Module 30 Precipitation from solid solution IInptel.ac.in/courses/113105023/Lecture30.pdf · ·...

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


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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.


3

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


4

∆ ∗∆

(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


5

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


6

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


8

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 ’’ → .


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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

→ ’’ → ’ → .


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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


11

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


12

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


13

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


15

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


19

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


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


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


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


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

Module 30 Precipitation from solid solution IInptel.ac.in/courses/113105023/Lecture30.pdf · ·...

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