,,, which suggests that Cu has no effect on the nucleation of Ag: however. the present analysis shows that Ag is nucleated catalytically by Cu. though the effect is not strong. Similarly, L;,,, < c’,, ,,,, for Pb-Ag and Sn-Bi. but the present analysis shows that Pb and Sn have no effect on the nucleation of Ag and Bi.

Thus far in the present analysis Southin and Chadwick’s data have been compared directly to Turnbull’s because that was the procedure adopted by Southin and Chadwick themselves. This made it possible to consider the relative ‘merits of the present analysis and Southin and Chadwick’s use of undercooling parameters. In fact. both methods rely on the implicit assumption that Turnbull observed true homogeneous nuckation, and this has recently been questioned [24-261. Attempts to understand why alloys with 0 = 180’ (i.e. in group B) have such high nucleation frequencies also tend to throw doubt on Turnbull’s data. For those alloys which nucleated homogeneously with 0 = 180’. the nucleation frequencies are consistently higher than the expected value by about ten orders of magnitude. This can only be explained if one of the parameters in equation (I 2) has been evaluated incorrectly and the most likely possibility is that the solid-liquid surface energies are all too low. Because the surface energy is cubed within the exponential in equation (12). only a very modest increase will reduce Zysl from the range 1021-1026 s-l me3 to the expected value of 5 1013s-’ me3. There are two possible explanations for such discrepancies in the surface energies. Firstly. the solid-liquid surface energies might be functions of composition so that Turnbull’s values for B are inappropriate for liquid droplets bf composition c,,. However. it seems unlikely that this would give such a consistent underestimate for the three alloys in group I3 (excluding Bi-Sn and Cd-Pb). In addition. this explanation seems particularly unsuitable for Zn-Bi which has cli = cL 1 1 so that the liquid droplets are virtually pure Bi. The second.-and more likely, explanation is that Turnbull’s experiments might not have shown true homogeneous nucleation. If Turnbull’s pure B had been nucleated heterogeneously by some unknown impurity, then his measured undercoolings would be less than those for true homogeneous nucleation. and his calculated solid-liquid surface energies would be

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43

consistently too small. Most of the metals studied by Turnbull had a consistent value of -0.2 for the undercooling parameter Whom(see Table II and this was taken to indicate that nucleation was indeed homogeneous. However. a complete confirmation of homogeneous nucleation has been performed for only two pure metals. Hg [I] and Ga [28]. and in both cases c‘,,,,,,,= 0.33. Moreover. recent workers have obtained higher undercoolings than Turnbull for pure Bi. Sn [24.25] and Pb [26]. giving undercooling parameters of 0.32. 0.34 and 0.40 respectively. Perepezko cr al. [24.25] used a technique similar to Turnbull’s with different emulsifying agents to obtain the dispersion of liquid droplets. and obtained undercooling parameters of 0.32 and 0.34 in pure Bi and Sn. They suggest that Turnbull‘s emulsifying agent could not have been completely inert so that he observed catalysed nucleation with much smaller undercooling parameters of 0.166 and 0.208, respectively. Using a slightly different technique. Stowell[26] has reported very large undercoolings in submicron liquid droplets of pure Pb. which again suggests that Turnbull’s lead droplets were nucleated heterogeneously. Stowell obtained an undercooling parameter of 0.40 instead of Turnbull’s value of 0.133. Thus it seems reasonable that the high values of nucleation frequency IXsI for alloys in group B are caused by too low a value of solid-liquid surface energy deduced from the erroneous assumption of homogeneous nucleation in Turnbull’s experiments. For all Southin and Chadwick’s alloys of the types A-Bi. A-Sn and A-Pb. the present calculations have been repeated using the homogeneous nucleation data of Perepezko cr al. [24.25] and Stowell[26]. This involved a re-calculation of the driving force for homogeneous nucleation AGVT to obtain a new value of contact angle 0 from equation (17). In Stowell’s experiments. the just-detectable nucleation frequency was IO-‘s- ’ for a 500 .A diameter particle so I$ was taken as 10” s-l m- 3: this gave a new value of 0.5 for the logarithmic ratio In(l4!c,B),ln(l*, A). Otherwise. the calculations were the same as those using Turnbull’s data. In addition, equation (16) was used to recalculate AG,. i.e. the driving force at which Southin and Chadwick’s liquid droplets begin to nucleate homogeneously: for calculating AGc. surface energies were taken from Perepezko et al. [24.X] and Stowell [26] instead of Turnbull. The results are shown in Table 6 with previous values in brackets. The higher undercoolings in these more recent experiments lead to lower values of contact angles. Thus for several alloys which Southin and Chadwick believed to have nucleated homogeneously. the later experiments indicate that this is not so. For instance. the original contact angle for Zn-Bi was 180’ suggesting no catalytic effect of Zn on the nucleation of Bi: using the data of Perepetko er al. for Bi. the contact angle is 86’ so that Zn is quite effective in catalysing the solidification of Bi. Similarly, the original contact angle for Sn-Pb was also 180’: using Stoweli’s data

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Table 6. Nucleatron data for alloys studied in Southin and Chadwick’s experiments, of the type A-Bi. A-Sn. and A-Pb. The data have been recalculated using recent results [24-271 on nucleation of pure Bi. Sn and Pb. The values in brackets are the previous data using Turnbull’s results. as quoted in Tables 2-5. For explanation of column headings see text or caption to Table 2

System A-B AI-Pb AI-Sn Cu-Pb Zn-Bi Zn-Pb Zn-Sn

AGYT (kJmol-‘1 1.910 2.40 I I.910 3.506 1.910 2.401

e Meg)

Nucleation type

~,L-~,s (mJm_‘)

68.8 70.6 68.8 79.3 68.5 70.6

.4(.-t) .4(A) .4(A) A(B) At.41 ,4(C)

59.6 24. I 67.3 -2.8 57.7 9.8

68.8 79.3

.4(A) “l(C)

67.X

.4(B) A(C) .4(B)

~vs, (s-l m-‘)

(mJ:-r)

12(19) 92( 180) 3Y6l) 82(126)

4.24 4.36 4.16 5.59 4.27 4.06

lo-17” 10-‘j IO-LWL’ IO’ to-‘186 10-s

3005)

mm

AGc (kJmol-r)

.48-Pb

I.910

lo(l4)

4.25

I()-‘40057

Cd-Bi

3.506

7q 130)

7.19

10-26

Bi-Sn** Sn-Bi Cd-Pb**

2.401 3.506 I.910

lOQl80) 72(121) 6t-J180)

3.28 7.13 4.27

IO9 lo-” lo-99

70.6 79.3 68.8

Sb-Pb**

I.910

51(119)

4.35

lo-102

68.8

Pb-Sn*

2.401

49(61)

4.81

70.6

A(C) AtAb

13.3 46.3

Sn-Pb+

I.910

59f180)

4.51

68.8

.4(B)

35.4

for Pb the contact angle is 59” and Pb is nucleated quite effectively by Sn. For a number of other alloys. the contact angle is reduced quite sharply so that the catalytic effect is considerably stronger than predicted by Turnbull’s data. Examples of this are Zn-Sn. Cd-Bi, and Sn-Bi for each of which the new contact angle is _ 75” compared to a previous value of _ 125”. When considered in general. the modified contact angles in Table 6 have a considerable effect on the grouping of alloys described in the last section. All of the alloys except two, Pb-Ag and Al-Ge. are in group .4. i.e. the liquid droplets are nucleated heterogeneously with f(6) < 1. 0 c 0 < 180-, AGNS< AGc, and IHsl < 10’3s-’ m-‘. The more recent homogeneous undercoolings in Bi, Sn and Pb must be regarded as more reliable than Turnbull’s data so the contact angles in Table 6 are good approximations to the true values and good indicators of catalytic efficiency. For the other alloys, there is no alternative to using Turnbull’s data and the contact angles in Tables 2-5 are the best available estimates. However. the increased undercoolings in Bi. Sn and Pb throw some doubt on Turnbull’s data for other pure metals; thus it would be valuable to have a redetermination of homogeneous undercoolings in pure elements such as Cu. Ag, Al, Ge and Sb. 5.3 Epitaxj

21.9 - 12.3 24.5 34.4

calculate values of the parameter ~,~--e,~. and the results are included in the tables. For a given j phase, i.e. a given value of 6. the contact angle is inversely proportional to the parameter b,L-u,s. Thus a good catalyst for nucleation of fl must have a surface energy in contact with liquid /I which is considerably greater than when in contact with solid /?. Conversely. there is no catalysis if the surface energy of x in contact with liquid /3 is considerably less than in contact with solid fi. Thus. Ag, Cu. Al, Zn and Sn have a decreasing catalytic effect on the nucleation of Pb. and the parameter u,~--u,~ decreases in the same order. Similarly, Al, Zn and Bi have a decreasing catalytic effect on nucleation of Sn: and NiAI,, ;(AIAg). CuAI,. Si and Ge have a decreasing effect on nucleation of Al. These results are again reflected in the values of uzL-rrzs. It can be argued that when there is an epitaxial orientation relationship between the catalyst and solid nucleus, uzs is small. bzL-uzs large, and z an effective catalyst [Z-4]. As noted previously [lo]. this argument is not supported by the experimental evidence: for instance, Cu and Ag are effective catalysts for nucleation of Pb, although the lattice misfits for Cu-Pb and Ag-Pb are large so that rrIs is expected to be large. The contact angles found in the present work have however been found to correlate with the ratio of the melting points. 7$,,,(TMB,for the fully metallic systems [29]. to

By balancing forces at the solid-liquid-catalyst triple point in Fig. 3(a). the contact angle can be given by the well-known

expression: 0 cos 0 = a,‘ - u,s

6. CONCLUSIONS (21)

where 0,‘. uIs are surface energies between the catalyst z and liquid and solid /?. respectively. The contact angles in Tables 2-6 have been used in equation (21)

By using solution models for both liquid and solid phases, it is possible in many cases to obtain reasonably accurate values for the driving force for solidification of alloys. The driving forces can be used to

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analyse Southin and Chadwick’s heterogeneous nucleation experiments. This approach is more fundamental than a simple comparison of maximum undertoolings which can sometimes lead to incorrect conelusions. In addition. it becomes possible to calculate contact angles and therefore quantify the catalytic process of heterogeneous nucleation. This is clearly a necessary first step towards understanding heterogeneous nucleation. The best values of contact angle are obtained in alloys for which the catalytic z phase is insoluble in both the liquid and nucleating solid. A comparison of Southin and Chadwick’s results with Turnbull’s data for nucleation in pure metals leads to mconsistencies which suggest that Turnbull’s data are inaccurate. This supports the evidence from more recent experiments on nucleation in pure metals. After making allowance for these more recent results, all of Southin and Chadwick’s alloys with two exceptions are shown to have nucleated heterogeneousiy with contact angles less than 180’.

NOMENCLATURE A. B pre-exponential frequency factors for the homogeneous and heterogeneous nucleation frequencies z. b phases in eutectic or monotectic alloy c0 bulk alloy composition c,. c,, equilibrium composttions of z and /I phases cL metastable equilibrium composition of undercooled liquid alloy /(tI) = $(2-3 cos e + co? 8) function of contact angle appearing in equation for heterogeneous nucleation frequency .f’((I. 0) 1 j(8 sin fj cos2 H - 3 sin3 (i) function of contact angle and curvature of catalyst surface appearing in equation for heterogeneous nucleation frequency AC, activation free energy for transporting an atom across the solid-liauid interface GLA. Gsr. GLB.Gs, molal free energies of pure liquid and solid A and B G,,. I&. G,,. Gsa partial molal free energies of .4 and B in the liquid and solid allov phases AC,.. _.. AC,,. _.. AG,,. A&n relative parttal molal free energies of .4-&d B-;n the liquid and solid allo! phases AG,‘!..,’ AZ;;‘);‘“’AG:$ . AGF,j.” ideal mrxing values of relaLA . tive partial molal free energies AGr driving force for solidification per mole of solid. at any temperature in a pure liquid AG,r value of AGr at the observed nucleation temperature AC;, driving force for solidification per mole of solid. at any temperature in a liquid alloy AG,s value of AG, at the observed nucleation temperature AGVs,. AGVs2 values or AGs which would be required to produce observable homogeneous and heterogeneous nucleation in hquid alloy droplets AGPs. AG;s. AC:, values of AGVs calculated from an activity coefficient. a regular solution model. and by extrapolatton AGc critical driving force above whtch homogeneous nucleation is faster than heterogeneous nucleation in liquid alloy droplets It Plank‘s constant ALLA. AHsA. AR,,. AH,, relative partial molal enthalpies of A and B in the hquid and solid alloy phases Ir homogeneous nucleation frequency per unit volume of liquid at any temperature in a pure hquid

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45

If just-detectable nucleation frequency per unit volume of specimen in a nucleation experiment on a pure liquid , Is, homogeneous and heterogeneous nucleation fresl’ quencies per unit volume of liquid at any temperature in a liquid allo) I: just-detectable nucleation frequency per unit volume of specimen in a nucleation experiment on liquid allo! droplets IKsl. l,52 values of I,, and Is, at the observ,ed nucleation temperature I, Boltzmann’s constant K shape factor. equal to 16n 3 for spherical nucle) 4. & latent heat of melting per mole of pure A and B )ly number of atoms per unit volume of liquid rrs number of atoms at the liquid-catalyst surface per unit volume of liquid .VOAvogadro’s number R gas constant AsL,,. A&“. As,. A.& relattve parttal molal entropies of A and B in the liquid and solid alloy phases 7 temperature TAannealing temperature T, liquidus temperature of undercooled liquid alloy droplets TMA. TM8 meltrng points of pure rl and B T,, observed nucleation temperature in pure liqutd TV, observed nucleation temnerature in liauid allo\ 7;,,,. TVs2 temperatures required to produce observable homogeneous and heterogeneous nucleation In liquid alloy droplets ATL = r, - T undercooling in liquid alloy droplets = (T,, - T,,) TMBhomogeneous undercooling par~h<,,l> ameter for solidification of pure B undercooling parC ,>,, = (T, - T,s) ‘T, heterogeneous ameter for solidification of liquid alloy droplets rr volume fraction of liquid alloy droplets ‘LI’ interaction parameter in regular solution model W(flat) work of nucleation on flat catalytic surface W(curved) work of nucleation on curved catalytic surface .Xnumber of liquid droplets per unit volume of liquid ;’ activity coefficient Rs. R, solid and liquid molar volumes ff contact angle 0 . 0”. BEcontact angles calculated from activity coefficient. regular solution model. and by extrapolation 4 angular measure of curvature of catalytic surface CTsolid-liqurd surface energy O,L. 0,s surface energies of solid and hquid m contact with 2 matnx. AcknoaIrdgemenrs-We would like to thank Professor R. W. Cahn. Dr M. G. Scott. and Dr P. Ramachandrarao. for valuable discusstons and critical reading of the manuscript. We would also like to acknowledge Professor G. A. Chadwick and Dr R. T. Southin both for helpful suggesttons and for making their results available m advance of publication.

REFERENCES 1. D. Turnbull. J. uppl. Phjs. 21, 1022 (1950). 2. D. Tumbull and R. E. Cech. J. appl. Phys. 21. 804 ( 1950). 3. M. D. Eborall. J. Inst. Merals 76. 295 (19491. 4. A. Cibula. J. Inst. Mnals 76, 321 (1949) 5. J. A. Revnolds and C. R. Tottle. J. Inst. Metals 80. 93 (1951). 6. F. A. Crossley and L. F. Mondolfo. J. Merols, .2’.Y 3. 1143 (1951). 7. M. E. Glicksman and W. J. Childs. Acra jnetull 10. 925 f1962J.

46

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8. J. H. Holloman A.I.M.E.

191.

803

DOHERTY:

and D. Turnbull.

HETEROGENEOUS Trotis. mrrull. Sec.

(1951).

9. F. J. Bradshaw, M. E. Gasper and S. Pearson. J. Inst.

Merals 87, I5 (1958). IO. B. E. Sundquist and L. F. Moodolfo. Trans. mernil. Sot. A.I.M.E. 221. 157 (1961). I I. P. B. Crossfey. A. W. Douglas and L. F. Mohdolfo, TItr Soiiditi~ufion of’ Mrruk Vol. I IO. p. IO. I.S.I. (1967). 12. B. E. Sundquist. Acru metall. 11, 630. 1002 (lQ63). 13. K. A. Jackson, lnd. Enuna ._ . Chem. 57. 29 (19651. 14. R. T. Southin and G. A. Chadwick. A& metal!. 26, 223 (1978). IS. C. C. Wang and C. S. Smith. Trans. tnerall. Sot. .4.I.IW.E. lB& I36 (1950). 16. M. E. Fine. fntrodwtion to Phase ~ru/ts~o~~?~arjo~ts in Condrt& Systems. Macmillan. New Yo;k (1964). 17. R. A. Swalin, Thermodynamics of So/ids. Wiley, New York (1966). 18. R. Hultgren, R. L. Orr. P. D. Anderson and K. K. Kelley, Selected Vaiues oj’ Thermodynntnic Properries of Me&s and Alloys. Wiley, New York (1963). 19. M. Hansen and K. Andcrko. Consfirurion of Binary Alloys. McGraw-Hill, New York (1958).

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20. R. P. Elliott, Consrirurion of Binary &lo,w-lsr Suppirmm. McGraw-Hill. New York (1965). 21. F. A. Shunk. Consriwriorl of Binur! .Allo,w-lr~d Supplement. McGraw-Hill. New York (1969). 22. B. Chalmers. Principles of Soiidijcuriorl. Wiley, New York ( 1964). 23. W. A. Miller and G. 4. Chadwick. Tier So/i~~j~~utio~~ of :Wrtcds. Vol. I IO. p. 49. I.S.I. (1967). 24. J. H. Perepezko, D. H. Rasmussen. 1. E. Anderson and C. R. Loper Jr.. Shefleld Iut. Con/: on Solidijicuriott (1976). In press. 25. 0. H. Rasmussen. J. H. Perepezko and C. R. Loper. Jr.. Second international Conference on Rapidly Quenched Metals (Edited by N. J. Grant and 8. C. Giessen), Vol. I. p. 5 I. MIT Press. Cambridge ( 1976). 26. M. J. Stowell. Phil. ‘way. 22 I (1970). 27. H&book of Chemistry und Physics. 53rd Edn (Edited by R. C’. Weast). Chemical Rubber Company Press. Cleveland (1972). 28. Y. Miyazawa and G. M. Pound. J. Cryst. Grwrh 23, 45 (1974). 29.

R. D. Doherty.

Scripru

nrrratl.

12, 59 I (1978).