A Lcu Overview

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Microstructure-Properties: II

Age Hardening

27-302

Lecture 9

Fall, 2002

Prof. A. D. Rollett

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Materials Tetrahedron

Microstructure Properties

Processing

Performance

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Objective

• The objective of this lecture is to describe

the relationship between precipitation and

hardness as an example of a key

microstructure-property relationship.

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References

• Phase transformations in metals and alloys, D.A.

Porter, & K.E. Easterling, Chapman & Hall.

• Materials Principles & Practice, Butterworth

Heinemann, Edited by C. Newey & G. Weaver.

• Mechanical Metallurgy, McGrawHill, G.E. Dieter, 3rd

Ed.

• Hull, D. and D. J. Bacon (1984). Introduction to

Dislocations. Oxford, UK, Pergamon.

• Courtney, T. H. (2000). Mechanical Behavior of

Materials. Boston, McGraw-Hill.

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Notation

a := lattice parameter

e := strain, misfit (or similar quantity to

describe a hardening mechanism)

G := shear modulus

b := Burgers vector

r := Particle size (radius)

f VV(a) := volume fraction (of precipitates)

s := stress (macroscopic)

t := shear stress (critical value, in some cases)

g := boundary energy, e.g. anti-phase boundary

<L3> := mean intercept length (of precipitates)

l := mean spacing (of precipitates)

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Age Hardening Curves

• The most quoted age

hardening curve is that for

Al-Cu alloys performed in the

late 40s. Keep in mind that

age hardening was known

empirically (Alfred Wilm) as

a technologically useful

treatment from the early

days of aluminum alloys.

• Higher Cu contents result in

higher maximum hardnesses

because larger volume

fractions of precipitate are

possible.

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Al-Cu precipitation sequence

• The sequence is:a0 a1 + GP-zones a2 + q“ a3 + q’ a4 + q

• The phase are:an - fcc aluminum; nth subscript denotes each equilibriumGP zones - mono-atomic layers of Cu on (001)Al

q“ - thin discs, fully coherent with matrixq’ - disc-shaped, semi-coherent on (001)q’ bct. q - incoherent interface, ~spherical, complex body-centered tetragonal (bct).

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Al-Cu ppt

structures

GP zone structure

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Al-Cu microstructures

• This tableau shows which of the different ppt types

are associated with which part of the hardening

curve.

GP zones

q”

q’

q

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Al-Cu driving forces

• Each precipitate has a different free energy curve w.r.t

composition. Exception is the GP zone, which may be

regarded as continuous with the alloy (leading to the

possibility of spinodal decomposition, discussed later).

• P&E fig. 5.27 illustrates the sequence of successively greater

free energy decreases and also successively greater ∆G*.

• P&E fig. 5.28 illustrates the point that the nucleation barriers

are much smaller for each individual nucleation step when the

next precipitate nucleates heterogeneously on the previous

structure.

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Al-Cu phase relationships

• The explanation of age hardening depends on

understanding the metastable phases that can

appear.

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Nucleation sites, reversion

• The nucleation sites vary depending on circumstances.

• q“ most likely nucleates on GP zones by adding additional

layers of Cu atoms.

• Similarly, q’ nucleates on q“ by in-situ transformation.

• However, q’ can also nucleate on dislocations, see P&E fig.

5.31a.

• The full sequence is only observable for annealing

temperatures below the GP solvus. Any of the intermediate

precipitates can be dissolved, reverted, by increasing the

temperature above the relevant solvus, fig. 5.32.

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Al-Ag: example 2

• The age hardening curve has the same double

peak as for the Al-Cu series, but the separation is

more pronounced.

Shewmon

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Al-Ag, contd.

• GP zones are spherical (Ag atom is larger than Al).

• g’ is hcp with OR (0001)g//(111)a and [1120]g //[110]a;

heterogeneously nucleated on the stacking faults of

dislocations which provide sites of local hexagonal packing.

• g is also hcp with the same OR; forms plate-like precipitates.

A cellular mechanism can also occur.

Shewmon

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Age hardening in steel: example 3

• It is important to understand that age hardening occurs in

almost any system in which the solid solubility decreases

appreciably with decreasing temperature. Ferrite has a very

low solubility for carbon and therefore age hardening (also

called quench hardening) occurs here too. To avoid it, the

soluble carbon levels must be reduced, which is a common

objective of the IF or interstitial-free steel grades. These have

additions of carbide formers such as Ti or Nb to sequester the

C.

Shewmon

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Cutting versus Bowing

• At small sizes, the dislocation cuts through the particle at a

lower stress than the Orowan bowing stress (and so this is

what is observed). Larger particles mean higher cutting

stresses.

• At large sizes, the dislocation bows around the particle more

easily than it cuts through it (so no cutting is observed).

Larger particles mean fewer particles (via coarsening) hence

lower flow stresses.

Particles becoming strongerFewer and fewer particles,

further apart

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Breaking Angle: fc

Strong Obstacles: f ~ 0° Weak Obstacles: f ~ 180°

t Gb

L cos

fc

2

Courtney

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relationships• In order to understand the relationship between microstructure

and hardness, we need to delve into the subject of hardening

mechanisms.

• The central concept is that the strength of a ductile material is

governed by dislocation flow past obstacles. Therefore

strength can be designed by controlling the density and nature

of the obstacles to dislocation motion. Most technological

(metallic) alloys rely on precipitation hardening in one form or

another to achieve high strengths. Ceramics, on the other

hand, are intrinsically harder and therefore the main objective

of strengthening is to increase their fracture toughness and

thereby increase their (reliable) load carrying capacity. The

objective of this discussion is therefore to bring your attention

to a number of ways in which we can understand and predict

the contributions to strength of different types of obstacle.

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Strengthening Methods

• Microstructural Feature:

strength dependence.

• Dislocations: strain/work hardening

(discussed in 301): (dislocation spacing)1/2.

• Internal Boundaries: grain boundaries can

have a strong strengthening effect, i.e. the

Hall-Petch effect (discussed in 301): (grain

size)-1/2.

• Dislocation Boundaries (low angle

boundaries): (subgrain size)-1.

• Second Phase Particles: particle spacing.

• Solutes: (concentration)1/2.

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Mechanisms of particle strengthening

1) Coherency Hardening: differences in density between the particle and

the matrix give rise to elastic stresses in the vicinity of the particle.

2) Chemical Hardening: creation of new surface when a particle is

sheared increases the area of the interphase boundary, which increases the

energy associated with the interface and hence an additional force must be

exerted on the dislocation to force it through the particle.

3) Order Hardening: passage of a dislocation through an ordered particle,

e.g. Ni3Al in superalloys, results in a disordered lattice and the creation of

antiphase boundaries.

4) Stacking-fault Hardening: a difference in stacking fault energy

between particle and matrix, e.g. Ag in Al, increases flow stress because of

the different separation of partial dislocations in the two phases.

5) Modulus Hardening: a large difference in elastic modulus results in

image forces when a dislocation in the matrix approaches a particle.

Consider, e.g., the difference between silver particles (nearly the same shear

modulus) and iron particles (much higher shear modulus) in aluminum.

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Dislocations

• A re-statement of the governing equation for

strength controlled by obstacle spacing:

s M t 0 aGb /l Parameter Description Comments

s flow stress Experimentally acce ssible through mechan ical

tests

<M> Average Taylor factor Magnitude ~3 fo r tension or compression;

depends on the nature of the deformation, the

texture and the crystal structure, e.g. < M>~1.73

for torsion (cubic metals)

t Athermal stress Contributions from grain size hardening,

solutes, etc.a Geometrical factor This term accoun ts for both geo metrical factors,

and for thermal activation

G Shear Modulus Must choose appropriate shear modulus for the

slip plane used;

Temperature dependen t

b Burgers vector Derived from the force on a dislocation (Peach -

Koehler Eq.)

¦ dislocation density Equivalent to the reciprocal of a mean obstacle

spacing; depends on work hardening

l obstacle spacing Given a number density of particles, the mean

spacing, l=N1/2

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crss versus density

Courtney

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Dislocation Boundaries

• At large strains and higher temperatures, low angle

boundaries appear as a subgrain network forms.

We distinguish this microstructural feature from the

first two categories because the [lattice]

misorientations are much smaller (2-5°) than grain

boundaries (15°+) and they are distinct from

statistically stored dislocations. This strengthening

method is most important at high temperatures

where other microstructural features such as

solutes are weak.

• The contribution to the flow strength is typically

found to be proportional to (grain size)-1; this is in

contrast to the 1/√d dependence of the Hall-Patch

effect.

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Solutes

• Solutes in a crystal act as obstacles to dislocation

motion through their elastic and/or chemical

interactions with dislocations. Most solutes are

weak hardeners except for the (technologically)

important class of interstitial solutes that induce

anisotropic distortions of the lattice, e.g. tetragonal

distortions of C in Fe.

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Substitutional solutes

• Most Solutes have only a rather weak effect on strength. In

other words, even if you put several per cent of a soluble atom

into another element, you will not see a dramatic increase in

flow stress. These remarks can be quantified by going back to

the Orowan equation, i.e. the force balance between the

forward motion and the resisting force:

tcrss = µb/l.

• For substitutional solutes, the numerator in the RHS, i.e. the

reaction force from the solute atoms is of order Gb2/120,

which is a small number. This is so because the small

differences in size between solute and matrix atoms results in

a small interaction energy with dislocations. Thus they are

weak obstacles and dislocations remain nearly straight when

interacting with solutes (“weak obstacles”, 7slides before this).

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Interstitial solutes

• Interstitials in bcc, however, can exert forces on the

order of Gb2/5 to Gb2/10, which are large values. In

this case, the dislocations bow out significantly

between the atoms, and the breaking angle

deviates significantly from 180°. In this case, the

concentration dependence is easy to obtain. The

spacing between interstitials is inversely

proportional to the (square root of the)

concentration, and so we can insert a spacing into

the standard (Orowan bowing) formula to get the

following, where A is a constant of order unity:

t = AGb(√c/b) = AG√c.

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Strength vs. solute content: examplesExamples:

a)

Substitutional

solutes in Cu

b) Interstitial

solutes in

Nb, Fe

c) Ca in NaCl

[Courtney]

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Second Phase Particles

• Whether introduced as insoluble particles in powder

compaction, or as precipitates in a solid state reaction, second

phase particles are generally the most potent strengthening

agent in practical high strength engineering materials. Iron-

base, aluminum, nickel, titanium alloys all employ second

phases to achieve high strength.

• Quantitative relationships: from previous stereological analysis

(301 - lecture 4):

L3 4r

3; l L3

1 VV

(a)

VV

(a ) ;

l 4r

3

1 VV

(a)

VV

(a) 4r 1 f

3 f

4r

3 f

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Mechanisms of particle strengthening1) Coherency Hardening: differences in density between the

particle and the matrix give rise to elastic stresses in the vicinity of the particle.

2) Chemical Hardening: creation of new surface when a particle is sheared increases the area of the interphase boundary, which increases the energy associated with the interface and hence an additional force must be exerted on the dislocation to force it through the particle.

3) Order Hardening: passage of a dislocation through an ordered particle, e.g. Ni3Al in superalloys, results in a disordered lattice and the creation of antiphase boundaries.

4) Stacking-fault Hardening: a difference in stacking fault energy between particle and matrix, e.g. Ag in Al, increases flow stress because of the different separation of partial dislocations in the two phases.

5) Modulus Hardening: a large difference in elastic modulus results in image forces when a dislocation in the matrix approaches a particle. Consider, e.g., the difference between silver particles (nearly the same shear modulus) and iron particles (much higher shear modulus) in aluminum.

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Coherency hardening

Differences in density between the particle and the matrix give

rise to elastic stresses in the vicinity of the particle. This has

been analyzed on the basis of the elastic stresses that exist in

the matrix adjacent to a particle that has a different lattice

parameter than the matrix. Ignoring differences in modulus for

now, and setting a parameter, e, that approximates a strain to

characterize the magnitude of the effect. For

e = (aparticle – amatrix )/ amatrix

t = 7|e|3/2 G(rf/b)1/2

• This mechanism applies to the early stages of precipitation,

e.g. strengthening by GP zones.

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Chemical hardening

• Cutting through a particle with a dislocation displaces one half

relative to the other by b, thereby creating new interfacial

energy of 2πrbg, where g is the interfacial energy between the

matrix and the particle. The distance over which this energy

has to be created occurs at the entry and exit points and so

the characteristic distance is of order b. Thus the force is

dE/dx, or,

F = 2πrbg/2b πrg

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Chemical hardening, contd.

• If the dislocations are straight, we can approximate the

spacing between particles as L=2r/f. Dividing the force by bL

to find the stress,

t = πfg/2b.

• A more realistic approach produces the following relationship.

t = 2G{g/Gr}3/2(fr/b)1/2

• Courtney defines a chemical hardening parameter, ech = g/Gr,

related to the interfacial energy, modulus and particle size.

This parameter is precisely analogous to the same parameter

used, e.g. in APB hardening. Chemical hardening applies

only in the early stages of precipitation.

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Order Hardening

• The hardening depends on the product of the antiphase-

boundary energy (APBE) and the area swept by a dislocation

in a particle. Thus the increase in flow stress is given by:

t = πf{APBE}/2b

• In general, low values of the APBE not only predict small

increments in hardness, but also the result that the

dislocations can move through the particles independently of

one another. A more detailed analysis, not presented here,

shows a square root dependence on volume fraction, with

particle size,

t = 0.7 Ge3/2 √(fr/b)

eord= APBE/Gb

• Important for

Ni-based superalloys

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Modulus hardening

• The line length in the particle is 2r and the change in tension

is (Gparticle-Gmatrix)b2/2, assuming the same Burgers vector in

matrix and particle. Multiplying the two together and dividing

by the distance, i.e. the radius, we get:

F= b2(Gparticle-Gmatrix) = Gb2e,

where e = (Gparticle-Gmatrix)/Gmatrix, a measure similar to that used in

solution hardening.

• More realistic estimates of modulus hardening lead to the

following formula:

t = 10-2 G e3/2 √{fr/b}

• Think of modulus hardening as being caused by a temporary

increase in dislocation line energy while it resides within a

particle.

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Summary

• A great variety of hardening mechanisms exist.

• Their functional dependence on parameters such

as particle size, spacing, volume fraction, are

similar. It is difficult therefore, to distinguish

experimentally between the mechanisms.

• Microscopy is required in order to determine which

mechanism is operative.

• Particle growth results in stronger particles; in most

cases, however, coarsening takes place

simultaneously which increases the particle

spacing. Orowan bowing takes over from particle

cutting at some point in the aging process.

• Particle hardening is essential to technological

alloys, at least for structural applications.

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Sample Problem• From Dieter, p219 (adapted):

• Question: Al-4%Cu (by wt.) has a yield stress of 600MPa. Estimate

the particle size and spacing.

• Solution: recognize that this stress relates to age hardening beyond

the peak hardness. Therefore use the Orowan bowing stress to

estimate the stress.

s<M> tcrss = <M> Gb/l

• G=27.6GPa; b=0.25nm; <M>=3.1:

spacing = 3.1*27,600*0.25.10-9/ 600= 35.7 nm

• Now we must estimate the volume fraction of particles for which we

use the phase diagram, assuming that we are dealing with the

equilibrium phase, q, which is 54 w/o Cu, and the a in equilibrium

with it, 0.5 w/o Cu.

• Wt. % Al = (54-4)/(54-0.5) = 93.5; wt. % q = 4-0.5/(54-0.5)=6.5

• Volume of a = 93.5gm/2.7 gm/cm3 =34.6 cm3

• Volume of q = 6.5/ 4.443 gm/cm3 = 1.5 cm3

• Volume fraction of a = 0.96; volume fraction of q = 0.04.

• Use l=4r(1-f)/3f (slide 22): r =3*0.04*35.7/4/(1-0.04) = 1.12 nm.