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The kinetics of discontinuous precipitation in copper-indium alloys
THE KINETICS
OF DISCONTINUOUS
PRECIPITATION
J. M. SHAPIRO?
IN COPPER-INDIUM
ALLOYS*
and J. S. KIRKALDYS
Supersaturated a phase copper-indium alloys decompose by both a continuous and discontinuous mode. An additional fine precipitate, called 6’, forms in the 10 at.% In alloy. The general precipitate does not The growth rate, v, and lamellar spacing, S, of the disusually interfere with the cellular reaction. continuous precipitate were measured, as well as the average a phase composition for some heat treat ments. For a given initial composition the reciprocal of spacing extrapolates to the value zero at a electron micrographs of the advancing interface region am temperature below the solvus. Transmission presented. Turnbull’s interface diffusion model yields a Various theories are evaluated using the kinetic data. reasonable growth rate, but the activation energy of the interface diffusion coefficient, bDB, appears The large S values support Cahn’s boundary diffusion and reaction control model. high, 36.5 kcal/mole. However, the boundary mobility required here is about l/1000 of that estimated from mobility-controlled reactions in other copper alloys. Kirkaldy’s proposal that discontinuous precipitation is a metastable An activation energy for bDB of 25 monotectoid reaction controlled by interface diffusion is tested. kcal/mole is obtained and a plausible explanation is given for the abnormally large spacings near the solvus temperature. Sluggish volume diffusion ahead of the interface (required by the local equilibrium hypothesis) may account for the relatively large spacings at all temperatures. CINETIQUE
DE LA PRECIPITATION DISCONTISUE LES ALLIAGES CUIVRE-INDIUM
DASS
Les alliages cuivre-indium sursatures en phase cc se decomposent suivant un mode it la fois continu et discontinu. Dans l’alliage a 10% at. In, il se forme un fin precipite supplementaire 6’. Le precipite continu ne gene habituellement pas la precipitation cellulaire. La vitesse de croissanoe Y et la distance lamellaire S du precipite discontinu ont Bte mesurees, ainsi que la composition moyenne de la phase a pour differents traitements thermiques. Pour une composition initial0 don&e, l’inverse de la distance s’annule par extrapolation pour une temperature inferieure L celle du solvus. Les auteurs presentent les micrographics Qlectroniques par transmission de la progression de la region de l’interface. Des theories variees ont et& utilisees pour Bvaluer les coefficients cinetiques. Le modele de diffusion a l’interface de Turnbull donne une vitesse de croissance raisonnable mais l’energie d’activation du coefficient de diffusion a l’interface bDs parait hlevee, 36,5 kcal/mole. Les valeurs Blevees de S confirment le modele de diffusion de la frontiere et de controle de la reaction de Cahn. Cependant, la valeur de la mobilite de la frontiere requise ici est environ Qgale a l/1000 de la valeur deduite des reactions controlees par la mobilite dans les autres alliages de cuivre. Les auteurs ont essay& d’utiliser l’hypoth&se de Kirkaldy suivant laquelle la precipitation discontinue est une reaction monotectoIde metastable contr8lee par la diffusion it l’interface. 11s ont ainsi obtenu pour bDs une Bnergie d’activation de 25 kcal/mole et, donnent une interpretation raisonnable pour les distances anormalement grandes correspondant aux temperatures voisines de la temperature du solvus. Une diffusion en volume lente, en avant de l’interface, (et necessitee par I’hypothese de l’equilibre local) pent just,ifier les distance* lamellaires relativement grandes it toutes les temperatures. DIE
KINETIK
DER
INHOMOGENES
AUSSCHEIDUNG
IN
KUPFER-INDIUM-LEGIERUNGEN Ubereiittigte Kupfer-Indium-Legierungen (a-Phase) zersetzen sich sowohl durch einen kontimuerlichen als such durch einen diskontinuierlichen Mechanismus. In der 10 At.% In-Legierung bildet sich ein zusiitzlicher feiner Niederschlag, genannt 6’. Die gewdhnliche Ausscheidung stiirt die Zellreaktion normalerweise nicht. Es werden sowhol die Wachstumsgeschwindigkeit 2) und der Lamellenabstand S der inhomogenen Ausscheidung als such die durchschnittliche Zusammensetzung der a-Phase fir einige Wiirmebehandlungen gemeseen. Der extrapolierte reziproke Abstand geht bei einer gegebenen Anfangszusammensetzung bei Temperaturen unterhalb der Solvustemperatur gegen Null. Es werden elektronenmikroskopische Durchstrahlungsaufnahmen der fortschreitenden Grenzflachenbereiche gezeigt. Verschiedene Theorien werden unter Verwendung der kinetischen Daten untersucht. Turnbulls Model1 der Grenzfliichendiffusion gibt verntinftige Werte der Wachstumsgeschwindigkeit, aber die bD B, erscheint hoch (36,5 kcal/Mol). Aktivierungsenergie des Koeffizienten der Grenzfliichendiffusion, Die grossen S-Werte sprechen fur Cahns Model1 der Grenzendiffusion und der Reaktionskontrolle. Die hierbei erforderliche Grenzen-Beweglichkeit ist jedoch nur etwa l/1000 der aus beweglichkeitskontrol. lierten Reaktionen in anderen Kupfer-Legierungen abgeschatzten Beweglichkeit der Grenzen. Kirkaldys Vorschlag, dass die inhomogene Ausscheidung eine metastabile, monotektische, von der Grenzflachen. diffusion kontrollierte Reaktion ist, wird gepriift. Man erhiilt eine Aktivierungsenergie von 25 kcal/Mol fur bDB und es wird eine plausible Erklarung fur die ungewohnlich grossen Abstlinde in der Niihe der Solvustemperatur gegeben. Eine langsame Volumendiffusion vor der Grenzflache (gefordert auf Grund der Hypothese des lokalen Gleichgewichts) konnte die Ursache der relativ grossen -4bstande bei allen Temperaturen sein.
* Received August 7, 1967; revised January 29, 1968. Extracted in part from the Ph.D. Thesis of McMaster University, 1966. t Research Department, Inland Steel Co., East Chicago, Ind. 46312. $ Department of Metallurgy and Materials Science, McMaster University, Hamilton, Ontario, Canada. ACTA
METALLURGICA,
VOL.
16, OCTOBER
1968
1239
J. M. Shapiro,
INTRODUCTION
AND
THEORY
DisconGnuous or cellular precipitation is said to occur when a supersaturated phase (CC’)decomposes to the structurally identical, but solute depleted cc phase and a new ,8 phase, by the growth of cells or nodules of the parallel Q + ,S mixture (usually ia~llellar) into the a’ phase. This react‘ion appears rnetallographically identical to cooperative eutectoid decomposition, e.g. pearlite growth in steels. as shown by t,hc example in Fig. 1. The cooperative side-byside growth of the a and /2phases results in regular kinetics for the isothermal reaction, i.e. constant lamellar spacing, 8’: and growth rate, c. This regularity has prompted theoretical d~liberat~io~lsin two main directions. The first seeks a~fundamental cause of t,he cellular rear&on. Most such theories cite a correlation of structural or mechanical properties of various alloy systems.c1-7) A thermodynamic basis for the reaction has been proposed by Kirkaldy.(s) The morphological similarity to eutcctoid decomposition suggests that the cellular reaction occurs in a metastable monotectoid, as indicated in Fig. 2. The other class of theories considers the mechallisnls whereby the precipitate achieves the observed growth rate and spacing. It is usually assumed that diffusion along the advancing interface accounts for the segregation of solute to the /?lamellae. This transport mode has been shown t,o be both sufficient2 and
FIG. 1. Typioal celluIar precipitate in G-7.5
necessary for the observed kinetics.@*rO) From a dimensional argument Turnbull proposed the following growth equation : 21=
(Xa’ X”’
X”) bDB --19
(I)
where b is the effec%ive boundary thickness, DB is the diffusion coefficient in the boundary, and X”’ and X” are the initial and equilibrium compositions respectively, of the cc phase. The spacing chosen is that predicted by Zener for pearlite~ol) viz.,
where 8 is the energy per unit area of c@ interface, I’ is the molar volume of the alloy, and AFo is the total free energy available from the transformat,ion. Cahn presented a model which includes both interface mobility and cell boundary diffusion contJroI.02) It was also assumed that the reaction proceeded with a maximum rate of free energy decrease. Various quantities could be calculated as a function of the parameter kM P( iPa)2 /3=_---6% AE;,
(3)
Mrherek is the rat,io of t~he~oncent’ration of solute in
a&y0 In, aged 10 hr at 420°C.
x 1530
SHAPIRO
AND KIRKALDY:
DISCOSTIKUOGS
PRECIPITATION
IX
Cu--In
ALLOYS
1241
where q and a are thermodynamic parameters and
(6) the departure of the composition of the product phases from their equilibrium values ; and the curvature of the a’a or a’p interface as a function of K. The determination of K is an optimization problem, and cannot be obtained through the application of equilibrium thermodynamics alone. The relative merits of each of the above theories will be examined later in the light of the experimental observations. Previous work on the w~fper-&&urn system
MOLE FRACTION FIG. 2. Free energy curves and phase diagrams for the metastable monotectoid reaction.
the interface to that in au cc lamella, and M is the mobility of the boundary, expressed as M
=
2
AF
(4)
where AF is the amount of free energy dissipated in the reaction. The factors in the paramet,er @ are specific to each alloy system at any given temperature. The quantities calculated are the growth speed, the spacing, the fraction of material precipitated, the fraction of AF,, which is not stored in the supersaturated product, and the fraction of AF, expended as a-@ surface energy. The assumption that discontinuous precipitation occurs in a metastable monotectoid allows one to apply to this reaction the results of the local equilibrium growth model derived for the symmetric eutcctoid reaction in a work henceforth referred to as Article A.03) The results are, briefly: an expression for the growth rate and spacing, vs3
=
48bhW - 1) 4r(-k - aI2
(5)
The solubility limit of the f.c.c. a phase shown in Fig. 3 is taken from Jones and Owen.(‘Q Hansen and Anderko show that the S phase, nominally C&In,, has a composition range from 29.0 to 30.6 at.% In at all temperatures below 613”C.(r5) The 6 phase precipitates from su~rsaturated a by both the continuous or general and discontinuous modes. Bohm observed that no general precipitate formed below 0.8 of the absolute solvus temperature, using light optical and X-ray metallography.d6*17) However, Corderoy and Honeycombe noted a general precipitate in 8.5 and 10 at.% In alloys at all temperatures above 250°C by means of electron microscopy. Nodule growth was eventually halted by the continuous depletion of the matrix. Biihm measured the growth rate of the discontinuous precipitate by direct optical microscopy, and the spacing by the method of Turnbull and Treaftis.d$) It was concluded from X-ray metallography that the a phase formed with the equilibriuln composit,ion.
9
-I -I 0
2
4
6
8
IO
12
INDIUM ATOMIC PERCENT FIG. 3. Section of the copper-indium phase diagram. A Temperature for which l/8 + 0 from Fig. li. 0
Cellular a phase composition m Cellulltr c( phase composition
for initially 7.5% In alloy. for init,lalty 10.1% In alloy.
ACTA
1242
METALLURGICA,
In the present continuous
contribution
precipitate
by light optical
may be considered
reaction
is determined.
spacing
and
of
The
a’/3 interfaces Alloys
shapes
purity.
a
of the other
speed, lamellar
composition the
are
phase
remeasured
composition
of the advancing
a’a
is and
are recorded.
were prepared
from
The composition
diffraction,
of
and
and thin foil
independent
The growth
a phase
uniformity
estimated.
is made
The extent to which the cellular
reaction
The
a study
and sequence of the discontinuous
electron microscopy.
materials
of 99.999%
was determined
using established
by X-ray
lattice parameter
versus
composition data. (14) Precipitation treatments were carried out’ in a stirred salt bath or in a forced convection
furnace.
were *l”C
in the salt bath during the longest times,
and from 10.5”C in
the
precipitation
temperature
variations
Water
quenching
in 40 days
followed
the
anneal.
Techniques employed
Typical
during short runs to *2”C
furnace.
standard
to prepare
for
copper
alloys
the rod specimens
were
for optical
and electron microscopy. (20) Some preferential electropolishing
of
the
Elmiskop
I, operated
tc phase
was
noted.
A
Siemens
speed
aries by
short
smaller
as the rate of
This method
of grain bound-
(generally
t,imes)
nodules
were
at lower
and late
was quite
the
super-
largest
cells
It was assumed starters
or
were
at a level other than the largest diameter, they also have been growing
coalesced distance :
occupation into
of
densities
slabs
advance
Several
solutions
have
to this problem
been suggested,(21~22~23)but the one chosen particularly
have seemed
suitable for the present situation,
the boundary A
been
to the plane
occupation
histogram
was
where
density increased with time.
plotted
of
the
length
boundary
occupied
in Fig. 4.
The tail of the distribution
of
grain
by a given slab width as shown
two parts, that due to obliqueness
is made up of of thinner
and that due to the fewer thick
slabs.
slabs
Beyond
a
certain distance we expect only the first contribution, and the estimation
of this distance is aided by noting
the appearance of the larger slabs. A ragged advancing front indicates an oblique grain boundary. Measurement
should
be made in the direction
of
cell growth, which in general, is neither normal to the original grain boundary,
nor in the plane of section.
If a slab is normal to the section plane then the growth distance
is larger
than
the slab width.
Therefore
this error tends to cancel the previous one.
One may
conclude that the inferred growth distance is accurate to within about 20:/,. The interlamellar spacing was measured in a manner somewhat
different
from
that
of
Turnbull
and
was measured
sectioned
higher
of polish.
may
oblique
showed
although ,4t
widest
selected for measurement,
visible were chosen for measurement. that
appearing
cell
when the occupation
or
slab
at 100 kV, was used for this
cells was low
saturations
the
growing from a grain boundary
For our work only those areas within a
increase of nodule diameter. straightforward
1968
Treaftis.(19)
study. The growth
16,
since
Experimental procedures the morphology
VOL.
making more
more slowly.
individual
cells
the
representative
difficult
bo determine,
which
values obtained were
normal
electron
fine
spacing
were
and the minimum
of the
was used to ensure that the lamellae to
the
polished
evidence of obliqueness microscopy,
section.
was available
e.g., interfacial
and an apparent
fraction
from application
of the lever rule.
1. Morphology
Additional in transmission
contrast, effects
of 6 greater hhan expected
EXPERIMENTAL
RESULTS
and sequence
qf
the discontinuous
and continuous precipitate a. Morphology
of the discontinuous precipitate.
cellular precipitate cooperative
6ol-7--l---r-l
a uniform
does not initially
growth.
exhibit
The
regular
It is usually preceded by a more
disorganized grain boundary precipitate whose average “spacing” exceeds that within the cell, as shown in Fig. 5. The a phase of the cell has the same orientation as the cx’ parent grain (light contrast),
and the region
between the two phases is populated
by a relatively
high
dislocation
The advancing
density, interface
rat,her than
a boundary.
and precipitate
distribution
is still irregular for this cell. precipitate with an associated 0
20 WIDTH
40 OF
60 SLAB
(p)
FIG. 4. Histogmm of cell growth distance.
80
visible.
Therefore we may reject the proposa1’1*3*24*2s)
t’hat discontinuous reaction
No coherent general elastic strain field is
precipitation
in the literal
sense.
is a recrystallization The
achievement
of
cooperatJian, coalescenca ul adjacent small cells, and the growth of cells from both sides of the original
of growth
grain
direction, i.e.: SOthat the CIphase map simultaneously
boundary
comes
later.
Figure
42O”C, 10 hr) shows the development
1 (7.5u/,
In:
wf cooperation
in Kg. 7, We believe the twins form either as accident#s or t’o enable the nodule to change growth
atlrl indicates that several growth
directions
can exist
have an incoherent int,erface with the tc’ mat’rix and a good fitting interface u:ith the S phase.
in a single cell.
t,hat only the cen-
In Fig. 1. the spacing appears quite uniform in the
tral portion
It is probable
of this cell is growing
in the plane of
the micrograph. That the boundary develop
precipitate;
by means of boundary
a,s well as t,he cells,
diffusion
is shown by
Fig. 6. _A 7.5 at. “/:, In alloy was held at 447’C for 23.5 hr, quenched and then held at 330°C for 10 hr. In the first heating the dark irregular boundary developed,
precipitate
in the second, the nodules grew.
It is seen
that’ t,he beginning of the nodules outlines the position of the KU’ boundary This
treatment. precipitate
particles
as it was at the end of the first boundary
to
bowed
supply
them
out
between
with
solute.
Hence the cells grew only on one side of the boundary, avoiding the deplet,ed region. This occurred although precipitate spacing (f 11 ,LC)is about
the boundary
three timca the cellular precipitate
lamellar
spacing
(3.8 ,~~cl) at the same temperat,ure (447”U). Under thcsc circumstances volume diffusion in the vi&&y of the fit& precipitate might have been expcctod to contribute to the solute flux. However, at 330°C the cells grew at the same rate as in samples with no prior treatment at
447’C,
showing
that
occurred in this region. One feature of nodule observation
no
appreciable
growth
is the
depletion frequent
of fine twins within the cells, as shown
of boundary diffusion mechanism. 7.676 In ulloy, aged 24 hr tit 447OC, then 10 hrs et 33U”C. x 4ou
Fla-. 6. Illustration
ACTA
124.1
METALLUKGJCA,
VOL.
16,
in the thinnest
1968 sections
of the foil of the lO.l”/b alloy
when the foil is near a (110) slightly
about that
of the foil. the
The interaction
dislocations
evident,
from
as is the
particles.
We
precipitate,
to 6.
been observed,
rates remained
spacing structure,
supersaturation. The matrix but
central
of
portion
microscopy indicated
seldom
more than
a
Individual
a factor
differ
cell.
reveals
in Fig. 8.
lamellae
the
However, uniform
electron spacing
of two, but groups of just three
by less than
20%,
so that
we may
the spacing
thoroughly
kinetic
by nodules.
a maximum
It was found that
of
and for a given composition
at a lower temperature
the growth rate. b. The continuous morphology
was not
increased with increasing supersaturation
ab a given temperature, reached
that
studied in this work is the rate of occupa-
tion of grain boundary this quantity
parameter
the
10 shows a greater than
spacing
of
only local relief of
between
6’ particles,
the same as the lamellar decrease
in growth
of 8’ is difficult,, phase,
spacing.
supersaturation, rate
was
despite
an
and electron
so that
we may
conclude that the 6’ precipitation
process
but a small amount
of solute segregation.
2. Growth speed, interlamellar depletion
and interface
discontinuous
than did
spacing,
distance
are plotted
time to yield the growth velocity initial delay time. with
the
solute
shape of the
precipitate against
as the slope and an
The initial delay times are associated
development
of cooperative
growth
from
the more widely spaced grain boundary
precipitates.
The values of growth speed are plotted
as a function
of temperature
in Fig.
11.
These
values
agree very
well with the results of B6hm.‘16) 6
and
precipitates.
The
6 precipitate
was
6’
continuous
The reciprocal
of spacing is plotted in Fig. 12. These
spacing values are about one eighth of those measured
by Corderoy and
by Bijhm.(16)
Honeycombe.(ls)
density
been extrapolated
precipitate
uniform except near the grain boundary about 1 ,u wide is free of precipitate.
appears
where a zone
This precipitate-
free zone does not appear to be caused by the loss of solute, but rather
Figure
significantly
in any high indium
involves
Nodule
for all but one alloy
indicating
detection
found t’o be similar to that described The
determined.
increase in the lattice parameter
tentatively
any
that the 8’ may
6 and b’ on the dis-
was
The values of growth
measurement. One interesting
b’ without
as
lamellae may vary by
safely take the average value in reporting
density
6’
a
and only a local disturbance
no corresponding
expected
with
should have a decreasing
The
feature
no early stage of S has
constant
The
is about
of
phase.
between S particles
t#he lamellar
is
two kinds
evidence of a close
temperature.
t#he lamellar
found.
new
size indicates
precipitate
precipitation
however,
FIG. 7. Twins in the a phase of a cell. 5.130/6 Jn alloqheld 18 hr at 345°C. x 10,000
Although
of the continuous
continuous
distance
this
with
6 precipitates
of the
its coexistence metastable
The effect
and
large
called
of intermediate
be a separate
growth
of these precipitates
the
coexistence
have
but tilted
lying in the plane
but have no structural
relationship particles
orientation,
[l 1l] direction
by the diffusion of vacancies
to the
boundary as proposed by Embury and Nicholson the system Al-5.9;/, Zn-2.9% Mg by weight.fz6)
for
Figure 9 shows a contrast effect which has not been seen previously in t,hese alloys. It is observed only
temperatures diagram, suggested In both
On the diagram so obtained
defining
the reciprocals
to zero (infinite plotted
the curve
closer observation a 3.0 at.%
“a”.
spacing),
have
and the
in Fig. 3, the phase This phenomenon
at low supersaturations.
In and the 7.5 at.%
In alloy,
temperatures could be chosen at which no cells formed, but at which the grain boundaries bowed with precipitate cipitation
growth
as shown
did not occur
in Fig.
6.
by the cellular
Thus
pre-
mode, even
SHAPIRO
AND
KIRKALDY:
DISCONTINUOUS
PRECIPITATION
IX
Cw
FIG. 8. Examples of t,he advancing interface of cells.
though
interface
diffusion
was still t,hr predominant’
t.ransport mechanism. Electron
microscopy
sho\+ing supersaturation
in t’he product
most, samples the CuK, doublet shows that a unique spacing
does not exist for a given
precipitation
treat,ment.
phase.
was not resolved for
t)he R phase. although it was for the lower angle lines of
CA’. We
t’herefore
estimate
t,he
composition
For example, we observed spacings of 0.091~ and 0.16 ,U
variat’ion of the product phase to be about ho.4
in different
In; consist’cnt
Therefore, values.
cells in the same 7.596 In, 249’C
alloy.
Fig. 12 refers to the finest’ of a range of It
is
not
known
whet,her
t,hcsc
values
correspond to the growth rates measured. If a spectrum of 2) exists as well, wc can only measure a value near the maximum. Nevertheless, in the discussion of the results it will bc assumed that the values of v and S do indeed correspond
t.o the same
The
Q phase compositions
are plott’ed
in Fig. 3,
with the blurring
of the doublet,
at,.% and
assuming all line broadening is due to lattice parameter change wit’h composition sbrain effect)s.
(Bohm
rat,her t,han particle size or assumed
the last cause after
first stating that the a phase was of equilibrium composit.ion.) The X-ray patterns from the tc’ phase on the same films verified the lack of any significant loss
of
supersaturation
continuous
growing cell.
For
by
the
relatively
copious
precipitation.
1Jigurc 8 shows several examples
of the advancing
Frc. 9. The 6’ prrcipit,ate.
interface.
The difkulties
in the quantitative
x 86,001)
lO.lV& In, aged 310 min at 330°C.
evalua-
concave
at the center.
This corresponds
t,o the case
tion of the shape are obvious, due partly to variations
where the spacing is larger t,han that which yields a
from one lamella to the next, and partly to the angle The interfaces appear similar t,o t’he section.
maximum
shapes calculated by Hill&
from the center t’o the edges of the lamellae.
of
and to t,hosc predicted in Article
A.
f2’) for pearlite equilibrium
for the symmckic
Altllougl~ some interfaces
toward t,he LX’phase all across the lamtllae,
eutectoid
are convex many are
In
growth
either
case,
rate for a symmetric the
curvature
eut,ectoid.
usually
increases At these
edges the radius of curvature
is as lit,tle as l/t0
spacing,
angles
and
the
with mechanical
resultant
equilibrium
are
the
consistent
requ~r~nlents.
DISCUSSION
The implications
of our observations
to some of the skuctural discont,inuous
theories
precipitation
with rcspcct
of the cause of
are clear.
There
is no
evidence in t,he Cu-In system t’hat, this is a rccrystallization reaction due to the strain of the general precipitate.
Also, since the two precipitaks
simultaneously, one
reaction
mismatch section three
a t,heory suggestzing the exclusion on account
cannot
apply.
WC shall
apply
growth
the merits
models
of
In the remainder our observations
the
of this to the
earlier, and discuss
conk01
versus
local
equi-
we
shall
interface.
1. Mobility and interface &flu&n evaluate
of
of the matrix-precipitate
presented
reaction
librium at the advancing
To
can develop
boundary
control mobility
calculate J?’ using our data and curves given by Cahn.(12) This can be done in two independent ways. The boundary
mobility
from /3 and compared Pia. IO. The effect cellular precipitate.
of the general precipit.ate on t,he 10.1% In, aged 230 min at 404°C. x 14,000
M
can then be calculated
with estimates from processes
t,hat involve only reaction control (and no diffusion). From t,he reliable values of the product tc phase composit,ion
(Fig. 3) we estimate
the fraction
of S
SHAPIRO
DISCOSTISTOCS
KIRKALDY:
-4%~
PRECIPITATIOS
IN
Cu
1247
111 .~LT~OTS
I
10-s
10-g Fro.
phase precipitated
to be 0.87 f
1
second
estimate
0.02, from which one
comes
from
the
SPEED,
cm /set
I. Growth speed vs. temperature. estimates
agree well for this alloy.
such consistency
obtains the first value ,!l = 0.0044 & 0.0010. The
10-G
10-7
GROWTH
variation
calculated
for Pb-Sn
alloys.
from either equation
C’ahn found
no
M can now be
(3) or (4) if we take
of K (defined in equation (6)) with /3. AF, is approxi-
hD, from the kinetic data using equation
mated by the expression
k to be about unity, and use Cahn’s curves to find AF + (1 -
1 -
xnj
from AF,,.
X”
111ci>
I
law for the solute. Raoult’s
law for bhe solvent and the equilibrium given
on the
phase
equat,ion (7) may be
a
diagram.
compositions
As discussed
fair approximation
later,
for moderate
supersaturations. AF,
For the 7.5% In alloy at 4OO”C, S = 0.314 p: B = 7.5 = 2.3 x 10s ergs/mole,
t’wo methods. the
allovs.‘28’
surface tensions in copper
Hence K = 13 and /3 = 0.005.
These bwo
At’ this point one may conclude
self-consistency
evidence model.
in favour Aaronson
of
Cahn’s
of the interface and Liu(2g) have
reaction
that strong
control
recenbly
shown
Since the values of M in Fig. 13 are of similar magni-
Three reactions which are mobility been chosen t,o provide a comparison of M just cit)ed. These arc:
P 015s 2 0) 0.2
z
03
z
0.4
-I
2
0.5
: 300
400 TEMPERATURE
500
“C
of lamellar spacing vs. temperature.
controlled
have
with t,he values
the grain growth kinet,ics
3.
FIG. 12. Reciprocal
is
tude. then such a calculation should yield a hD, value close to that’ obtained from equation (1).
0.1
200
analysis
6 D,f can be calculated directly from Cahn’s equations.
cm3/molc and 88 is taken as 400 ergs/cm2 which is of the order of grain boundary
The results are plotted in Fig. 13, where it
is seen that, t,here is again fair agreement’ between the (7)
which is based on Henry’s
(l), assume
ACTA
1248
METALLURGICA,
VOL.
little
16,
effect
1968
on
the
boundary
diffusion
coefficient
between phases of different composition. The curve for the 7.5% higher temperatures. to be associated
alloy rises less steeply
This behaviour
with the anomolous
lamellar spacing, and accompanying rate. I/S
at
was suspected increase in the
drop in the growth
If we assume that. t’he temprratJures at, which extrapolates
boundary
to
zero
define
a virtual
phase
in Fig. 3, then new values of X” may be used
in equation
(1) from which bD,
may be recalculated
and plotted, as shown in Fig. 14. earity of t’he curve is thereby pretation
of
this
virt’ual
It is seen t)hat lin-
attained.
phase
The inter-
boundary
will
be
given in the next section. The activation
energy calculated from the curves in
Fig. 14 is 36.5 kcal/mole.
This value is considerably
higher than that’ found by Bohm. 17 06
I 12
/ 14
1000/T
(OK-‘)
IO
the difference
! 16
IE
FIG. 13. The interface mobility calculated for various react,ions. a. Discontinuous precipitation from equation (4). i h. Discontinuous
M = -
(
c. Gram
bDsAF, growth
equation
volume
(3).
kcal/mole
1
in copper
600°C. and the solidification The somewhat Appendix
I.
lengthy
alloys
cipitation If
evaluated
from
independent’ly
copper
indium
the
greater t’han
discont8inuous
alloys,
bho effect
system
at 5O/o Sn. reflecting the rapid increase of D,. Thevaluefor
boundary
self-diffusion
On this basis one might
‘4
to-
‘5
73
mobilities
are
to cellular precipitation
and
if
Cahn
of simultaneous
cannot, be sustained
WC are therefore discontinuous
IO-
has
correctly
mobility
at local
equilibrium
;
IO- I6
0
and 0” D
IO- ”
that’
must occur in the Cu-In and
t’hereforc
from Turnbull’s
IO- ‘8
in the
presence of a metastable const’itution which is consi&ent- with this cooperative mode of transformation. 2. Interfuce d$usivity
:: r
across the interface.
led to the strong implication
precipitation
5
in
dimension~al
nrgumen,t
Values of bD,. found from equation (I), are plotted in Fig. 14. It is seen that initial composition has
IO-19
in
at 1YJ, Sn t’o 35
pre-
diffusion control, then a significant chemical potent,ial difference
energy for
is 47 kcal/mole;
may be found in
evaluat,ed
similar to those pertinent calculated
The act,ivation
in copper
<,
experimentfs using Cahn’s theory.
the
energy
The results? plotted in Fig. 13, are seen
to be about three orders of magnitude mobilities
be from
at, about
of pure tin at 232°C.
calculations
alloys.
in silver is 11 kcal/mole.
n soli~liflc:~tionof till
gallium
energies
alloys it is 58 kcal/molc
wit)h tin content.
of brasses from 475°C to 700°C. the massive /l--f transformation
activation
self-diffusion
copper-tin
in Cu-Zn.
l massive renction in Cu-Ga
to
diffusion
for the same syst,em, but neither data is available for
kMa2 V2
p=
One expect’s
boundary
x copper-indium
from
and
spacing
values obtained in the two experiments.
0.45 to 0.65 of the volume diffusion activation
V TF
precipitation
21 kcal/mole,
must be due to the different
-L I.4
1.5
FIG. 14. Boundary
diffusion Turnbull‘s
1.7
1.6 1000
/T
I.8
1.9
OK
coefficient equation.
calculated
from
SHAPIItO
AND
KIRKALDY:
DISCOSTISUOCS
PRECIPITATION
IS
Cu-In
ALLOYS
1249
The slope of Fig. 15 yields an activation energy of 37 _rt 3 kcal/mole. The activation energy for diffusion is estimated from this value and the temperature variation of q and (l/2 - a). From equation (5)
IO-IQ
2E li”
II
w
where E bD, = m “,
d In (bD,) d(lIRT)
,
etc.
(9)
The necessary reinterpretations of q and l/2 - n and the calculation of E, and El,2_a are given in Appendix II. At 400°C the results are
lo-=
E, = -9
10-23
& 2 kcal/mole
2 E,/, __a = - 3 kcal/mole IO-24
/
I
I
I
I
I
Hence
E hBB
=
25
&
5
kcaljmole
This is a reasonable value for a boundary diffusion controlled process. The extrapolation l/S to temperaPIa. 15. vSS ~8. reciprocal temperature. tures below the solvus (Fig. 3) can be explained by the l 10.1 ,It.yo In c 5.1 at.4, 111 a 7.5 nt.% In metastable monotectoid model. Figure 2 shows that the free energy of the metastable monotectoid reaction, expect a value of 20-25 kcal/mole for phase boundary diffusion in copper-indium alloys, rather than the AF,, is less than that of the ordinary precipitation reaction, AF, + AF,. If the reciprocal of spacing is 36 k&/mole from equation (1). Recent improvements assumed proportional to the free energy available to Turnbull’s equation deal mainly with the temperafrom the reaction, then l/S should vanish below rather ture dependent product phase eompositions~ and may than at the solvus tem~rature. yield more reasonable values of the activation energy. The interpretation of the large observed spacings 3. Metastable monotectoid, loud equilibrium model in relation to the theoretical minimum 2 o’p V/AF, for growth is difficult, We found K to have a value 13 instead Certain of our observations suggest adherence to of 1.5 or 2 obtained by applying simple optimal local equilibrium for the copper-indium system, for principles to equation (5). However. since K is calculated assuming no supe~aturation in the product example, the increasing curvature of the advancing interface, from the center of an ct lamella te the 6 phases, then a more accurate calculat.ion would give a higher value. phase. The main argument for this model follows A further consideration admits of larger spacings. from an analysis of the kinetics using an adaptation of As mentioned in Article A, the establishment of local equation (5) to the non-symmetric monotectoid reaction. equilibrium requires that the composition of the u’ Equation (5) relates the kinetic variables v and S of phase at the interface takes values that may differ the symmetric eut,ectoid reaction to the temperature from the bulk composition. It is also recalled that dependent parameters of the alloy system: bD,, q1 the nominal volume diffusion distance ahead of a movand I/%-o. A reinterpretation of q and l/2 - a ing interface is D&, which is estimated to be from should make the equation suitable for the non5 8, to 20 A in Cu-In alloys at the temperatures of symmetric reaction. interest. (This estimate is based on copper-tin diffuTherefore log VP is plotted against reciprocal temperature in Fig. 15. This curve sion data.) If this is not sufficient to establish the differs from Fig. 14 in absolute magnitude (due to the required composition in the u’ phase, then a decrease omission of the factors relating wS3 to bD,) and in of v increases the distance. From equat,ion (5) the the linearity of the curve at the highest temperatures spacing can also increase, taking advantage of the for t’he 7.5% in alloy. However bDB in Fig. 14 and longer time available for boundary effusion. We canvS3 in Fig. 15 are both independent of the initial no6 state unequivocally at this time the extent to alloy composition. which the factors cited influence the value of K, or 1.4
1.6
I5
IO00
/T
1.7
OK.
1.8
1.9
ACTA
1 P50
whether
K is indeed
RIETALLURGI(‘_A.
high and indicates
a departure
from local equilibrium. SUMMARY 1. Precipitation,
AND
CONCLUSIONS
morphology
CI Cu-In
from
alloys
Supersaturated ‘J. phase copper-indium alloys decompose bv bot’h conbinuous and discontinuous modes.
In addition
a fine precipitate,
in 10 at. “6 In alloys. interfere
called A’, forms
The general precipitatSe does not
with the cellular reaction
indium cont’cnts during moderate
in alloys with lon precipitation
t,imes.
\-Old. 4. W. the 5. H. 6. M.
(19641. C. ZE~ER, Trans Am. Irrst. M&r. Etqrs 167, 550 (1946). 12. .J. W. CAHN, Acta Met. 7, 18 (1959). and .1. R. K~RK~LDV. .-1cttr Nrt. 16, 579 13. .J. M. SHAPIRO (196s). II. H. 0. JONES and E. A. OXVET, .I. Inst. :II~fnZs 82, 445 119541.
uous precipit#at’e has been measured, as well as theavcr-
16. 17. IX.
a given initial extrapolat,ed
composition
the solvus.
Transmission
advancing
interface
electron
equilibrium.
2. Evaluation
of various
evaluation
Turnbull’s the
kinetic
diffusion
rate.
belo\%
19.
of the
“0.
a close approach
theories
of the
interface
growth
boundary
of spacing
micrographs
region indicate
t&omechanical
An
the reciprocal
t)o the value zero at, a temperature
The
data
showed
that
model can account
activation
diffusion coefficient
energy
of
the value of boundary agreement
that
calculated
reactions The
“5. 26.
kinetic
activation
equations
energy
of
for a symmetric
t’emperature
than
with the metastable
controlled
33. 34.
eutectoid
35.
model.
an excessively diffusion
large
metastable hypothesis,
will only be attained
to which The larger of of
of the interface rate.
with the recording
with the solution to the stability growbh reactions.
The local
model
but an unequivocal change
from
estimate
monotectoid
value for the free energy
is
an
conclusion of a precise
of the reaction
and
ANDERKO, C’onstitutiou
of
E&wry
APPENDIX
1. F. W. JONES, P. LEECH and C. SSKES, Proc. R. SW. 181. 154 (1942). 2. H. K. HARDY, J. Inst. MetaZs 75. 707 (1948-g). 3. A. H. GEISLER, Phase Transforkationd in Solids, p. 432.
I
Calculations of Interface Mobility The
interface
mobility
transformations alone, first
i.e., without example
the average
growth.
radius
is M = M’/4aV,
in the kinetic
Feltham
three
reaction
of a grain
The the
approxi-
at any instant
the mobility
where
that
previously
M’ is t,he constant
equation, r2 -
kinetics
Assuming
r, of a grain boundary
then it is easily shown that defined
from
by the interface
long range solute diffusion.
is grain
radius of curvature, mates
is calculated
controlled
problem for lamellar
REFERENCES
Wiley (1951).
36.
the
or the existence
ahead
K.
M. HILLERT, Jernkont. Annlr 141, 757 (1957). M. C. IN&IAN and H. R. TIPLER, Metall. Rec. 8, 105 (1963). H. I. AARONSON and Y. C. Lru, Scripta Met. 2, 1 (1968). P. FELTHAM and G. J. COPLEY, Acta Met. 6, 593 (1958). T. B. MASSALSKI, Acta Met. 6, 243 (1958). D. A. RIGNEY and J. M. BLAKELU, Acta AWet. 14, 1375 (1966). W. A. TILT~ER,Acta Met. 14, 1383 (1966). D. A. RIGNEY and J. M. BLAKEL~, Acta Net. 14, 1385 (1966). K. HULTGREN, R. L. ORR, P. D. ANDERSON and K. K. KELLEI’, Selected Values of Thermodynamic Properties of Metals and Alloys, p. 260. Wiley (1963). A. H. COTTRELL, Theoretical Structural Metallurgy, p. 158. Arnold (1955).
to zero is consistent
to reduce the nodule growth
equilibrium
interface
between
may reflect a departure
of the reaction,
volume
for
temperature
monotectoid
spacings
equilibrium,
the free energy substantial
kcal/mole
and the
27. 28. 29. 30. 31. 32.
system yield an
large difference
spacing extrapolates
than predicted
attractive
25
The observed
the reciprocal
tending
for quantismaller
mobility
applied to the copper-indium
diffusion.
local
purely
H’ANSEN and
A&us, 1). 590. McGraw-Hill (19581. H. I%H~, 2. MetaUk. 50, 87 {1959j. H. B~~HDI,2. Met&k. 52, 564 (1961). 1). d. H. CORDEROY and R. W. K. H~~-E~(,~>IBE, J. Inst. Metals. 92, 65 (1963-4). D. TI-RNBIJLL and H. N. TREAFTIS. Tmns. ,4m. Id. Nin. Enqrs. 212, 33 (1958). G. TnoMAs, Transmission Electrotr Microscopy of Netale, p. 153. Wiley (1962). .J. W. CAHN and W. C. HAGEL, Decompositio?l of Amtenite by D~~us~onaZ Processes, p. 132. Interscience (1962). M. SVLONEN, Suomal. Tiedeakaf. Terim. A VI, 1 (1957). ,J. W. CanN and W. C. HAREL, Acta Met. 11, 561 (1963). J. B. NE~V’KIRK, Precipitation from, Solid Solution, p. 22, ASM (1959). H. K. HARDY and T. J. ~;EAI+ Prog. Xetal Phys. 5, 143 (1954). J. D. EIBOR’~ and R. B. SIcnor.sox, Acta Xet. 13, 403 (1965).
in other copper alloys.
reaction
solvus
from
required
1000 times
22. 23. 24.
for
is 36.5 kcal/mole.
mobility
is about
21.
the
The large values of spacing support Cahn’s model of boundary diffusion and reaction control. However tative
C. HAGEL and H. J. BEATIE, JR., Special Report 64 of
I I.
15. iit.
For
1968
Iron and Steel Institute, p. 98 (1959). B&M, Z. Metallk. 52, 518 (1961). 8. STJLONES, Acta Met. 12, 748 (1964). 7. M. 8. SI-LONEN, Acta polytech. stand. Ch 28, 1 (1964). x. J. S. KIRKALDE~, Decomposition of Austenitt by Diflusional Processrs, p. 39. Interscience (1962). 0. 11.TURNBTTL, Acta Met. 3. 55 (1955). IO. J. S. HIHSCHHORX and R..A. &RE&, _4ct~ Net. 12, 120
The growth rat*e and lamellar spacing of the discontinage CIphase compositjion for some heat treatments.
16,
r,,2 = M’t.
and Copley(30) measured the grain growth Cu-Zn alloys (lo%-35% Zn)
in concentrated
in the temperature range 475°C to 700°C. M’ was found to be a function of concentration, and so for comparison
with
the
values
of mobility
found
for
8H;ZPIRO
Cu-7.5:/,
KIRKALDY:
DISC’OKTIN~O17S
In we choose data for 22.5%
M
Zn. assuming
of indium has the same effect as 3 at.“,;
that 1 at.% zinc.
AND
obtained
in this
way
is presented
straight line on Fig. 14 and extrapolated t#he lower temperatures The
massive
In the Cu-Ga the [, phase,
from the temperature is stable.
for Cu-7.5%
is
a
appears
some
In.
12.51
ALLOTS
Excellent’
APPENDIX
agreement
massive
even on quenching
II
Calculation of E, and E1,2_r,
diffusionless
of the growth of t,he
systemf3n
Cu-In
as a
slightly to
new phase appears to be t’he passage of an incoherent product,,
appropriate
IS
with the two previous estimates of mobilit,y is found.
of interest.
t,ransformation
react,ion for which the mechanism boundary.
PRECIPITBTION
We assume that q need be evaluated product
tc phase at’ the solubility
symmetric
only for t,he
limit in this non-
From equat,ion (14a) of t,hr theoreti-
case.
cal papcro3) wc tribe
range where the parent’ 1 phase
ay
i
Thus the growth rates are very high, and we
(10)
q=iax”i
e&matte from micrographs’31) t,hat the minimum value is V = 10e2 cm/see
We choose as the thermodynamic
model of the solid
solution
interaction
t’he nearest
found in standard
neighbour
texts
AF for this reaction cannot be large, for the reaction
The free energy of formation
is observed in Cu-Ga alloys at 6OO”C, only 16°C below the temperature where the /3 phase is stable. Since the
sed as
of -AF
we estimate
a maximum
of the solution is expres-
f” = 2RT, X (1 -- X) + RT [X In X
5, phase can only be stable with respect to /l for some finite undercooling,
value
+ (1 -
Therefore
20 Cal/mole +
a conservative
and the solubility
log erg/mole
estimate of the mobility
at,
M = lo-l1 cm.mole.sec-i.erg-l
limit is approximated
X,‘(T)=exp
about, 600°C is where
T, is the temperature
fair
(11) and equation
with
the value
obtained
for
grain
Finally,
we consider
an example
alloys is somewhat undercooling
more remote.
used
at the interface
directions.‘s2)
suggest
that
for which
Solidification
for various
The experimental
the recorded
mobilities
may
be
(4) and to
th e value
for AHJT,
of
0.01
-3.31
atomic
mobilities
of
systems are of the same magnitude ing solidus temperatures, Fig.
14 at the
We shall metal
of
1
(1 -
1 X,2
XJ2
11
(13)
= 0.027, so that and E, = -9
5 2
arises from the oversimplified
We interpret phase
occurs
l/2 -
a as l/2 -
3X,” because the S
at X A 0.3 instead follows
symmetrical
model
of E, with temperature.
phase
of X = 1.
from the discussion of an analysis diagram.
which For
This
in Article the
assumes Cu-In
system 2 E1,2_a = -3
kcal/mole
at the correspond-
and hence plot this value in
temperature
(T) in
kcal/mole.
a
different
(
q = 20.6 kcal/mole,
A on the limitations
For pure tin this value applies at 505°K.
xe
X,” (673°K)
T, = 1215’K,
interpretation
M = 7.5 X lo-l1 cm.mole.sec.-lerggl
that
For a Cu-In,
and the variation
Cal/mole oK(35) yields
assume
X = X,”
X,)
2To
The uncertainty
to have a value of about
-
+ T
techniques
v=wAT,
Using
Y
crystallo-
agree with the empirical expression
cm/sec.“K.
(12)
1
12X,(1
of the
AF = AH, AT/T, may be combined
where w was found
.
indicated by equation
(10) (considering
R2T2r E, = -
rates
as a function
low by an order of magnitude.(33s34) Equation the relation
the
controlled reaction in Cu-In
in pure tin have been measured
by
the latter operation) results in the following expression :
growth at the same temperature. relationship to a mobility
1
-WI (11)
as the top of the mis-
This single estimate is plot’ted in Fig. 14 and is seen in agreement
-‘:
(
cibility gap. The differentiation
graphic
X) In (1 -
to be - AF s
model
such as that of Cottrell.(a@
1090”K,
which
is
at 4OO”C, and it must be remembered also temnerature
denendent.
that Eliz+ is
ACTA
1252
METALLURGICA,
ACKNOWLEDGMENTS
The authors assistance:
gratefully
the National
for a grant-in-aid University
of
acknowledge Research
the following
Council of Canada
of research
Graduate
(JSK) ; the McMaster the Research Scholarship,
Ontario Graduate Fellowship,
VOL.
and the St,eel Company
16,
Canada
cooperation Steel thanks
1968
Fellowship
in Metallurgy
of the Research
Company
is sincerely
are due to Dr. J. W.
Corderop
for their comments
(JMS).
The
Department
of Inland
appreciated.
Finally,
Cahn and Dr. J. A.
and suggestions.
OF DISCONTINUOUS
PRECIPITATION
J. M. SHAPIRO?
IN COPPER-INDIUM
ALLOYS*
and J. S. KIRKALDYS
Supersaturated a phase copper-indium alloys decompose by both a continuous and discontinuous mode. An additional fine precipitate, called 6’, forms in the 10 at.% In alloy. The general precipitate does not The growth rate, v, and lamellar spacing, S, of the disusually interfere with the cellular reaction. continuous precipitate were measured, as well as the average a phase composition for some heat treat ments. For a given initial composition the reciprocal of spacing extrapolates to the value zero at a electron micrographs of the advancing interface region am temperature below the solvus. Transmission presented. Turnbull’s interface diffusion model yields a Various theories are evaluated using the kinetic data. reasonable growth rate, but the activation energy of the interface diffusion coefficient, bDB, appears The large S values support Cahn’s boundary diffusion and reaction control model. high, 36.5 kcal/mole. However, the boundary mobility required here is about l/1000 of that estimated from mobility-controlled reactions in other copper alloys. Kirkaldy’s proposal that discontinuous precipitation is a metastable An activation energy for bDB of 25 monotectoid reaction controlled by interface diffusion is tested. kcal/mole is obtained and a plausible explanation is given for the abnormally large spacings near the solvus temperature. Sluggish volume diffusion ahead of the interface (required by the local equilibrium hypothesis) may account for the relatively large spacings at all temperatures. CINETIQUE
DE LA PRECIPITATION DISCONTISUE LES ALLIAGES CUIVRE-INDIUM
DASS
Les alliages cuivre-indium sursatures en phase cc se decomposent suivant un mode it la fois continu et discontinu. Dans l’alliage a 10% at. In, il se forme un fin precipite supplementaire 6’. Le precipite continu ne gene habituellement pas la precipitation cellulaire. La vitesse de croissanoe Y et la distance lamellaire S du precipite discontinu ont Bte mesurees, ainsi que la composition moyenne de la phase a pour differents traitements thermiques. Pour une composition initial0 don&e, l’inverse de la distance s’annule par extrapolation pour une temperature inferieure L celle du solvus. Les auteurs presentent les micrographics Qlectroniques par transmission de la progression de la region de l’interface. Des theories variees ont et& utilisees pour Bvaluer les coefficients cinetiques. Le modele de diffusion a l’interface de Turnbull donne une vitesse de croissance raisonnable mais l’energie d’activation du coefficient de diffusion a l’interface bDs parait hlevee, 36,5 kcal/mole. Les valeurs Blevees de S confirment le modele de diffusion de la frontiere et de controle de la reaction de Cahn. Cependant, la valeur de la mobilite de la frontiere requise ici est environ Qgale a l/1000 de la valeur deduite des reactions controlees par la mobilite dans les autres alliages de cuivre. Les auteurs ont essay& d’utiliser l’hypoth&se de Kirkaldy suivant laquelle la precipitation discontinue est une reaction monotectoIde metastable contr8lee par la diffusion it l’interface. 11s ont ainsi obtenu pour bDs une Bnergie d’activation de 25 kcal/mole et, donnent une interpretation raisonnable pour les distances anormalement grandes correspondant aux temperatures voisines de la temperature du solvus. Une diffusion en volume lente, en avant de l’interface, (et necessitee par I’hypothese de l’equilibre local) pent just,ifier les distance* lamellaires relativement grandes it toutes les temperatures. DIE
KINETIK
DER
INHOMOGENES
AUSSCHEIDUNG
IN
KUPFER-INDIUM-LEGIERUNGEN Ubereiittigte Kupfer-Indium-Legierungen (a-Phase) zersetzen sich sowohl durch einen kontimuerlichen als such durch einen diskontinuierlichen Mechanismus. In der 10 At.% In-Legierung bildet sich ein zusiitzlicher feiner Niederschlag, genannt 6’. Die gewdhnliche Ausscheidung stiirt die Zellreaktion normalerweise nicht. Es werden sowhol die Wachstumsgeschwindigkeit 2) und der Lamellenabstand S der inhomogenen Ausscheidung als such die durchschnittliche Zusammensetzung der a-Phase fir einige Wiirmebehandlungen gemeseen. Der extrapolierte reziproke Abstand geht bei einer gegebenen Anfangszusammensetzung bei Temperaturen unterhalb der Solvustemperatur gegen Null. Es werden elektronenmikroskopische Durchstrahlungsaufnahmen der fortschreitenden Grenzflachenbereiche gezeigt. Verschiedene Theorien werden unter Verwendung der kinetischen Daten untersucht. Turnbulls Model1 der Grenzfliichendiffusion gibt verntinftige Werte der Wachstumsgeschwindigkeit, aber die bD B, erscheint hoch (36,5 kcal/Mol). Aktivierungsenergie des Koeffizienten der Grenzfliichendiffusion, Die grossen S-Werte sprechen fur Cahns Model1 der Grenzendiffusion und der Reaktionskontrolle. Die hierbei erforderliche Grenzen-Beweglichkeit ist jedoch nur etwa l/1000 der aus beweglichkeitskontrol. lierten Reaktionen in anderen Kupfer-Legierungen abgeschatzten Beweglichkeit der Grenzen. Kirkaldys Vorschlag, dass die inhomogene Ausscheidung eine metastabile, monotektische, von der Grenzflachen. diffusion kontrollierte Reaktion ist, wird gepriift. Man erhiilt eine Aktivierungsenergie von 25 kcal/Mol fur bDB und es wird eine plausible Erklarung fur die ungewohnlich grossen Abstlinde in der Niihe der Solvustemperatur gegeben. Eine langsame Volumendiffusion vor der Grenzflache (gefordert auf Grund der Hypothese des lokalen Gleichgewichts) konnte die Ursache der relativ grossen -4bstande bei allen Temperaturen sein.
* Received August 7, 1967; revised January 29, 1968. Extracted in part from the Ph.D. Thesis of McMaster University, 1966. t Research Department, Inland Steel Co., East Chicago, Ind. 46312. $ Department of Metallurgy and Materials Science, McMaster University, Hamilton, Ontario, Canada. ACTA
METALLURGICA,
VOL.
16, OCTOBER
1968
1239
J. M. Shapiro,
INTRODUCTION
AND
THEORY
DisconGnuous or cellular precipitation is said to occur when a supersaturated phase (CC’)decomposes to the structurally identical, but solute depleted cc phase and a new ,8 phase, by the growth of cells or nodules of the parallel Q + ,S mixture (usually ia~llellar) into the a’ phase. This react‘ion appears rnetallographically identical to cooperative eutectoid decomposition, e.g. pearlite growth in steels. as shown by t,hc example in Fig. 1. The cooperative side-byside growth of the a and /2phases results in regular kinetics for the isothermal reaction, i.e. constant lamellar spacing, 8’: and growth rate, c. This regularity has prompted theoretical d~liberat~io~lsin two main directions. The first seeks a~fundamental cause of t,he cellular rear&on. Most such theories cite a correlation of structural or mechanical properties of various alloy systems.c1-7) A thermodynamic basis for the reaction has been proposed by Kirkaldy.(s) The morphological similarity to eutcctoid decomposition suggests that the cellular reaction occurs in a metastable monotectoid, as indicated in Fig. 2. The other class of theories considers the mechallisnls whereby the precipitate achieves the observed growth rate and spacing. It is usually assumed that diffusion along the advancing interface accounts for the segregation of solute to the /?lamellae. This transport mode has been shown t,o be both sufficient2 and
FIG. 1. Typioal celluIar precipitate in G-7.5
necessary for the observed kinetics.@*rO) From a dimensional argument Turnbull proposed the following growth equation : 21=
(Xa’ X”’
X”) bDB --19
(I)
where b is the effec%ive boundary thickness, DB is the diffusion coefficient in the boundary, and X”’ and X” are the initial and equilibrium compositions respectively, of the cc phase. The spacing chosen is that predicted by Zener for pearlite~ol) viz.,
where 8 is the energy per unit area of c@ interface, I’ is the molar volume of the alloy, and AFo is the total free energy available from the transformat,ion. Cahn presented a model which includes both interface mobility and cell boundary diffusion contJroI.02) It was also assumed that the reaction proceeded with a maximum rate of free energy decrease. Various quantities could be calculated as a function of the parameter kM P( iPa)2 /3=_---6% AE;,
(3)
Mrherek is the rat,io of t~he~oncent’ration of solute in
a&y0 In, aged 10 hr at 420°C.
x 1530
SHAPIRO
AND KIRKALDY:
DISCOSTIKUOGS
PRECIPITATION
IX
Cu--In
ALLOYS
1241
where q and a are thermodynamic parameters and
(6) the departure of the composition of the product phases from their equilibrium values ; and the curvature of the a’a or a’p interface as a function of K. The determination of K is an optimization problem, and cannot be obtained through the application of equilibrium thermodynamics alone. The relative merits of each of the above theories will be examined later in the light of the experimental observations. Previous work on the w~fper-&&urn system
MOLE FRACTION FIG. 2. Free energy curves and phase diagrams for the metastable monotectoid reaction.
the interface to that in au cc lamella, and M is the mobility of the boundary, expressed as M
=
2
AF
(4)
where AF is the amount of free energy dissipated in the reaction. The factors in the paramet,er @ are specific to each alloy system at any given temperature. The quantities calculated are the growth speed, the spacing, the fraction of material precipitated, the fraction of AF,, which is not stored in the supersaturated product, and the fraction of AF, expended as a-@ surface energy. The assumption that discontinuous precipitation occurs in a metastable monotectoid allows one to apply to this reaction the results of the local equilibrium growth model derived for the symmetric eutcctoid reaction in a work henceforth referred to as Article A.03) The results are, briefly: an expression for the growth rate and spacing, vs3
=
48bhW - 1) 4r(-k - aI2
(5)
The solubility limit of the f.c.c. a phase shown in Fig. 3 is taken from Jones and Owen.(‘Q Hansen and Anderko show that the S phase, nominally C&In,, has a composition range from 29.0 to 30.6 at.% In at all temperatures below 613”C.(r5) The 6 phase precipitates from su~rsaturated a by both the continuous or general and discontinuous modes. Bohm observed that no general precipitate formed below 0.8 of the absolute solvus temperature, using light optical and X-ray metallography.d6*17) However, Corderoy and Honeycombe noted a general precipitate in 8.5 and 10 at.% In alloys at all temperatures above 250°C by means of electron microscopy. Nodule growth was eventually halted by the continuous depletion of the matrix. Biihm measured the growth rate of the discontinuous precipitate by direct optical microscopy, and the spacing by the method of Turnbull and Treaftis.d$) It was concluded from X-ray metallography that the a phase formed with the equilibriuln composit,ion.
9
-I -I 0
2
4
6
8
IO
12
INDIUM ATOMIC PERCENT FIG. 3. Section of the copper-indium phase diagram. A Temperature for which l/8 + 0 from Fig. li. 0
Cellular a phase composition m Cellulltr c( phase composition
for initially 7.5% In alloy. for init,lalty 10.1% In alloy.
ACTA
1242
METALLURGICA,
In the present continuous
contribution
precipitate
by light optical
may be considered
reaction
is determined.
spacing
and
of
The
a’/3 interfaces Alloys
shapes
purity.
a
of the other
speed, lamellar
composition the
are
phase
remeasured
composition
of the advancing
a’a
is and
are recorded.
were prepared
from
The composition
diffraction,
of
and
and thin foil
independent
The growth
a phase
uniformity
estimated.
is made
The extent to which the cellular
reaction
The
a study
and sequence of the discontinuous
electron microscopy.
materials
of 99.999%
was determined
using established
by X-ray
lattice parameter
versus
composition data. (14) Precipitation treatments were carried out’ in a stirred salt bath or in a forced convection
furnace.
were *l”C
in the salt bath during the longest times,
and from 10.5”C in
the
precipitation
temperature
variations
Water
quenching
in 40 days
followed
the
anneal.
Techniques employed
Typical
during short runs to *2”C
furnace.
standard
to prepare
for
copper
alloys
the rod specimens
were
for optical
and electron microscopy. (20) Some preferential electropolishing
of
the
Elmiskop
I, operated
tc phase
was
noted.
A
Siemens
speed
aries by
short
smaller
as the rate of
This method
of grain bound-
(generally
t,imes)
nodules
were
at lower
and late
was quite
the
super-
largest
cells
It was assumed starters
or
were
at a level other than the largest diameter, they also have been growing
coalesced distance :
occupation into
of
densities
slabs
advance
Several
solutions
have
to this problem
been suggested,(21~22~23)but the one chosen particularly
have seemed
suitable for the present situation,
the boundary A
been
to the plane
occupation
histogram
was
where
density increased with time.
plotted
of
the
length
boundary
occupied
in Fig. 4.
The tail of the distribution
of
grain
by a given slab width as shown
two parts, that due to obliqueness
is made up of of thinner
and that due to the fewer thick
slabs.
slabs
Beyond
a
certain distance we expect only the first contribution, and the estimation
of this distance is aided by noting
the appearance of the larger slabs. A ragged advancing front indicates an oblique grain boundary. Measurement
should
be made in the direction
of
cell growth, which in general, is neither normal to the original grain boundary,
nor in the plane of section.
If a slab is normal to the section plane then the growth distance
is larger
than
the slab width.
Therefore
this error tends to cancel the previous one.
One may
conclude that the inferred growth distance is accurate to within about 20:/,. The interlamellar spacing was measured in a manner somewhat
different
from
that
of
Turnbull
and
was measured
sectioned
higher
of polish.
may
oblique
showed
although ,4t
widest
selected for measurement,
visible were chosen for measurement. that
appearing
cell
when the occupation
or
slab
at 100 kV, was used for this
cells was low
saturations
the
growing from a grain boundary
For our work only those areas within a
increase of nodule diameter. straightforward
1968
Treaftis.(19)
study. The growth
16,
since
Experimental procedures the morphology
VOL.
making more
more slowly.
individual
cells
the
representative
difficult
bo determine,
which
values obtained were
normal
electron
fine
spacing
were
and the minimum
of the
was used to ensure that the lamellae to
the
polished
evidence of obliqueness microscopy,
section.
was available
e.g., interfacial
and an apparent
fraction
from application
of the lever rule.
1. Morphology
Additional in transmission
contrast, effects
of 6 greater hhan expected
EXPERIMENTAL
RESULTS
and sequence
qf
the discontinuous
and continuous precipitate a. Morphology
of the discontinuous precipitate.
cellular precipitate cooperative
6ol-7--l---r-l
a uniform
does not initially
growth.
exhibit
The
regular
It is usually preceded by a more
disorganized grain boundary precipitate whose average “spacing” exceeds that within the cell, as shown in Fig. 5. The a phase of the cell has the same orientation as the cx’ parent grain (light contrast),
and the region
between the two phases is populated
by a relatively
high
dislocation
The advancing
density, interface
rat,her than
a boundary.
and precipitate
distribution
is still irregular for this cell. precipitate with an associated 0
20 WIDTH
40 OF
60 SLAB
(p)
FIG. 4. Histogmm of cell growth distance.
80
visible.
Therefore we may reject the proposa1’1*3*24*2s)
t’hat discontinuous reaction
No coherent general elastic strain field is
precipitation
in the literal
sense.
is a recrystallization The
achievement
of
cooperatJian, coalescenca ul adjacent small cells, and the growth of cells from both sides of the original
of growth
grain
direction, i.e.: SOthat the CIphase map simultaneously
boundary
comes
later.
Figure
42O”C, 10 hr) shows the development
1 (7.5u/,
In:
wf cooperation
in Kg. 7, We believe the twins form either as accident#s or t’o enable the nodule to change growth
atlrl indicates that several growth
directions
can exist
have an incoherent int,erface with the tc’ mat’rix and a good fitting interface u:ith the S phase.
in a single cell.
t,hat only the cen-
In Fig. 1. the spacing appears quite uniform in the
tral portion
It is probable
of this cell is growing
in the plane of
the micrograph. That the boundary develop
precipitate;
by means of boundary
a,s well as t,he cells,
diffusion
is shown by
Fig. 6. _A 7.5 at. “/:, In alloy was held at 447’C for 23.5 hr, quenched and then held at 330°C for 10 hr. In the first heating the dark irregular boundary developed,
precipitate
in the second, the nodules grew.
It is seen
that’ t,he beginning of the nodules outlines the position of the KU’ boundary This
treatment. precipitate
particles
as it was at the end of the first boundary
to
bowed
supply
them
out
between
with
solute.
Hence the cells grew only on one side of the boundary, avoiding the deplet,ed region. This occurred although precipitate spacing (f 11 ,LC)is about
the boundary
three timca the cellular precipitate
lamellar
spacing
(3.8 ,~~cl) at the same temperat,ure (447”U). Under thcsc circumstances volume diffusion in the vi&&y of the fit& precipitate might have been expcctod to contribute to the solute flux. However, at 330°C the cells grew at the same rate as in samples with no prior treatment at
447’C,
showing
that
occurred in this region. One feature of nodule observation
no
appreciable
growth
is the
depletion frequent
of fine twins within the cells, as shown
of boundary diffusion mechanism. 7.676 In ulloy, aged 24 hr tit 447OC, then 10 hrs et 33U”C. x 4ou
Fla-. 6. Illustration
ACTA
124.1
METALLUKGJCA,
VOL.
16,
in the thinnest
1968 sections
of the foil of the lO.l”/b alloy
when the foil is near a (110) slightly
about that
of the foil. the
The interaction
dislocations
evident,
from
as is the
particles.
We
precipitate,
to 6.
been observed,
rates remained
spacing structure,
supersaturation. The matrix but
central
of
portion
microscopy indicated
seldom
more than
a
Individual
a factor
differ
cell.
reveals
in Fig. 8.
lamellae
the
However, uniform
electron spacing
of two, but groups of just three
by less than
20%,
so that
we may
the spacing
thoroughly
kinetic
by nodules.
a maximum
It was found that
of
and for a given composition
at a lower temperature
the growth rate. b. The continuous morphology
was not
increased with increasing supersaturation
ab a given temperature, reached
that
studied in this work is the rate of occupa-
tion of grain boundary this quantity
parameter
the
10 shows a greater than
spacing
of
only local relief of
between
6’ particles,
the same as the lamellar decrease
in growth
of 8’ is difficult,, phase,
spacing.
supersaturation, rate
was
despite
an
and electron
so that
we may
conclude that the 6’ precipitation
process
but a small amount
of solute segregation.
2. Growth speed, interlamellar depletion
and interface
discontinuous
than did
spacing,
distance
are plotted
time to yield the growth velocity initial delay time. with
the
solute
shape of the
precipitate against
as the slope and an
The initial delay times are associated
development
of cooperative
growth
from
the more widely spaced grain boundary
precipitates.
The values of growth speed are plotted
as a function
of temperature
in Fig.
11.
These
values
agree very
well with the results of B6hm.‘16) 6
and
precipitates.
The
6 precipitate
was
6’
continuous
The reciprocal
of spacing is plotted in Fig. 12. These
spacing values are about one eighth of those measured
by Corderoy and
by Bijhm.(16)
Honeycombe.(ls)
density
been extrapolated
precipitate
uniform except near the grain boundary about 1 ,u wide is free of precipitate.
appears
where a zone
This precipitate-
free zone does not appear to be caused by the loss of solute, but rather
Figure
significantly
in any high indium
involves
Nodule
for all but one alloy
indicating
detection
found t’o be similar to that described The
determined.
increase in the lattice parameter
tentatively
any
that the 8’ may
6 and b’ on the dis-
was
The values of growth
measurement. One interesting
b’ without
as
lamellae may vary by
safely take the average value in reporting
density
6’
a
and only a local disturbance
no corresponding
expected
with
should have a decreasing
The
feature
no early stage of S has
constant
The
is about
of
phase.
between S particles
t#he lamellar
is
two kinds
evidence of a close
temperature.
t#he lamellar
found.
new
size indicates
precipitate
precipitation
however,
FIG. 7. Twins in the a phase of a cell. 5.130/6 Jn alloqheld 18 hr at 345°C. x 10,000
Although
of the continuous
continuous
distance
this
with
6 precipitates
of the
its coexistence metastable
The effect
and
large
called
of intermediate
be a separate
growth
of these precipitates
the
coexistence
have
but tilted
lying in the plane
but have no structural
relationship particles
orientation,
[l 1l] direction
by the diffusion of vacancies
to the
boundary as proposed by Embury and Nicholson the system Al-5.9;/, Zn-2.9% Mg by weight.fz6)
for
Figure 9 shows a contrast effect which has not been seen previously in t,hese alloys. It is observed only
temperatures diagram, suggested In both
On the diagram so obtained
defining
the reciprocals
to zero (infinite plotted
the curve
closer observation a 3.0 at.%
“a”.
spacing),
have
and the
in Fig. 3, the phase This phenomenon
at low supersaturations.
In and the 7.5 at.%
In alloy,
temperatures could be chosen at which no cells formed, but at which the grain boundaries bowed with precipitate cipitation
growth
as shown
did not occur
in Fig.
6.
by the cellular
Thus
pre-
mode, even
SHAPIRO
AND
KIRKALDY:
DISCONTINUOUS
PRECIPITATION
IX
Cw
FIG. 8. Examples of t,he advancing interface of cells.
though
interface
diffusion
was still t,hr predominant’
t.ransport mechanism. Electron
microscopy
sho\+ing supersaturation
in t’he product
most, samples the CuK, doublet shows that a unique spacing
does not exist for a given
precipitation
treat,ment.
phase.
was not resolved for
t)he R phase. although it was for the lower angle lines of
CA’. We
t’herefore
estimate
t,he
composition
For example, we observed spacings of 0.091~ and 0.16 ,U
variat’ion of the product phase to be about ho.4
in different
In; consist’cnt
Therefore, values.
cells in the same 7.596 In, 249’C
alloy.
Fig. 12 refers to the finest’ of a range of It
is
not
known
whet,her
t,hcsc
values
correspond to the growth rates measured. If a spectrum of 2) exists as well, wc can only measure a value near the maximum. Nevertheless, in the discussion of the results it will bc assumed that the values of v and S do indeed correspond
t.o the same
The
Q phase compositions
are plott’ed
in Fig. 3,
with the blurring
of the doublet,
at,.% and
assuming all line broadening is due to lattice parameter change wit’h composition sbrain effect)s.
(Bohm
rat,her t,han particle size or assumed
the last cause after
first stating that the a phase was of equilibrium composit.ion.) The X-ray patterns from the tc’ phase on the same films verified the lack of any significant loss
of
supersaturation
continuous
growing cell.
For
by
the
relatively
copious
precipitation.
1Jigurc 8 shows several examples
of the advancing
Frc. 9. The 6’ prrcipit,ate.
interface.
The difkulties
in the quantitative
x 86,001)
lO.lV& In, aged 310 min at 330°C.
evalua-
concave
at the center.
This corresponds
t,o the case
tion of the shape are obvious, due partly to variations
where the spacing is larger t,han that which yields a
from one lamella to the next, and partly to the angle The interfaces appear similar t,o t’he section.
maximum
shapes calculated by Hill&
from the center t’o the edges of the lamellae.
of
and to t,hosc predicted in Article
A.
f2’) for pearlite equilibrium
for the symmckic
Altllougl~ some interfaces
toward t,he LX’phase all across the lamtllae,
eutectoid
are convex many are
In
growth
either
case,
rate for a symmetric the
curvature
eut,ectoid.
usually
increases At these
edges the radius of curvature
is as lit,tle as l/t0
spacing,
angles
and
the
with mechanical
resultant
equilibrium
are
the
consistent
requ~r~nlents.
DISCUSSION
The implications
of our observations
to some of the skuctural discont,inuous
theories
precipitation
with rcspcct
of the cause of
are clear.
There
is no
evidence in t,he Cu-In system t’hat, this is a rccrystallization reaction due to the strain of the general precipitate.
Also, since the two precipitaks
simultaneously, one
reaction
mismatch section three
a t,heory suggestzing the exclusion on account
cannot
apply.
WC shall
apply
growth
the merits
models
of
In the remainder our observations
the
of this to the
earlier, and discuss
conk01
versus
local
equi-
we
shall
interface.
1. Mobility and interface &flu&n evaluate
of
of the matrix-precipitate
presented
reaction
librium at the advancing
To
can develop
boundary
control mobility
calculate J?’ using our data and curves given by Cahn.(12) This can be done in two independent ways. The boundary
mobility
from /3 and compared Pia. IO. The effect cellular precipitate.
of the general precipit.ate on t,he 10.1% In, aged 230 min at 404°C. x 14,000
M
can then be calculated
with estimates from processes
t,hat involve only reaction control (and no diffusion). From t,he reliable values of the product tc phase composit,ion
(Fig. 3) we estimate
the fraction
of S
SHAPIRO
DISCOSTISTOCS
KIRKALDY:
-4%~
PRECIPITATIOS
IN
Cu
1247
111 .~LT~OTS
I
10-s
10-g Fro.
phase precipitated
to be 0.87 f
1
second
estimate
0.02, from which one
comes
from
the
SPEED,
cm /set
I. Growth speed vs. temperature. estimates
agree well for this alloy.
such consistency
obtains the first value ,!l = 0.0044 & 0.0010. The
10-G
10-7
GROWTH
variation
calculated
for Pb-Sn
alloys.
from either equation
C’ahn found
no
M can now be
(3) or (4) if we take
of K (defined in equation (6)) with /3. AF, is approxi-
hD, from the kinetic data using equation
mated by the expression
k to be about unity, and use Cahn’s curves to find AF + (1 -
1 -
xnj
from AF,,.
X”
111ci>
I
law for the solute. Raoult’s
law for bhe solvent and the equilibrium given
on the
phase
equat,ion (7) may be
a
diagram.
compositions
As discussed
fair approximation
later,
for moderate
supersaturations. AF,
For the 7.5% In alloy at 4OO”C, S = 0.314 p: B = 7.5 = 2.3 x 10s ergs/mole,
t’wo methods. the
allovs.‘28’
surface tensions in copper
Hence K = 13 and /3 = 0.005.
These bwo
At’ this point one may conclude
self-consistency
evidence model.
in favour Aaronson
of
Cahn’s
of the interface and Liu(2g) have
reaction
that strong
control
recenbly
shown
Since the values of M in Fig. 13 are of similar magni-
Three reactions which are mobility been chosen t,o provide a comparison of M just cit)ed. These arc:
P 015s 2 0) 0.2
z
03
z
0.4
-I
2
0.5
: 300
400 TEMPERATURE
500
“C
of lamellar spacing vs. temperature.
controlled
have
with t,he values
the grain growth kinet,ics
3.
FIG. 12. Reciprocal
is
tude. then such a calculation should yield a hD, value close to that’ obtained from equation (1).
0.1
200
analysis
6 D,f can be calculated directly from Cahn’s equations.
cm3/molc and 88 is taken as 400 ergs/cm2 which is of the order of grain boundary
The results are plotted in Fig. 13, where it
is seen that, t,here is again fair agreement’ between the (7)
which is based on Henry’s
(l), assume
ACTA
1248
METALLURGICA,
VOL.
little
16,
effect
1968
on
the
boundary
diffusion
coefficient
between phases of different composition. The curve for the 7.5% higher temperatures. to be associated
alloy rises less steeply
This behaviour
with the anomolous
lamellar spacing, and accompanying rate. I/S
at
was suspected increase in the
drop in the growth
If we assume that. t’he temprratJures at, which extrapolates
boundary
to
zero
define
a virtual
phase
in Fig. 3, then new values of X” may be used
in equation
(1) from which bD,
may be recalculated
and plotted, as shown in Fig. 14. earity of t’he curve is thereby pretation
of
this
virt’ual
It is seen t)hat lin-
attained.
phase
The inter-
boundary
will
be
given in the next section. The activation
energy calculated from the curves in
Fig. 14 is 36.5 kcal/mole.
This value is considerably
higher than that’ found by Bohm. 17 06
I 12
/ 14
1000/T
(OK-‘)
IO
the difference
! 16
IE
FIG. 13. The interface mobility calculated for various react,ions. a. Discontinuous precipitation from equation (4). i h. Discontinuous
M = -
(
c. Gram
bDsAF, growth
equation
volume
(3).
kcal/mole
1
in copper
600°C. and the solidification The somewhat Appendix
I.
lengthy
alloys
cipitation If
evaluated
from
independent’ly
copper
indium
the
greater t’han
discont8inuous
alloys,
bho effect
system
at 5O/o Sn. reflecting the rapid increase of D,. Thevaluefor
boundary
self-diffusion
On this basis one might
‘4
to-
‘5
73
mobilities
are
to cellular precipitation
and
if
Cahn
of simultaneous
cannot, be sustained
WC are therefore discontinuous
IO-
has
correctly
mobility
at local
equilibrium
;
IO- I6
0
and 0” D
IO- ”
that’
must occur in the Cu-In and
t’hereforc
from Turnbull’s
IO- ‘8
in the
presence of a metastable const’itution which is consi&ent- with this cooperative mode of transformation. 2. Interfuce d$usivity
:: r
across the interface.
led to the strong implication
precipitation
5
in
dimension~al
nrgumen,t
Values of bD,. found from equation (I), are plotted in Fig. 14. It is seen that initial composition has
IO-19
in
at 1YJ, Sn t’o 35
pre-
diffusion control, then a significant chemical potent,ial difference
energy for
is 47 kcal/mole;
may be found in
evaluat,ed
similar to those pertinent calculated
The act,ivation
in copper
<,
experimentfs using Cahn’s theory.
the
energy
The results? plotted in Fig. 13, are seen
to be about three orders of magnitude mobilities
be from
at, about
of pure tin at 232°C.
calculations
alloys.
in silver is 11 kcal/mole.
n soli~liflc:~tionof till
gallium
energies
alloys it is 58 kcal/molc
wit)h tin content.
of brasses from 475°C to 700°C. the massive /l--f transformation
activation
self-diffusion
copper-tin
in Cu-Zn.
l massive renction in Cu-Ga
to
diffusion
for the same syst,em, but neither data is available for
kMa2 V2
p=
One expect’s
boundary
x copper-indium
from
and
spacing
values obtained in the two experiments.
0.45 to 0.65 of the volume diffusion activation
V TF
precipitation
21 kcal/mole,
must be due to the different
-L I.4
1.5
FIG. 14. Boundary
diffusion Turnbull‘s
1.7
1.6 1000
/T
I.8
1.9
OK
coefficient equation.
calculated
from
SHAPIItO
AND
KIRKALDY:
DISCOSTISUOCS
PRECIPITATION
IS
Cu-In
ALLOYS
1249
The slope of Fig. 15 yields an activation energy of 37 _rt 3 kcal/mole. The activation energy for diffusion is estimated from this value and the temperature variation of q and (l/2 - a). From equation (5)
IO-IQ
2E li”
II
w
where E bD, = m “,
d In (bD,) d(lIRT)
,
etc.
(9)
The necessary reinterpretations of q and l/2 - n and the calculation of E, and El,2_a are given in Appendix II. At 400°C the results are
lo-=
E, = -9
10-23
& 2 kcal/mole
2 E,/, __a = - 3 kcal/mole IO-24
/
I
I
I
I
I
Hence
E hBB
=
25
&
5
kcaljmole
This is a reasonable value for a boundary diffusion controlled process. The extrapolation l/S to temperaPIa. 15. vSS ~8. reciprocal temperature. tures below the solvus (Fig. 3) can be explained by the l 10.1 ,It.yo In c 5.1 at.4, 111 a 7.5 nt.% In metastable monotectoid model. Figure 2 shows that the free energy of the metastable monotectoid reaction, expect a value of 20-25 kcal/mole for phase boundary diffusion in copper-indium alloys, rather than the AF,, is less than that of the ordinary precipitation reaction, AF, + AF,. If the reciprocal of spacing is 36 k&/mole from equation (1). Recent improvements assumed proportional to the free energy available to Turnbull’s equation deal mainly with the temperafrom the reaction, then l/S should vanish below rather ture dependent product phase eompositions~ and may than at the solvus tem~rature. yield more reasonable values of the activation energy. The interpretation of the large observed spacings 3. Metastable monotectoid, loud equilibrium model in relation to the theoretical minimum 2 o’p V/AF, for growth is difficult, We found K to have a value 13 instead Certain of our observations suggest adherence to of 1.5 or 2 obtained by applying simple optimal local equilibrium for the copper-indium system, for principles to equation (5). However. since K is calculated assuming no supe~aturation in the product example, the increasing curvature of the advancing interface, from the center of an ct lamella te the 6 phases, then a more accurate calculat.ion would give a higher value. phase. The main argument for this model follows A further consideration admits of larger spacings. from an analysis of the kinetics using an adaptation of As mentioned in Article A, the establishment of local equation (5) to the non-symmetric monotectoid reaction. equilibrium requires that the composition of the u’ Equation (5) relates the kinetic variables v and S of phase at the interface takes values that may differ the symmetric eut,ectoid reaction to the temperature from the bulk composition. It is also recalled that dependent parameters of the alloy system: bD,, q1 the nominal volume diffusion distance ahead of a movand I/%-o. A reinterpretation of q and l/2 - a ing interface is D&, which is estimated to be from should make the equation suitable for the non5 8, to 20 A in Cu-In alloys at the temperatures of symmetric reaction. interest. (This estimate is based on copper-tin diffuTherefore log VP is plotted against reciprocal temperature in Fig. 15. This curve sion data.) If this is not sufficient to establish the differs from Fig. 14 in absolute magnitude (due to the required composition in the u’ phase, then a decrease omission of the factors relating wS3 to bD,) and in of v increases the distance. From equat,ion (5) the the linearity of the curve at the highest temperatures spacing can also increase, taking advantage of the for t’he 7.5% in alloy. However bDB in Fig. 14 and longer time available for boundary effusion. We canvS3 in Fig. 15 are both independent of the initial no6 state unequivocally at this time the extent to alloy composition. which the factors cited influence the value of K, or 1.4
1.6
I5
IO00
/T
1.7
OK.
1.8
1.9
ACTA
1 P50
whether
K is indeed
RIETALLURGI(‘_A.
high and indicates
a departure
from local equilibrium. SUMMARY 1. Precipitation,
AND
CONCLUSIONS
morphology
CI Cu-In
from
alloys
Supersaturated ‘J. phase copper-indium alloys decompose bv bot’h conbinuous and discontinuous modes.
In addition
a fine precipitate,
in 10 at. “6 In alloys. interfere
called A’, forms
The general precipitatSe does not
with the cellular reaction
indium cont’cnts during moderate
in alloys with lon precipitation
t,imes.
\-Old. 4. W. the 5. H. 6. M.
(19641. C. ZE~ER, Trans Am. Irrst. M&r. Etqrs 167, 550 (1946). 12. .J. W. CAHN, Acta Met. 7, 18 (1959). and .1. R. K~RK~LDV. .-1cttr Nrt. 16, 579 13. .J. M. SHAPIRO (196s). II. H. 0. JONES and E. A. OXVET, .I. Inst. :II~fnZs 82, 445 119541.
uous precipit#at’e has been measured, as well as theavcr-
16. 17. IX.
a given initial extrapolat,ed
composition
the solvus.
Transmission
advancing
interface
electron
equilibrium.
2. Evaluation
of various
evaluation
Turnbull’s the
kinetic
diffusion
rate.
belo\%
19.
of the
“0.
a close approach
theories
of the
interface
growth
boundary
of spacing
micrographs
region indicate
t&omechanical
An
the reciprocal
t)o the value zero at, a temperature
The
data
showed
that
model can account
activation
diffusion coefficient
energy
of
the value of boundary agreement
that
calculated
reactions The
“5. 26.
kinetic
activation
equations
energy
of
for a symmetric
t’emperature
than
with the metastable
controlled
33. 34.
eutectoid
35.
model.
an excessively diffusion
large
metastable hypothesis,
will only be attained
to which The larger of of
of the interface rate.
with the recording
with the solution to the stability growbh reactions.
The local
model
but an unequivocal change
from
estimate
monotectoid
value for the free energy
is
an
conclusion of a precise
of the reaction
and
ANDERKO, C’onstitutiou
of
E&wry
APPENDIX
1. F. W. JONES, P. LEECH and C. SSKES, Proc. R. SW. 181. 154 (1942). 2. H. K. HARDY, J. Inst. MetaZs 75. 707 (1948-g). 3. A. H. GEISLER, Phase Transforkationd in Solids, p. 432.
I
Calculations of Interface Mobility The
interface
mobility
transformations alone, first
i.e., without example
the average
growth.
radius
is M = M’/4aV,
in the kinetic
Feltham
three
reaction
of a grain
The the
approxi-
at any instant
the mobility
where
that
previously
M’ is t,he constant
equation, r2 -
kinetics
Assuming
r, of a grain boundary
then it is easily shown that defined
from
by the interface
long range solute diffusion.
is grain
radius of curvature, mates
is calculated
controlled
problem for lamellar
REFERENCES
Wiley (1951).
36.
the
or the existence
ahead
K.
M. HILLERT, Jernkont. Annlr 141, 757 (1957). M. C. IN&IAN and H. R. TIPLER, Metall. Rec. 8, 105 (1963). H. I. AARONSON and Y. C. Lru, Scripta Met. 2, 1 (1968). P. FELTHAM and G. J. COPLEY, Acta Met. 6, 593 (1958). T. B. MASSALSKI, Acta Met. 6, 243 (1958). D. A. RIGNEY and J. M. BLAKELU, Acta AWet. 14, 1375 (1966). W. A. TILT~ER,Acta Met. 14, 1383 (1966). D. A. RIGNEY and J. M. BLAKEL~, Acta Net. 14, 1385 (1966). K. HULTGREN, R. L. ORR, P. D. ANDERSON and K. K. KELLEI’, Selected Values of Thermodynamic Properties of Metals and Alloys, p. 260. Wiley (1963). A. H. COTTRELL, Theoretical Structural Metallurgy, p. 158. Arnold (1955).
to zero is consistent
to reduce the nodule growth
equilibrium
interface
between
may reflect a departure
of the reaction,
volume
for
temperature
monotectoid
spacings
equilibrium,
the free energy substantial
kcal/mole
and the
27. 28. 29. 30. 31. 32.
system yield an
large difference
spacing extrapolates
than predicted
attractive
25
The observed
the reciprocal
tending
for quantismaller
mobility
applied to the copper-indium
diffusion.
local
purely
H’ANSEN and
A&us, 1). 590. McGraw-Hill (19581. H. I%H~, 2. MetaUk. 50, 87 {1959j. H. B~~HDI,2. Met&k. 52, 564 (1961). 1). d. H. CORDEROY and R. W. K. H~~-E~(,~>IBE, J. Inst. Metals. 92, 65 (1963-4). D. TI-RNBIJLL and H. N. TREAFTIS. Tmns. ,4m. Id. Nin. Enqrs. 212, 33 (1958). G. TnoMAs, Transmission Electrotr Microscopy of Netale, p. 153. Wiley (1962). .J. W. CAHN and W. C. HAGEL, Decompositio?l of Amtenite by D~~us~onaZ Processes, p. 132. Interscience (1962). M. SVLONEN, Suomal. Tiedeakaf. Terim. A VI, 1 (1957). ,J. W. CanN and W. C. HAREL, Acta Met. 11, 561 (1963). J. B. NE~V’KIRK, Precipitation from, Solid Solution, p. 22, ASM (1959). H. K. HARDY and T. J. ~;EAI+ Prog. Xetal Phys. 5, 143 (1954). J. D. EIBOR’~ and R. B. SIcnor.sox, Acta Xet. 13, 403 (1965).
in other copper alloys.
reaction
solvus
from
required
1000 times
22. 23. 24.
for
is 36.5 kcal/mole.
mobility
is about
21.
the
The large values of spacing support Cahn’s model of boundary diffusion and reaction control. However tative
C. HAGEL and H. J. BEATIE, JR., Special Report 64 of
I I.
15. iit.
For
1968
Iron and Steel Institute, p. 98 (1959). B&M, Z. Metallk. 52, 518 (1961). 8. STJLONES, Acta Met. 12, 748 (1964). 7. M. 8. SI-LONEN, Acta polytech. stand. Ch 28, 1 (1964). x. J. S. KIRKALDE~, Decomposition of Austenitt by Diflusional Processrs, p. 39. Interscience (1962). 0. 11.TURNBTTL, Acta Met. 3. 55 (1955). IO. J. S. HIHSCHHORX and R..A. &RE&, _4ct~ Net. 12, 120
The growth rat*e and lamellar spacing of the discontinage CIphase compositjion for some heat treatments.
16,
r,,2 = M’t.
and Copley(30) measured the grain growth Cu-Zn alloys (lo%-35% Zn)
in concentrated
in the temperature range 475°C to 700°C. M’ was found to be a function of concentration, and so for comparison
with
the
values
of mobility
found
for
8H;ZPIRO
Cu-7.5:/,
KIRKALDY:
DISC’OKTIN~O17S
In we choose data for 22.5%
M
Zn. assuming
of indium has the same effect as 3 at.“,;
that 1 at.% zinc.
AND
obtained
in this
way
is presented
straight line on Fig. 14 and extrapolated t#he lower temperatures The
massive
In the Cu-Ga the [, phase,
from the temperature is stable.
for Cu-7.5%
is
a
appears
some
In.
12.51
ALLOTS
Excellent’
APPENDIX
agreement
massive
even on quenching
II
Calculation of E, and E1,2_r,
diffusionless
of the growth of t,he
systemf3n
Cu-In
as a
slightly to
new phase appears to be t’he passage of an incoherent product,,
appropriate
IS
with the two previous estimates of mobilit,y is found.
of interest.
t,ransformation
react,ion for which the mechanism boundary.
PRECIPITBTION
We assume that q need be evaluated product
tc phase at’ the solubility
symmetric
only for t,he
limit in this non-
From equat,ion (14a) of t,hr theoreti-
case.
cal papcro3) wc tribe
range where the parent’ 1 phase
ay
i
Thus the growth rates are very high, and we
(10)
q=iax”i
e&matte from micrographs’31) t,hat the minimum value is V = 10e2 cm/see
We choose as the thermodynamic
model of the solid
solution
interaction
t’he nearest
found in standard
neighbour
texts
AF for this reaction cannot be large, for the reaction
The free energy of formation
is observed in Cu-Ga alloys at 6OO”C, only 16°C below the temperature where the /3 phase is stable. Since the
sed as
of -AF
we estimate
a maximum
of the solution is expres-
f” = 2RT, X (1 -- X) + RT [X In X
5, phase can only be stable with respect to /l for some finite undercooling,
value
+ (1 -
Therefore
20 Cal/mole +
a conservative
and the solubility
log erg/mole
estimate of the mobility
at,
M = lo-l1 cm.mole.sec-i.erg-l
limit is approximated
X,‘(T)=exp
about, 600°C is where
T, is the temperature
fair
(11) and equation
with
the value
obtained
for
grain
Finally,
we consider
an example
alloys is somewhat undercooling
more remote.
used
at the interface
directions.‘s2)
suggest
that
for which
Solidification
for various
The experimental
the recorded
mobilities
may
be
(4) and to
th e value
for AHJT,
of
0.01
-3.31
atomic
mobilities
of
systems are of the same magnitude ing solidus temperatures, Fig.
14 at the
We shall metal
of
1
(1 -
1 X,2
XJ2
11
(13)
= 0.027, so that and E, = -9
5 2
arises from the oversimplified
We interpret phase
occurs
l/2 -
a as l/2 -
3X,” because the S
at X A 0.3 instead follows
symmetrical
model
of E, with temperature.
phase
of X = 1.
from the discussion of an analysis diagram.
which For
This
in Article the
assumes Cu-In
system 2 E1,2_a = -3
kcal/mole
at the correspond-
and hence plot this value in
temperature
(T) in
kcal/mole.
a
different
(
q = 20.6 kcal/mole,
A on the limitations
For pure tin this value applies at 505°K.
xe
X,” (673°K)
T, = 1215’K,
interpretation
M = 7.5 X lo-l1 cm.mole.sec.-lerggl
that
For a Cu-In,
and the variation
Cal/mole oK(35) yields
assume
X = X,”
X,)
2To
The uncertainty
to have a value of about
-
+ T
techniques
v=wAT,
Using
Y
crystallo-
agree with the empirical expression
cm/sec.“K.
(12)
1
12X,(1
of the
AF = AH, AT/T, may be combined
where w was found
.
indicated by equation
(10) (considering
R2T2r E, = -
rates
as a function
low by an order of magnitude.(33s34) Equation the relation
the
controlled reaction in Cu-In
in pure tin have been measured
by
the latter operation) results in the following expression :
growth at the same temperature. relationship to a mobility
1
-WI (11)
as the top of the mis-
This single estimate is plot’ted in Fig. 14 and is seen in agreement
-‘:
(
cibility gap. The differentiation
graphic
X) In (1 -
to be - AF s
model
such as that of Cottrell.(a@
1090”K,
which
is
at 4OO”C, and it must be remembered also temnerature
denendent.
that Eliz+ is
ACTA
1252
METALLURGICA,
ACKNOWLEDGMENTS
The authors assistance:
gratefully
the National
for a grant-in-aid University
of
acknowledge Research
the following
Council of Canada
of research
Graduate
(JSK) ; the McMaster the Research Scholarship,
Ontario Graduate Fellowship,
VOL.
and the St,eel Company
16,
Canada
cooperation Steel thanks
1968
Fellowship
in Metallurgy
of the Research
Company
is sincerely
are due to Dr. J. W.
Corderop
for their comments
(JMS).
The
Department
of Inland
appreciated.
Finally,
Cahn and Dr. J. A.
and suggestions.