NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering ||
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Module 30
Precipitation from solid solution II
Lecture 30
Precipitation from solid solution II
NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering ||
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Keywords : Meta‐stable precipitates, GP zones, Cu‐4.5%Cu alloy, critical thickness of coherent
precipitates, kinetics of precipitation, precipitation hardening: mechanism, precipitate
coarsening, spinodal decomposition
Introduction In the last module we looked at the stability of super saturated solid solution. Terminal solid
solutions where the solubility decreases with temperature can be transformed to a
supersaturated state by homogenization and subsequent quenching. The process is known as
solutionizing (solution treatment). The strength of the alloy increases on subsequent aging if
the precipitates that form initially are coherent. The process is known as age hardening. There
are two modes of precipitation. These are homogeneous and heterogeneous. Heterogeneous
precipitation occurs at the grain boundaries, edges and corners. It does not contribute to the
strengthening of the grains. This needs to be suppressed. Strengthening is more effective if the
precipitates are uniformly distributed within the grains. This occurs in the case of homogeneous
precipitation. It is promoted by low surface energy () and low coherency strain (Es). The precipitates that fulfill these conditions are either coherent or semi‐coherent precipitates.
Normal precipitates are incoherent. The surface energy of such precipitates is high. It needs
higher thermal activation for nucleation. We are also familiar with the characteristics of various
types of precipitate that could form during age hardening. We would see in a little more detail
the stability of such precipitates and the mechanism of strengthening. Besides nucleation there
is an entirely different mode by which precipitates could form. It is known as spinodal
decomposition. We would see under what conditions it is likely to take place. We would also
learn about a few commercial age hardenable alloys.
Meta‐stable precipitates: As the name suggests these are not the most stable forms of precipitates. They form because
the nucleation of stable precipitates needs higher thermal activation to overcome the energy
barrier for the creation new high energy grain boundaries.
NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering ||
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Meta-stable precipitate
G
’
12
Meta-stable precipitates have higher volume free energy but lower surface free energy because of
favorable crystal structure.
A B A % B
TA
(a) (b)
T
1
2
L
+ L
The sketch (b) in slide 1 gives a part of the phase diagram of a hypothetical alloy made of two
metals A & B. Let us consider the thermodynamic stability of an alloy at a temperature T shown
by the horizontal dotted line in this sketch. This is best described by the free energy
composition diagram for the different phases that may be present at this temperature. These
are given in the sketch (a). Under normal equilibrium we can have only two solids (which is rich in A) & (which is rich in B) in this system. Let us consider the case of meta‐stable
precipitate ’. Its free energy composition plot has also been included in this sketch. Note that
its free energy at all compositions is higher than that of the stable phase . The common
tangents to the free energy plots of , and , ’ are shown by two dashed lines. Note the compositions of the matrix that can coexist with or ’ at this temperature. 1 is the
composition of the solvus representing the equilibrium between & . 2 is the composition of
the solvus representing the equilibrium between & '. Note that % B in matrix which is in equilibrium with the meta‐stable phase is higher than that of which is in equilibrium with
stable . It means that the phase diagram should have two solvus curves one representing
composition of in equilibrium and the other representing the composition of in equilibrium with ’. The former is shown with a firm line and the latter with a dashed line. If
there are three met‐stable precipitates there should be two additional solvus curves.
Recall that the driving force for precipitation is the volume free energy (Gv) which depends on
the degree of super saturation or the extent of super cooling. This is used up to overcome the
activation barrier. In the last module the expressions for critical radius and the activation
barrier were derived. These are as follows:
∗∆
(1)
Slide 1
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∆ ∗∆
(2)
Note that �Gv is negative (< 0) however Es > 0. The magnitude of �Gv is greater than Es so that
r* is positive. Figure 1 illustrates the physical significance of equation1 & 2 with respect to
stable and meta‐stable precipitates.
Note that the critical radius (r*) and the activation barrier (G*) for meta‐stable precipitate (’) is lower than that of the stable precipitate (). This is because the meta‐stable precipitate is
either coherent or semi‐coherent. Therefore its surface energy () is significantly lower than that of the incoherent stable precipitate.
Precipitation from super saturated solid solution of Cu in Al: Let us take a specific case of Al ‐ 4.5% Cu. Slide 2 shows the relevant portion of the Al‐Cu phase
diagram and a free energy composition diagram at a given ageing temperature. Note that is the stable phase. It is an inter‐metallic compound whose chemical formula is CuAl2.
Precipitation in this alloy begins with the formation of clusters of Cu atoms. This is commonly
known as Guiner Preston Zone (GPZ). Later it dissolves and a new precipitate denoted as ’’ nucleates. Later it is replaced by ’ and in successive stages. Amongst these is the most
stable phase it has the lowest free energy whereas GPZ has the highest fee energy. Note the
relative position of the free energy composition plots of the four different phases. The free
energy composition plot given in slide 2 has a set of dotted lines that are tangent to the free
energy composition plots of different precipitates and that of the matrix . This gives the composition of that coexists with the respective precipitates. For example 4 is the
composition of the solvus curve for , 3 is the composition of the solvus for ’ & so on. This gets reflected in the form of additional solvus curves in the phase diagram. Note that these
plots are schematic. These are not to scale.
rrr
G
r0
’
r*’ r*
G*’
G*
Fig 1
NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering ||
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Al-Cu terminal solid solution
Al 4.5 Cu
L+L
660
520
= CuAl2
T
G
4.5
’
’’
GPZ
0 → 1+ GPZ→ 2+’’→3+’ →4+
GPZ’’
’
1
The main features of the precipitates are given in a tabular form in slide 3. GP Zones are disk
shaped clusters consisting of a few Cu atoms within the matrix of . These are too thin and small to be considered as precipitates having a specific crystal structure. These are perfectly
coherent with the matrix. ’’ too is a disk shaped precipitate. Its crystal structure is tetragonal. One of its lattice parameters (a) is nearly same as that of aluminum which is FCC. Therefore the
cube planes of are its habit plane. The precipitate is coherent with the matrix. The lattice
parameter in the direction perpendicular to the habit plane is large. Therefore its growth in this
direction is restricted. It remains coherent as long as it is small. The third met‐stable precipitate
’ too has tetragonal crystal lattice. Its lattice parameter ‘a’ is nearly same as that of the matrix
. This too is coherent along the crystal directions [100] & [010]. Its lattice spacing along [001] is not as large as that of ’’. It is plate shaped and partially coherent with the matrix. Note that
the stable precipitate too has tetragonal lattice. Its lattice parameters are very much different
from that of the matrix. Therefore these are totally in coherent.
Slide 2
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Precipitates in Al-4.5 Cu alloy
GP zones coherent disk Cu cluster
’’ coherent disk Tetragonal a=4.04 c=7.68
’ Partially coherent
plates Tetragonal a=4.04 c=5.80
Incoherent Tetragonal a=6.07 c=4.87
Al: FCC a=4.04 Angstrom
Critical thickness of coherent / semi‐coherent precipitate: The driving force for precipitation is the degree of super saturation or the extent of super
cooling. The forces opposing it are the strain energy (Es) and the surface energy (). The energy balance would critically depend on the size of the precipitate. The total volume free
energy increases as the cube of the radius (r). Therefore in the initial stages the net increase in
the strain energy is likely to be less than that of the net surface energy. However at a later stage
when the precipitate becomes large the net surface energy is likely to be greater than that due
to strain. Let us try to estimate the total strain energy and the surface energy of a disk or a
plate shaped precipitate. Figure 2 gives the shape of the two common forms of coherent
precipitates. Slide 4 gives the expressions for the two forms of energy.
Slide 3
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Precipitate shape & size
22 2
2 2
2
1 3
1 2
2 2
4 11
3
S
crit
E V E At t E
Area At At
tE A
Factors inhibiting precipitation: strain energy: coherent ppt & surface energy for incoherent ppt
En
erg
yES
Area
coherent
incoherent
Coherent precipitates tend to have disk or plate morphology because of strain energy associated with
lattice mismatch.
t
The key to the symbols used in slide 4 for Es & Area are as follows: = average volume of the
disk shaped precipitates, effective elastic modulus, = lattice miss‐match, = Poisson ratio = 0.33, t = thickness of the precipitate, r = radius of the disk shaped precipitate, and A = r/t.
Note that the expressions are valid for disk shaped precipitates. Similar expressions can be
derived for plate shaped precipitate as well. The sketch in slide 4 gives the plots for strain
energy and surface energy. Initially the surface energy is less. Beyond a critical thickness the
surface energy becomes greater than the strain energy. The expression for tcrit is obtained by
rearranging the terms in equation Es = Area. Note that it is proportional to and inversely proportional to the square of lattice miss‐match (). Coherent precipitate has low surface energy. It is extremely thin unless the lattice miss‐match is negligible. In most commercial alloys
such precipitates are thin and these are either disk shaped or plate shaped.
Slide 4
Fig 2: Shows the two common shapes of coherent
precipitates. Volume & surface area can be represented in
terms of thickness (t) and a dimensionless term A = r/t for
t
2r
t
NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering ||
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Kinetics of precipitation: Thermodynamics gives only the stability of precipitates. However the kinetics of the process is
governed by nucleation and growth. The former depends on the magnitude of the activation
barrier whereas the latter depends on the diffusivity of the solute in the matrix. This is why the
rate of nucleation is higher at lower ageing temperature since it corresponds to higher super
cooling whereas the rate of growth is higher at higher ageing temperature because of higher
diffusivity. Therefore the time it takes for the precipitate to form initially decreases with
decreasing ageing temperature until it reaches a minimum thereafter it starts increasing. The
shape of the time temperature plot corresponds to that of a ‘C’. Slide 5 gives a set of typical
time – temperature plots for four different precipitates for an alloy having X % Cu. Its location is
shown in the phase diagram of the alloy given in slide 5. The phase diagram includes a set of
four solvus curves for four different precipitates (GP zone, ’, ’’, ). Each of the four precipitates has different activation barrier. GP zone has the lowest activation barrier.
Therefore the nose of the ‘C’ shaped plot occurs at the shortest time. The stable precipitate has the highest activation barrier. It takes the longest time to form. This clearly suggests that
the sequence of precipitation would depend on the temperature of ageing. Let us consider the
ageing of the alloy having X% Cu after it is homogenized at a temperature within the field and subsequently quenched to retain excess Cu in solid solution. Note that the dashed vertical line
at %Cu = X intersects the solvus curve for GP zone at T1. This means it can form only if it is aged
at a temperature lower than T1. Imagine a horizontal line at temperature below T1 on the time
temperature transformation diagram given in slide 5. It would first intersect the ‘C’ curve for
the GP zone and subsequently intersect the extended parts of the ‘C’ curves for ’, ’’ and . It means, the precipitation would occur in the sequence GPZ → ’ → ’’ → if the alloy is aged at a temperature below T1. The sequence of precipitation at a temperature above T2 but below T3
would be ’’ → .
NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering ||
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Solubility of meta stable precipitates
T1
T2
T3T4
GP
’’’
% Cu
T1
T4
Log (time)
Precipitation sequence depends on composition and ageing temperature
L
+ L
GP
’
’’
x
Figure 3 explains the effect of composition on the sequence of precipitation at a temperature
close to room temperature. GP zone forms only if % Cu is greater than a specific solubility limit.
Slide 5
% Cu
T
GPZ
’’’
+ L
X1 X2
Fig 3: Gives a schematic phase diagram of Al – Cu showing
a set of solvus curves for GPZ, ’’, ’, and . Note that the vertical line representing alloy X1 intersects the solvus
curves for ’ and . Therefore the expected sequence of precipitation here is ’ → The vertical line representing alloy X2 intersects the solvus curves for GPZ, ’’, ’ and . The sequence of precipitation for this alloy should be GPZ
→ ’’ → ’ → .
NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering ||
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Effect of % Cu on pptn hardening
HV
40
140
Log (time) 100 days
GPZ’’’
Cu
2%
4.5%
The plots in slide 6 show the effect of composition on the hardness versus ageing time plots at
a low (room) temperature. The shape of the hardness versus time plots is a direct reflection of
the structural changes that takes place within the matrix. The increase in hardness in the
portion of the plots denoted by solid lines is due to the formation of GP zones or clusters of Cu
atoms. The dashed line represents the effect of ’’ precipitates and the dotted line denotes the stage at which precipitation of ’ takes place.
Effect of temperature on ageing
HV
40
140
Log (time) 100 days
GPZ’’’
Temp
Slide 6
Slide 7
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The plots in slide 7 show the effect of ageing temperature on the hardness versus ageing time
plots of an alloy having a fixed composition (~4.5% Cu). The shape of the hardness versus time
plots is a direct reflection of the structural changes that takes place within the matrix. The
increase in hardness in the portion of the plots denoted by solid lines is due to the formation of
GP zones or clusters of Cu atoms. The dashed line represents the effect of ’’ precipitates and the dotted line denotes the stage at which precipitation of ’ takes place.
Precipitation hardening: The strength of a metal or an alloy depends on the density and the mobility of dislocations
present in the matrix. Dislocation density increases with the extent of cold work. This results in
an increase in its strength. This is known as strain (or work) hardening. The atoms of alloying
elements in a metal are surrounded by elastic stress fields. The interaction between the stress
fields of foreign atoms with those of the dislocations result in solid solution strengthening. The
presence of precipitates in the matrix offers additional resistance to the movement of
dislocation that result in further strengthening. This is why on ageing which is associated with
the formation of an array of fine precipitates the hardness of Al‐4.5% Cu alloy increases
significantly from around 40HV to about 140HV. Let us look at the interactions between
precipitates and dislocations in a little more details.
Precipitation hardeningStrength of a metal / alloy depends on the number & mobility of dislocation
Loop around a ppt & move
Shear & move
Cross slip
Slide 8 illustrates what could happen when a dislocation encounters a precipitate during its
glide on a slip plane by a set of sketches. The strength of the precipitate is higher than the
Slide 8
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strength of the matrix. It would need higher applied stress to overcome resistance offered by
the precipitate. The first sketch shows how it bends around the precipitate. Usually there would
be an array of precipitates on a slip plane. With increasing stress the dislocation would bend
even more and move further by leaving a dislocation loop around the precipitate. This is shown
with the help of the second sketch in slide 8. The dislocation can also glide through the
precipitate if it is not strong enough. This is illustrated with the help of the third sketch. Note
that as the dislocation moves through the precipitate it leaves behind shear steps on the
precipitate. Dislocation can also avoid the resistance offered by the precipitate by cross slip or
climb. However only screw dislocations can cross slip. This is favored by low stacking fault
energy. The presence of alloying element is accompanied by lowering of stacking fault energy.
Therefore it would need additional stress to overcome the barrier. Climb on the other hand
needs thermal activation. This is possible only at high temperature and it is restricted to edge
dislocations only.
Particle looping
2
min 0 0
0.5ln ln
2 22
T Gb d Gb ddbR r d rb
0
1 1 3
2
3ln
2 2
f
d r
Gb f d
r r
d
f
r
Slide 9 shows the interaction between dislocation and a linear array of precipitates. It also gives
the steps involved in the derivation of the expression relating the increase in the shear strength
of the alloy due to presence of precipitates. The force acting on the dislocation is given
by where is the resolved shear stress on the glide plane and b is the Burgers vector of the dislocation. This tends to bend the dislocation between the precipitates. Because of the
increase in the length of the dislocation there will be restoring force acting on the dislocation.
This would tend to make it straight. This is given by T/R where T is the line tension of the
dislocation and R is the radius of curvature of the dislocation. The elastic stored energy per unit
Slide 9
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length of the dislocation is a measure of the line tension. This is given by
0.5 where d is the inter particle spacing and r0 is the radius of the dislocation core.
Note that the minimum radius of curvature is equal to half of the inter particle spacing (Rmin =
d/2). What happens when the dislocation is forced to bend further is explained with the help of
a set of sketches in fig 4. The arrows on the dislocation line represent the direction. The arrow
labeled as b is its burgers vector. When the dislocation bends between the precipitates, a part
of it becomes a positive screw and a part becomes a negative screw dislocation. The two
parallel segments having opposite characters would attract each other. The two would meet at
a point and get annihilated leaving behind loops around each of the precipitates and the main
dislocation would glide further and become straight.
Look at the expressions given in slide 8. The increase in shear strength of the alloy is given
by∆ . It is inversely proportional to the average spacing between the
precipitates which is a function of the shape, the size (r = radius of the precipitates) and the
amount of precipitates (f = volume fraction). Assuming the shape of the precipitates to be
spherical it is possible to show that it is given by . Thus the final expression for the
increase in the shear strength of the alloy is as follows:
∆ (3)
The extent of strengthening is directly proportional to the square root of the volume fraction of
precipitates and inversely proportional to the size of the precipitates.
b
Positive
screw
Negative
screw
b
Positive
screw Negative
screw
(b) (a)
Fig 4: Shows what happens when a dislocation
is forced to loop around precipitates. Note that
the initial straight dislocation is an edge
dislocation. Its character changes when it
bends. A part of it around the precipitate
becomes a positive screw and the other part
becomes a negative screw dislocation. The two
would join at the constriction (b). This leaves
behind a set of loops around the precipitates
when it moves further.
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Particle cutting
Cutting of precipitate generates
•APB
•New surface
1/3 1/2
1/2 1/2
:
:s
APB f r
f r
Particle before shearing
Particle after shearing
Top view of the Particle after
shearing
Slide 10 explains with the help of a set of sketches what happens when the dislocation moves
through the precipitate. It would result in the creation of new precipitate matrix interface. It
can also form anti phase domain boundary (APB) within the precipitate if it has an ordered
structure. Therefore shearing of precipitate needs additional energy for the creation of new
interfaces. The increase in strength depends on the size and the volume fraction of precipitates.
Note that unlike looping the stress to cut a precipitate increases with its size.
Particle looping vs cutting
Smaller precipitate: cutting more likely
Initial stage: small coherent particles form & f keeps increasing till its limiting
value. Strength increases due to both f & r
Once f reaches its limit particles coarsen & become large enough to allow looping
r
Looping f1/2/r
cutting f1/2 r1/2
CRSS
f
Slide 10
Slide 11
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The strength of particles or the precipitates are much higher than that of the matrix. The overall
strength of an age harden‐able alloy will certainly be lower than that of the precipitate. The
presence of such hard particles on the slip planes of the matrix resists dislocation glide. The
resistance offered can be overcome either by looping or shearing. The sketch in slide 11
illustrates the effect of particle size on the extent of strengthening due to the two competing
strengthening mechanism. In the initial stage the increase in strength is due to the formation of
tiny coherent precipitates. It is proportional to / / . It means the strength increases with
the increase in the volume fraction of precipitates and their size. The effect of volume fraction
(f) may cease once it reaches its limit. However the strength would continue to increase as the
precipitate grows. This is shown by the dashed line. The upper limit is set by the strength of the
array of precipitates in the matrix to resist the formation of dislocation loops. This is
represented by the solid line. It decreases as the radius of the precipitate increases. The CRSS
denotes the critical resolved shear strength of the hard precipitate. It is the upper limit of the
contribution from hard precipitates towards strengthening. Note that there is an optimum size
at which the alloy attains its maximum strength due to precipitation. The contribution from the
volume fraction of precipitate towards strengthening is also important. However there is a limit
to which it could grow. This is determined by the compositions of the precipitate and the alloy.
The volume fraction of precipitate can be increased by increasing solute concentration. Its
contribution is shown by set of dashed line in slide 11.
Let us now look at the effect of the size of precipitate on its thermodynamic stability. This is
illustrated with the help of a set of free energy composition plots for the matrix and the
precipitate in slide 12. The matrix is denoted as and the precipitate is denoted as . There are three plots marked as , r and . The suffix r and have been used with to denote the size of the precipitate. r represents phase having radius r. represents phase having a very large radius approaching infinity. Note that a coarse precipitate () is more stable than a fine
precipitate (r).
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Over ageing: particle coarsening
3 3o
e
r r kt
k D X
r
XB
Low : Nimonic
Low Xe: TD Nickel (W)
Low D: Low alloy steel
G
0 112
Note that the common tangent to the plots labeled & r meets the plot for at 2. This
represents the composition of the phase surrounding the precipitate r. The common
tangent to the plots for and shows that the composition of surrounding is 1. Note
that 2 > 1. This shows that there is a concentration gradient within the matrix. Therefore
solute would continue to diffuse from smaller precipitates to larger precipitates. In other words
coarser precipitates would continue to grow at the expense of the finer ones. This phenomenon
is known as particle coarsening or Ostwald ripening. This is responsible for over‐ageing. It is
accompanied by loss of strength.
From the above physical concept it is possible to derive expressions describing the kinetics of
the growth of precipitates during ageing process. The average size of precipitates (r) at any
instant (t) is given by:
Slide 12
r
1 2
2 > 1
Solute flow Fig 5: Illustrates the flow of solute down the concentration
gradient from a fine precipitate to a coarse precipitate. This
is responsible for the dissolution of finer precipitates and
the growth of coarser ones. The phenomenon is known as
Ostwald ripening.
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(4)
Note that r0 is the radius of the precipitate at t = 0.The rate constant k depends on the
diffusivity (D) of the solute in the matrix, the energy of the precipitate matrix interface () and the solubility limit (Xe). This suggests that the most stable precipitate that does not grow is the
one which is perfectly coherent. The interface energy of such a precipitate is negligible ( = 0). The ’ phase in Ni base super‐alloy is an excellent example. The – ’ interface in this alloy is coherent. Therefore it is extremely stable. It can withstand prolonged thermal exposure with
minimal loss of high temperature load bearing capacity. It is one of the most popular high
temperature alloys. Dispersion strengthened alloys such as TD nickel is an example where the
structural stability is derived from the low solubility of the constituents in the matrix. It has
ThO2 particles dispersed a matrix of Ni. Both thorium and oxygen have low solubility in Ni.
Therefore the growth rate of the dispersed phase is extremely low.
Spinodal decomposition: There are alloys where precipitation occurs by a process which is entirely different from what
we have learnt so far. The composition of a precipitate is significantly different from that of the
matrix. The atoms in an alloy are in constant motion. The composition at a point may keep
changing or fluctuate from time to time. When the amplitude exceeds a critical value a stable
precipitate forms. The probability of a large fluctuation in composition is low. However it
increases with increasing driving force (degree of super cooling or super saturation). This is the
normal mode of precipitation. It is associated with a discontinuity in the composition distance
plot. As against this there is a mode of precipitation where the composition changes in a
manner shown in slide 13. It is called spinodal decomposition. It occurs under conditions where
a small fluctuation in composition is thermodynamically stable. There is no critical amplitude
barrier to be overcome for the transformation to take place.
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Spinodal decomposition
G
(a) (c)
(b) A B
T
A B
x1 x2y1 y2
zy
z
z
z
t
y
y
y
1 2
Tc
y1 - y y1 + y
’y2 - y y1 + y
’
The sketch (a) in slide 13 shows the phase diagram of an alloy exhibiting a miscibility gap. The
alloy consists of homogeneous above a critical temperatureHowever it breaks down into 1
(rich in A) and 2 (rich in B) below a critical temperature (Tc). The sketch (b) shows the free
energy composition diagram at a temperature T. Note that the compositions of 1 and 2 that
can coexist at this temperature are x1 and x2 respectively.
Consider an alloy having a composition Y1 that has been quenched to a temperature T after it
has been homogenized at a temperature above Tc. The alloy therefore, is in a super saturated
state (’). It would have a natural tendency to decompose into a mixture of 1 & 2. Normal
precipitation occurs through large fluctuation in composition. However let us examine whether
a small fluctuation in this alloy is stable. Let the amplitude of small fluctuation in composition
be y. Assume that ’ decomposes into a mixture of alloys where atom fractions of B are y1‐y and y1+y respectively. The free energy of the mixture is given by the point of intersection of
the line joining the two points marked as y1‐y and y1+y (in sketch b of slide 13) and the vertical line at y1. This is higher than the free energy of the super saturated’. Therefore small
fluctuation in composition in this case is not stable.
Consider an alloy having a composition Y2 that has been quenched to a temperature T after it
has been homogenized at a temperature above Tc. The alloy is therefore is in a super saturated
state (’). It would have a natural tendency to decompose into a mixture of 1 & 2. Let us
examine whether a small fluctuation in this alloy is stable. Let the amplitude of small fluctuation
in composition be y. Assume that ’ decomposes into a mixture of alloys where atom
Slide 13
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fractions of B are y2‐y and y2+y respectively. The free energy of the mixture is given by the
point of intersection of the line joining the two points marked as y2‐y and y2+y (in sketch b of slide 13) and the vertical line at y2. This is lower than the free energy of the super saturated’. Therefore small fluctuation in composition in this case is stable. The composition of this alloy
therefore, can change continuously as shown in sketch (c) of slide 13. It shows that the
composition of the alloy as a function of distance at different times (t). Note that initially the
amplitude of fluctuation is less but it grows with time. In the end the compositions of the peak
and the trough corresponds to x2 and x1 respectively. It means that the solutes atoms are
diffusing from regions having low concentration (trough) to regions having high concentration
(peak). This is a case of uphill diffusion. Under normal conditions diffusion takes place down the
concentration gradient. However alloys that exhibit miscibility gap are not ideal solid solution.
There is a difference between activity that represents effective concentration and the actual
concentration. The direction of solute atoms due to diffusion is governed by activity gradient.
There is a composition range within the miscibility gap where small fluctuation in composition
in a super saturated solid solution is stable. Figure 6 illustrates how this could be determined
from the free energy (G) composition (XB) diagram at a given temperature (T). There is an
inflection point on the G versus XB plot where the change over takes place. Figure 6 (b) shows
that the slope of the plot is initially negative. It becomes positive at XB = X1 and it keeps
increasing till it reaches a peak at XB = X’1. Subsequently it decreases passes through a minimum
at XB = X’2. Figure 6 (c) shows the second differential of G is equal to zero at the two points of
inflection. The exact location can be determined from the following equation.
0 (5)
The solution of equation 5 gives a set of values for the inflection points as a function of
temperature. Slide 14 shows the location of these in the phase diagram.
0
0
G
XB
XB
XB
(a)
(b)
(c)
Fig 6
X1 X2 X’1 X’2
NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering ||
20
T
A B
Miscibility gap
Spinodal •Incoherent ?
•Coherent ?
Spinodal decomposition
Al 22.5 at % Zn 0.1 at% Mg
CuNiFe
The inflection points of the free energy composition diagram are shown in the form of a dashed line in slide 14. It defines the boundary of the domain where spinodal decomposition can take place. Two of the most common alloys where such a transformation can take place are Al‐Zn‐Mg & CuNiFe. It gives a fine homogeneous distribution of precipitates within the matrix. The nature of the precipitate can be either coherent or incoherent. We know that the solvus curve for a coherent precipitate is different from that of an incoherent precipitate. Therefore the spinodal curve for a coherent precipitate is likely to be different from that of an incoherent precipitate. Precipitation hardenable alloys:
Slide 14
NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering ||
21
A few age harden-able alloy system
Aluminum Al-Ag Zones → ’ : Ag2Al
Al-Zn-Mg Zones → M’ M: MgZn2
Al-Mg-Si Zones →’ : Mg2Si
Al-Cu GP→’’→’ : CuAl2Al-Mg-Cu Zones →S’ S: Al2CuMg
Copper Cu-Be Zones →’ : CuBe
Cu-Co Zones
Nickel Ni-Cr-Ti-Al ’ (cubes) : Ni3Ti/Al
Slide 14 gives a list of common alloys whose strength can be significantly increased by
precipitation. The table includes the sequence in which precipitation takes place during ageing.
In most of these the incoherent precipitates are inter metallic compounds having definite
chemical formula. It is given in the last column. The intermediate precipitates are coherent. We
looked at the ageing behavior of Al‐Cu alloy in details. The behavior of others is expected to
follow the same trend.
Summary: In this module we looked at the precipitation behavior of age hardenable alloys a little more
critically. All super saturated (terminal) solid solutions where precipitation occurs in stages
through at least one (meta‐stable) coherent precipitate exhibit age hardening behavior. The
precipitates should be uniformly distributed within the grains. This is promoted by
homogeneous precipitation. It is facilitated by low surface energy () and low strain energy (Es). The strength of the alloy is a function of the volume fraction and the size of the precipitates.
This depends on the composition of the alloy, ageing temperature and ageing time. The
precipitates act as barriers to dislocation glide. This results in strengthening. A dislocation can
overcome the barrier either by shearing or by looping. There is an optimum size of the
precipitate that gives maximum strengthening. If an alloy is aged beyond this stage the
precipitates that are coarse continue to grow. Beyond this the average particle size increases
although the volume fraction of the precipitates remains constant. This is accompanied by loss
of strength or hardness. The phenomenon is known as over‐ageing. There are several
Slide 15
NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering ||
22
commercial alloys where age hardening is the main mechanism of strengthening. Some of these
have precipitates that are extremely stable and do not grow even after prolonged thermal
exposure. These are suitable high temperature applications.
Exercise:
1. Why non‐age hardenable aluminium alloys are chosen for beverage can?
2. When can you get more than one peak in the hardness versus aging time plot of a given
alloy at a given temperature?
3. If in an alloy 1nm thick disk of ’’ has formed, estimate the critical diameter at which its
coherency is likely to be completely lost. Given = 10%, = 500 mJ/m2 and E = 70000MPa
4. Ag rich GP zones can form in a dilute Al‐Ag alloy. Given that the lattice parameters of Al
and Ag are 0.405nm and 4.09nm respectively. What is the likely shape of these zones?
5. Under what heat treatment condition an age harden‐able alloy can be machined?
6. Show that the shear stress to move a dislocation in a matrix by cutting dispersed
spherical particles is proportional to the cube root of f where f represents volume
fraction of particles.
Answer:
1. Beverage cans are maufactured by cold working. The alloy must have good ductility. Age
hardenable alloys have relatively poor ductility. More over these are more expensive.
2. It happens when more than one coherent meta‐stable precipitates form during aging.
This illustrated in the following sketch where there are two intermediate coherent
precipitates ’ & ’’:
Note if x2 is aged at T1 we get one peak for ’ another for ’’ with formation of which is incoherent hardness starts dropping. Alloy x1 lies beyond ’ solvus here only ’’ precipitate could form therefore
only one peak is expected. Similar situation would
arise for a given alloy when aged at different
temperatures.
A
TA
’
’’
x x
NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering ||
23
3. Coherent precipitates have disc shape. Its elastic stored energy is given by:
. Assuming Poisson ratio =1/3 and volume of disc shaped precipitate
where t is thickness of disc and A is its aspect ratio. Therefore radius of
disc = At. On substitution of these in the expression for elastic stored energy:
. Likewise stored surface energy is given by: 2
Coherency is lost when Es exceeds S. By equating the two one gets an expressions for
critical thickness: 1 or; 1 10.
1 on soling this
A = 20. This means critical diameter = 2*20 =40nm
4. Al & Ag both have face centered close packed structure. Their atomic radii should be
proportional to their lattice parameters. Therefore lattice mismatch = 100 x (0.409‐0.405)/0.405
= 0.99%. If mismatch is less than 5%, the shape is determined by its surface energy. Spherical
shapes have less surface energy. If it is greater than 5% it is likely to be disc shaped.
5. It is easy to machine a material alloy when it is soft. Age hardenable material can be easily
machined either when it is over aged to a low hardness or under solution treated condition.
Some solution treated alloy would age harden during machining in those cases the former
option is better.
6. Imagine spherical particles of radius r0 are arranged in a regular fashion where the inter‐
particle spacing is x. The work done to move a dislocation through a distance 2r0 to cut a
precipitate is given by 2 This is used up in creating new surface having energy =
.
Equating these two you get . Volume fraction can be expressed in terms of
precipitate size and spacing assuming that these are arranged in the form of a cube
lattice with spacing x such that . On substituting this in the expression for
shear stress you get or ∝
b x
2r0
2At
t
Volume of the disc =
Surface area = 2