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Formation of modulated structures in copper-nickel-iron alloys
FORMATION
OF MODULATED
STRUCTURES
M. HILLERTt,
M.
COHEN3
IN COPPER-NICKEL-IRON
ALLOYS*
and B. L. AVERBACHS
By means of X-ray diffraction, the formation of modulated structures with periodic variations in composition due to a simple interchange of atoms has been studied in copper-nickel-iron alloys as a function of composition, temperature and time. The results are compared with two models suggested for this type of “exchange transformation”. The reaction characteristics seem to favor a kinetic model rather than the Guinier model. The activation energies for the transformation, evaluated at the first detectable stage and for a later stage, show that both these stages are diffusion-controlled inside as well as outside the spinodal. This indicates that the spinodal is of no real significance with regard to the nucleation process. The variation with temperature and composition of the first wavelength to form in the modulated structure is in agreement with the kinetic model. FORMATION
DUNE STRUCTURE PERIODIQUEMENT MODULEE DANS LES ALLIAGES Cu-Ni-Fe Les auteurs ont Btudii: par diffract,ion de rayons X, en fonction de la composition, de la temperature et du temps, la formation dans les alliages cuivre-nickel-fer d’une structure periodique avec variation alternative de la composition due a un simple &change entre atomes. 11s ont compare les resultats obtenus avec deux modeles imagnines pour rendre compte de cette transformation par Bchange. Les caracteristiques de la reaction semblent pencher en daveur d’un mod& base sur une equation cinetique plutot qu’en faveur du modele de Guinier. L’energie d’activation de cette transformation determinee au premier stade detectable et B un stade ulterieur montre que ces deux &tats sont regis par le phenomene de diffusion aussi bien a l’interieur qu’& l’exterieur du domaine spinodal. Ceci indique que la courbe spinodale n’a pas de signification reelle en ce qui concerne la germination. La variation des premieres longueurs d’ondres avec la temperature et la composition pour former une structure periodique est en bon accord avec le mod&e base sur l’equation cinetique. DIE
BILDUNG VON MODULLIERTEN STRUKTUREN BE1 KUPFER-NICKEL-EISENL-EGIERUNGEN An Kupfer-Nickel-Eisen-Legierungen wurde rontgenographisch die Bildung von modulierten Strukturen mit periodischen Anderungen der Zusammensetzung als Folge eines einfachen Austauschs van Atomen untersucht in Abhiingigkeit van Zusammensetzung, Temperatur und Zeit. Die Resultate werden verglichen mit zwei Modellen, die fur diese Art van “Austauschumwandlung” vorgeschlagen worden sind. Die Reaktion scheint mehr im Einklang mit einem kinetischen Model1 zu verlaufen, als mit dem Guinier-Modell. Die Aktivierungsenergien fur die Umwandlung, ausgewertet bei dem ersten nachweisbaren Stadium und bei einem spateren Stadium, zeigen, da13 diese beiden Stadien durch die Dies zeigt, da6 die Spinodale fur den Diffusion bedingt sind und zwar beiderseits der Spinodalen. Die Temperaturund KonzentrationsabhangigKeimbildungsprozel3 keine wshre Bedeutung besitzt. keit der ersten Wellenliinge, die sich in der modulierten Struktur bildrt, stimmt mit dem kinetischen Model1 iiberein.
1. INTRODUCTION
transformations
Phase
the formation
sometimes
in metals commonly
however,
by a rearrangement
In the present paper, this type of reaction simply
usually
exchange
transformation positions
are often
found
importance is
because the
reactions,
of
the
within a misci-
of Guinier-Preston
zones
of the latter category. theories were first applied
interfacial
who emphasized energy
between
the the
nucleus and matrix.
He derived an expression for this
interfacial
by
energy
considering
an
exchange
transformation.
Such transform-
in ordering
reactions
to metallic systems by Becker”),
with one another,
with the aid of vacancies.
ations
The formation
The classical nucleation
taking place
of the atoms on the parent-lattice
referred to as an exchange
in precipitation
gap.
may be an example
there is only a composi-
tional change, the whole transformation
atoms
bility
of a new lattice and a new composition.
In some instances,
sites.
involve
In a different approach
and
* This
Borelius@) changes
pointed
out
in composition
to the nucleation the importance from
that
problem, of gradual
of the
original
paper is based on a thesis submitted in partial fulflllment of the requirements for the degree of Sc.D. in Metallurgy at the Massachusetts Institute of Technology, Cambridge, Mass., in May 1956. The research was sponsored by the U.S. Atomic Energy Commission. Received July 1,
matrix to that of the final precipitate. When doing so, he also was considering exchange transformations. He concluded that there should be no activation energy
1960. t Formerly Research Assistant,
barrier
Department of Metallurgy, Massachusetts Institute of Technology. Now at the Swedish Institute for Metal Research, Stockholm. z Department of Metallurgy, Massachusetts Institute of Technology, Cambridge, Mass. ACTA
METALLURGICA,
VOL.
9,
JUNE
1961
536
for
nucleation
inside
the
spinoclal§
in
a
QThe spinodal is the locus of points within a miscibility gap where 6*F/Sx2= 0. F is the free energy of mixing and x the composition.
HILLERT, miscibility
COHEN
gap.
AND AVERBACH:
This was in agreement
thermodynamics
but in opposition
Becker’s
The two approaches
theory.
MODULATED
with classical
as does Borelius’
S 0.9
treat-
In
when applied
view
of
the
fact
theories have actually transformations,
to exchange that
it seemed
z- 0.7 3 G 0.6
+0x
exchange
.; 0.3
nucleation
0. I
with exchange
and
the
are given in this paper.
tests of the significance
of the spinodal
by
kinetic experiments have usually been attempted on transformations involving changes in lattice as well
ABC
E
I
0 0. CU
0.2
homogenized
the Vycor
for inhomogeneous the available
systems seems capable of explainHowever, of such structures.
experimental
information
on modulated
structures was not sufficient for a thorough test of the model.
The X-ray
on the exchange
data to be presented in this paper transformation
in Cu-Ni-Fe
alloys
1.0 N ia Feo.3
at 950°C for 10 days, and quenched into
brine by breaking
cally from contaminating
formation
0.9
0.0
FIG. I. Compositions and temperatures of annealing experiments. Miscibility gap and spinodal calculated according to first approximation of the nearestneighbor model.
with a Carborundum
the
J
0.3 0.4 0.5 0.6 0.7 Composition, x
as in composition. Exchange transformations often lead to modulated structures. A solid-solution model recently proposedc6) ing
FGHI
0
appropriate
transformation,
no
a” 0.2 5
study on the kinetics of a
results of such an investigation Previous
these
particularly
to carry out an experimental straight-forward
all
+OXAO
: 0.4
transformations.
been concerned
,= 0.0
2 ; 0.5
a more recent theory of Furthermore, ment. nucleation due to one of the present authors,(5>6) also leads to the same prediction about the role of the spinodal
surface.
The alloys
powder
passing
selected
for
into powder
wheel, and separated Carborundum.
through
the
tubes under the liquid
were then ground
a
X-ray
magneti-
The metallic
325-mesh
sieve
experiments
and
Quartz capillaries of about 0.7 mm o.d. and 0.4 mm i.d. were filled with metallic helium.
2. EXPERIMENTAL
DETAILS
was
coarser
powder was used for chemical analysis, Table 1. powder
to a length
about 7 mm and sealed under an atmosphere
will be used for such a test.
537
ALLOYS
1.0
by Becker and
were later combined by Hobstetterc3) and shown(5) that the Scheilc4), and it was recently combination leads to the same conclusion about the of the spinodal
IN Cu-Ni-Fe T,,,=1116”K
to the result of
Borelius
significance
STRUCTURE
These
capsules
were dropped
of
of 50 mm
down
into a
quartz tube in a furnace held at a temperature
100°C
Although the relevant nucleation theories have been worked out for binary alloys, it was found advan-
above the peak (1116°K)
tageous
capillaries were slid into another quartz tube leading
to study
present purposes. with tie-lines
the ternary
Cu-Ni-Fe
for
This system has a miscibility
gap
that point
system
almost
corner in the phase diagram.c7) prepared, section Ni,.,Fea.,
having
their
toward
the copper
A series of alloys were
compositions
on
a binary
a solution treatment into
a
second
temperature
of the miscibility
of about 2 hr at temperature,
furnace
held
at
within the miscibility
were allowed
gap. After
to transform
a
the
predetermined
gap.
for different
The samples lengths
time, after which they were quenched into brine.
of The
the point
thin capsules could then be broken and the metallic
on the opposite side of the ternary diagram.
powder, which had sintered into a rod, was examined
through
this corner
They could thus be considered
and through
as binary alloys in the
quasi-binary system Cu-M where M stands for Ni,,,Fe,.3 (Fig. 1) and they will be treated as such in the following
discussion.
The stable phases within the
miscibility gap, as well as the parent solid solution, are all f.c.c. The alloys were vacuum melted from carbonyl nickel, spectrographic copper and a specially purified iron received
here from Battelle
Memorial
The melts were chill cast under vacuum
Institute,
in a copper
mold to minimize segregation. The castings (5 mm in dia.) were vacuum sealed in separate tubes of Vycor,
by X-rays in a 19-cm Debye-Scherrer camera. exposure time of 4 hr was usually adopted. TABLE 1. Compositions Alloy
A R
c E F B H :
cu 84.9 79.7 74.3 64.4 56.0 48.3 39.6 33.8 24.9
of dloys
(at. %)
I
Ni
Fe
,
10.5 14.2 18.0 24.8 30.7 36.6 42.2 46.2 53.0
4.6 6.1 7.7 10.8 13.3 15.1 18.2 20.0 22.1
An
538
ACTA
________
-
TABLE 2. Wavelength,
___~ _____
Transformation temp. (“K)
Time (min)
1111
METALLURGICA,
A
VOL.
9,
in number of atomic planes for several alloys
C
B
E
P
60
:17 0:s 2
96 112 122 150
0 0.15 0.5 2
72 82 102 122
0 0.2 0.5 2 16
1050
0 0.33 1.5 5 16 30 180 0 15
895
136)
800
126 202
122 140 202 256
;s 118 154 220
100 116 146 162 242
82 102 122 152 242
126 136 238
104 124 156
:: 94 118 162
62 74 82 104 180
54
54
72 100 140
88 144
56 68 90 102 138 162 300
46
46
46
74 92 118 136 204
76 90 114 140 204
70 90 120 148 222
68 86 108 156 186
56 76 100 138 168
44 68
44
44
:: 88 122 148
1;: 144
86 120 148
82 108 144
64 78
50 62 68 86 94 128
42
42
42
72 88 120
74 84 120
88 106 152
46 60
60 68
70 84 122
No satellites
are uncertain
from
J
40 56 72
because of lack of resolution between satellite and Bragg reflection.
previous
of these alloys reflections.
and the reaction
work@)
within
side-bands,
that
the
expressed in number of unit cells in the [loo] direction.
the miscibility
For
or satellites to
expressed in number of atomic planes.
These side-bands
closer to the main reflections proceeds,
68
68
gap gives rise to so-called the main X-ray
96
76
18720
known
(370)
100 118 256
98 120
662
was
136 174
0 240 480 960 2880 11880
Values in parentheses
It
78
(220) (290)
0 2880 11880
decomposition
114
2:: 720
720
I
H
G
(1000)
1100
1080
1961
move
as the transformation
can thus be followed
by
f.c.c.,
significance
1 = 2L
where
1 is the
same
distance
The physical
of L and 1 will be discussed in Section 3.1.
All the measured
data are compiled
in Table
terms of 1, together with values extrapolated
2 in
to zero
studying the distance between each pair of side-bands.
time of transformation
Most of the present measurements
The Z-values yielded by this method are fairly close to
(200) pair of side-bands.
were made on the
The evolution
of the side-
the first values actually
by the method shown in Fig. 2. observed,
and were taken as
bands was examined as a function of time, temperature and composition.
representative
The data obtained were evaluated Daniel-Lipson relation(s)
The distance between a satellite and its Bragg reflection could not be measured very accurately with
by means of the
L = h . tan 6J/[h2+ k2 + P) . 8191
(1)
of the first structure to form on trans-
formation.
the present technique, mainly appreciable width of the satellite.
because of the Due to this effect,
where 6 is the angle of the Bragg reflection (h, k, I) and
it is estimated
66 is the difference in angle between the center of a side-band and its Bragg reflection. L is a distance
Table 2 is at least &2 for 1 = 100, 17 for 1 = 200 and f15
that the uncertainty
for 1 = 300, the variation
in the Z-values in
with magnitude
of 1
HILLERT,
COHEN
MODULATED
AVERBACH:
AND
STRUCTURE
IN
Cu-Ni-Fe
ALLOYS
539
section through the ternary phase diagram was determined; it was found that in this section the miscibility
gap had almost
the same shape as pre-
dicted by the zeroth and first approximations of the nearest-neighbor model for binary solutions (the latter sometimes
called the quasi-chemical
slight asymmetry located this
unusually
assume
model
temperature bility
good
gap
is independent
according
Figure
1
indicates
20
IO (Annealing
m
Time,Minutes)
marks indicate These
FIG. 2. Parabolic plot of growth of wavelength.
being caused by the reciprocal
relationship
between 1
and 80. In some cases, the measurements particularly difficult because the side-bands hardly
be
reflection.
distinguished
The corresponding
parentheses develop
into
copper-poor,
f.c.c.
l-values
Bragg
are given
sharp reflections
characteristic
which
are tetragonal
the two
Eventually,
transformation
The latter compositions
of two
and the other
instead
of cubic
the coherency products
were evaluated,
is
become using
the relation between lattice parameter and composition determined
by Bradley et aZ.(g). In this way, the binary
annealing
asymmetry
mentioned
and
It
thus
.E -I
.c
2 ;
5 .
r
:
2.05
Distance
in Reciprocal
outside
reflect
actua.1
the
the
very
slight
above.
was first observed
by
Bradley(l”)
that
some
alloys, which are solution treated at a high
temperature
and then aged inside the miscibility
gap,
develop satellites or side-bands to t,he Bragg reflections during the early stage of precipitation. Lipson’s)
suggested
by a periodic variation sponding also
variation
a periodic
or modulation
of
the distance
to
the
(There is
X-ray
scattering
for practical
components
purposes
Cu, Ni and Fe are
Daniel and Lipson showed
in reciprocal
space
between
the
L
the
periodic
expressed in number of unit cells.
is the
Accord-
ingly, the observation indicates
in a corre-
and the main reflection should be equal to
where
variation
in composition
spacing.
the
power, which can be neglected
were caused
resulting
of the lattice
variation
the three
Daniel and
that the side-bands
along a cube axis of the lattice,
l/L
al
fall
3.1. X-ray theory of side-bands
side-bands
I
a
3. DISCUSSION
that
1 i
and
of the side-
was found.
probably
nearly alike in this respect.)
for
compositions
runs where no reaction
gap,
because
Positions of Lines CoherentTetrogonal Phoses
the
The check
miscibility
in
long reaction times, the side-bands
of coherency.
and
main
first-approximation
Each point usually represents
conditions
Cu-Ni-Fe
phases (Fig. 3), one copper-rich
because lost
the
in Table 2.
After relatively f.c.t.
from
and misci-
times.
were could
of composition
the
all
series of four different
I 30
safely
The symmetric
at which the development
bands was studied. I
to
may
in the nearest-
is shown in Fig. 1.
temperatures /
one
energy
for this system.
calculation
0 _0
symmetry,
that the interaction
neighbor
Only a
The peak of the gap was and 53 at.% copper. Because of
1116°K
at
theory).
was found.
main
wavelength
that the side-bands
reflection
that
the
of
as
the
wavelength
move closer
reaction of
the
proceeds modulation
increases during annealing. Hargreavedll) calculated the intensities of the sidebands for a periodic model in which regions of a Ro
Lattice
FIG. 3. Calculated intensity distribution in side bands for asymmetric alloy. Y is the number of Guinier zones of the type shown in Fig. 4, situated one after the other.
definite composition alternate with regions of another composition. He assumed that (a) these “lamellae” were all of the same thickness (i.e. the composition of the regions was symmetric composition
of the
alloy)
with respect to the mean and
(b)
the alternating
540
ACTA
compositions
METALLURGICA,
were identical to the compositions
two stable phases obtained As a consequence,
of the
after very long annealing.
Hargreaves’
calculation
VOL.
9,
1961
s;i
Composition
Lattice
A
lpyziq
could apply
only to the symmetric alloy at the center of the miscibility gap. He found experimentally that the two side-bands intensity
of a main reflection
are different
when the alloy composition
and suggested
in
is asymmetric,
that this is due to the difference
in
thickness of the alternating regions.
On the basis of this model, Balli and Zakharova(121 were able to compute the asymmetry of the
side-band length ing;
of the side-bands
alloy
composition,
asymmetry
should
of the periodic
concluded
increase
structure
this was verified experimentally.
demonstrated electron
by Biedermann.
microscopy.
Geisler
and
work showed that the tetragonal present
in the as-quenched
decomposition
in-
but
their
phases were already
alloy.
are evidently
subject recently
A
of modulated
structures
However,
an
review
slow
of
the
out that the observed width
whereas
a
periodic structure should result in sharp satellites. order to account isolated
zones
for the width, were
formed
composition
between
in
two lamellae
In this model
side-bands
from
proportional
he suggested
the
the
In that
homogeneous
of another
(Fig. 4), the distance main
reflection
is
comof the
inversely
to the width of the zone, L in unit cells or
1 in atomic planes. The case of a finite number of Guinier zones like the one in Fig. 4, situated
one after
in reciprocal
Planes,n
space,
N is the total
The ot,her quantities
Fig. 4. The amplitude proportional
G(R),
to
are defined
by
of the scattered X-radiation
is
and
the
intensity
obtained as the square of the amplitude ima~ary vicinity
part.
A
is simply
since it has no
was
calculation
of the (200)-reflection
made
in the
for the case er = 0.02,
parison among the different u values. Apparently
another
within
not being aware of the work by Balli
a
Tiedema
et aI.
a.lso treated t,he
model suggested by Hargreave@).
Their
calculation
again confirmed that’ the asymmetry
side-bands
is due
composition. they
concluded
bands.
to the
When
appreciable
a central lamella of one
matrix, each zone comprising position.
R is the vector
and Zakharova(r2),
has been published
appreciable
:n
ot Atomic
number of planes in the system and v is the number of
asymmetric
Guinier(r”) pointed have
Number
es = 0.002, 1 = 44, b = 20, c = 2 and v = 1, 2, 3 or
by Guinie+).
side-bands
Spacing=0
infinity, respectively. Fig. 3 presents the calculated intensities divided by u2 in order to fa,cilitate a com-
sufficiently
comprehensive
Lattice
of
to allow a study of the kinetics of the earlier stages of decomposition.
J I!
b
2C
alloys used
The kinetics
in the copper-nickel-iron
in this investigation
using
Newkirk alloy
b
( I-•cp)
Spacing=a.
I
, 1
-
Guinier zones.
The formation
alloys has also been
a copper-nickel-cobalt
vestigated
on anneal-
and Knelleru3)
in’
Lattice
the
as the wave-
grows
of periodic structures in Cu-N&Fe
that
i
FIG. 4. Model of Guinier zone in an asynmetxiricdfoy.
from the asymmetry
and
Spacing= 0 (I+E,)
t,hat the
diffuse
asymmetry
considering latter
intensity
for explaining expression
should
they discarded
intensity
Fig. 3.
It may be that Tiedema for
of this kind,
G(R)
and
predict
experimentally However,
the
(2) does not give a as demonstrated
in
et al. neglected
the
z-Ra, in the
rRalv/sin
there
an side-
Guinier’s model
the observed side-bands.
for G(R) in equation
effect of the second term,-sin
alloy
model,os)
between t’he two
diffuse
expression
of the
Guinier’s
This has not been observed
and, as a consequence,
of the
appears
to
be no
homogeneous matrix was treated by one of the present authors.(5) The following expression was found for the
reason for discarding Guinier’s model on these grounds.
part
3.2. Theories of formation
of
denoted G(R) =
X
the
structure
factor
which
James’17)
has
C(R) : sin 17 RUN sin rr Ra
The periodic -
sin v Ralv sin ir Ra
sin R Ral sin r Ra(1 -
+
sin 7~Ra2v
sin VTRa[t es) -
2b(l -
sin QTRa(1 -
es)]
es)
+
sin rrRa(l
-/-
E&
1
‘by the
(2)
was originally
rather than for elucidating the transforma-
tion characteristics of systems giving rise to this kind of X-ray scattering. GuinieP) pointed out two shortcomings of the model. First, it, predicts sharp satellites whereas the observed side-bands are quite broad.
sin rr Ra Zc(1 + &,)I II
suggested
Daniel and Lipson(8f for the purpose of explaining side-bands,
sin rr Ral
model
of modulated structures
Secondly,
in terms of the periodic model, the
movement of the side-bands toward the main reflection on continued aging would imply a drastic
HILLERT,
rearrangement length
COHEN
This process
zone,
as suggested
operative
outside
the spinodal
Guinier put forward
for the width of the side-bands.
the zone model
which
individual
matrix.
The movement
then implies that the width of the
zones increases with cont)inued reaction,
process which is readily
visualized
particle.
central lamella in Fig. 4 is thus equivalent and
the
depleted
two
side-lamellae
regions
of matrix
are
of the matrix
down-hill
its boundary,
to
the
the particle.
to diffuse from the rest
into
from them be transferred
The
to a particle
equivalent
surrounding
Material can thus be expected
a
and can be com-
pared with the growth of a precipitate
the side-lamellae
and
into the central lamella at
thus causing an increase
of its width.
On the other hand, the Guinier model does not account for the original
formation
of a zone,
or predict
its
A solid-solution periodic
in
model for inhomogeneous
a preceding
structures
the light
thus
paper,‘“)
seems
the formation quite
systems,
indicates
are in fact metastable
of this model,
structures
that
states.
In
of periodic A kinetic
natural.
treatment
based on the solid-solution
developed
and was found to yield specific information
about
the
characteristics matrix
model was also
of an exchange
mation such as is involved complete
can receive
the spinodal,
possible mechanism
here.
transfor-
It predicts that the
can be transformed
541
ALLOYS
by
Guinier,
can
be
only, where the side-
material
the Guinier
by down-hill
diffusion
For compositions model
instead
within
provides
a
of the periodic The structure suggested by Daniel and Lipson. kinetic model, on the other hand, which predicts a periodic
for the formation
structure with a spectrum
applicable
of wavelengths,
both inside and outside the spinodal,
so long as there is no effective
is just
barrier to nucleation.
This condition is certainly realized inside the spinodal, and probably
in a considerable
part of the miscibility
gap outside the spinodal where the barrier is relatively 10w.(~) Close to the solubility
limit,
should be a range of compositions barrier is of appreciable Guinier
model
might
height. be
miscibility
gap.
nucleation interest
As
process,
be
there
In this region,
applicable,
whereas
intimately
and it would
the t,he
for the rest of the
a consequence,
may
to establish
however,
where the nucleation
kinetic model would be appropriate transformation
initial width. presented
lamellae
from the rest of the matrix.
According
to this model, the transformation starts at widely separated positions only, resulting in individual zones, by the unchanged
Cu-Ni-F’o
individual
accounts
of the side-bands
IN
atoms as the wave-
Instead,
separated
STRUCTURE
seems most unlikely.
of the component
increases.
MODULATED
AVERBACH:
AND
the
mode
related
to
of the
be of considerable
the different
regions where t’he
two models apply. 3.3. Kinetics The increase
during
Guinier
model
growth
(or thickening)
aging
of zone
can be compared
of a plate-like
which leads to a kinetic equation
spontaneously,
2” -
width
in the
with the sideways precipitate,
of the form
I,” = k * (t -
to)
(3)
giving a wide spectrum of wavelengths. The average wavelength increases progressively as the reaction
with the value m = 2.
proceeds.
width of the zone when first formed and t, the time of
behavior
Hence, this model is able to account for the of the side-bands.
more critically, variation
In order to test the model
one should examine
with composition
in detail (a) the
and temperature
of the
1, according
compared involves area
sections. be calculated(6)
of a Guinier zone can also
by means of the kinetic treatment just
described, and for the symmetric alloy composition it was found that a Guinier zone should induce the formation
of
a periodic
structure
with
a
rather
uniform wavelength which, in turn, should give rise to quite sharp side-bands. The explanation of this situation is that a local fluctuation will give rise to up-hill diffusion if the alloy composition is within the spinodal.
This fact was not taken into account
Guinier when he described and, as a consequence, 2
a decrease
between
the development the growth
by
of a zone
in width
of an
spherical
to the kinetic
with general coalescence
Greenwood
The further development
the
On the other hand, the increase in average wavelength,
length.
in subsequent
I, denotes
its formation.
average wavelength in the first stage of transformation, and (b) the kinetics of growth of the average waveThis test will be undertaken
The quantity
of
regions has
particles,
diffusion-controlled,
the of
situation
the
of
interfacial
compositions. coalescence
of
(3) but with the value
In this case, 1 stands for the mean radius of
the particles. growth
can be
assuming that the process is and has derived a kinetic equation
of the same form as equation m = 3.
amount different
analysed
model,
because it merely
It seems reasonable
of the lamellar
to assume that the
wavelength
can be fairly well described
and an evaluation
of the exponent
important information transformation.
in the
present
by equation
(3),
m might give some
concerning
the
mode
of
Unfortunately, m cannot be rigorously obtained from equation (3) unless the value of t, - l,m/k is
ACTA
542
METALLURGICA,
VOL.
9,
1961
bands sharpen as the reaction continues, indicating that the spectrum narrow.
of wavelengths
In the beginning,
corresponds decrease
becomes more and more the width of the satellites
to the value u = 1 in Fig. 3;
in width
corresponds
later, the
to v = 2 or 3.
situation may lead to a more serious competition the remaining wavelengths. at any
stage
average
value
equation
may
value, i.e. 1 20
30 40 50
200 300
100 Reaction
Time,
growth
500
described
minutes
known.
However,
measurements
this quantity
at reaction
can be neglected
for
and a plot of log 1 versus log t can be used to evaluate m.
Fig. 5 presents a typical
temperature
plot of this kind for a
of 895”K, giving m = 4.8 for all the alloys.
by an equation
for
the by
six
alloys
straight
E-J
lines
are in
z(Jn = I??* (t -
equation
it was found
very
for a determination
to)
(4) becoming
The findings
accurate
data
are
of the two constants
can be equally
for by n = 2, m = 0 (which
satisfactorily
polation
with
be better
(4) with the experimental
that
the measured
accordance
may
of the form
I,“) . (I -
On comparing results,
axis represent the uncertainty
represented
it is possible that the
wavelength
too narrow as the process proceeds.
and n.
values
of the
where n is a measure of the extent to which the growth
required
in the data according
by
distribution
is retarded by the speotrum of wavelengths
The three vertical ranges marked close to the ordinate to Section 2. In view of this uncertainty,
of the
has increased from its initial
I,. Consequently,
(1” -
times much larger than t,,
only
which in turn depends on how much
process
5. Logarithmic plot of growth of wavelength.
FIG.
not
as demanded
(3), but also of the momentary
the average wavelength 60
Thus, t’he growth rate of 1
be a function
of the wavelength
of wavelengths,
This
among
m
well accounted
was utilized
for extra-
purpose in Fig. 2) and n = 1, m = 1 as well
as n = 0, m = 4 or 5 (Fig. 5).
Thus, although
the
previous value of m = 4 or 5 may be characteristic
of
ment for each alloy also falls close to its straight line,
a coalescence
as
indicating
already discussed,
it is also possible that the process
is at least partly
controlled
equation
(3).
It may be noted that the first measure-
that the neglected
quantity
t, -
is
l,“lk
much smaller than the time of the first measurement. A,
For the alloys
B and C, the data scatter
more
process
in a modulated
range of wavelengths.
structure,
by the availability
of a
In any event, the data cannot
widely about the straight lines shown and thus provide
be reconciled
no real evidence
because the sideways growth of a zone is not dependent
Nevertheless,
for the applicability
of equation
(3).
if one assumes that it holds for these
on the availability
alloys as well as for the others, the m value comes out
one
to be of the same magnitude
applicable
studied.
for all compositions
When the data at the several temperatures
are analysed,
the plots comply
with equation
are too large for the
Guinier model but come closer to the kinetic model. It should be emphasized, ments on alloys difficult
to carry
however,
that the measure-
very near the solubility out,
limit were
and the development
of the
side-bands could not be followed long enough under these conditions to make the results conclusive in this range. In his treatment of the coalescence process, Greenwood assumed that the same relative distribution of particle size always existed at any stage of the process.
This is not a valid assumption
the present case because it is observed
may
with t’he Guinier
of a range of wavelengths.
conclude
that
the
in the region
position examined
Guinier
model Hence,
model
of temperature
and
is not com-
in this study.
(3) with
m values lying between 4 and 5. The m values thus obtained
in this way
in
that the side-
3.4. Activation
energy
The preceding wavelength drawn although
analysis of the isothermal
does
not
concerning
allow
the
details
it seems to favor
often useful to evaluate
any
growth in
conclusions of
the
the kinetic
an activation
to be
mechanism model.
It is
energy for the
process, which can be done in the following way. For an isothermal reaction, there must be a relationship between time of reaction, t, the temperature, T, and the extent of reaction, the latter being here defined by the wavelength,
1. It is usually
relation can be represented
assumed
that this
by the equation:
t(L T) = f(J) * exp (Q/W
(5)
HILLERT,
COHEN
where all the temperature
dependence
in terms of the activation
energy,
hand, if one chooses to identify of the reaction
MODULATED
AND AVERBACH:
is accounted Q.
I
dependence
Cu-Ni-Fe
I
I
I
energy
ALLOYS
543
I
I
I
energy for the
rate-controlling mechanism (for instance, diffusion), it may be advantageous to take into account the temperature
IN
On the other
the activation
with the activation
for
STRUCTURE
of the free-energy
Slope
Corresponding
to 55
Kcol/mol-
change, _
AF, by writing: t(Z, T) = f(2) . AF(T)
* exp (Q/RT)
(6)
which gives for a specified value of 1:
log--
-
t(L T)
Alioy
F,
Exptl.
(7)
t(L T,,) where T, is any reference temperature
(here taken as
800°K). The first term in equation plotting
(7) was determined
log 1 versus log t for each temperature
composition,
and noting how much each curve had to along the log t-axis in order to achieve with the 800°K-curve
for
force for the reaction
the second term of equation decrease
in
interfacial
(7) can be regarded as the
energy
growth in wavelength
which enters in
accompanying
the
of the lamellar structure.
The
specific int’erfacial energy is not known experimentally but can be evaluated
theoretically
models that have been proposed. AF(T)/AF(T,) interfaces
enters, the simple model of coherent
proposed
by
sufficient for evaluating (7). According
from the various Since only the ratio
Becker(l)
was
considered
the second term of equation
to this treatment, the int,erfacial energy
First
is proportional thus
The second term of equation
calculated
example,
as
As an was 1.2 at T = 1100°K and
this quantity
was taken as 800°K.
is plotted versus l/T in Fig. 6, and the straight line drawn through the points has a slope corresponding to an activation noting
energy
of 66 kcal/mole.
It is worth
that the value would have come out 10 kcal/
mole lower if the more primitive equation used instead of equation The activation
(5) had been
(6) or (7).
energy for diffusion in the Cu-Ni-Fe
Therefore,
SC 0
by Danielt20) as 66 kcal/
it seems reasonable
It was also of considerable activation reaction nucleation
to assume that
that
since
this
process.
produced
logarithmically
could
have
Although
a bearing these
t,he shortest
detectable versus l/T
for two of the alloys.
stage of on
the
measurements
time of reaction
side-bands
was
plotted
as shown in Figs. 7 and 8
At low temperatures,
seem to fall on a straight
66 Kc011 ino1
interest to evaluate the
energy for the very first detectable
were not very precise,
Energy
T,
The left-hand side of equation (7)
the growth of the wavelength is controlled by diffusion, which is in accordance with the kinetic model.
0.9
Aclivotion
(7) was
2 * log [Ax( T)/Az( T,)].
0.6 at T = 1050”K, when the reference temperature
mole.
B c
of Side-Bonds
to (Ax)~ where Ax is the width of the
gap.
system has been determined
Alloy
Appeoronce
FIG. 7. C-curve for first stage of transformation in alloy P.
miscibility
the same a,lloy. The driving
LoQTime(Minutes)for
and
be displaced
the best possible coincidence
1
by
these data
line corresponding
to an
apparent activation energy of 55 kcal/mole. However, the temperature dependence of the free-energy change
1.4 L
I -5
I -4
-3 Left-
I hond
I -2 Side
I of
-1 Es.(T)
0
I
FIG. 6. Evaluation of activation energy for the growth of wavelength of modulated structure in Cu-Ni-Fe alloys.
should be taken into account as before, and the shortest time of reaction should thus be expected to be proportional to A F * exp (QIRT). This expression was evaluated for alloy F assuming that Q = 66 kcal/mole
and that AF is the difference in free energy
544
ACTA
I
I
I
I
METALLURGICA,
l-
I
0.9
VOL.
9,
preceding
1961
section
operative
here.
that
the
Guinier
kinetic model, which is effective 1.0
Slope Correspondinq 55 Kcal /mol
in this exchange
y I.1
to the
when the activation
has
r.
where
the
free
energy
of
goes to zero for a certa,in 8110~ composition
previously
temperature
Temperature
transformation.
temperature
nucleation
Spinodal
is not
energy barrier for nucleation is low or absent. One may conclude that the @nodal is of little significance
to
The
E: 0 1.2
model
It also lends further support
been
ident,ified(zl)
of the transformation
a temperature
wit,h the
nose
curve or even with
below the nose.
This does not seem
justified in view of the fact that the entire C-shape of the dashed 1.3
curve
in Fig.
7 was calculated
for the
growth process alone.
Alloy C,Exptl.
3.5. Initial wavelength
I
I
1.4
I
I
I 0 2 -I Log Time (Minutes) for First Appeoronce FIG.
between brium
8. C-curve
L
3
4
special
in alloy C.
(one-phase state.
state) and the equili-
with temperature the diffusion
of t~he wavelength,
distance,
should also
be taken into account, but this effect was quite small in the present investigation. The result for alloy F is represented in Fig. 7 by the dashed
curve which has been displaced
give the best fit with the experimental the proportionality (QIRT)
constant
between
interest
capable,
predicting
the magnitude
its variation
with
A comparison experime~ltally considerable
subsequent
growth
dependent
such comparisons
might
values
quantities.
Unfortu-
are not decisive
because:
grounds
but was chosen
because
it yielded
fairly close to the first wavelength values
closer
method
actually
values
observed.
of extrapolation
to the predicted
would
quantities.
It
on a growth
process
precise
mechanism
as the
of wavelength,
which also gave
experimental
the observation
~~hniques
of satellites
which
would
allow
at an earlier stage of
transformation. (2) Table 2 and Figs. 7 and 8 show that the reaction
the spinodal
inside the m~cibility to
showed
experimental
appears that this point can only be clarified by more
This finding is not surprising for alloy F because it
barrier
the
as obtained
model
the initial wavelength is not, justified on any theoretical
Q = 66 kcal. falls within
by
for the
of appearance
is mainly
composition.(6)
was evaluated
the kinetic
discrepancy,
yield
by the same diffusion
and
of
as well as
of t,he initial wavelengths
A less conservative
controlled
conditions,
since
t and AP * exp
process in this alloy is so rapid at all
side-bands
certain
temperature
and from
the nucleating
the time
is in
of this quantity
being higher than the predicted nately,
is of
model
in Fig. 2 and presented in Table 2.
The agreement is quite good over the entire temperature range. This indicates that that
under
product,
kinetic
(1) The method of extrapolat,ion used for evaluating
is not known.
temperatures
the
to
laterally points,
because
principle
extrapolation
for solid solutions was used for this calculation. The variation
or more properly the average
of the first transformation
This is why the initial wavelength
The zeroth approximation
and consequently
The initial wavelength, wavelength
of Side-Bands
for first stage of transformation
the initial
(two-phase)
I
gap,
nucleation.
be expected
as soon as it is quenched and there should
Possibly,
at the higher
a different
is very rapid at high temperatures, exchange
transformation
be no
before
result
been reached.
temperatures
for
alloy C, which does not enter within the spinodal until
star&
the predetermined
quench has
The initial l-values, as measured,
may
predictions
reaction
the
temperature
thus be characteristic (3) The
suggesting that the during
of a higher temperature. from
the kinetic
model
are
below 1000/T = 1.1. However, as shown in Fig. 8, the experimental points for alIoy C do not indicate the
based on an interaction energy Y, which has been evaluated from the shape of the miscibility gap,
presence spinodal,
applying However,
of any retarding factor just above the suggesting that the nucleation step is still
the nearest-neighbor interaction model. there is a difference in atomic size among
rapid enough to be obscured before the side-bands become
by the growth process detectable. This result
the components in the Cu-N-Fe system, giving rise to strains during the early stages of transformation.
is in
the
These strains are reduced in the final state where the
full
agreement
with
conclusion
in
the
HILLERT,
alloy
COHEN
contains
phases.
large regions
Consequently,
smaller
at
indicated
the
MODULATED
AVERBACH:
AND
IX
Cu-Ni-Fe
ALLOYS
545
of the two equilibrium
the
“effective”
beginning
of
v may
transformation
by the shape of the miscibility
equilibrium
STRUCTURE
be than
gap in the
diagram.
Therefore, it does not appear possible at present to base a test of the kinetic model on the magnitude of the initial wavelength. instead to examine length
It may be more appropriate
the variation
with temperature
of the initial wave-
and composition.
done in Figs. 9 and 10. The measured here compared
with the optimum
defined as the wavelength between
free-energy
This calculation
wavelength,
which maximizes
ture
the ratio
change and diffusion distance.c6) In principle,
to show the same variation
and
is lope
composition
lopt
with tempera-
as the kinetic
model
0.5 Composition,
whether the alloy composition
From Figs. 9 and 10, it is found that the variation temperature
agreement
and
composition
with theory,
is roughly
the main point
there is no sharp change at the spinodal (x = 0.34 and 0.66) in Fig. 10.
in
being that compositions
In view of this fact
The
value
of the
exponent
describing
the progressive
annealing
favors
range of compositions
the kinetic
model
offers a fairly good description
the very first detectable
the
stages
of
transformation
in
Cu-N-Fe
alloys. Two
models
structures
for
resulting
discussed.
the
formation exchange
It is concluded
only
when
of that
the
have
Guinier
only outside the spinodal
there is an appreciable
energy for nucleation.
modulated
reactions
The kinetic
operative when the activation
activation
effective
on
examined.
evaluated
for
stage of reaction as well as for indicates that in accordance
This implies that there is no
barrier to nucleation
in these experiments.
were studied on both sides of the
spinodal and it thus appears that the spinodal is of no real significance with regard to nucleation
in this type
of transformation.
model should be
barrier is low or absent,
The initial wavelength the kinetic model. by a number wavelength
1.0 z
and temperatures
increase of wavelength,
The transformations
equation
in the complete
energy for the reaction,
with the kinetic model.
from
model should be operative and
model
both these stages are diffusion-controlled 4. SUMMARY
been
the subsequent
m in the
increase of wavelength
the kinetic
The activation
of
is outside or inside the
spinodal.
as well as the findings in Section 3.4, it appears that early
I.0 Ni 0.7 Feo.3
X
would
predict. with
0
cu
FIG. 10. Wavelength of first transformation product (plotted points) compared with the calculated optimum wavelengths (drawn curve). Letters indicate alloy designations.
is much easier to carry out than one
based directly on the kinetic model. is expected
This is
wavelength
is larger than predicted
The discrepancy
of factors,
but the variation
with temperature
by
may be explained of initial
and composition
is in
general agreement with the kinetic model.
0.9
t
+ 0.8
ACKNOWLEDGMENTS
The financial support Energy
Commission
of the United States Atomic is
gratefully
acknowledged.
Thanks are also due to Mr. R. Goss for valuable in preparing Steel Institute
the
alloys.
kindly
The
supplied
American
Iron
the high-purity
help and iron.
REFERENCES 0
IO 20 30 40 50 60 70 00 90 100 II0 120 Wovelength
,.& Atomic
Plones
Fm. 9. Variation of initial wavelength with temperature in alloy P, compared with calculated optimum values, lost, for symmetric composition.
1. R. BECKER, 2. MetaUk. 29, 245 (1937). G. BORELIUS,Ann. Phys. Lpz., 28, 507 (1937). 3. J. N. HOBSTETTER, Trans. Amer. Inst. Min. (Metall.) Emgrs 180, 121 (1949). 4. E. SCREIL,2. MetaZZk. 42, 40 (1952). 5. M. HILLERT, A Theory of Nucleation for Solid MetaZZic Solutions, SC. D. Thesis, Mass. Inst. Tech. (1956).
2.
546
ACTA
6. M. HILLERT, Acta Met. 9, 525 (1961).
METALLURGICA,
7. W. K~STER and W. DANNBRL, 2. Metallk. 27, 220 (1935). 8. V. DANIEL and H. LIPSON, Proc. Roy. Sm. 182, 378 (1943). 9. A. J. BRADLEY, W. F. Cox and H. J. GOLDSCHMIDT, J. Inst. Met. 67, 189 (1941). 10. A. J. BRADLEY, Privah communication to DANIEL and LIPSON (Ref. 8). 11. M. E. HARGREAVES, Acta Cyst., Camb. 4, 301 (1951). 12. D. BALLI and M. I. ZAKHAROVA, Dokl. Akad. Nuuk. SSSR 96, 453, 737 (1954). 13. E. B~EDERXANN and E. KNELLER, 2. Metallk. 47, 289, 760 (lQ56).
VOL.
9,
1961
14. A. H. GEISLER and J. B. NEWKIRK,
15. 16. 17. 18. 19. 20. 21.
Trans. Amer. Inst. Min. (Metall.) Engrs 189, 101 (1949). A. GUINIER, Solid State Phys. 9, 293 (1959). A. GUINIER, Acta Met. 3, 510 (1955). R. W. JAMES, Optical Principles of the Diffraction of X-Rays. George Bell, London (1954). T. J. TIEDEMA, J. BOUMAN and W. G. BURGERS, Acta Met. 5, 310 (1957). G. W. GREENWOOD, Acta Met. 4, 243 (1956). V. DANIEL, Proc. Roy. Sot. 192, 575 (1947). G. BORELIUS, J. Metals, N. Y. 3, 477 (1951).
OF MODULATED
STRUCTURES
M. HILLERTt,
M.
COHEN3
IN COPPER-NICKEL-IRON
ALLOYS*
and B. L. AVERBACHS
By means of X-ray diffraction, the formation of modulated structures with periodic variations in composition due to a simple interchange of atoms has been studied in copper-nickel-iron alloys as a function of composition, temperature and time. The results are compared with two models suggested for this type of “exchange transformation”. The reaction characteristics seem to favor a kinetic model rather than the Guinier model. The activation energies for the transformation, evaluated at the first detectable stage and for a later stage, show that both these stages are diffusion-controlled inside as well as outside the spinodal. This indicates that the spinodal is of no real significance with regard to the nucleation process. The variation with temperature and composition of the first wavelength to form in the modulated structure is in agreement with the kinetic model. FORMATION
DUNE STRUCTURE PERIODIQUEMENT MODULEE DANS LES ALLIAGES Cu-Ni-Fe Les auteurs ont Btudii: par diffract,ion de rayons X, en fonction de la composition, de la temperature et du temps, la formation dans les alliages cuivre-nickel-fer d’une structure periodique avec variation alternative de la composition due a un simple &change entre atomes. 11s ont compare les resultats obtenus avec deux modeles imagnines pour rendre compte de cette transformation par Bchange. Les caracteristiques de la reaction semblent pencher en daveur d’un mod& base sur une equation cinetique plutot qu’en faveur du modele de Guinier. L’energie d’activation de cette transformation determinee au premier stade detectable et B un stade ulterieur montre que ces deux &tats sont regis par le phenomene de diffusion aussi bien a l’interieur qu’& l’exterieur du domaine spinodal. Ceci indique que la courbe spinodale n’a pas de signification reelle en ce qui concerne la germination. La variation des premieres longueurs d’ondres avec la temperature et la composition pour former une structure periodique est en bon accord avec le mod&e base sur l’equation cinetique. DIE
BILDUNG VON MODULLIERTEN STRUKTUREN BE1 KUPFER-NICKEL-EISENL-EGIERUNGEN An Kupfer-Nickel-Eisen-Legierungen wurde rontgenographisch die Bildung von modulierten Strukturen mit periodischen Anderungen der Zusammensetzung als Folge eines einfachen Austauschs van Atomen untersucht in Abhiingigkeit van Zusammensetzung, Temperatur und Zeit. Die Resultate werden verglichen mit zwei Modellen, die fur diese Art van “Austauschumwandlung” vorgeschlagen worden sind. Die Reaktion scheint mehr im Einklang mit einem kinetischen Model1 zu verlaufen, als mit dem Guinier-Modell. Die Aktivierungsenergien fur die Umwandlung, ausgewertet bei dem ersten nachweisbaren Stadium und bei einem spateren Stadium, zeigen, da13 diese beiden Stadien durch die Dies zeigt, da6 die Spinodale fur den Diffusion bedingt sind und zwar beiderseits der Spinodalen. Die Temperaturund KonzentrationsabhangigKeimbildungsprozel3 keine wshre Bedeutung besitzt. keit der ersten Wellenliinge, die sich in der modulierten Struktur bildrt, stimmt mit dem kinetischen Model1 iiberein.
1. INTRODUCTION
transformations
Phase
the formation
sometimes
in metals commonly
however,
by a rearrangement
In the present paper, this type of reaction simply
usually
exchange
transformation positions
are often
found
importance is
because the
reactions,
of
the
within a misci-
of Guinier-Preston
zones
of the latter category. theories were first applied
interfacial
who emphasized energy
between
the the
nucleus and matrix.
He derived an expression for this
interfacial
by
energy
considering
an
exchange
transformation.
Such transform-
in ordering
reactions
to metallic systems by Becker”),
with one another,
with the aid of vacancies.
ations
The formation
The classical nucleation
taking place
of the atoms on the parent-lattice
referred to as an exchange
in precipitation
gap.
may be an example
there is only a composi-
tional change, the whole transformation
atoms
bility
of a new lattice and a new composition.
In some instances,
sites.
involve
In a different approach
and
* This
Borelius@) changes
pointed
out
in composition
to the nucleation the importance from
that
problem, of gradual
of the
original
paper is based on a thesis submitted in partial fulflllment of the requirements for the degree of Sc.D. in Metallurgy at the Massachusetts Institute of Technology, Cambridge, Mass., in May 1956. The research was sponsored by the U.S. Atomic Energy Commission. Received July 1,
matrix to that of the final precipitate. When doing so, he also was considering exchange transformations. He concluded that there should be no activation energy
1960. t Formerly Research Assistant,
barrier
Department of Metallurgy, Massachusetts Institute of Technology. Now at the Swedish Institute for Metal Research, Stockholm. z Department of Metallurgy, Massachusetts Institute of Technology, Cambridge, Mass. ACTA
METALLURGICA,
VOL.
9,
JUNE
1961
536
for
nucleation
inside
the
spinoclal§
in
a
QThe spinodal is the locus of points within a miscibility gap where 6*F/Sx2= 0. F is the free energy of mixing and x the composition.
HILLERT, miscibility
COHEN
gap.
AND AVERBACH:
This was in agreement
thermodynamics
but in opposition
Becker’s
The two approaches
theory.
MODULATED
with classical
as does Borelius’
S 0.9
treat-
In
when applied
view
of
the
fact
theories have actually transformations,
to exchange that
it seemed
z- 0.7 3 G 0.6
+0x
exchange
.; 0.3
nucleation
0. I
with exchange
and
the
are given in this paper.
tests of the significance
of the spinodal
by
kinetic experiments have usually been attempted on transformations involving changes in lattice as well
ABC
E
I
0 0. CU
0.2
homogenized
the Vycor
for inhomogeneous the available
systems seems capable of explainHowever, of such structures.
experimental
information
on modulated
structures was not sufficient for a thorough test of the model.
The X-ray
on the exchange
data to be presented in this paper transformation
in Cu-Ni-Fe
alloys
1.0 N ia Feo.3
at 950°C for 10 days, and quenched into
brine by breaking
cally from contaminating
formation
0.9
0.0
FIG. I. Compositions and temperatures of annealing experiments. Miscibility gap and spinodal calculated according to first approximation of the nearestneighbor model.
with a Carborundum
the
J
0.3 0.4 0.5 0.6 0.7 Composition, x
as in composition. Exchange transformations often lead to modulated structures. A solid-solution model recently proposedc6) ing
FGHI
0
appropriate
transformation,
no
a” 0.2 5
study on the kinetics of a
results of such an investigation Previous
these
particularly
to carry out an experimental straight-forward
all
+OXAO
: 0.4
transformations.
been concerned
,= 0.0
2 ; 0.5
a more recent theory of Furthermore, ment. nucleation due to one of the present authors,(5>6) also leads to the same prediction about the role of the spinodal
surface.
The alloys
powder
passing
selected
for
into powder
wheel, and separated Carborundum.
through
the
tubes under the liquid
were then ground
a
X-ray
magneti-
The metallic
325-mesh
sieve
experiments
and
Quartz capillaries of about 0.7 mm o.d. and 0.4 mm i.d. were filled with metallic helium.
2. EXPERIMENTAL
DETAILS
was
coarser
powder was used for chemical analysis, Table 1. powder
to a length
about 7 mm and sealed under an atmosphere
will be used for such a test.
537
ALLOYS
1.0
by Becker and
were later combined by Hobstetterc3) and shown(5) that the Scheilc4), and it was recently combination leads to the same conclusion about the of the spinodal
IN Cu-Ni-Fe T,,,=1116”K
to the result of
Borelius
significance
STRUCTURE
These
capsules
were dropped
of
of 50 mm
down
into a
quartz tube in a furnace held at a temperature
100°C
Although the relevant nucleation theories have been worked out for binary alloys, it was found advan-
above the peak (1116°K)
tageous
capillaries were slid into another quartz tube leading
to study
present purposes. with tie-lines
the ternary
Cu-Ni-Fe
for
This system has a miscibility
gap
that point
system
almost
corner in the phase diagram.c7) prepared, section Ni,.,Fea.,
having
their
toward
the copper
A series of alloys were
compositions
on
a binary
a solution treatment into
a
second
temperature
of the miscibility
of about 2 hr at temperature,
furnace
held
at
within the miscibility
were allowed
gap. After
to transform
a
the
predetermined
gap.
for different
The samples lengths
time, after which they were quenched into brine.
of The
the point
thin capsules could then be broken and the metallic
on the opposite side of the ternary diagram.
powder, which had sintered into a rod, was examined
through
this corner
They could thus be considered
and through
as binary alloys in the
quasi-binary system Cu-M where M stands for Ni,,,Fe,.3 (Fig. 1) and they will be treated as such in the following
discussion.
The stable phases within the
miscibility gap, as well as the parent solid solution, are all f.c.c. The alloys were vacuum melted from carbonyl nickel, spectrographic copper and a specially purified iron received
here from Battelle
Memorial
The melts were chill cast under vacuum
Institute,
in a copper
mold to minimize segregation. The castings (5 mm in dia.) were vacuum sealed in separate tubes of Vycor,
by X-rays in a 19-cm Debye-Scherrer camera. exposure time of 4 hr was usually adopted. TABLE 1. Compositions Alloy
A R
c E F B H :
cu 84.9 79.7 74.3 64.4 56.0 48.3 39.6 33.8 24.9
of dloys
(at. %)
I
Ni
Fe
,
10.5 14.2 18.0 24.8 30.7 36.6 42.2 46.2 53.0
4.6 6.1 7.7 10.8 13.3 15.1 18.2 20.0 22.1
An
538
ACTA
________
-
TABLE 2. Wavelength,
___~ _____
Transformation temp. (“K)
Time (min)
1111
METALLURGICA,
A
VOL.
9,
in number of atomic planes for several alloys
C
B
E
P
60
:17 0:s 2
96 112 122 150
0 0.15 0.5 2
72 82 102 122
0 0.2 0.5 2 16
1050
0 0.33 1.5 5 16 30 180 0 15
895
136)
800
126 202
122 140 202 256
;s 118 154 220
100 116 146 162 242
82 102 122 152 242
126 136 238
104 124 156
:: 94 118 162
62 74 82 104 180
54
54
72 100 140
88 144
56 68 90 102 138 162 300
46
46
46
74 92 118 136 204
76 90 114 140 204
70 90 120 148 222
68 86 108 156 186
56 76 100 138 168
44 68
44
44
:: 88 122 148
1;: 144
86 120 148
82 108 144
64 78
50 62 68 86 94 128
42
42
42
72 88 120
74 84 120
88 106 152
46 60
60 68
70 84 122
No satellites
are uncertain
from
J
40 56 72
because of lack of resolution between satellite and Bragg reflection.
previous
of these alloys reflections.
and the reaction
work@)
within
side-bands,
that
the
expressed in number of unit cells in the [loo] direction.
the miscibility
For
or satellites to
expressed in number of atomic planes.
These side-bands
closer to the main reflections proceeds,
68
68
gap gives rise to so-called the main X-ray
96
76
18720
known
(370)
100 118 256
98 120
662
was
136 174
0 240 480 960 2880 11880
Values in parentheses
It
78
(220) (290)
0 2880 11880
decomposition
114
2:: 720
720
I
H
G
(1000)
1100
1080
1961
move
as the transformation
can thus be followed
by
f.c.c.,
significance
1 = 2L
where
1 is the
same
distance
The physical
of L and 1 will be discussed in Section 3.1.
All the measured
data are compiled
in Table
terms of 1, together with values extrapolated
2 in
to zero
studying the distance between each pair of side-bands.
time of transformation
Most of the present measurements
The Z-values yielded by this method are fairly close to
(200) pair of side-bands.
were made on the
The evolution
of the side-
the first values actually
by the method shown in Fig. 2. observed,
and were taken as
bands was examined as a function of time, temperature and composition.
representative
The data obtained were evaluated Daniel-Lipson relation(s)
The distance between a satellite and its Bragg reflection could not be measured very accurately with
by means of the
L = h . tan 6J/[h2+ k2 + P) . 8191
(1)
of the first structure to form on trans-
formation.
the present technique, mainly appreciable width of the satellite.
because of the Due to this effect,
where 6 is the angle of the Bragg reflection (h, k, I) and
it is estimated
66 is the difference in angle between the center of a side-band and its Bragg reflection. L is a distance
Table 2 is at least &2 for 1 = 100, 17 for 1 = 200 and f15
that the uncertainty
for 1 = 300, the variation
in the Z-values in
with magnitude
of 1
HILLERT,
COHEN
MODULATED
AVERBACH:
AND
STRUCTURE
IN
Cu-Ni-Fe
ALLOYS
539
section through the ternary phase diagram was determined; it was found that in this section the miscibility
gap had almost
the same shape as pre-
dicted by the zeroth and first approximations of the nearest-neighbor model for binary solutions (the latter sometimes
called the quasi-chemical
slight asymmetry located this
unusually
assume
model
temperature bility
good
gap
is independent
according
Figure
1
indicates
20
IO (Annealing
m
Time,Minutes)
marks indicate These
FIG. 2. Parabolic plot of growth of wavelength.
being caused by the reciprocal
relationship
between 1
and 80. In some cases, the measurements particularly difficult because the side-bands hardly
be
reflection.
distinguished
The corresponding
parentheses develop
into
copper-poor,
f.c.c.
l-values
Bragg
are given
sharp reflections
characteristic
which
are tetragonal
the two
Eventually,
transformation
The latter compositions
of two
and the other
instead
of cubic
the coherency products
were evaluated,
is
become using
the relation between lattice parameter and composition determined
by Bradley et aZ.(g). In this way, the binary
annealing
asymmetry
mentioned
and
It
thus
.E -I
.c
2 ;
5 .
r
:
2.05
Distance
in Reciprocal
outside
reflect
actua.1
the
the
very
slight
above.
was first observed
by
Bradley(l”)
that
some
alloys, which are solution treated at a high
temperature
and then aged inside the miscibility
gap,
develop satellites or side-bands to t,he Bragg reflections during the early stage of precipitation. Lipson’s)
suggested
by a periodic variation sponding also
variation
a periodic
or modulation
of
the distance
to
the
(There is
X-ray
scattering
for practical
components
purposes
Cu, Ni and Fe are
Daniel and Lipson showed
in reciprocal
space
between
the
L
the
periodic
expressed in number of unit cells.
is the
Accord-
ingly, the observation indicates
in a corre-
and the main reflection should be equal to
where
variation
in composition
spacing.
the
power, which can be neglected
were caused
resulting
of the lattice
variation
the three
Daniel and
that the side-bands
along a cube axis of the lattice,
l/L
al
fall
3.1. X-ray theory of side-bands
side-bands
I
a
3. DISCUSSION
that
1 i
and
of the side-
was found.
probably
nearly alike in this respect.)
for
compositions
runs where no reaction
gap,
because
Positions of Lines CoherentTetrogonal Phoses
the
The check
miscibility
in
long reaction times, the side-bands
of coherency.
and
main
first-approximation
Each point usually represents
conditions
Cu-Ni-Fe
phases (Fig. 3), one copper-rich
because lost
the
in Table 2.
After relatively f.c.t.
from
and misci-
times.
were could
of composition
the
all
series of four different
I 30
safely
The symmetric
at which the development
bands was studied. I
to
may
in the nearest-
is shown in Fig. 1.
temperatures /
one
energy
for this system.
calculation
0 _0
symmetry,
that the interaction
neighbor
Only a
The peak of the gap was and 53 at.% copper. Because of
1116°K
at
theory).
was found.
main
wavelength
that the side-bands
reflection
that
the
of
as
the
wavelength
move closer
reaction of
the
proceeds modulation
increases during annealing. Hargreavedll) calculated the intensities of the sidebands for a periodic model in which regions of a Ro
Lattice
FIG. 3. Calculated intensity distribution in side bands for asymmetric alloy. Y is the number of Guinier zones of the type shown in Fig. 4, situated one after the other.
definite composition alternate with regions of another composition. He assumed that (a) these “lamellae” were all of the same thickness (i.e. the composition of the regions was symmetric composition
of the
alloy)
with respect to the mean and
(b)
the alternating
540
ACTA
compositions
METALLURGICA,
were identical to the compositions
two stable phases obtained As a consequence,
of the
after very long annealing.
Hargreaves’
calculation
VOL.
9,
1961
s;i
Composition
Lattice
A
lpyziq
could apply
only to the symmetric alloy at the center of the miscibility gap. He found experimentally that the two side-bands intensity
of a main reflection
are different
when the alloy composition
and suggested
in
is asymmetric,
that this is due to the difference
in
thickness of the alternating regions.
On the basis of this model, Balli and Zakharova(121 were able to compute the asymmetry of the
side-band length ing;
of the side-bands
alloy
composition,
asymmetry
should
of the periodic
concluded
increase
structure
this was verified experimentally.
demonstrated electron
by Biedermann.
microscopy.
Geisler
and
work showed that the tetragonal present
in the as-quenched
decomposition
in-
but
their
phases were already
alloy.
are evidently
subject recently
A
of modulated
structures
However,
an
review
slow
of
the
out that the observed width
whereas
a
periodic structure should result in sharp satellites. order to account isolated
zones
for the width, were
formed
composition
between
in
two lamellae
In this model
side-bands
from
proportional
he suggested
the
the
In that
homogeneous
of another
(Fig. 4), the distance main
reflection
is
comof the
inversely
to the width of the zone, L in unit cells or
1 in atomic planes. The case of a finite number of Guinier zones like the one in Fig. 4, situated
one after
in reciprocal
Planes,n
space,
N is the total
The ot,her quantities
Fig. 4. The amplitude proportional
G(R),
to
are defined
by
of the scattered X-radiation
is
and
the
intensity
obtained as the square of the amplitude ima~ary vicinity
part.
A
is simply
since it has no
was
calculation
of the (200)-reflection
made
in the
for the case er = 0.02,
parison among the different u values. Apparently
another
within
not being aware of the work by Balli
a
Tiedema
et aI.
a.lso treated t,he
model suggested by Hargreave@).
Their
calculation
again confirmed that’ the asymmetry
side-bands
is due
composition. they
concluded
bands.
to the
When
appreciable
a central lamella of one
matrix, each zone comprising position.
R is the vector
and Zakharova(r2),
has been published
appreciable
:n
ot Atomic
number of planes in the system and v is the number of
asymmetric
Guinier(r”) pointed have
Number
es = 0.002, 1 = 44, b = 20, c = 2 and v = 1, 2, 3 or
by Guinie+).
side-bands
Spacing=0
infinity, respectively. Fig. 3 presents the calculated intensities divided by u2 in order to fa,cilitate a com-
sufficiently
comprehensive
Lattice
of
to allow a study of the kinetics of the earlier stages of decomposition.
J I!
b
2C
alloys used
The kinetics
in the copper-nickel-iron
in this investigation
using
Newkirk alloy
b
( I-•cp)
Spacing=a.
I
, 1
-
Guinier zones.
The formation
alloys has also been
a copper-nickel-cobalt
vestigated
on anneal-
and Knelleru3)
in’
Lattice
the
as the wave-
grows
of periodic structures in Cu-N&Fe
that
i
FIG. 4. Model of Guinier zone in an asynmetxiricdfoy.
from the asymmetry
and
Spacing= 0 (I+E,)
t,hat the
diffuse
asymmetry
considering latter
intensity
for explaining expression
should
they discarded
intensity
Fig. 3.
It may be that Tiedema for
of this kind,
G(R)
and
predict
experimentally However,
the
(2) does not give a as demonstrated
in
et al. neglected
the
z-Ra, in the
rRalv/sin
there
an side-
Guinier’s model
the observed side-bands.
for G(R) in equation
effect of the second term,-sin
alloy
model,os)
between t’he two
diffuse
expression
of the
Guinier’s
This has not been observed
and, as a consequence,
of the
appears
to
be no
homogeneous matrix was treated by one of the present authors.(5) The following expression was found for the
reason for discarding Guinier’s model on these grounds.
part
3.2. Theories of formation
of
denoted G(R) =
X
the
structure
factor
which
James’17)
has
C(R) : sin 17 RUN sin rr Ra
The periodic -
sin v Ralv sin ir Ra
sin R Ral sin r Ra(1 -
+
sin 7~Ra2v
sin VTRa[t es) -
2b(l -
sin QTRa(1 -
es)]
es)
+
sin rrRa(l
-/-
E&
1
‘by the
(2)
was originally
rather than for elucidating the transforma-
tion characteristics of systems giving rise to this kind of X-ray scattering. GuinieP) pointed out two shortcomings of the model. First, it, predicts sharp satellites whereas the observed side-bands are quite broad.
sin rr Ra Zc(1 + &,)I II
suggested
Daniel and Lipson(8f for the purpose of explaining side-bands,
sin rr Ral
model
of modulated structures
Secondly,
in terms of the periodic model, the
movement of the side-bands toward the main reflection on continued aging would imply a drastic
HILLERT,
rearrangement length
COHEN
This process
zone,
as suggested
operative
outside
the spinodal
Guinier put forward
for the width of the side-bands.
the zone model
which
individual
matrix.
The movement
then implies that the width of the
zones increases with cont)inued reaction,
process which is readily
visualized
particle.
central lamella in Fig. 4 is thus equivalent and
the
depleted
two
side-lamellae
regions
of matrix
are
of the matrix
down-hill
its boundary,
to
the
the particle.
to diffuse from the rest
into
from them be transferred
The
to a particle
equivalent
surrounding
Material can thus be expected
a
and can be com-
pared with the growth of a precipitate
the side-lamellae
and
into the central lamella at
thus causing an increase
of its width.
On the other hand, the Guinier model does not account for the original
formation
of a zone,
or predict
its
A solid-solution periodic
in
model for inhomogeneous
a preceding
structures
the light
thus
paper,‘“)
seems
the formation quite
systems,
indicates
are in fact metastable
of this model,
structures
that
states.
In
of periodic A kinetic
natural.
treatment
based on the solid-solution
developed
and was found to yield specific information
about
the
characteristics matrix
model was also
of an exchange
mation such as is involved complete
can receive
the spinodal,
possible mechanism
here.
transfor-
It predicts that the
can be transformed
541
ALLOYS
by
Guinier,
can
be
only, where the side-
material
the Guinier
by down-hill
diffusion
For compositions model
instead
within
provides
a
of the periodic The structure suggested by Daniel and Lipson. kinetic model, on the other hand, which predicts a periodic
for the formation
structure with a spectrum
applicable
of wavelengths,
both inside and outside the spinodal,
so long as there is no effective
is just
barrier to nucleation.
This condition is certainly realized inside the spinodal, and probably
in a considerable
part of the miscibility
gap outside the spinodal where the barrier is relatively 10w.(~) Close to the solubility
limit,
should be a range of compositions barrier is of appreciable Guinier
model
might
height. be
miscibility
gap.
nucleation interest
As
process,
be
there
In this region,
applicable,
whereas
intimately
and it would
the t,he
for the rest of the
a consequence,
may
to establish
however,
where the nucleation
kinetic model would be appropriate transformation
initial width. presented
lamellae
from the rest of the matrix.
According
to this model, the transformation starts at widely separated positions only, resulting in individual zones, by the unchanged
Cu-Ni-F’o
individual
accounts
of the side-bands
IN
atoms as the wave-
Instead,
separated
STRUCTURE
seems most unlikely.
of the component
increases.
MODULATED
AVERBACH:
AND
the
mode
related
to
of the
be of considerable
the different
regions where t’he
two models apply. 3.3. Kinetics The increase
during
Guinier
model
growth
(or thickening)
aging
of zone
can be compared
of a plate-like
which leads to a kinetic equation
spontaneously,
2” -
width
in the
with the sideways precipitate,
of the form
I,” = k * (t -
to)
(3)
giving a wide spectrum of wavelengths. The average wavelength increases progressively as the reaction
with the value m = 2.
proceeds.
width of the zone when first formed and t, the time of
behavior
Hence, this model is able to account for the of the side-bands.
more critically, variation
In order to test the model
one should examine
with composition
in detail (a) the
and temperature
of the
1, according
compared involves area
sections. be calculated(6)
of a Guinier zone can also
by means of the kinetic treatment just
described, and for the symmetric alloy composition it was found that a Guinier zone should induce the formation
of
a periodic
structure
with
a
rather
uniform wavelength which, in turn, should give rise to quite sharp side-bands. The explanation of this situation is that a local fluctuation will give rise to up-hill diffusion if the alloy composition is within the spinodal.
This fact was not taken into account
Guinier when he described and, as a consequence, 2
a decrease
between
the development the growth
by
of a zone
in width
of an
spherical
to the kinetic
with general coalescence
Greenwood
The further development
the
On the other hand, the increase in average wavelength,
length.
in subsequent
I, denotes
its formation.
average wavelength in the first stage of transformation, and (b) the kinetics of growth of the average waveThis test will be undertaken
The quantity
of
regions has
particles,
diffusion-controlled,
the of
situation
the
of
interfacial
compositions. coalescence
of
(3) but with the value
In this case, 1 stands for the mean radius of
the particles. growth
can be
assuming that the process is and has derived a kinetic equation
of the same form as equation m = 3.
amount different
analysed
model,
because it merely
It seems reasonable
of the lamellar
to assume that the
wavelength
can be fairly well described
and an evaluation
of the exponent
important information transformation.
in the
present
by equation
(3),
m might give some
concerning
the
mode
of
Unfortunately, m cannot be rigorously obtained from equation (3) unless the value of t, - l,m/k is
ACTA
542
METALLURGICA,
VOL.
9,
1961
bands sharpen as the reaction continues, indicating that the spectrum narrow.
of wavelengths
In the beginning,
corresponds decrease
becomes more and more the width of the satellites
to the value u = 1 in Fig. 3;
in width
corresponds
later, the
to v = 2 or 3.
situation may lead to a more serious competition the remaining wavelengths. at any
stage
average
value
equation
may
value, i.e. 1 20
30 40 50
200 300
100 Reaction
Time,
growth
500
described
minutes
known.
However,
measurements
this quantity
at reaction
can be neglected
for
and a plot of log 1 versus log t can be used to evaluate m.
Fig. 5 presents a typical
temperature
plot of this kind for a
of 895”K, giving m = 4.8 for all the alloys.
by an equation
for
the by
six
alloys
straight
E-J
lines
are in
z(Jn = I??* (t -
equation
it was found
very
for a determination
to)
(4) becoming
The findings
accurate
data
are
of the two constants
can be equally
for by n = 2, m = 0 (which
satisfactorily
polation
with
be better
(4) with the experimental
that
the measured
accordance
may
of the form
I,“) . (I -
On comparing results,
axis represent the uncertainty
represented
it is possible that the
wavelength
too narrow as the process proceeds.
and n.
values
of the
where n is a measure of the extent to which the growth
required
in the data according
by
distribution
is retarded by the speotrum of wavelengths
The three vertical ranges marked close to the ordinate to Section 2. In view of this uncertainty,
of the
has increased from its initial
I,. Consequently,
(1” -
times much larger than t,,
only
which in turn depends on how much
process
5. Logarithmic plot of growth of wavelength.
FIG.
not
as demanded
(3), but also of the momentary
the average wavelength 60
Thus, t’he growth rate of 1
be a function
of the wavelength
of wavelengths,
This
among
m
well accounted
was utilized
for extra-
purpose in Fig. 2) and n = 1, m = 1 as well
as n = 0, m = 4 or 5 (Fig. 5).
Thus, although
the
previous value of m = 4 or 5 may be characteristic
of
ment for each alloy also falls close to its straight line,
a coalescence
as
indicating
already discussed,
it is also possible that the process
is at least partly
controlled
equation
(3).
It may be noted that the first measure-
that the neglected
quantity
t, -
is
l,“lk
much smaller than the time of the first measurement. A,
For the alloys
B and C, the data scatter
more
process
in a modulated
range of wavelengths.
structure,
by the availability
of a
In any event, the data cannot
widely about the straight lines shown and thus provide
be reconciled
no real evidence
because the sideways growth of a zone is not dependent
Nevertheless,
for the applicability
of equation
(3).
if one assumes that it holds for these
on the availability
alloys as well as for the others, the m value comes out
one
to be of the same magnitude
applicable
studied.
for all compositions
When the data at the several temperatures
are analysed,
the plots comply
with equation
are too large for the
Guinier model but come closer to the kinetic model. It should be emphasized, ments on alloys difficult
to carry
however,
that the measure-
very near the solubility out,
limit were
and the development
of the
side-bands could not be followed long enough under these conditions to make the results conclusive in this range. In his treatment of the coalescence process, Greenwood assumed that the same relative distribution of particle size always existed at any stage of the process.
This is not a valid assumption
the present case because it is observed
may
with t’he Guinier
of a range of wavelengths.
conclude
that
the
in the region
position examined
Guinier
model Hence,
model
of temperature
and
is not com-
in this study.
(3) with
m values lying between 4 and 5. The m values thus obtained
in this way
in
that the side-
3.4. Activation
energy
The preceding wavelength drawn although
analysis of the isothermal
does
not
concerning
allow
the
details
it seems to favor
often useful to evaluate
any
growth in
conclusions of
the
the kinetic
an activation
to be
mechanism model.
It is
energy for the
process, which can be done in the following way. For an isothermal reaction, there must be a relationship between time of reaction, t, the temperature, T, and the extent of reaction, the latter being here defined by the wavelength,
1. It is usually
relation can be represented
assumed
that this
by the equation:
t(L T) = f(J) * exp (Q/W
(5)
HILLERT,
COHEN
where all the temperature
dependence
in terms of the activation
energy,
hand, if one chooses to identify of the reaction
MODULATED
AND AVERBACH:
is accounted Q.
I
dependence
Cu-Ni-Fe
I
I
I
energy
ALLOYS
543
I
I
I
energy for the
rate-controlling mechanism (for instance, diffusion), it may be advantageous to take into account the temperature
IN
On the other
the activation
with the activation
for
STRUCTURE
of the free-energy
Slope
Corresponding
to 55
Kcol/mol-
change, _
AF, by writing: t(Z, T) = f(2) . AF(T)
* exp (Q/RT)
(6)
which gives for a specified value of 1:
log--
-
t(L T)
Alioy
F,
Exptl.
(7)
t(L T,,) where T, is any reference temperature
(here taken as
800°K). The first term in equation plotting
(7) was determined
log 1 versus log t for each temperature
composition,
and noting how much each curve had to along the log t-axis in order to achieve with the 800°K-curve
for
force for the reaction
the second term of equation decrease
in
interfacial
(7) can be regarded as the
energy
growth in wavelength
which enters in
accompanying
the
of the lamellar structure.
The
specific int’erfacial energy is not known experimentally but can be evaluated
theoretically
models that have been proposed. AF(T)/AF(T,) interfaces
enters, the simple model of coherent
proposed
by
sufficient for evaluating (7). According
from the various Since only the ratio
Becker(l)
was
considered
the second term of equation
to this treatment, the int,erfacial energy
First
is proportional thus
The second term of equation
calculated
example,
as
As an was 1.2 at T = 1100°K and
this quantity
was taken as 800°K.
is plotted versus l/T in Fig. 6, and the straight line drawn through the points has a slope corresponding to an activation noting
energy
of 66 kcal/mole.
It is worth
that the value would have come out 10 kcal/
mole lower if the more primitive equation used instead of equation The activation
(5) had been
(6) or (7).
energy for diffusion in the Cu-Ni-Fe
Therefore,
SC 0
by Danielt20) as 66 kcal/
it seems reasonable
It was also of considerable activation reaction nucleation
to assume that
that
since
this
process.
produced
logarithmically
could
have
Although
a bearing these
t,he shortest
detectable versus l/T
for two of the alloys.
stage of on
the
measurements
time of reaction
side-bands
was
plotted
as shown in Figs. 7 and 8
At low temperatures,
seem to fall on a straight
66 Kc011 ino1
interest to evaluate the
energy for the very first detectable
were not very precise,
Energy
T,
The left-hand side of equation (7)
the growth of the wavelength is controlled by diffusion, which is in accordance with the kinetic model.
0.9
Aclivotion
(7) was
2 * log [Ax( T)/Az( T,)].
0.6 at T = 1050”K, when the reference temperature
mole.
B c
of Side-Bonds
to (Ax)~ where Ax is the width of the
gap.
system has been determined
Alloy
Appeoronce
FIG. 7. C-curve for first stage of transformation in alloy P.
miscibility
the same a,lloy. The driving
LoQTime(Minutes)for
and
be displaced
the best possible coincidence
1
by
these data
line corresponding
to an
apparent activation energy of 55 kcal/mole. However, the temperature dependence of the free-energy change
1.4 L
I -5
I -4
-3 Left-
I hond
I -2 Side
I of
-1 Es.(T)
0
I
FIG. 6. Evaluation of activation energy for the growth of wavelength of modulated structure in Cu-Ni-Fe alloys.
should be taken into account as before, and the shortest time of reaction should thus be expected to be proportional to A F * exp (QIRT). This expression was evaluated for alloy F assuming that Q = 66 kcal/mole
and that AF is the difference in free energy
544
ACTA
I
I
I
I
METALLURGICA,
l-
I
0.9
VOL.
9,
preceding
1961
section
operative
here.
that
the
Guinier
kinetic model, which is effective 1.0
Slope Correspondinq 55 Kcal /mol
in this exchange
y I.1
to the
when the activation
has
r.
where
the
free
energy
of
goes to zero for a certa,in 8110~ composition
previously
temperature
Temperature
transformation.
temperature
nucleation
Spinodal
is not
energy barrier for nucleation is low or absent. One may conclude that the @nodal is of little significance
to
The
E: 0 1.2
model
It also lends further support
been
ident,ified(zl)
of the transformation
a temperature
wit,h the
nose
curve or even with
below the nose.
This does not seem
justified in view of the fact that the entire C-shape of the dashed 1.3
curve
in Fig.
7 was calculated
for the
growth process alone.
Alloy C,Exptl.
3.5. Initial wavelength
I
I
1.4
I
I
I 0 2 -I Log Time (Minutes) for First Appeoronce FIG.
between brium
8. C-curve
L
3
4
special
in alloy C.
(one-phase state.
state) and the equili-
with temperature the diffusion
of t~he wavelength,
distance,
should also
be taken into account, but this effect was quite small in the present investigation. The result for alloy F is represented in Fig. 7 by the dashed
curve which has been displaced
give the best fit with the experimental the proportionality (QIRT)
constant
between
interest
capable,
predicting
the magnitude
its variation
with
A comparison experime~ltally considerable
subsequent
growth
dependent
such comparisons
might
values
quantities.
Unfortu-
are not decisive
because:
grounds
but was chosen
because
it yielded
fairly close to the first wavelength values
closer
method
actually
values
observed.
of extrapolation
to the predicted
would
quantities.
It
on a growth
process
precise
mechanism
as the
of wavelength,
which also gave
experimental
the observation
~~hniques
of satellites
which
would
allow
at an earlier stage of
transformation. (2) Table 2 and Figs. 7 and 8 show that the reaction
the spinodal
inside the m~cibility to
showed
experimental
appears that this point can only be clarified by more
This finding is not surprising for alloy F because it
barrier
the
as obtained
model
the initial wavelength is not, justified on any theoretical
Q = 66 kcal. falls within
by
for the
of appearance
is mainly
composition.(6)
was evaluated
the kinetic
discrepancy,
yield
by the same diffusion
and
of
as well as
of t,he initial wavelengths
A less conservative
controlled
conditions,
since
t and AP * exp
process in this alloy is so rapid at all
side-bands
certain
temperature
and from
the nucleating
the time
is in
of this quantity
being higher than the predicted nately,
is of
model
in Fig. 2 and presented in Table 2.
The agreement is quite good over the entire temperature range. This indicates that that
under
product,
kinetic
(1) The method of extrapolat,ion used for evaluating
is not known.
temperatures
the
to
laterally points,
because
principle
extrapolation
for solid solutions was used for this calculation. The variation
or more properly the average
of the first transformation
This is why the initial wavelength
The zeroth approximation
and consequently
The initial wavelength, wavelength
of Side-Bands
for first stage of transformation
the initial
(two-phase)
I
gap,
nucleation.
be expected
as soon as it is quenched and there should
Possibly,
at the higher
a different
is very rapid at high temperatures, exchange
transformation
be no
before
result
been reached.
temperatures
for
alloy C, which does not enter within the spinodal until
star&
the predetermined
quench has
The initial l-values, as measured,
may
predictions
reaction
the
temperature
thus be characteristic (3) The
suggesting that the during
of a higher temperature. from
the kinetic
model
are
below 1000/T = 1.1. However, as shown in Fig. 8, the experimental points for alIoy C do not indicate the
based on an interaction energy Y, which has been evaluated from the shape of the miscibility gap,
presence spinodal,
applying However,
of any retarding factor just above the suggesting that the nucleation step is still
the nearest-neighbor interaction model. there is a difference in atomic size among
rapid enough to be obscured before the side-bands become
by the growth process detectable. This result
the components in the Cu-N-Fe system, giving rise to strains during the early stages of transformation.
is in
the
These strains are reduced in the final state where the
full
agreement
with
conclusion
in
the
HILLERT,
alloy
COHEN
contains
phases.
large regions
Consequently,
smaller
at
indicated
the
MODULATED
AVERBACH:
AND
IX
Cu-Ni-Fe
ALLOYS
545
of the two equilibrium
the
“effective”
beginning
of
v may
transformation
by the shape of the miscibility
equilibrium
STRUCTURE
be than
gap in the
diagram.
Therefore, it does not appear possible at present to base a test of the kinetic model on the magnitude of the initial wavelength. instead to examine length
It may be more appropriate
the variation
with temperature
of the initial wave-
and composition.
done in Figs. 9 and 10. The measured here compared
with the optimum
defined as the wavelength between
free-energy
This calculation
wavelength,
which maximizes
ture
the ratio
change and diffusion distance.c6) In principle,
to show the same variation
and
is lope
composition
lopt
with tempera-
as the kinetic
model
0.5 Composition,
whether the alloy composition
From Figs. 9 and 10, it is found that the variation temperature
agreement
and
composition
with theory,
is roughly
the main point
there is no sharp change at the spinodal (x = 0.34 and 0.66) in Fig. 10.
in
being that compositions
In view of this fact
The
value
of the
exponent
describing
the progressive
annealing
favors
range of compositions
the kinetic
model
offers a fairly good description
the very first detectable
the
stages
of
transformation
in
Cu-N-Fe
alloys. Two
models
structures
for
resulting
discussed.
the
formation exchange
It is concluded
only
when
of that
the
have
Guinier
only outside the spinodal
there is an appreciable
energy for nucleation.
modulated
reactions
The kinetic
operative when the activation
activation
effective
on
examined.
evaluated
for
stage of reaction as well as for indicates that in accordance
This implies that there is no
barrier to nucleation
in these experiments.
were studied on both sides of the
spinodal and it thus appears that the spinodal is of no real significance with regard to nucleation
in this type
of transformation.
model should be
barrier is low or absent,
The initial wavelength the kinetic model. by a number wavelength
1.0 z
and temperatures
increase of wavelength,
The transformations
equation
in the complete
energy for the reaction,
with the kinetic model.
from
model should be operative and
model
both these stages are diffusion-controlled 4. SUMMARY
been
the subsequent
m in the
increase of wavelength
the kinetic
The activation
of
is outside or inside the
spinodal.
as well as the findings in Section 3.4, it appears that early
I.0 Ni 0.7 Feo.3
X
would
predict. with
0
cu
FIG. 10. Wavelength of first transformation product (plotted points) compared with the calculated optimum wavelengths (drawn curve). Letters indicate alloy designations.
is much easier to carry out than one
based directly on the kinetic model. is expected
This is
wavelength
is larger than predicted
The discrepancy
of factors,
but the variation
with temperature
by
may be explained of initial
and composition
is in
general agreement with the kinetic model.
0.9
t
+ 0.8
ACKNOWLEDGMENTS
The financial support Energy
Commission
of the United States Atomic is
gratefully
acknowledged.
Thanks are also due to Mr. R. Goss for valuable in preparing Steel Institute
the
alloys.
kindly
The
supplied
American
Iron
the high-purity
help and iron.
REFERENCES 0
IO 20 30 40 50 60 70 00 90 100 II0 120 Wovelength
,.& Atomic
Plones
Fm. 9. Variation of initial wavelength with temperature in alloy P, compared with calculated optimum values, lost, for symmetric composition.
1. R. BECKER, 2. MetaUk. 29, 245 (1937). G. BORELIUS,Ann. Phys. Lpz., 28, 507 (1937). 3. J. N. HOBSTETTER, Trans. Amer. Inst. Min. (Metall.) Emgrs 180, 121 (1949). 4. E. SCREIL,2. MetaZZk. 42, 40 (1952). 5. M. HILLERT, A Theory of Nucleation for Solid MetaZZic Solutions, SC. D. Thesis, Mass. Inst. Tech. (1956).
2.
546
ACTA
6. M. HILLERT, Acta Met. 9, 525 (1961).
METALLURGICA,
7. W. K~STER and W. DANNBRL, 2. Metallk. 27, 220 (1935). 8. V. DANIEL and H. LIPSON, Proc. Roy. Sm. 182, 378 (1943). 9. A. J. BRADLEY, W. F. Cox and H. J. GOLDSCHMIDT, J. Inst. Met. 67, 189 (1941). 10. A. J. BRADLEY, Privah communication to DANIEL and LIPSON (Ref. 8). 11. M. E. HARGREAVES, Acta Cyst., Camb. 4, 301 (1951). 12. D. BALLI and M. I. ZAKHAROVA, Dokl. Akad. Nuuk. SSSR 96, 453, 737 (1954). 13. E. B~EDERXANN and E. KNELLER, 2. Metallk. 47, 289, 760 (lQ56).
VOL.
9,
1961
14. A. H. GEISLER and J. B. NEWKIRK,
15. 16. 17. 18. 19. 20. 21.
Trans. Amer. Inst. Min. (Metall.) Engrs 189, 101 (1949). A. GUINIER, Solid State Phys. 9, 293 (1959). A. GUINIER, Acta Met. 3, 510 (1955). R. W. JAMES, Optical Principles of the Diffraction of X-Rays. George Bell, London (1954). T. J. TIEDEMA, J. BOUMAN and W. G. BURGERS, Acta Met. 5, 310 (1957). G. W. GREENWOOD, Acta Met. 4, 243 (1956). V. DANIEL, Proc. Roy. Sot. 192, 575 (1947). G. BORELIUS, J. Metals, N. Y. 3, 477 (1951).