Kinetics of precipitation of tin from lead-tin solid solutions

KINETICS OF PRECIPITATION OF TIN FROM LEAD-TIN SOLID SOLUTIONS*

D. TURNBULL and H. N. TRBAFTISt

At all temperatures, T, and initial solute concentrations, X0, investigated the first rapid stage of precipi- tation of tin, j3, from lead, oi, is effected by the nucleation and growth of cells. Each cell consists of a multi- plicity of @-plates, originating from a single nucleus, interspersed in the depleted a. Independently of T and X0, the kinetic law of precipitation is, with rare exceptions:

x=1-exp(-l#).

This kinetic law is consistent with the hypothesis that all nuclei originate at the beginning of precipitation within a limited number of singular elements of the structure. Precipitation proceeds lo7 to lo* times faster than calculated from Seith and Laird’s data on the rate of diffusion of tin into lead. The precipitation rate is greatly increased by cold-working the supersaturated 01.

LA CINl?TIQUE DE LA PRl%IPITATION DE L’l?TAIN A PARTIR DES SOLUTIONS SOLIDES PLOMB-6TAIN

A toutes les tempdratures, T, et concentrations initiales, X0, consid&+es, le premier stade rapide de precipitation d’&ain, 8, a partir de plomb, (Y, s’bffectue par la germination et la croissance de cellules. Chaque cellule consiste en une multiplicit6 de lamelles de 0 provenant d’un seul germe, qui parshment le O! appauvri. Independamment de T et X0 la loi cinetique de la precipitation est, avec de rares exceptions:

x=1-exp(-W).

Cette loi cinetique est compatible avec l’hypothese que tous les germes naissent au debut de la precipitation dans un nombre limit& d’&ments simples de la structure. La pr&ipitation est 10’ 5_ lo* fois plus rapide que ne l’indique le calcul basC sur les don&es de Seith et Laird concernant la vitesse de diffusion de l’&ain dans le plomb. La vitesse de la prCcipitation est considerablement accrue par 1’6crouissage de l’a! sursatur&

DIE KINETIK DER AUSSCHEIDUNG VON ZINN AUS FESTEN BLEI-ZINN LijSUNGEN

Das erste, schnelle Stadium der Ausscheidung von Zinn, j3, aus Blei, 01, ist bei allen untersuchten Tempera- turen T und Ausgangskonzentration von XO der gel&ten Komponente, durch die Keimung und das Wach- stum von Zellen beeinflusst. Jede Zelle besteht aus einer Vielzahl von fl PlLttchen, die von einem einzigen Keim ausgehen und Zwischen denen sich das verarmte (Y befindet. UnabhLngig von T und X0 ist das kinet- ische Gesetz der Ausscheidung mit wenigen Ausnahmen

x=1-exp(-_bt3).

Dieses kinetische Gesetz steht im Einklang mit der Hypothese, dass alle Keime beim Beginn der Ausscheid- ung innerhalb einer begrenzten Anzahl spezieller stellen im Gefiige entstehen. Die Ausscheidung erfolgt 10’ bis lo* ma1 schneller als nach den Angaben von Seith und Laird iiber die diffusionsgeschwindigkeit fiir Zinn und Blei berechnet. Die Ausscheidungsgeschwindigkeit wird durch Kaltbearbeitung des iibersgttigten n stark erh6ht.

INTRODUCTION

Borelius and co-worker&JJ have measured the rate of precipitation of tin (P-phase) from lead-rich solid solu- tions (a-phase). Their results were shown to be ap- parently in agreement with the Borelius-Konobeyevski (B-K) theory of nucleation. On the other hand, Hardy4 showed that the results also appeared to be consistent with the Becker theory of nucleation. (See review of Hollomon and Turnbull for detailed discussion of these theories.) The interpretations of Borelius et al. and Hardy presupposed that nucleation is effected in struc- turally pure (excepting for point defects) a-crystals. No satisfactory explanation was offered for the kinetic law of precipitation. By the kinetic law we mean x=f(t) where x is the fraction of precipitation (to be quanti- tatively defined later) that has taken place isothermally in time t.

* Received June 25, 1954.

Y!rZ eneral Electric Research Laboratory, Schenectady, New

Inferences about the rate of nucleation in a precipita- tion process are of doubtful validity in the absence of a satisfactory theory for the kinetic law. Also, there are some other important characteristics of the precipitation behavior in lead-tin that are not explained by the Borelius or Hardy interpretations.

Seith and Laird6 have reported that the interdiffusion coefficient, D, of lead and tin in (Y containing 2 atom per cent tin can be described as a function of absolute temperature, T, by the equation

D= 4.0 exp[- 26,00O/RT] cm2 set-‘. (1)

If it is assumed that the rate of diffusion of tin during precipitation is described by D, it follows that in a lo- minute period at 300°K tin could be drained from an a-region having a linear extent of 2 to 3 A. Actually, for an initial atom fraction of tin in (Y of -0.15, the first rapid stage of precipitation is nearly completed in 10 minutes and microstructural observations indicate that the thickness of the depleted (Y region is of the order of a micron. Hence, during precipitation the diffusion coeffi-

ACTA METALLURGICA, VOL. 3, JANUARY 19.55 43

4-l ACTA METALLURGICA, VOL. 3, 1955

cient of tin atoms is apparently 10’ to lo* times larger than is predicted by Eq. (1). The precipitation measure- ments were made on solutions initially much more concentrated than used in Seith and Laird’s measure- ments; nevertheless it would be very surprising if the rate of diffusion of tin in (Y increased by a factor of lo7 when the atom fraction, X, increases from 0.02 to 0.10. Thus it is likely that the rate of precipitation of tin is many orders of magnitude larger than is predicted from the appropriate volume diffusion data. This rapid rate of precipitation might be accounted for either (a) by the presence of a much greater than equilibrium number of point defects during precipitation, as suggested by Zener’ and Seitz,8 or (b) by the drainage of tin from solution along diffusion short circuits sweeping the solid solution in advance of the precipitate. It has been demonstrated9 that elements of incoherent grain bound- aries and dislocation channels are about equally effective short circuits in self-diffusion.

Borelius et ~2.’ report the time, 7, required for the isothermal half completion of precipitation as a function of T and the initial atom fraction, X0, of tin in a. They find that d ln(l/T)/dT is dependent on X0 when the temperature exceeds a characteristic value T’, but inde- pendent of X0 for T<T’. T’ is a function of X0. To explain this result, Borelius and Hardy assume that the thermodynamic part, AG*, of the potential energy barrier to nucleation becomes negligible at T= T’. How- ever, for T< T’, l/~ is still strongly dependent upon X0; for example, at T= 273°K (a temperature well below T’ for X,=0.100) 1,‘~ is 500 times larger for X0=0.235 than for X0=0.100. For precipitation limited only by diffusion, we expect 1,‘~ to be at most directly pro- portional to X0, so there seems to be a factor of at least 100 in the variation of l/~ with X0 not accounted for by the Borelius and Hardy interpretations.

Because of these inadequacies in the Borelius and Hardy interpretations, we have reinvestigated the ki- netics of precipitation of tin from lead-tin solid solutions for the purpose of establishing the kinetic law and its relation to the individual rates of nucleation and growth of the precipitate particles.

EXPERIMENTAL

Precipitation was followed by measuring the resist- ance of the alloys as a function of time at constant temperature.

Preparation of Alloys

Alloys were made from American Smelting and Re- fining Co., 99.999 per cent pure lead, and Vulcan Detinning Co., 99.997 per cent pure tin. Wires 20-30 mils in diameter and 2-4 inches in length were made for the resistance measurements. RI&t of the wires (cast wires-C) were cast in Pyrex tubes of uniform capillary bore 20 to 25 mils in diameter. In thi’s operation specified amounts of the pure metals are placed in bulb A of the

IO, ,b)

FIG. 1. Schematic diagram of (a) wire casting vessel, and (b) specimen mounting.

vessel shown in Fig. la. The vessel is evacuated (to 10 to 100 micron pressure of mercury) and partly immersed in a furnace held at a temperature exceeding the melting point of the alloy. The melt is then run from bulb A to bulb B and a portion of it forced into the capillary by the pressure of an inert gas. The entire apparatus is then rapidly cooled so that the alloy in the capillary solidifies. The cast wire is easily removed from a capillary tube that is clean initially. After each casting operation the vessel was thoroughly cleaned and a new capillary sealed in. The alloy remaining after preparation of the wire was analyzed for tin (to fl part in 100). Before beginning a series of measurements the wire was sealed in an evacuated Pyrex tube and homogenized at 225°C for a period of at least 15 hours.

A few of the wires were extruded (extruded wire-E) from l/2-inch diameter specimens chill cast in a graphite coated copper mold after melting in air.

Resistance Measurements

An approximately 2-inch length of lead-tin wire was mounted across two silver tipped l/16 inch diameter constantan rods as shown in Fig. lb. The wire and copper potential leads were secured to the silver tips by solder having the same composition as the resistance specimen. The nichrome rods were held on a transite backing secured to a l/2 inch diameter supporting rod that could be displaced vertically in either direction. (The axes of the nichrome and supporting rods were vertical .)

To determine resistance, a current of 1.000 ampere (held constant to within 1 part in 105) was passed through the wire and the resulting drop in potential (between the copper potential leads) was measured as a function of time. The potential determinations were ac- curate to about lop5 volt corresponding to 10e5 ohm in resistance. The magnitude of the resistance change, AR, during the rapid stage of precipitation ranged from approximately 0.002 to 0.006 ohm out of a total resist- ance of 0.04 to 0.08 ohm. Hence, the uncertainty in AR is about 1 part in 200 to 600.

TURNBULL AND TREAFTIS: KINETICS OF PRECIPITATION 45

Homogenization

For heat treating the wires, a rectangular nichrome wound furnace, having chamber dimensions 1X5 inch and 3 inch high, was positioned about 3 inch directly above the constant temperature bath. The furnace chamber was closed excepting for a small slot at the bottom that admitted the wire and its supporting rods. Argon gas, preheated by circulation through a copper coil wound around the inner surface of the furnace, was passed through the chamber during heat treatment. For temperature measurement the hot junction of a thermo- couple was positioned in the vicinity of the wire mid- point. Temperatures along the whole length of the wire between the silver terminals deviated no more than 10” from the chamber temperature at 250°C. Prior to a precipitation rate measurement, the wire was heated at 250°C (about 80-100 degrees higher than the equi- librium homogenization temperature) for one hour.

Procedure

All the necessary electrical connections for measuring R= f(t), where R is the wire resistance, were made be- fore homogenization. Hence, after the heat treatment the wire was plunged directly into a constant tempera- ture bath; the standard current started and timing begun. The bath was thermostated to fO.l”C. Silicone oil served as the bath fluid excepting for 0°C an ice bath (with an arrangement to keep pieces of ice from con- tacting the wire) was used.

The rate of resistance change with time, dR/dt, at first increases from zero to a maximum (dR/dt), and then decreases as time increases. Resistance was re- corded until cZR/dt had fallen to a value approximately 1 per cent of (dR/dt),. The points to be shown in the figures were calculated from resistances read from con- tinuous recorder curves.

We have given in Table I the initial resistance, -Ro, (rounded off at the fourth place for tabulation) of a particular wire (3C) as a function of temperature. Also given is AR which is defined by:

AR=Ro-RI, (2)

where R, is the resistance at the time that (dR/dt) has fallen to (l/SO) (dR/dt),.

TABLE I. Variation of resistance change, AR, for a single wire during precipitation with T. X0=0.147 (Wire 3C).

7-K Ko (ohms)

AR (ohms) E?Q. Calc. AR/R0

273 0.0728 0.087 288 0.0763 0.083 303 0.0806 0.0062 0.0059 0.077 320 0.0852 0.0057 0.0056 0.067 333 0.0887 0.0051 0.0052 0.057 348 0.0926 0.0045 0.0046 0.049 358 0.0952 0.0039 o.Oa41 0.041 373 0.0992 0.0029 0.0032 0.029

~____

It is interesting that for a given wire:

ARa: X0-X,, (3)

where X, is the equilibrium atom fraction of tin in CL From the data of Borelius et al.,’ we deduce the ap- proximate dependence of 171, on T to be:

1210 log x,= - ---+2.11. . (4

T

Table I also compares the experimental AR at various temperatures with the value of AR calculated assuming the validity of Eq. (3). For all temperatures the agreement between the two values of AR is excellent.

RESULTS

Much exploratory work was done in this investigation and it is not practicable to report all of the results in detail. Therefore, we shall describe the representative findings of most significance and state the largest deviations from these that were found.

The Kinetic Law

We now give a more precise definition of the fraction, x, of precipitation that has occurred in time, t, as follows :

x0-x2 x=----

x0-x,’ (5)

where Xt is the “apparent” atom fraction of tin in cy at time t, Xf is the “final” value of X but is not necessarily identical with X,. In order to obtain x=f(t) we must relate some property of the precipitating system, such as resistance, to x.

It is expected that the specific energy, El, evolved isothermally during precipitation should be approxi- mately proportional to the amount of precipitate per unit mass of alloy. Nystrom3 measured Et= J(f) for the precipitation of tin from lead using an isothermal calorimeter. Assuming x m Et, he finds for x+0 that

x=&n, (6)

where n=3 for low temperatures and n=4 for “high” temperatures and initial concentrations. Equation (6) probably derives from the more general law of pre- cipitation :

x=1-exp(-6t”). (7)

Borelius et al.’ have shown one isothermal relationship R= j”(t) for X0= 0.192 and T= 359. If it is assumed that x is related to Rt by the following equation:

S-R, x= __

Ro- R,’ (8)

where Rt is the wire resistance at time 1, x=j(f) is described by Eq. (7) with n=3.0 for xqO.5. For the

46 ACTA METALLURGICA, VOL. 3, 195.5

lowest initial concentrations and temperatures reported by Nystrom, and assuming the validity of Eq. (8), our results are best described by Eq. (7) with n= 3.0. Hence, there is an excellent correspondence between the kinetic law calculated from calorimetric data and that obtained from resistance data with the use of Eq. (8).

On the basis of this agreement we have used Eq. (8) to convert all of our results from Rt= f(t) to x= f(t). In this calculation Rf is defined, as before, to be the resistance at the time that (dR/dt) has fallen to (l/SO) (dR/df),. Hence, Xf must exceed X,. The as- sumed relation between R and x is probably a poor approximation for ~$0.5.

We find that Rf is related to R,, the resistance at t= m, by:

(Ro-R,)m0.6(Ro-RR,).

The very marked decrease in precipitation rate when ARm0.6(Ro-R,) was also observed by Borelius et al.’ This effect is not due to a peculiarity of the electrical

TABLE II. Summary of kinetic data on some cast wires. T,=/(Xo) is temperature at which, according to Borelius, AG* should disappear.

Temp.% n ;, 2 (min-1) -Logb

Wire No. 17C X0=0.112 (6.8 wt. per cent) 7’,=298

273 3.2 “o:% 6.26 298 2.8 5.04 315 2.9 0.0185 4.91 323 2.9 330 3.0

ES 4.94 5.21

3.53 3.0 0.0041 6.98 6=3.0

W’ire No. 2C X0=0.147 (9.0 wt. per cent) T,=320

273 2.9 0.035 4.20 282 0.052 3.68 288 0.066 3.37 294 0.074 3.22 303 3.1 0.089 2.98 319 2.7 0.096 2.88 333 3.1 0.089 2.98 348 2.7 0.057 3.56 363 0.026 4.58 373 3.0 0.0063 6.43

fi=2.9

Wire No. 3C X0=0.147 (9.0 wt. per cent) T,=320

273 3.1 0.042 3.96 288 3.2 0.086 3.03 303 0.119 2.60 320 0.135 2.44 333 2.9 0.125 2.53 348 2.8 0.089 2.98 358 0.062 3.45 373 3.1 0.0159 5.23

?i =3.0

Wire No. 15C

273 309 321 334 353 373 393

X0=0.170 (10.5 wt. per cent) T,.=330

3.05 0.130 2.49 0.412 0.98

3.1 0.426 0.93 0.44 0.89

2.9 0.292 1.44 3.4 0.109 2.71 3.3 0.0079 6.14

A=3.15

00

- 005

080 090 100 I10 I 20 IM LOG t

FIG. 2. Dependence of Log ln[l/l-x] on logt for two cast wires. T=273”K, X0=0.147.

behavior of the precipitating system, for calorimetric data, kindly sent to us by Professor Borelius, indicated a similar sharp decrease in dXJdt at XO- Xt~0.7 (XO- XJ.

Therefore, we conclude that precipitation occurs in two stages possibly described by two kinetic laws. The first rapid stage drains 60 per cent of the thermodynamic excess of tin from solution and the remainder is drained out in a second very slow stage. It may be that drainage of tin in the fast reaction is effected primarily by diffusion short circuits but the slow reaction is limited by lattice diffusion.

We shall now describe our results on the first stage of precipitation.

Rates of precipitation, dq’dt=k, in different cast wires or in a succession of experiments on one wire for a constant degree of precipitation x= c, temperature, and initial atom fraction of tin generally agreed to within 12 percent [ (dx/dt),=&]. However, there was a single cast wire in which f, differed as much as 33 per cent in a succession of experiments. Among different cast wires of the same composition the precipitation rates (shown in Table II) in wires 2C and 3C exhibit the widest disa- greement (a total error of 29 parts per 100 in the tin analyses would account for this disagreement). The maximum scatter in f, for the extruded wires (T, XO constant) was larger than for the cast wires and often exceeded 20 per cent. The rate measurements of Borelius et al apparently scatter as widely as ours for extruded wires.

The differences in i, for a given specimen are often substantially larger than the measurement errors and seem to indicate changes in the specimen. A wire might change in a succession of experiments in (a) initial homogeneity of composition, (b) average composition, (c) microstructure-including such fine structural de- tails as the dislocation distribution. We found that for a particular wire Ro was reproduced to 1 part in 4000 in a succession of experiments where z& differed as much as 33 per cent. Considering that the total change in R during

TURNBULL AND TREAFTIS: KINETICS OF PRECIPITATION 47

precipitation was 320 parts in 4000 and no trend in i, was discerned in successive experiments, we conclude that the scatter in i’, is probably due primarily to microstructural changes that occur in successive homog- enization and precipitation treatments.

To indicate the order of reproducibility of the data the results of three experiments at 273’K on two cast wires of the same composition (X0= 0.147) are shown as log ln[l/(l -x)] vs logt plots in Fig. 2. The linearity of the curve is consistent with Eq. (7) and the slope of the composite line fi= 2.9.

If a single kinetic law (i.e., one value of n) applies, it should be possible to superpose x=f(t) relations by multiplying the unit of time for each isotherm by a suitable factor, f, “superposition factor.” All of the isothermal relations were thus superposable. Figures 3a and 3b show the superposition of representative iso- therms for two cast wires. The curves were made to coincide at x=0.5 with f= 1.00 at 273°K.

For xyO.5 all isotherms are satisfactorily described by a straight line relation between log ln[l/ (1 -x)] and logt as predicted by Eq. (7). The slopes, n, of these lines for each of four cast wires, having X0 ranging from 0.112 to 0.170, at a series of temperatures are given in Table II. fi, the average n value, is also shown. The relation between log ln[l/ (1 -x)] and logt at repre- sentative temperatures for three of the wires are shown in Figs. 4a, 4b, and 4c.

The average of all the n-values for cast wires is 3.0 with a mean deviation of ~1~0.15 and, excepting for one specimen, the maximum deviation is 0.4. In a single cast

FIG. 3a. Superposition of kinetic relations for different temperatures. Wire 3C, X0=0.147.

FIG. 3b. Superposition of kinetic relations for different temperatures. Wire 15C, X0=0.170.

1 0.40

030

A 0.20

FIG. 4a. Dependence of Log ln[l/l--xl on logt at various temperatures. Wire 17C, X0=0.112.

0.0 -

-a5

;

$1.0~

3 -1.1)

-2.0

FIG. 4b. Dependence of Log ln[l/l--x] on logt at various temperatures. Wire 3C, X0=0.147.

o"l---------1050 - 030

- 020

x

- 010

005

001 -030 0.0 020 0.40 060 080

LOG 1

FIG. 4c. Dependence of Log ln[l/l--21 on logt at various temperatures. Wire 15C, X0=0.170.

specimen (21C, X0=0.170) the maximum deviation ex- ceeded 0.4, n being 2.3 in two successive experiments at 273°K. There is no discernible trend of n with either X0 or T.

Isotherms for the extruded wires also were well described by Eq. (7) as is shown in Fig. 5, but the slopes of these lines (see, for example, Table III) tended to scatter more widely than did those for the cast wires.

Dependence of Precipitation Rate on Temperature

From Eq. (7) with n=3, we obtain

~=3b1’3(1-x)[-ln(l-x)]2/3. (9)

48 AACTA METALLURGICA, VOL. 3, 1955

FIG. 5. Dependence of Log In[l/l--x] on logt at various temperatures for an extruded wire. X0=0.147.

Hence, li: =A,&/3 c , (10)

whereA,=3(1-c)[-ln(l-c)]2~3. Let t=~, when z=c. It then follows:

l/rc= B,b”3. (11)

where B,=[ln(l/l-c)]-1’3. Combining (10) and (11) we get :

kc= (Ac,‘&)W~c). (12)

Hence it follows for constant X0:

d ln(l/T,)/d(l/T)=d ln&/d(l/T)

and for constant T: =d ln(b)‘/3/d(l/T), (13)

d ln(1/7,)/dXo=d ln&/dXo=d ln(N3)/dX0. (14)

Thus, the shapes of the In& vs l/T or XO and ln( l/Tr) z’s l/T or X0 curves will be identical for all values of c. Borelius” curves of ln(l/T& vs l/T and X0 therefore also describe the shapes of the ln(b)1’3 vs l/T or XO curves as well as the shapes of In& vs l/T or X0 for any value of c.

Logb for each of the representative isotherms is given in Tables II and III. However, for purposes of inter- preting the results of thermal cycling experiments, it is more convenient to consider the variation of precipita- tion rate with temperature and initial tin concentration. Therefore, we have also listed in Tables II and III &,.2 for each of the isotherms.

In Fig. 6, In&.2 is plotted against l/T for each of the initial concentrations. These points are shown as circles and the curves through them are solid. The shapes of these curves are qualitatively very like the shapes of the

TABLE III. Summary of kinetic results on an extruded wire (X0=0.147).

Temp.‘K ?I -Logb

273 3.4 0.062 3.45 333 2.9 0.229 1.71 363 3.8 0.068 3.33 373 2.9 0.032 4.45

%=3.25

d ln(l/& vs l/T relations found by Borelius ef al’ at the corresponding initial concentrations. Below some temperature T’ characteristic of X0, d ln&.a/d(l/T) be- comes virtually independent of X0. Our relationship between T’ and XO agrees within experimental error with T’=f(Xo) found by Borelius et ~2.

Although we are in fair agreement with Borelius et al on the dependence of precipitation rate on temperature and initial concentration, the absolute magnitudes of our rates are considerably less than they report. In Fig. 7, ln(l/TIJ vs l/T found for a cast wire (25C, X0=0.124) in this investigation is compared with In (l/& = f( l/T) reported for the same composition by Borelius et ~2. Our l/71,2 values are about a factor of 2 to 3 smaller than the corresponding values of Borelius et al. We found that the rates in extruded wires are significantly larger than in cast wires (cf. Tables II and III). Our conclusion is that the absolute magnitude of the precipitation rates are considerably dependent upon the concentration of structural impurities in the solid

FIG. 6. Dependence of logarithm of precipitation rate (at .~=0.2), after various temperature cycling treatments, on l/T.

solution. These results further demonstrate the inva- lidity of interpretations of existing kinetic data for this system assuming homogeneous nucleation and growth of precipitate particles in structurally pure crystals.

Results of Thermal Cycling Experiments

Thermal cycling experiments sometimes provide in- formation that is useful in separating the nucleation and growth components of precipitation rates.* In such ex- periments precipitation is completed at a temperature Tz after having been begun at a different temperature TI. The initial rate of precipitation, Si at the second temperature, where (x= xi) Ts, should be due only to the growth of p particles nucleated at T1. Thermal cycling experiments in which the first temperature T1 is greater than the second temperature Tz will be designated as down-cycling experiments and the inverse type of ex-

* See, for example, Wert’sio experiments on the precipitation of carbide and nitride from or-iron.

TURNBULL AND TREAFTIS: KINETICS OF PRECIPITATION 49

periment in which Tz> T1 as up-cycling experiments. In either event the precipitation rate at the second temper- ature Tz and degree of precipitation X=G, will be denoted by the symbol k’,. Tests in which precipitation begins and goes to completion at the same temperature will be designated direct precipitation experiments and the corresponding precipitation rate denoted by kc. It may be possible to relate k’J& to N’/N where &v’ and *%’ are the number of nuclei operative at Tz as a result of the thermal cycling and direct precipitation processes, respectively.

Let x= f(t) for direct precipitation at Tz, and x= fl(l) at Tz after some prior precipitation at T1. ji(t) is defined to coincide with f(t) at x=xi. Initially we calculated f*(t) on the assumption that the resistance, R, of any wire would be a single function of x, at constant temper- ature, independently of thermal history. Figure 8 com- pares j(t) and fi(t) so calculated from a series of experiments in which X0=0.147, T1=373 and TZ =273”K. fl(t) calculated in this way approaches a

0 I

IIwaIus El.&

FIG. 7. Comparison between precipitation rates obtained by Horelius el al and in this investigation for X0=0.124. (Our wire 25C).

limiting value less than unity, XI, that is smaller the larger is xi. Table IV gives l--xl for each zi.

This failure of fi(t) to approach unity and the ap- parent dependence of 1 -xl on xi is accounted for if it is assumed that :

1. Precipitation is effected by the growth of “cells.” Each cell consists of a multiplicity of p-particles, all originating from a single nucleus, interspersed in (Y.

2. The atom fraction, Xf, of tin within a cell formed at temperature, T, is related to the equilibrium concen- tration, X,, of tin at that temperature by the equation:

x0-x/ -------Cc, X0-X,

where C is temperature independent. The basis of this relation is not yet understood.

3. Upon quenching to Tz there is no appreciable precipitation of tin within the parts of cells grown at T1.

Until swept over by a cell boundary, the concentra- tion of tin in the supersaturated solution remains at X0.

FIG. 8. Dependence of x on t at 273°K after prior precipitation at 373°K to apparent X=X<. Wire 18C, X0=0.147.

Thus, j(t) must be identified with the volume fraction of the specimen transformed by the growth of the cells. In a thermal cycling experiment, cells nucleated at one temperature will develop a two layer structure upon growth at a second temperature. This structure will consist of a “core” formed at T1, wherein the final tin concentration Xf=Xo-C[Xo- (X,)&J, and a “shell” formed at Tz, wherein X,=X0--C[Xo- (X,)Q]. X, is given by Eq. (4). Since the final tin concentration within

TABLE IV. Comparison between 1 I XI and f”(t) -~~ a).

xi (uncorrected) (1 --II) f” @I -h (0

0.0 0.0 0.04 0.04 Z& 0.08 0.09 0:OS 0.18 0.19 0.18 0.23 0.22 0.23 0.30 0.36 0.30 0.46 0.44 0.46

a cell element is determined by the temperature at which it formed it is evident that the specimen resist- ance cannot be a single function of x at constant temperature.

With the use of the assumptions explained above, we can calculate the correction that must be applied to convert jl(t) to transformed volume fraction, jU (t). This correction, ju(t)- ji(t), should be equal to l-xl. l--xl and jv(t) - Jo, obtained from the results on wire 18C, are in excellent agreement as may be seen from Table IV. jv(t) from this series of experiments is plotted against t in Fig. 9. Here the initial transformed volume fractions,

FIG. 9. Same results as Fig. 8 but with x and xi corrected to correspond to transformed nolume fraction.

ACTA METALLURGICA, VOL. 3, 195.5

TlYLlYlH,

FIG. 10. Dependence of x on t at Tz after prior precipitation to x=xi at Tt. TI<T’ and Tz<T’. Extruded wire, X0=0.147.

xi, at Tz have been corrected by the addition of

m-.flW

Where both T1 and Tz were less than a characteristic temperature, T,, the precipitation rate proved to be independent of the thermal cycling treatment, at least for xiqO.2. In this event, f,,(t)=f(t). Figure 10 shows the excellent correspondence between fV(t) and f(t) for an extruded wire cycled between 333°K and 273°K. Similar results from experiments with a cast wire are given in Fig. 11. T, corresponds, within experimental error, to the temperature T’ at which d ln&/d(l/T) becomes independent of X0. These results indicate that the distribution of cell number with size at constant x is independent of temperature when T < T’.

When T>T’, the precipitation rate is clearly de- pendent upon thermal history (see Fig. 8). Figure 12 compares the kinetic relations, fV(t) and f(t), obtained from cycling an extruded wire from a low (Tl= 273’K) to a higher temperature (TX= 373’K). For this composi- tion (X0=0.147) 273<T’<373”. At 373°~‘~.z/&~= 1.7.

A series of up-cycling experiments were carried out with T1=273’K and the initial volume fraction pre- cipitated at this temperature, xi, approximately equal to 0.02. The logarithms of the resulting precipitation rates at x= 0.2 [logY”.n = logp&.t] are shown (triangular points) as a function of l/T and X0 in Fig. 6. p is defined by the equation :

p=LlY&:,. (16)

When Tz is less than the characteristic temperature, the rates at T2 resulting from up-cycling and direct pre- cipitation are equal. However, s?o.z>&.z when Tz> T’.

I fl I 06

x /

Frc. 1 I Dependence of x on I at 273°K after prior preril)it:tl ion In x=x; at 333°K. Wire 4C, X0=0.147.

Down-cycling experiments were carried out with Tz=2730K and TI varied between 393” and 298°K. Figures 13a and 13b show several down-cycling iso- therms x=f,,(t) at 273°K corresponding to various temperatures T1 of initial precipitation for cast wire 2OC, X0=0.147. In these experiments p was evaluated from the initial rate of precipitation at 273”K, i.e., at x= xi. When T1 exceeded the characteristic temperature the initial precipitation rate at 273°K was less than the corresponding direct precipitation rate but these rates (&and 2:‘s) were equal when Tl< T’. In Fig. 6 log(&&p) (square points) from the down-cycling experiments is plotted against l/T for different X0. Here p is evaluated at x=xi and (i~.~/p)=f(T~).

Effect of Cold Working

It is well known” that cold-working a supersaturated solid solution greatly increases the rate of precipitation. We did not attempt a complete evaluation of the effect of cold-working on the precipitation of p from (Y but made a few experiments on this effect that are pertinent to the evolution of our interpretation.

I 4 P I

FIG. 12. Dependence of x on 1 at 373°K after prior precipitation to I=xi at 273°K. Extruded wire, X0=0.147.

The change of resistance of a cold-worked super- saturated solution is the sum of two factors, one due to recovery and the other to precipitation. However, we found that the displacement of resistance due to cold- working the a! at a low temperature is negligible in comparison with the total change in resistance due to precipitation at a low temperature, e.g., 273°K. There- fore, it appears that x= f(t), for a cold-worked specimen, calculated with the use of Eq. (8) should have at least qualitative validity.

Figure 14 shows the effect of straining an extruded wire at liquid nitrogen temperature 9 per cent in tension prior to precipitation at 273°K. Figure 1.5 compares s=f(t) for a cast wire at 273°K (a) without prior strain and (b) hammered flat at 273°K immediately after quenching from the homogenization temperature (in a period short in comparison with the precipitation time).

From these and other of our results we conclude the following :

1. Cold-working the cy greatly increases the rate of tin precipitation. The effect increases with strain and :ti,

TURNBULL AND TREAFTIS: KINETICS OF PRECIPITATION 51

for a hammered specimen was as much as four times larger than & for a specimen that had not been strained.

2. There is no such simple functional relationship between x and t as Eq. (7) for strained specimens. If we attempt to describe the data with Eq. (7), n is less than 3.0 and decreases with increasing x.

3. The effect of strain on C& is most pronounced in the early stages of precipitation. For x>O.6 there appears to be little difference between kc values for the hammered and unstrained specimen (see Fig. 15).

Microstructural Observations

We deduced from the kinetic results that the pre- cipitate must form by the growth of cells containing a

FIG. 13a. Dependence of x on t at 273°K after prior precipitation to X=Xi at Ti. Wire 2OC, X0=0.147.

FIG. 13b. Dependence of x on t at 273°K after prior precipitation to X=X< at Ti. Wire 2OC, X0=0.147.

multiplicity of @-particles dispersed in (Y. Recently Tiedema and Burger$ established that such cells do indeed form and grow in CY lead-tin alloys. It appeared that P-lamellae were dispersed within the cell (Y. Mr. Donald Wood of our laboratory independently verified the cell-like mode of precipitation in this system (see Fig. 16a) and clearly resolved (see Fig. 16b) the lamellar P-precipitate. Wood found that the cells nucleate prefer- entially at grain boundaries. Tiedema and Burgers’ alloy specimens were single crystals but the cell nuclei formed at dendrite boundaries where the composition segregation would be highest. There is some possibility that these nuclei came from occluded crystals.

This cellular mode of precipitation has been observed in many systems and is known as the “recrystallization” reaction (we shall call it cellular transformation).

TlYE WINI

FIG. 14. Dependence of x on t at 273°K for unstrained wire and for same wire strained 9 per cent in tension at boiling temperature of liquid nitrogen. Extruded wire, X0=0.147.

Geisler13 has reviewed its essential features. Usually the cells nucleate at grain boundaries and a reorientation of the ~1 matrix accompanies their growth. In the lead-tin system there is no evidence that any mode of precipita- tion other than the cellular reaction is operative.

INTERPRETATION

The Kinetic Law

The kinetic law for nucleation and growth trans- formations is usually of the form of Eq. (7) with n ranging between 1 and 4. How n is related to the mechanism of nucleation and growth has been reviewed in other publications.10J4v’5

We have to account for n-3 with rare departures that appear to be greater than the experimental error. From microstructural results, we know that (a) the cells have a form that is nearly spherical, (b) the cell dimensions are negligible in comparison with the wire diameter, and (c) the interlamellar spacing of fi within the cells is approximately constant. We picture a cell growing uniformly in three dimension with a constant density (as evidenced by the constant interlamellar spacing) of tin diffusion sinks (F-edges) in its surface. Therefore, the cells should advance linearly at a rate independent of time. How the constant surface density of P-edges is maintained will be discussed in a following publication.

If the cell grows linearly at a constant rate, we can only account for n= 3 if the number, A+, of cells/volume remains constant throughout the precipitation. In this event the constant b of Eq. (7) is given by:

b = (47r/3)A7G3, (17)

FIG. 15. Dependence of x on 1 at 273OK for unstrained wire and for same wire hammered flat at 273°K. Wire 6C, X0=0.147.

52 ACTA METALLURGICA, VOL. 3, 1955

FIG. 16a. Cast film after 16 minutes at 300°K. X0=0.124. Magnification looOX.

where G is the linear rate of cell growth. The constancy of N implies that all the cell nuclei are formed in the earliest stages of precipitation. Since the initial super- saturation of the o-matrix is not reduced until swept by a cell boundary, it follows that the nuclei must originate at a limited number of singular sites. Activation of nuclei at these sites occurs either athermally during quenching or isothermally at the beginning of transfor- mation. N and G will depend upon X0, T, and strain.

Wood finds that the cells nucleate preferentially at grain boundaries. If N is of the same order of magnitude or smaller than the number, g, of solid solution grains per unit volume, we should expect n=3 (for x not too large) as though nuclei had formed randomly. However, it is possible for the specific number of nuclei at grain boundaries to be so great (that is, N>>g) that all the grain boundary area is coated with cells while the amount of precipitation is still very small. From this point the total transformed volume should increase at a

FIG. 16b. Cast fdm after 16 hours at 300°K. X0=0.112. Magnification 2000X.

constant rate so that n would be unity. For the more general case, n should fall between 1 and 3 approaching 3 for x-+0 but decreasing below 3 as x increases. In our lead-tin wire, ‘17 must have been of the same order of magnitude as (or less than) g since n-3 for ~~0.5.

Occasionally n found in experiments with extruded wire was significantly larger than 3.0. This result may indicate that thermal nucleation in some instances persists throughout the transformation period. Relief of the transformation strain by plastic flow of random elements of the matrix would also lead to n>3.0.

Kinetic Law in Thermal Cycling Experiments

In the following discussion 2 will represent the rate of precipitation that results from direct quenching to Tz from the homogenization temperature while k’ will denote the rate at TB after some precipitation at Ti. In interpreting these results it will be assumed that G is independent of thermal history.

Let precipitation proceed for time t at TS after transformation to x=xi at T1 (t=O, x=xi). Then:

~‘=4n(1--)[N1Gz(ri+Gzt)2+N2G23tZ], (18) where

Nr= number of cells nucleated at T1. Nz= number of cells nucleated at Tz. ri=[(3/47rNl) ln{l/(l-xi)}]1/3; (19)

that is, ri=radius of cells at x=x;.

When NC+Ni and for x very large it may be that Eq. (18) does not adequately compensate for impingement ; however, we shall draw conclusions from (18) only in limiting cases where (18) should be a satisfactory ap- proximation. Integration of (18) gives :

ln[(1--x~)/(l-~)]=4?mNl(G~ri2t+r,G22t2)

+ (4r/3)Gz3t3(Nl+X2). (20)

When t = 0, (18) becomes :

k’i=47r(1-xi)NrGzri2.

For xi=O, we have:

(21)

~=47r(1--3C)N‘&23t~. (22)

We deduce from Eqs. (19), (21) and (22) that

Ci?i,/ki= (Nr/N2)1’3Sp. (23)

Equation (23) is valid if there is no precipitation or resolution of the solute in the cells formed at T,. Provided xi is small and Ni>>AT2, it follows:

(k’e/&) = (Nr/M#‘3. (24)

Figure 17 shows the decomposition of f?(t) for a down- cycling (373-273”) experiment into two components. One of the components derives from nuclei formed at 273“ and is calculated from the f(t) curve for 273” also shown in Fig. 17. The second component results from nuclei formed at 373O and is obtained by subtracting the

TURNBULL AND TREAFTIS: KINETICS OF PRECIPITATION 53

first component from fV(l). The second component is approximately linear over the intermediate range of x values as expected on the basis of Eq. (20). The decomposition was not carried further because the impingement correction becomes more uncertain as

fd0 increases. From Eqs. (13), (17), and (23) with X0 constant we

can relate the rate of precipitation to the rate of cell growth as follows :

and @c/P) T+T~=Y~G, (254

(&p) T84T= YLG. (25b)

Here yc and T, are constants and S& and G correspond to T which is variable. The subscripts refer to the thermal cycle; in (25a) T1= T, Tz= T,, while in (25b) T1=T,y and Tz= T.

Figure 6 shows 10g(~i.o.z/p)T~Ts= fi(l/T) or 10g(fo.$I)T8+T= fi(l/T) plotted against l/T for four XO values with T,=273” and T> T,. For T less than the characteristic temperature, the dependence of cell growth rate on T is identical whether it is deduced from down-

TlYE ‘YW,

FIG. 17. Decomposition of z=/“(t) from a down-cycling experiment into two components. Wire MC, X0=0.147.

cycling or up-cycling experiments ; that is, fl( l/T) = fi(l/T) when T < T’. However, when T> T’ the growth rate inferred from down-cycling experiments is larger than the corresponding rate inferred from up- cycling experiments [fi(l/T)> fi(l/T) when T>T’]. In the down-cycling experiments (T-+T,)p was calcu- lated from (23) but in the up-cycling experiments (T,+T) there was some tendency for ,9 to redissolve at the highest temperatures so xi was kept small (~0.02) and p evaluated from (24) with c=O.2. Because of this resolution in up-cycling, we believe that fi(l/T), (Eq. 25a), may be a more satisfactory representation of the dependence of log(yG) on l/T than is fi(l/T).

In any event we can draw the following conclusions from the thermal cycling experiments:

1. At constant X0, N increases with decreasing T until T= T’. For T< T’, N is constant.

2. With X0 constant, 1nG is not a linear function of l/T, possibly excepting the temperature range T<273”. G increases with increasing T to a maximum G=G,,, at T= T,, and then decreases sharply with further in-

crease in T. At T= T,,, the supersatruation ratio (X,/X,) is of the order of 5 or 6.

Further Remarks

As an electrical conductor, our precipitating system consists of spherical elements (cells) having a charac- teristic conductivity, X,, imbedded in (Y matrix of con- ductivity, X,? Landauer I5 has discussed the dependence of the resultant conductivity of such a system upon the concentration and size of cells. From Landauer’s treat- ment we infer that while the validity of our Eq. (8) is questionable for z very large, it ought to satisfactorily describe the dependence of R on x when x20.4.

We are in essential agreement with Borelius and co- workers on the facts, so far as they determined them, of precipitation in the lead-tin system. We have shown that their l/71,2 is proportional to &, where c may range from 0 to 0.5. We are in good agreement with them on the variation of 1/~~,2 with X0 and T though in absolute magnitude our l/r112 values on cast wires are con- siderably smaller than theirs. Also, we find that d log&/d(l/T) or d log(l/r&/d(l/T) becomes inde pendent of X0 at some temperature T’= f (X0)-that is, about the same as found by Borelius et al. Their limited results on the kinetic law appear to be consistent with our findings. However, Borelius et al report no micro- structural or thermal cycling experiments that have given us basic information on the precipitation mechanism.

There are some important questions not resolved by our formal interpretation. These are:

1. Why, as shown by the microstructural and kinetic evidence, precipitation of tin from lead occurs at a rate 10’ to lOa times greater than calculated from Seith and Laird’s results on the diffusion of tin into dilute solutions of tin in lead.

2. What are the reasons for the complexity of the dependence of 1ogG on l/T.

3. Why does N become temperature independent below some critical temperature T’.

4. What is the mechanism of the slow reaction.

These questions will be dealt with in the following paper entitled, “Theory of Cellular Precipitation.”

CONCLUSIONS

1. Precipitation of tin from lead takes place in two distinct stages. The first stage is a “fast” reaction that drains about 60 per cent of the thermodynamic excess of tin from solution. The remaining 40 per cent of the tin is drained at a rate (slow reaction) about 2 orders of magnitude slower than the fast reaction.

2. The kinetic law for the fast reaction is:

x=1-exp(-bt”).

With rare exceptions, n=3.0 independently of tempera- ture or initial tin concentration.

54 ACTA METALLURGICA, VOL. 3, 19.55

3. Precipitation is effected by the nucleation and growth of cells. Each cell consists of a multiplicity of @- particles, all originating from a single nucleus, inter- spersed in (Y. This mode of precipitation has been re- ferred to in the literature as the “recrystallization reaction.”

4. We deduced that b is related to the number N of cell nuclei and their rate of growth G as follows:

b= (4r/3)NG3.

5. The precipitation rate &, at constant transformed volume and temperature, is independent of thermal cycling treatments (wherein precipitation is begun at one temperature and completed at a different tempera- ture) when the cycling temperatures are less than a characteristic temperature, T’. & is strongly dependent on the thermal history if one of the cycling temperatures exceeds T’.

6. Even for T < T’ the precipitation rate is strongly dependent upon the initial concentration, X0, of tin; for example, when XO increases from 0.112 to 0.170, Z& increases by a factor of about 15.

7. The kinetic law is satisfactorily interpreted if it is assumed that all cells have nucleated at the beginning of precipitation and originate from a limited number of singular sites mostly in the vicinity of grain boundaries. Therefore, interpretations of the precipitation kinetics in this system that have presupposed nucleation in structurally pure crystals have to be abandoned.

8. At constant initial tin concentration, the number of nuclei increases with decreasing T until the charac- teristic temperature is reached. N is constant at temper- atures equal to or less than T’.

9. G increases with increasing T to a maximum G,,,

at T,,, and then decreases sharply with further increase in T. At T,,, the supersaturation ratio is 5 or 6.

10. Precipitation takes place about 10’ to lo8 times faster than calculated from Seith and Laird’s results on the diffusion of tin into dilute solutions of tin in lead.

11. Cold-working the supersaturated solid solution greatly increases the precipitation rate.

ACKNOWLEDGMENT

We express our thanks to E. W. Balis and W. W. Welbon for the tin analyses and to Donald Wood for his work on the microstructures. We are pleased to acknowl- edge valuable discussions with A. H. Geisler on the nature of the “recrystallization reaction.” Also, we are indebted to Professor G. Borelius for sending us his unpublished calorimetric results on the isothermal pre- cipitation of tin from lead.

1.

2.

3. 4. 5.

6. 7. 8.

1:: 11. 12.

13.

:“;:

REFERENCES

G. Borelius, F. Larris, and E. Ohlsson, Arkiv Matematik, Astronomi och Fysik 31A, No. 10 (1944). G. Borelius and K. M. S%fsten, Arkiv Matematik, Astronomi och Fysik 36B, No. 5 (1948). J. Nystrom, Arkiv for Fysik. 1, 359 (1949). H. K. Hardy, J. Inst. Metals 77, 457 (1950). J. H. Hollomon and D. Turnbul!, Progress in Metal Physics, Vol. 4: 333, Pergamon Press (1933). W. Serth and J. G. Laird, Z. Metallkunde 24, 193 (1932). C. Zener, Private communication. F. Seitz, L&at Solide, Institut International de Physique Solvay, pp. 401-405 (1952). D. Turnbull and R. E. Hoffman, Acta Met. 2,419 (1954). C. A. Wert, J. Appl. Phys. 20, 943 (1949). B. L. Averbach, Trans. A.S.M. 41, 262 (1949). T. J. Tiedema and W. G. Burgers, Applied Scientific Research 4A. 243 (1954). A. H. Geisler, “Phase Transformations in Solids” (John Wiley 1951), pp. 387-545. C. A. Wert and C. Zener, J. Appl. Phys. 21, 5 (1950). R. Landauer, J. Appl. Phys. 23, 779 (1952).