Heterogeneous nucleation in entrained Sn droplets

HETEROGENEOUS

NUCLEATION Sn DROPLETS

IN ENTRAINED

P. G. BOSWELL Dental

School,

University

of Queensland,

Turbot

Street, Brisbane

4000, Australia

G. A. CHADWICK Department

of Mechanical

Engineering,

(Recrired

University of Southampton, England

Highfield,

Southampton

SO9 5NH,

27 Nooember 1978; in recised form 3 July 1979)

Abstract-Differential thermal analysis has been used in conjunction with entrained droplet specimens to measure the nucleation rates of uncontaminated Sn-rich liquids in contact with solid Bi, Zn and Al. The Bi(Sn) system gave a linear nucleation rate plot and an analysis of the data in terms of the classical theory of heterogeneous nucleation at variable composition, yielded a pre-exponential frequency factor, J,“, that was in good agreement with the predicted value for pure Sn. On the other hand, the nucleation rates plots for Zn(Sn) and AI were irregular and could only be analysed to give approximate estimates of J,” f(>r these two systems. The origin of the irregularity was discussed in terms of the surface structure of entrained droplets. In the case of Bi(Sn), the nucleation rate data yielded a catalyst surface energy difference of -SO + 26 mJm_’ which agreed with a theoretical estimate for pure Sn nucleating in contact with pure Bi. R&umb-Nous avons mesurt: la vitesse de germination de liquides non contaminCs riches en Sn au contact de Bi, Zn et Al solides, en combinant l’analyse thermique diff&entielle et l’entrainement de gouttelettes. Dans le cas du systime Bi(Sn), le diagramme des vitesses de germination est IinCaire; l’analyse des rt‘sultats B l’aide de la thtorie classique de la germination h&tCrog&e B composition variable conduit a une valeur du facteur prCexponentiel J: en bon accord avec la valeur prkvue pour le Sn pur. Par contre, les vitesses de germination variaient irrCgulitrement dans les systkmes Zn(Sn) et AI( de sorte que I’on n’a pu obtenir que des valeurs grossitrement approchCes de Ji dans ces deux systemes. Nous avons discutk ces irrtgularitts a partir de la structure superficielle des gouttelettes entrainees. Dans le cas de Bi(Sn), la vitesse de germination conduisait B une diffbrence d‘Cnergie superficielle catalytique de -50 f 26 mJm_‘, en bon accord avec une estimation thkorique dans le cas de la germination du Sn pur au contact de Bi pur. Zusammenfassung -Die differentielle thermische Analyse wurde in Verbindung mit mitgerissenen TrGpfchenproben benutzt, urn die Keimbildungsraten nichtkontaminierter Sn-reicher Fliissigkeiten, die im Kontakt mit festem Bi, Zn und Al stehen, zu messen. Das System Bi(Sn) ergab ein lineares Keimbildungsratendiagramm; die Analyse der Daten mit der klassischen Theorie der heterogenen Keimbildung bei variabler Zusammensetzung ergab einen prgexponentiellen Frequenzfaktor Ji, der mit dem vorausgesagten Wert fiir reines Sn gut iibereinstimmen. Andererseits waren die Keimbildungsratendiagramme fiir Zn(Sn) und Al(Sn) unregelmiil3ig und konnten nur fiir NIherungswerte von J; ausgewertet werden. Der Ursprung der UnregelmPBigkeit wird unter Beriicksichtigung der OberflHchenstruktur der mitgerissenen T&pfchen diskutiert. Be; Bi(Sn) ergaben die Daten der Keimbildungsraten eine Differenz der Katalysator-Oberflichenenergie von -50 k 26mJ*m-‘. Dieser Wert stimmt mit der theoretischen Abschgtzung fiir die Keimbildung von reinem Sn im Kontakt mit reinem Bi iiberein.

INTRODUCTION has been used by Wang and Smith [l] and by Chadwick and coworkers [2,3] to measure the nucleation temperatures of clean liquids in contact with uncontaminated solids. The method involves equilibrating entrained droplets at a temperature that is just above the equilibrium solidification temperature (the eutectic temperature. TE. in the case of binary eutectic alloys, as illustrated in Fig. I). The droplet specimen is then slowly cooled and the compdsition- of the droplet The

entrained

t Note:

droplet?

technique

the term ‘droplet’ usually denotes a spherical liquid particle. In the present study the term also refers to a small, generally non-spherical, volumes of entrained liquid.

liquid follows the extrapolated liquidus. Nucleation of the solid phase occurs at a nucleation temperature TN, corresponding to an undercooling, AT, = T, - TN, below the equilibrium melting (liquidus) temperature, T,, of the nucleating phase. Both Wang and Smith [I] and Southin and Chadwick [2] observed large AT, values for some eutectic systems. These measured values can be compared with reported maximum undercoolings for nucleation in single component systems. The latter undercoolings have been determined using volumetric [4-61, calorimetric [3, 7791 and

microscopic

obtained

using

undercoolings

[lo, these

1 l]

methods.

three

techniques

Recent gave

results maximum

of 0.32, 0.34, 0.33, 0.36 and - 0.6 T, for

210

Fig.

BOSWELL

AND

1. Schematic

CHADWICK:

phase

diagram

entrained

droplets.

HETEROGENEOUS

for

nucleation

in

NUCLEATION

IN ENTRAINED

Sn DROPLETS

measured rates can be used to test the predictions of the classical theory of heterogeneous nucleation; and to investigate the catalytic potency of different nucleating substrates. The main advantage of the internal droplet configuration is that nucleation takes place in clean liquid in contact with solid surfaces having a known structure, composition and macroscopic shape. These features are generally not present in Vonnegut’s [ 131 classical suspended-drop technique used previously to measure heterogeneous nucleation rates. Southin and Chadwick [Z] observed large variations of the undercooling required for nucleation in entrained Sn droplets. In particular, the AT’, values for Bi(Sn), Zn(Sn) and Al(Sn) are 0.25, 0.23 and 0.18 T, reflecting significant differences in the relative nucleation potency of the various entraining solids. The present study deals exclusively with this series of Sn droplet alloys. EXPERIMENTAL

Bi [S], Sn [S], Ga [6], Hg [9] and Pb [lo] respectively. In only one instance did Southin and Chadwick’s [2] internal droplet measurements yield an undercooling of > 0.3 T,; this was for liquid Ge in contact with Al solid, designated Al(Ge), having AT’, = 0.33 T,. Of the remaining eutectic alloys investigated only three systems gave undercoolings of between 0.2 and 0.3 T,, these systems being Zn(Bi), Bi(Sn), and Zn(Sn) having AT, = 0.27, 0.25 and 0.23 T’,‘,respectively. Nucleation rates, and not merely nucleation temperatures, have been determined experimentally for Hg [4], Sn [S] and Ga [6] using dilatometry, for Sn using optical microscopy [ 1 l] and for a Sn-10 at.% Bi alloy using calorimetry [7]. The measurements for suspended Hg droplets coated with mercurous laurate and for suspended Ga droplets yielded pre-exponential frequency factors, Jz (expt), in the classical rate equation for homogeneous nucleation of 104s and 1O46 mm3 s-l respectively (the superscript ‘u’ implies volume, or homogeneous, nucleation while the superscript ‘a’ denotes heterogeneous nucleation on the droplet surface). These two values of .I: (expt) were 1O7’2 times larger than the theoretical value, J,” (theo), of 104’ mm3 s- ’ predicted by classical nucleation theory and interpretations for this discrepancy have been presented [6, 123. The nucleation rate measurements for oxide-coated Sn droplets [5] and for Hg droplets with mercurous acetate or mercurous sterate surface films [4] gave .I: (expt) factors of 1O34 and 1031 me2 s-r respectively, if it was assumed that the entire droplet surface acted as a nucleating substrate. These experimental values were approximately equal to the predicted .Ji (theo) values of 103*.’ and 1031 mm2 s-l respectively. In this communication we wish to describe measurements of heterogeneous nucleation rates obtained using differential thermal analysis (DTA) and specimens of the entrained droplet type. The

Procedures identifical to those described by Chadwick and co-workers [2,3,14] were used to prepare = 50 g ingots of the Bi(Sn), Zn(Sn) and AI eutectic alloys. Each ingot contained a large number of uniformly distributed Sn-rch droplets entrained in a solid matrix. Cylinders (5 mm dia. by 5 mm long) were spark-machined from the ingots and examined using a Rigaku-Denki DTA unit, with supplementary facilities for measuring the specimen temperature accurate to within f 0.2 K. A pan-type sample holder was employed together with a stagnant, inert (argon) atmosphere which prevented oxidation of the droplet specimens. A Bi, Zn or Al cylinder, having dimensions equal to those of the specimens, was used as a reference and the differential temperature between the reference and specimen was, in all cases, measured using a sensitivity of 5 mK, equivalent to 10 pv fullscale deflection. Several DTA thermograms were recorded for each alloy by repeatedly cooling the sample at 1.25 K min- ’ from a temperature that was 5 f 1 K above the eutectic temperature. After a reproducible droplet exotherm had been obtained, the specimens were cooled at 1.25 K min-’ to the start of the droplet peak and quenched into water. The microstructures of the mechanically polished and unetched droplets were examined, within 30min of quenching, by scanning electron microscopy. The composition profiles across individual droplets were then investigated using both energy dispersive and wavelength dispersive electronbeam microanalysis. Finally, the droplet size distributions were obtained from an optical metallographic study of lightly etched specimens. EXPERIMENTAL

RESULTS

1. DifSerential thermal analysis Figure 2 shows droplet solidification

typical DTA thermograms for in the Bi(Sn), Zn(Sn) and Al(Sn)

BOSWELL

AND

CHADWICK:

Fig. 2. DTA thermograms

NUCLEATION

HETEROGENEOUS

for solidification droplets.

in entrained

Sn

systems (see Ref. [23 for descriptions of complete thermal analysis spectra for these alloys). Also included in the figure are bar markers indicating 7;,, the nucleation temperatures (accurate to i 1 K) reported by Southin and Chadwick [2]. These authors took TN to be the temperature at the peak of the inverse cooling rate curve. In every system examined there was found to be excellent agreement between this temperature and the peak position of the DTA droplet exotherm. The DTA traces, however, are reasonably smooth, continuous and reproducible unlike the earlier inverse cooling rate curves which were oniy plotted over a limited number of points spanning the nucleation interval. The DTA thermogram for Al(Sn) also demonstrates that the droplet peak described by Southin and Chadwick could, in fact, be resolved into two peaks. This observation illustrates the superior resolution of the present technique.

It was found that quenched droplets in the three alloys under investigation usually developed fine-scale structures of the types shown in Fig. 3(a) for Bi(Sn) and in Fig. 3(b) for Zn(Sn). These structures may have arisen via solidification with solute segregation followed perhaps by spheroidisation and coarsening in both the liquid and solid states. Alternatively, composition-invariant (‘massive’) solidification may have been followed by solid-state decom~sition of a supersaturated solid solution. There was some evidence in favour of solidification with solute segregation in Bi(Sn) specimens. Specifically, it was observed that sections through a few of the larger Bi(Sn) drop-

IN ENTRAINED

Sn DROPLETS

211

lets located near the centres of DTA specimens occasionally revealed Sn dendrites (see Fig. 4). It is reasonable to argue that these dendrites formed during quenching from the start of the DTA droplet exotherm rather than during slow cooling to the quench temperature. This is because the majority of dendrites that may have formed during slow cooling would be expected to coarsen prior to quenching. On the other hand, those that formed during quenching would not coarsen to any great extent. The dendrite imaged in Fig. 4 does not show evidence for extensive coarsening therby indicating that the Bi(Sn) droplet solidified with solute segregation upon quenching the DTA specimen through the droplet exotherm. Furthermore, given that the Bi(Sn) droplet of Fig. 4 solidified with solute segregation we note that a decrease in the imposed cooling rate to the cooling rate (1.25 K min- ‘) used in the DTA measurements would not favour composition-invariant solidification. In other words, if a droplet solidified dendritically on quenching it, most certainly, would not solidify massively on slow cooling. Aside from this argument. it should be appreciated that the undercooled droplets were self-quenched. By this, we mean that the latent heat of solidification was largely dispersed into the entraining solid and not into the quenching medium. Hence, cooling the entire DTA sample at different rates has little effect on the cooling rate for an individual droplet. particuIarly if it is located near the centre of the sample. The solidi~cation mechanisms responsible for the development of the droplet struc-

Fig. 3. Scanning electron micrograph of microduplex droplet structures observed in as-quenched (a) Bi(Sn); (b) Zn(Sn) alloys. (Magn. x 4000).

212

BOSWELL

AND

CHADWICK:

HETEROGENEOUS

NUCLEATION

IN ENTRAINED

Sn DROPLETS

ture shown in Fig. 4(a) are therefore likely to be representative of those that take place during slow cooling through the nucleation range. Attention is also drawn to several important microstructural features displayed by the Bi(Sn) droplet imaged in Fig. 4(a). First, the droplet has a thin (< 1 pm) surface layer of Sn. This layer appeared to be a single crystal implying that the growth of the Sn dendrite into the bulk was accompanied by the growth of an interface dendrite. The latter grew very rapidly and evidently encircled the droplet surface before the bulk dendrite reached the droplet boundary lying opposite the point of nucleation. This solidification model is illustrated schematically in Fig. 4(b). Second, there was no evidence for the existence of an extensive ‘pre-dendritic’ or ‘massive’ [ 151 region formed as a result of composition-invariant solidification. This observation implies that any massive solidification, with a planar or slightly curved interface, was terminated within a growth distance of less than 1 pm. This would be the case if the heat sink provided by the solid matrix was unable to prevent rapid recalescence of the solid-liquid interface to a temperature close to T,. The T, temperature is the temperature at which solid and liquid phases of the same composition have the same free energy: it is also approximately equal to the maximum temperature for massive solidification. 3. Electron beam microanalysis bulk dendrita

Fig. 4. Dendrrtic structure observed in an as-quenched Bi(Sn) alloy: (a) scanning electron micrograph; (b) schematic representation of solidification mode; (c) microprobe traces along line indicated in (a). (Micrograph magn. x Swo).

Figure 4 shows microprobe traces of the BiLl and SnKl characteristic X-ray emissions obtained on passing a 20 nm dia. electron beam across the Bi(Sn) droplet imaged in Fig. 4(a). The count rates were measured using an energy dispersive system and an acceleration voltage of 50 kV. The two traces do not show any detectable segregation within the Sn droplet. Thus, it would appear that the fine-scale microstructure imaged in the SEM micrograph cannot be resolved by the microanalysis of bulk specimens. A wavelength dispersive microanalysis system was used to measure i?, the average droplet composition, for each of the three alloys. Microanalyses were carried out by comparing the ratios of the numbers of Sn and solvent (Bi, Zn and Al) counts obtained on positioning the electron probe at the centres of droplets, and within the solidified grain boundary liquid. Readings taken from the grain boundary material were used as standards because this material solidified at the eutectic temperature and had an average composition equal to CD the eutectic composition. Readings from different droplets were repeated until the highest observed ration of Sn to substrate counts, Ns,fN,, had been recorded several times. This ratio was then used to compute ? from the formula E/(1 - 2) = k*(Ns,INJ*(Fs,IF,),

(1)

where k* is a standardisation constant and Fs, and F, are the total correction factors for Sn and solvent atoms. The factors are products of separate factors

BOSWELL ANDCHADWICK:

HETEROGENEOUS NLJCLEATfON IN ENTRAINED Sn DROPLETS

213

due to absorption, fluorescence, back scattering and stopping power [16]. The standardisation constant effectively corrects for other factors and it is obtained using the formula k* = EWV - ~e)l/E(~~~;JN:)~(~S”/~:~l,(21 where the asterisks refer to grain boundary material. The ex~rimentally determined droplet compositions, ‘i; are detailed in Table 1. Not inconsiderable importance can be attached to these compositions because they locate extrapolated liquidi for the three alloys investigated. These extrapolations not only have appreciable theoretical and practical significance but they also permit the computation of AT, vafues for nucleation from ~ui~ibrium phase diagrams. These AT, values are given in Table 1 and they can be compared with the AT, values obtained by making linear extrapolations of equilibrium liquidi (as illustrated in Fig 1).

Optical metallography confirmed that the entrained Sn droplets in the quenched specimens displayed their equilibrium shapes. These shapes can be approximated to cubes, spheres and round-edged discs for the Bi(Sn), Al(Sn) and Zn(Sn) droplet systems [14j, Droplet size distributions were obtained in order to calculate S, the mean droplet surface area. In the case of AI( a frequency histogram for D, the droplet diameter, observed on random sections, was measured. This distribution was subsequently manipulated using Saltykov’s [ 171 ‘section diameter’ method to give the true particle size distribution plotted In Fig. 5(a). This latter distribution is bimodal with a small peak adjacent to a large peak having a sharp cut-off. It is likely that one of the peaks (probably the smaller)

Fig. 5. Droplet size distributions: [a) AIISn): (bl Bi(Sn); fcl znjsnr

r---J

I

corresponded to droplets lying on low angle (‘cell’) boundaries and that these droplets resisted coarsening. Furthermore, intersections of the droplet surfaces with the cell boundaries may have produced slightly more potent sites for heterogeneous nucleation. This would account for the two DTA droplet exotherms for AI shown in Fig. 2. Table t gives iI_ the mean surface diameter (as defined in the Table) and S, the mean surface area, calculated using the droplet size distribution. Two sets of values for each of these parameters are given. The first set (denoted by a single prime), is for the entire distribution whereas the

10.6 + 0.2 at.% Bi 0.8 f 0.4 at.% Zn

0.02 + 0.01 at.% Al

1.4 f 0.2

E

2.93 f 0.07 2.0 + 0.3

(x l$K3) Cube (side a) Round edged disc (asuect ratio A) (Dia. Dl Sphere (Dia. Dj

Shape a, = 8.97 [A = 23 fi. = 4.95 II; = 6.3811 tr; = 6.9011

Mean surface size5 (pm)

t [JJ, (expt) computed assuming entire droplet surface active. $ Corrected for specific heat changes. 9:Mean surface size, x = [Z(N& Yf]t, where (NLk is the observed number fraction of droplets having a size yi. )/ri; for entire size distribution: ti: for large peak of distribution. B As determined by Southin and Chadwick [2].

Notes

1031.2’1.5 1027+3

Bi(Sn) Zn(Sn)

)

(m-

CJ:l, @W)tS zs-1

System

10-9.89 10-9.83

10-9.85

10-9.31

(ms’)

415

361 389

( :;:K)

Table 1. Summary of nucleation rate, composition and surface measurements for entrained Sn droplets

90 f 1

124 & 2 114 f 2

AT,(K) (for E)

117 116

ALWK) (extrap. to TN)

BOSWELL

AND

CHADWICK:

HETEROGENEOUS

second (denoted by a double prime) is for the larger peak of the size distribution. In the case of Bi(Sn), having cube-shaped droplets, the areas, Ai, of four-sided sections were measured in order to generate a histogram describing the relative fractions, (N& of the cross-sectional areas of different groups. i. The appropriate Saltykov area analysis equation was then derived and used to calculate the true size distribution of the cube-shaped droplets. The true droplet size distribution, plotted in Fig. 5(b), is strongly peaked indicating the successful development of a Bi(Sn) droplet array having uniform droplet sizes. Table I gives values of U,, the mean surface size and S, the mean surface area, calculated using the true droplet size distribution. Finally. for Zn(Sn), having round-edged droplets, measurements were made of, D, the diameters of circular sections (i.e. sections parallel to the Zn basal plane); and of, A, the ratio of the section length to section thickness for sections normal to the basal plane. The aspect ratio distribution displayed a broad peak at 2 2.0. This A value was then used to calculate a series of Saltykov coefficients with which to compute the droplet diameter distribution from the observed diameters of circular sections. The droplet diameter distribution, obtained for an A value of 2, is plotted in Fig. 5(c) and Table 1 gives the average droplet surface area. It should be noted that the distribution does not display a narrow asymmetric peak which would characterise an array of equally-sized droplets. This failure of Zn(Sn) droplets to attain local equilibrium accounts for the large spread in A values observed both in this study and in Miller and Chadwick’s [ 141 earlier investigation of droplet shapes. The latter authors have ascribed the slow rate of equilibration to the low driving forces attending the coarsening of Zn(Sn) droplets having highly anisotropic surface energies.

NUCLEATION

IN ENTRAINED

Sn DROPLETS

215

have pointed out that at some time, t,, during the time interval, 6t, the nucleation rate computed using equation (3) will equal the true nucleation rate at ti. In the present study, the nucleation ranges were z 8 K at a cooling rate of 1.25 K min-‘. The ranges were therefore divided into 13 or more segments each having a time interval of < 37 s. Using Wood and Walton’s analysis it can be shown that ti for this time interval is < 0.62 ht. This value of ti is sufficiently close to 0.5 6t for us to identify ti with the mid-point of the time interval. It can be shown that ignoring this small displacement in ti typically introduces an error of less than 0.5% in the gradients and intercepts of the nucleation rate plots (see below). In order to relate the nucleation kinetics with the observed thermal effects it will be assumed that the DTA specimen is everywhere at a temperature only infinitesimally greater than the measured surface temperature. A heat balance for the specimen during the time interval St therefore gives [ 191 -AH.v.h’,(6x)

= u(6T’) + h(U)(&).

(4)

where AH is the latent heat of fusion per unit volume, N, is the total number of droplets (average volume, 8) in the specimen, e is the specific heat and h is the heattransfer coefficient between the specimen and the furnace chamber. It can be assumed that e, h and AH are independent of temperature over the small temperature range corresponding to the droplet exotherm. Thus, we can integrate equation (4) between the start, tl, and finish, t2, times of the DTA peak illustrated schematically in Fig. 8. Integration yields AH.c.N, where A is the area under

= hA, the DTA exotherm.

(5) The

THEORETICAL

1. Meusurement

of the nucleation rute

If it is assumed that the solidification of each droplet required only one nucleation event per droplet, then it can be shown that the total nucleation rate per droplet during cooling from a time t to a time t + 6t is given by [7, 181 Js = - (l/&)ln[x(t

+ &)/x(t)],

(3)

where, for surface nucleation, J is the nucleation rate per unit area, s is the droplet surface area (all surface sites are equally active), x(t) and x(t + at) are the fractions of unsolidified droplets at t and at t + ht. The assumption of a single nucleus per droplet is reasonable because, following nucleation, the droplet liquid recalesces rapidly and the rate of nucleation decreases catastrophically. Equation (3) gives Js as a function of time whereas we need to measure the nucleation rate as a function of the specimen temperature. Wood and Walton [18]

Fig. 6. Schematic free energy diagram for nucleation in an undercooled eutectic alloy showing the various approximations for the composition of the critical nucleus.

216

BOSWELL

AND

CHADWICK:

HETEROGENEOUS

NUCLEATION

IN ENTRAINED

Sn DROPLETS

dependence ascribed to the exponential factor, Q(T), appearing in the classical nucleation rate expression J =

JoexpC-Q(T)I.

(8)

The factor Q(T) is given by Q = Ka3/kT(AG)‘,

(9)

where k is Boltzmann’s constant, K is a geometric constant, Q is the liquid-nucleus surface energy and AG is the free-energy change, per unit volume, obtained on forming a critical nucleus. As a first approximation we shall assume that u is the temperature independent and that AG is given by I

I

0

I

Cylld.r

I

I

2

AG = ASAT

. R.dl”.

Fig. 7. Results of the enthalpic determination heat-transfer coefficient.

of the DTA

substitution of equation (5) into equation by integration from t to tz gives

(4) followed

x(r) = C6A - (elk)

In

between

the specific heats

(6) (at constant pressure) of the solid and liquid phases at

(7)

where the unprimed variables, 6Tand 6A are calculated at time t and the primed variables, 6T’ and 6A’, are calculated at t + 6t. Hence, the total nucleation rate at the temperature corresponding to the midpoint of the time interval can be determined from the DTA thermogram by computing the function [6A - (e/h)6Tl at t and at t + ht. 2. Nucleation kinetics Several nucleation rate laws have been proposed and they are largely distinguished by the temperature

I

I

I

*a

t

1.c

s

Fig. 8. Schematic representation of a DTA droplet therm showing alternative baseline constructions.

CAC/A&lCl - G’i/AT,MT,IT,)Ib (11)

where AC is the difference

(WI/A,

6A’ - (e/h)(GT’) 6A - (e/h)(GT) ’

TInu.

where AS is the volume entropy of fusion at the nucleation temperature and AT is the undercooling. The entropy term can be corrected for the heat capacity effect by multiplying it by a factor such that [6,20] AS = A&(1 -

where 6A is the area under the peak from t to tj?, and GTis the temperature difference at t. Substituting equation (6) into equation (3) gives Js = -(l/&)

(IO)

, mm

exo-

the nucleation temperature, T, is the equilibrium temperature, T, is the mid-point of the observed nucleation range, AT, = T, - T, and AS, is the uncorrected entropy of fusion. Combining equations (8-l 1) it can be shown that the gradient, p, of plot of In Js vs [l + c(T - T,)]/(AT)‘Tis given by p = Ku3/k(AS)‘,

(12)

where c = (AC/AS) C(MAT,)‘)ln(T,/T,)

-

l/U.

(13)

For small values of c, the quantity p can be obtained by evaluating the gradient of a log Js vs l/(AT)‘T plot and correcting this gradient for the heat capacity effect. With regard to the equilibrium temperature T,, in the case of nucleation at variable composition T, is set equal to T,, the equilibrium liquidus temperature. In the case of nucleation at constant composition it is normally assumed that T, = TO.It will be convenient to distinguish between nucleation at variable and constant composition by using the subscripts ‘m’ and ‘0’ (i.e. T,, AT,, AS,,,, [J],; T,, AT,, etc.). By extrapolating reported [21,22] measurements of AC for pure Sn we estimate that AC = 2.5, 1.7 and 1.3 JK- ’ mole- ’ over the temperature ranges of the nucleation rate measurements for Bi(Sn), Zn(Sn) and Al(Sn). There have been few published investigations of the specific heats of undercooled liquid alloys [Zl, 221. However, none of the Sn-rich droplet (liquid) alloys appears to display evidence for atomic clustering in the composition ranges of interest so, for these alloys, it is reasonable to use the specific heat values for pure Sn. In order to calculate the contact angle Q, at the nucleus-liquid-catalyst triple junction it will be assumed that the critical nucleus comprises a spheri-

BOSWELL

AND

CHADWICK:

HETEROGENEOUS

cal cap in contact with a planar solid surface. The geometric factor K appearing in equation (9) is therefore given by K = 167$(@/3,

(14)

where f(0) = (2 - 3 case + cos38)/4. The difference between the catalyst-liquid and catalyst-solid surface energies can be estimated using (Tcj_- acs = g cos 9.

(15)

In calculating AS, for nucleation at variable composition it is usually assumed that the liquid has a uniform composition, particularly at the nucleus interface [23]; and the composition of the nucleus is equal to the composition yielding the maximum free energy change, AG’ [24]. For homogeneous nucleation and for heterogeneous nucleation on an inert substrate it is generally also assumed that AC’ is given approx. by AG ‘I’, the free energy change obtained on forming unit volume of the bulk nucleating phase, p, in equilibrium with the liquid, L[23]. In the present study nucleation occurred in a liquid which was in equilibrium with the substrate material. In this case it is more accurate to equate AC’ and AGaP, the free energy change resulting from the solidification of unit volume of /I in equilibrium with the entraining solid, TX1251. The differences between AG’, AGLO and AG@, and between the corresponding nucleus compositions Cb, Cgi’ and C;i”, are illustrated in Fig. 6 for a binary eutectic alloy. The figure shows schematic free energy curves for the L, a and /3 phases at a temperature below the eutectic temperature. The maximum free energy change, AS’, is obtained approx. [26] by applying the parallel-tangent construction to p and L whereas AC”” and AGLfi are obtained by applying common-tangent constructions. The free energy change AGLfl is given approx. by ASLOALzT,where A.SLli is the weighted average of the volume entropies of fusion of the pure components [27]. This approximation can only be used if both /j’ and L are markedly solute-rich, or solvent-rich, regular sotutions having small interaction parameters [24]. Similarly, AG”’ 2 AS”“*ATm where AG”” values can be estimated by extropolating thermodynamic data to the nucleation temperature. Cantor and Doherty 1251 carried out these extrapolations for Bi(Sn) and their estimated AC”” value can be used to evaluate As’“. Similar estimates were also obtained for AI(%) and Bn(Sn) but for reasons that will become apparent, they will not be required. Table 2 gives the AS,,, value for Bi(Sn) which was calculated by setting AS, equal to As1” in equation (11). For nucleation at constant composition it will be assumed that AS,, the volume entropy of fusion (uncorrected for a non-zero AC) at r, is given by the difference between the weighted averages of the pure component volume entropies of fusion for the liquid and solid phases.

NUCLEATION

IN ENTRAINED

Sn DROPLETS

217

Table 2. Heterogeneous nucleation at variable composition in entrained Bi(Sn) droplets -Sn-10.6 at.“/,,RI Composition, E ,0X,.2 t 1.5m-2 s-I Pre-exponent, [Jg], (expt)? (2.93 + 0.07) x 10” K3 Exponent. p,,,i Equilibrium temperature, ‘Q Nucleation temperature, TN Specific heat change, AC: Entropy change, A&,$

485 * 1K 361 * iK 2.5 JK I mole ..’ (11.4 1t:0.2)10~Jm-~K-’

Nucleus interracial free energy, o// Contact angle function, f(0) Contact angle, t, Free energy difference, (T<~,- cr,

72 + 4mJm-’ 0.83 i_ 0.17 145 + 35 - 50 + 26 mJm_’

Notes t Corrected for specific heat changes. 3 Extrapolated: see [21]. $ AS, = A.Szilcalculated by extrapolation; see [ZS]. i/ From TMfor suspended Sn-10.6 at.“;, Bi droplets; see

PI. 4. Interfacial free energy With regard to the interfacial free energy 0, we note that the measured [S] maximum undercooling is 0.34 T, for pure Sn, The surface energy calculated using Turnbull’s classicial homogeneous nucleation rate expression is therefore 71 mJm_‘. If the LothePound [ 123 correction factor is included then the calculations summarised by Stowell [lo] can be carried out to give 0 = 78 mJm_‘. The small specific heat correction for AS,,, reduces these u values to 70 and 77 mJm_’ respectively so a lower limit range for D (pure Sn) is 74 k 4 mJm_‘. As a first approximation we can use this value of o‘ for each of the Sn-rich droplet liquids. In the case of Bi(Sn) however, it is possible to use the observed [S] nucleation tem~rature for suspended Sn-Bi droplets of the same composition as the entrained droplets to estimate a lower limit cr value for Bi(Sn). The nucleation temperature for suspended Sn-10.6 at.“,, Bi droplets is 320 K and AS, = ASL” = 9.0 x IO’ Jm-” K- I. Repeating Stowell’s calculations (with and without the Lothe-Pound correction) we find that a lower limit range for ci in the Bi(Sn) droplets is 71 5 4mJm-* (corrected for the specific heat change on nucleation).

ANALYSIS

OF EXPERIME~AL

RESULTS

In order to calculate Js using equation (7) it is necessary to estimate the heat-transfer coefficient, h, for the entrained droplets specimens. Equation (5) shows that h is given by the total heat of reaction divided by the DTA peak areas. An enthalpic calibration of the DTA unit was therefore conducted by measuring the exothermic peak areas for the eutectoid transformation in a Zn-59.4 at.% Al alloy and for the grain-boundary solidification reaction in the droplet

218 BOSWELL

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NUCLEATION

IN ENTRAINED

Sn DROPLETS

specimens. In the former case, cylinders, having diameters equal to their heights, of the alloy were cooled at 1.25 K min- ’ from 560 K and the area of the DTA peak produced by the exothermic eutectoid reaction was measured using a planimeter. The enthalpy of the eutectoid transformation, AlYE,is given by [28] AHs = 46.03 - 4.822 x 10-4P Jg-‘, where P is in Zn-Al interfacial area per g of alloy. Metallographic observations indicated that extensive coarsening of the Zn-Al lamellar eutectoid structure took place during the DTA exotherm. Thus, we can ignore the small contribution of the interfacial enthalpy to the enthalpy of transformation, so that AHE = 46 J g-‘. Figure 7 gives the experimental h (= AH,/A) values as a function of the cylinder radius. It can be seen that h is relatively constant for cylinders having radii that are approx. equal to the radius (2.5 mm) of the entrained droplet specimens. The figure also includes h values obtained by measuring the DTA peak area for the grain-boundary solidification reaction prior to the extensive development of equilibrated droplets. Under these circumstances, the total volume of entrained liquid, which was measured using quantitative metallography, can be assumed to solidify during the grain boundary reaction. There is good agreement between the h values obtained for different droplet specimens and for the eutectoid Zn-Al specimens. The average h value is 0.018 J K-i s- 1 for cylinders having a radius of 2.5 mm and this value was used in calculating the total nucleation rate. 2. Nucleation kinetics (a) Nucleation at variable composition. The DTA droplet exotherms were converted into a series of x-y coordinates and these coordinates were read into a computer. A computer program then numerically integrated the area under the peak and calculated Js using equation (7). The calculated nucleation rates are plotted in Fig. 9 as a function of l/(AT,)‘T. In obtaining the x-y coordinates it was necessary to draw a baseline across the DTA peak. Such a baseline should have an inflexion, of the type shown in Fig. 8, corresponding to the specific heat change on droplet solidification. In the case of Bi(Sn) and Zn(Sn), it was found that the high temperature and low temperature segments of the DTA traces on either side of the droplet peaks were parallel, but offset. The baselines under the peaks were therefore assumed to display inflexions in accordance with the theoretical variation illustrated in Fig. 8. There are, in principle, means of evaluating the approximate shape of the DTA baseline between the parallel offsets. However, these considerations proved to be unwarranted because the assumption of a step-change baseline or a linearchange baseline, as demonstrated in Fig. 8, gave only small changes in the calculated Js values. These changes were, moreover, outweighed by integration and curve-fitting errors. Hence, Js values were com-

Fig. 9. Nucleation rate plots for (a) Bi(Sn); (b) Zn(Sn); (c) Al(Sn).

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puted by averaging the values obtained using a step change baseline and a linear-change baseline. In the case of Al(Sn), the presence of a high-temperature peak which overlapped with the low-temperature peak meant that it was not possible to construct a reliable baseline using the principles outlined above. Instead, we were obliged to make a linear extension of the low-temperature segment in the manner shown in Fig. 2. Reference to the nucleation rate plot of log Js vs l/(AT,)‘Tfor Bi(Sn) demonstrates that the majority of the data points lay on a straight line (integration errors account for the round-off effect at the ends of the plot). The nucleation rate plots for Al(Sn) and Zn(Sn) on the other hand, show the only a limited number of points lay on, or close to, the straight lines drawn through the points. In the case of Al(%) we attribute this to peak overlap and problems associated with the construction of the baselines. These complications were apparently not present for Zn(Sn) so the origin of the irregular nucleation rate plot for this system should perhaps be sought for elsewhere. The only distinctive metallographic feature shown by the Zn(Sn) specimens was poor equilibration of the droplet arrays. This tendency probably resulted from pinning of the droplet interfaces by the intersection of the interfaces with low-energy basal planes. It is also possible that the nucleation rate, at a given temperature, on the flatter, basal portions of the interfaces was lower than on the curved, higher energy portions of the interfaces. In other words, entrained droplet specimens having highly anisotropic interfacial free energies might be expected to display smeared out droplet exotherms. Moreover, the irregular nucleation rate plot for Zn(Sn) may represent a transition from nucleation at large undercoolings on low-energy, basal planes to nucleation at lower undercoolings on high-energy, non-basal planes. The intercepts and gradients of the log Js vs l/(AT,)‘Tplots were evaluated and the specific heat corrections were then carried out using c values obtained by substituting the appropriate estimates for AS,, AC, T, and T, into equations (11) and (13). The corrected exponents, p,,,, are detailed in Table 1 together with the average droplet compositions and the equilibrium liquidus temperatures used in calculating AT,. In the case of Bi(Sn), the specific heat corrections reduced the gradient of the log [Js], vs l/(AT,)‘T plot by 3% and the intercept by 0.6. For the two remaining systems the corresponding corrections were 2% and 0.3. Table 1 also gives the pre-exponential factors, [Ji],,, (expt) obtained by dividing the measured [Jis],,, values by the average droplet surface areas, S. In the case of Bi(Sn), [Ji],,, (expt) = 1031.2’ ‘.’ mm2 s-l which compares favourably with the Ji (theo) value of 1032.5”.5 rn-‘s-i predicted [S] by the classical theory of heterogeneous nucleation. The [J,“],(expt) values for Al(Sn) and Zn(Sn) are significantly smaller than Jz(theo). However, we do not attach much sig-

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219

nificance to these results because the two [J;], (expt) values were estimated by fitting straight lines to very irregular Js plots. Indeed, these two systems will not be given any further consideration. Using equations (12) (14) and (15) together with the measured exponent p,,, and the estimated AS,,,, T, and c values for Bi(Sn), we can calculate the contact angle function f(S), the contact angle 0, and the surface energy difference, oc,. - ocs: the results of these calculations are given in Table 2. (b) Nucleation at constant composition. The intercept obtained on plotting log Js as a function of l/(A.T,)‘Tfor Bi(Sn), when corrected for the specific heat effect, gave a [Ji], (expt) value of 1O2812 me2 s-l for nucleation at constant composition (the ‘& temperature for the droplet liquid was assumed to be 475 K). This experimental estimate of the preexponential frequency factor is about an order of magnitude smaller than the theoretical value of 1032.5’ 1.5 me2 s- ’ if the experimental and theoretical errors are taken into consideration. Consequently, while the nucleation rate data for Bi(Sn) are best described in terms of nucleation at variable composition, nucleation at constant composition may provide a viable interpretation if, in the future. it is established that some of the assumptions used in the analysis are inaccurate. DISCUSSION 1. Nucleation at variable and constant composition In the case of Bi(Sn) entrained droplets there was found to be some metallographic evidence for solidification with solute segregation in that dendritic structures were occasionally observed in larger droplets which had been quenched from a temperature corresponding to the start of the droplet exotherm. It was also argued that the droplets solidified with solute segregation during slow cooling through the nucleation interval. The results showing that the nucleation rate data for Bi(Sn) were best described in terms of nucleation at variable composition would therefore appear to be consistent with the inferred solidification behaviour. Clearly, it will be necessary to conduct further detailed examinations of the structures of quenched and slowly-cooled Bi(Sn) droplets in order to examine whether the fine-scale microstructures observed in the smaller droplets were produced by decomposition of a massively solidified phase or by precipitation during solidification with solute segregation. The possibility that nucleation in Bi(Sn) entrained droplets at an undercooling of 124 K may not involve solute partitioning cannot be overlooked because Perepezko et al. [S] have shown that suspended Sn-Bi droplets with < 30 at.% Bi solidified massively following nucleation at undercoolings of between 144 and 175 K. Unfortunately, there appears to be no theoretical analysis with which to discriminate between nucleation at variable and constant composi-

220

BOSWELL

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tion in alloys. Classical nucleation theory invokes microscopic equilibrium and necessarily requires that nucleation takes place predominantly at variable composition. Even at large undercoolings the composition of the classical nucleus does not differ greatly from the composition of the solid nucleating phase in equilibrium with the liquid [29]. Baker and Cahn [30] have demonstrated that the more tractable problem of planar growth at constant composition cannot be analysed using existing thermodynamic concepts. This finding does not augur well for any future attempts at rigourously describing nucleation at constant composition. The problem of reconciling nucleation at variable composition with subsequent growth at variable composition has been considered by Kamenetskaya [29]. This author argued that growth of a critical nucleus of variable composition could proceed with solute segregation until the interfacial concentration became equal to the alloy composition. At this point massive growth would take place provided the interface was at a temperature below T,. This solidification model has been developed by Hillert and Sundman [31] who showed that a supercritical cluster, of variable composition, could evolve into a ‘massive’ dendrite, i.e. a dendrite having a tip concentration equal to the liquid composition. It is likely that Hillert and Sundman’s analysis can be used to describe solidification in undercooled droplets. However, it appears to have limited application in the study of massively solidified grains in rapidly quenched alloys. Microstructural investigations have shown that these grains do not appear to grow dendritically, excepting possibly under thermal conditions which lead to the development of thermal dendrites [32]. In the case of solidification during rapid cooling it would seem that we require an analysis for the evolution of non-dendritic massive grains following nucleation at variable and constant composition. 2. Contact angles It is perhaps unfortunate that of the three Sn droplet systems examined only Bi(Sn) gave useful results. In the case of Al(Sn), it was likely that the nucleation rate measurements were complicated by the presence of an overlapping exotherm. Using quantitative metallography, the Al(Sn) droplet size distribution was shown to be bimodal so a simple model based upon the intersections of cell boundaries with droplets was invoked in order to account for the observed size distribution and thermal effects. According to the model, droplets lying on cell boundaries were smaller (because they resisted coarsening due to pinning of the interfaces) and tended to solidify at higher temperatures (because their surfaces were more potent catalysts for nucleation). Hence the narrow, low-temperature exotherm of Fig. 2 should correspond to the larger portion of the droplet size distribution of Fig. 5. However, the ratio of the volumes of droplet liquid contained in pinned and unpinned droplets was

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Sn DROPLETS

estimated to be ~0.05 using the droplet size distribution, and % 1.0 using the relative areas of the DTA peaks for Al(Sn). The large discrepancy between these two values suggests that a simple model based on two types of droplets is not entirely satisfactory. Any refinements to the model that would decrease the discrepancy are highly conjectural so they will not be discussed. In the case of Zn(Sn) it is reasonable to assume that equilibration of the droplets was resisted by pinning of the droplet surfaces as a result of their intersection with low-energy basal planes. Moreover, nucleation would tend to take place at a higher temperature on the basal portions of the droplet surfaces than on the curved, higher energy portions. The irregular nucleation plot for Zn(Sn) may therefore correspond to a transition from nucleation on basal planes to nucleation on the remaining parts of the droplet surfaces. The nucleation rate measurements for Bi(Sn) gave an interfacial free energy difference (Q.,, - acs) of -50 f 26 mJm_’ and it is interesting to compare this experimental estimate with theoretical estimates. Miedema and den Broeder [33] in a recent treatment of interfacial free energies that was based on Turnbull’s [34] analysis, argued that r~ was made up of structural and chemical contributions whereby I, OCL

0(-s

=

2

h.l4

+

&Lb

+

&m>

0.15 (a”c + 0;) + &pm,

(16)

(17)

where primed values refer to enthalpy contributions, double primed values to entropy contributions and 8 values to the surfaces of pure component solids. As a first approximation we can equate the two chemical contributions in order to obtain an expression for (ecL - a&. Using Miedema and den Broeder’s values for the various non-chemical terms appearing in equations (16) and (17) we find that (act. - acs) X - 100mJm-2 for liquid Sn in contact with solid Bi. This estimate is in good agreement with the experimental value for an entrained Sn-10.6 at.% Bi liquid alloy nucleating a %-rich phase on a Bi-rich solid phase. CONCLUSIONS Differential thermal analysis was used in conjunction with entrained-droplet specimens to compare the nucleation kinetics of Sri-rich liquids in contact with Bi, Zn and Al solid solutions. Assuming that the entire droplet surface acted as a nucleating catalyst and that nucleation occurred at variable composition, it was established that the pre-exponential frequency factor, [.I:],,, (expt) describing the experimentally determined nucleation kinetics was 1031.2’ 1.5 m-* s-l for Bi(Sn) and 1027~3m~2s~1 for Zn(Sn) and AI( The value for Bi(Sn) compares favourably with the J; (theo) value of 103*.’ ’ I.’ m-’ s- ’ predicted by the classical theory of heterogeneous nucleation for pure Sn.

BOSWELL

AND

CHADWICK:

HETEROGENEOUS

The relatively large errors associated with the [J&, (expt) factors for Zn(Sn) and AI were due to the irregular nature of the nucleation rate plots for these two systems. In the case of Al(Sn), the irregularity may have been brought about by the presence of an overlapping DTA exotherm corresponding to the solidification of droplets located at cell boundaries. In the case of Zn(Sn), the nucleation plot possibly reflected a transition between nucleation on more active and less active portions of the droplet surfaces. The gradient, pm, of the (variable composition) nucleation rate plot for Bi(Sn) was (2.93 + 0.07) x lO* K3. By assuming a nucleus-liquid surface free energy (T.of 72 i 4 mJm_’ and by calculating the free energy change accompanying the formation of a critical nucleus having a composition equal to the composition of the bulk phase in equilibrium with the entraining solid Bi, it was calculated that f(0) = 0.83 & 0.17. In the case of a spherical-cap nucleus, this geometric factor corresponded to a surface energy difference (oc, - cc.1 of - 50 + 26 mJm_‘. This experimental estimate of(nc,, - r~s) was in good agreement with a recently derived, theoretical estimate for pure St1 in contact with solid Bi. A comparison between analyses of the nucleation kinetics for Bi(Sn) based on nucleation at variable and constant composition indicated that the measurements were best described in terms of nucleation at variable composition. Scanning electron metallography showed that droplets quenched from the start of the nucleation range generally contained a fine-scale structure. However, sections through relatively large Bi(Sn) droplets revealed the presence of dendrites. Using these observations. it was reasoned that the entrained Sn droplets solidified with solute partitioning on slow cooling through the DTA droplet exotherms.

NUCLEATION

tralian Research research.

Grants

Committee

for supporting

this

I. C. C. Wang and C. S. Smith. Trans. AIME 188, 136 (1950). 2. R. T. Southin and G. A. Chadwick, Actu metull. 26, 223 (19781.

221

R. Elliot and F. R. int. Conf on So~~d~~c~~fj~~ and Cnsting. Preprint, p. 611 (1977). D. Turnbull, J. &cm. Phys. 20, 411 (1952). G. M. Pound and V. K. La Mer, J. Am. chenz. Sot. 74, 2323 (1952). Y. Miyazawa and G. M. Pound, J. Crysta/ Growth 23, 45 (1974). D. H. Rasmussen and C. R. Loper, Acttf me&$. 24, I17 (1976). J. H. Perepezko, D. H. Rasmussen. I. E. Anderson and C. R. Loper, Sheffield ConfY on Solidifcation und Cusrif~g, Preprint, p. 289 (19773. J. H. Pereoezko and D. H. Rasmussen, Me&r//. %n.s. 919, 1490 il978). M. J. Stowell, Phil. Mug. 22, 1 (1969). V. P. Skrioov. V. P. Koverda and G. T. Butorin. Stx. Plr:s. c~~~r~~~l~~~r‘15, 1065 (197 1). J. Lothe and G. M. Pound, in Nucieution (edited by A. C. Zettlemoyer), p. 109. Dekker, New York (1969). B. Vonnegut, J. C&id Sci. 3, 563 (1948). W. A. Miller and G. A. Chadwick, Proc. R. Sec. A. 312, Sale, Shefield

4. 5. 6. I. 8.

9. 10. Il. 12. 13. 14.

256 (1969). 15. IL Biloni and B. Chalmers, 16. 17. 18. 19. 20. 21. 22. 23 24 25

27. 28. 29. 30.

REFERENCES

Sn DROPLETS

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26.

A~l\fio~~~~rlyrtnrt?rThe authors are indebted to the Aus-

IN ENTRAINED

31. 32. 33. 34.

Truns. Met. Sot. AIME 233, 373 ( 1965). S. J. B. Reed, ~lectro~~ h~i~ro~~ob~ Anulgsis. p. 18X. Cambridge Univ. Press, Cambridge (1975). S. A. Saltykov, Stereometric Metallogruphy. Metallurgizat, Moscow (1958). G. R. Wood and A. G. Walton, J. uppl. Phy.5. 41, 3027 (1970). H. J. Borchard and F. Daniels. J. Am. &em. SM. 79, 41 (19573. D. R. H. Jones and G. A. Chadwick, Phii. Mug. 24, 995 (1971). H. S. Chen and D. Turnbull, Actu mural/. 16, 369 (1968). G. Schick and K. L. Komarek, Z. Metullk. 65, 112 (1974). H. I. Aaronson, K. R. Kinsman and K. C. Russell, Scripfa defame,4, 101 (1974). R. B. Heady and J. W. Cahn, J. chum. P!-)x 58, 896 (1973). B. Cantor and R. D. Doherty, Acto mettrll. 27, 33 (1979). H. Reiss and M. Shugard, J. &em. Phyx. 65, 5280 (1977). W. V. Youdelis, Me&l Sci. J. 9, 464 (1975). D. Cheetham and F. R. Sale, Acra ~~t~~/. 22. 333 (1974). D. S. Kamenetskaya, Growth ofCrystals8, 271 (1969). J. C. Baker and J. W. Cahn, in Solidifcrrtion, p, 23, ASM, Metals Park, Ohio (1971). M. Hillert and B. Sundman. Actu metcdf. 25. I1 (1977). P. Ramachandrarao, M. G. Scott and G. A. ‘Chadwick, Phil. May. 25, 961 (1972). A. R. Miedema and F. J. A. den Broeder. Z. ~~~rf~~~~. 70, 14 (1979). D. Turnbull, fmpurities and Impet$ction.s, p. 121. ASM, Metals Park, Ohio (1955).