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What is microstructure?

Microstructure originally meant the structure inside a

material that could be observed with the aid of a microscope.

Since the invention of prefixes for units, the micrometer(1

m) happens to correspond to the wavelength of light. As

visible light is used to form images in a light/optical

microscope, microstructure has come to be accepted as those

elements of structure with length scale of order 1 m.

Elements of Microstructure

Most observable elements of microstructure are

discontinuities, or defects in the material.

Grain boundaries are discontinuities in the crystal lattice -

differences in orientation.

Phase boundaries are discontinuity in composition and,

commonly, crystal structure.

Dislocations are local discontinuities in the lattice, thus line

defects.

Point defects (very difficult to observe!) are missing atoms(vacancies) or extra (interstitial) atoms.

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How to Observe Microstructure

Observation of microstructure requires us to acquireimages.

In order of increasing effort, the standard methods are (1)optical microscopy, (2) scanning electron microscopy(SEM), (3) scanning probe microscopy (SPM), (4)transmission electron microscopy (TEM).

Microscopies that rely on topographic contrast typicallyrequire some form of specimen preparation in order toreveal the microstructure.

Metallography

Metallography/ceramography is the art of specimen

preparation for microscopy. The aim is to maximize contrast

for the microstructural elements of interest while minimizing

artifacts.

Not all imaging methods require topographic relief.

Channeling contrast in the SEM uses variations in

crystallographic orientation to affect image brightness giving

a gray-scale image of grain structure, for example.

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Taxonomy of Microstructure

Different types of materials exhibit different characteristic

microstructures that are often easily recognized.

The characteristic microstructures are more closely related

to phase relationships than to elemental composition.

Example: eutectics typically exhibit lamellar two-phase

structures as a result of cooperative growth from a single

phase melt.

Various microstructures result from processing someare characteristic of the processing technique

Solidification

Powder Consolidation

Thin Film deposition

Mechanical Working

Annealing

Phase Transformation

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Columnar versus Equiaxed Solidification Microstructures

Columnar at low nucleation

density on walls w/high growth

rates

Equiaxed at high nucleation

density w/low growth rates.

Spatial distribution of nuclei

varies.

Affects mechanical properties.

1.4: 99.99% Al, R=6in/min; 1.5: 99.8% Al, R=6in/min

1.6: 99.5% Al, R=6in/min; 1.7 99.2% Al, R=8 in/min.

Solidification Microstructures: effect of growth conditions

High temperature gradient w/low

growth rate (low G/R) for plane

front.

Low gradient w/high growth rate

for cellular (1.10), or dendritic

(1.13) solidification fronts.

Constitutional Supercooling

Affects electronic properties (in Si,

e.g.)

Nutting & Baker: plate III

1.8: Sn-0.006%Pb, G/R=2000C/cm2/s; 1.9:

G/R=2000; 1.10: G/R=1000; 1.11: G/R=400; 1.12:

G/R=350; 1.13: Sn-0.2%Pb, G/R=200.

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Solidification Microstructures: chemical segregation, coring

Phase relationships predict that

segregation will occur during

solidification.

Segregation observable as

coring, 1.14; persists after

deformation, 1.15

Affects uniformity of

mechanical, other properties.

1.14: Cu-7Ni-3Al, chill cast.

1.15: same, forged and recrystallized.

Solidification Microstructures: primary, eutectic phases

Phase relationships predict that

off-eutectic compositions will

solidify first with a primary

phase (in dendritic form) and

then eutectic after sufficient

segregation has occurred.

2.1: Sn, subgrain structure in one grain

2.2: Sn-10Pb, eutectic between primary dendrites

2.3: Sn-30Pb, more eutectic

2.4: Sn-37Pb, fully eutectic

2.5: Sn-60Pb, primary Pb dendrites

2.6: Pb, etched grain boundaries

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Eutectoid, Peritectoid Reactions

More complex phase

relationships lead to morecomplex microstructures, not

surprisingly.

Example: Cu-12.3Al, cooled

slowly from 800 to 500C, then

quenched (to avoid formation of

phase). 2particles(precipitation), surrounded by

1 (peritectoid reaction).

Effect of Cold Work

As the amount of cold work

(plastic deformation) is

increased, so the density of slip

bands (twins also in some

materials) increases, and the

aspect ratio of the grain shape

increases.

6.1 (top left): annealed 70:30 brass

6.2 (top right): 40% reduction in thickness

6.3 (top right): 70% reduction in thickness

6.4 (top right): 90% reduction in thickness

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Grain size as a function of prior deformation level

The grain size after

recrystallization decreases with

increasing prior strain, i.e. the

nucleation density increases.

Example of commercial purity

Al, recrystallized at 600C

(1.5h) after 2 (top), 6, 8 & 10%

(bottom) reduction in tensile

strain.

Eutectoid reactions in Fe-C system

Varying the carbon content in

medium-carbon steels leads to

the expected variation in

eutectoid (pearlite) content.

16.3: 0.1%C, nital etch

16.4: 0.35% C

16.5: 0.55% C

16.6: 0.80% C

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Ceramics

Pores trapped in grainsTwo-phase ceramics. (a) As sintered

and (b) heat treated at 1600C for 30hours. ZTA 30% (zirconia-toughened

alumina with 30 vol% YSZ containing

10 molar% yttria). - the phases remain

dispersed after a long anneal!

Thin Films

Typical PVD coating characterized by columnar growth

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Free Surfaces

(P&E Chap 3 Crystal interfaces & Microstructure)

Surface Tension

Surface Tension vs Surface Energy

Surface tension: when a force is applied to a surface, the surface

atoms stretch their bonds in response to the force

If the force is strong enough, atoms at the surface will break

up allowing atoms from below to come to the surface and

create additional surface sites

The interatomic forces present between the surface atoms

(which resists the applied force) is surface tension

Surface tension is a force/unit length: units of dynes/cm, N/m

Surface EnergySurface energy: when a surface is reversibly increased, work is

done against the surface and energy is spent

At the same time, the energy of the surface is increased due to

the increased area.

This increased energy is the surface energy

is the free energy/unit area (J/m2=N/m)free energy of system(bulk + surface)= G=Go + A

Go=molar free energy in bulk

=excess energy due to being on surfaceA=surface area/mole of surface atoms

is mainly due to broken/ missing bonds with contributionsfrom bond direction and length differences and entropy

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Surface energy and surface tension should

be related

Consider a liquid film in a wire frame

a force F is exerted to move the film

The surface stretches and the free energy

increases

dG= dA + AdThe work done to move the wire frame

W=FdA should be equal to the change in

energy of the film

dG= dA + Ad F dA

F= +AddA

in liquids ddA=0, so F= (surfacetension and surface energy are the same

thing in liquids)

In solids, atomic migration is not feasible at most temperaturesThe surface structure therefore changes as the material is

stretched and ddA is not zero.

For solids can be significantly orientation dependent due todifferent atomic arrangements on different (hkl)s

Gibbs free energy

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Estimating Surface Energy of SolidsBy quantifying , surface energy of solids, as a function of bonding(heat of sublimation), surface orientation, and crystal structure, wewill be able to quantify how other phases (gases, liquids, anothersolid) will wet and react with the surface. The energy of the surfacebecomes important for nucleation of another material on to it.

The surface energy of a solid is a function of the broken bond energy ofexposed atoms (i.e. how the energy of the surface differs from theenergy of the bulk)

Energy per bond

s/0.5ZNAHs: enthalpy of sublimation - vaporize a solid by breaking all the

bonds) (Ls in Porter & Easterling)Z coordination number (bonds/atom), 0.5ZNA=bonds/mol

Work per surface atom (this is the work required to form a surface-cleave/break the bonds that are left broken on a surface)

w= #(bonds/atom) s/2 0.5ZNA

factor 2 added because only half the work to break a bond isapportioned to that atom (other half to the other side of the bond)

Surface energy (work/area):w N/A= #(bonds/atom) s/2 0.5ZNA (N/A)

N/A=number of atoms/area

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Equation for surface energy for solids

The surface energy of a solid is a function of the broken bond

energy of exposed atoms (i.e. how the energy of the surface differs

from the energy of the bulk)

= #(broken bonds/atom) N/A = (s/0.5ZNA) #(broken bonds/atom) N/A2 2

Hs: (Ls in P&E) enthalpy of sublimation (vaporize a solid by breaking allthe bonds)

Z : coordination number (bonds/atom), 0.5ZNA=bonds/mol (1/2 so youdont double count bonds)

N/A: number of atoms/area

# broken bonds/ atom is divided by two because when you break thebonds (theoretically) you form two surfaces- we are only calculating thesurface energy of one surface

Example Calculation of Surface Energy

Example: calculate the surface energy of (111) Cu.Cu is FCC, heat of sublimation is 170 kJ/(g mol) and ao=3.615

The (111) plane is the close packed plane- so six of the 12 bonds lie in theplane.

How many bonds lie above plane in question? These are the bonds that willbe broken when making the surface.For (111) 3 bonds lie below and 3 bonds lie above, so there will be threebroken bonds per atom.

Determine the area of the plane relative to ao the lattice parameter.

A=3/2ao2

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Find N: the number of atoms per area (how many effective atoms arecontained in the cross section of the plane). The cross section of theplane within the unit cell contains 6 atoms. However, the 3 edge atoms

are all only halfway in the cell. The three corner atoms are only 1/6 in theunit cell. The effective atoms in that area is 2. N=2

This compares relatively well with the measured value of 1600 ergs/cm2

(doesnt take into account 2nd nearest neighbors)

Shows that varies with crystal structure and surface orientation wesee this when we etch materials; different planes etch at different rates.

e.g. the chemical etch rates for Ge on the (100), (110), and (111) planes are1.00, 0.89, and 0.62, respectively.

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For real surfaces close to low index, atomically flat surfaces (i.e.vicinal) the

surface is made up of flat terraces of singular orientations bounded by ledges

Assuming a simple model, can tabulate the number of missing bonds for

different arrangements of atom on a surface

Site missingbonds

Interior atom 0

surf. Atom 1

atom next to vac. 2

atom on ledge 2

atom at kink 3

atom ads. on ledge 4

atom ads. on terrace 5

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If deviation from singular orientation occurs by tilting about axis ll

to ledges, the ledge density increases, so increases. If also tiltabout another axis, get an increased kink density on ledges.

If orientation is far from singular, get a max. ledge/kink density and

This is a diffuse surface. This leads to :

ESV= (cos sin ||) /2a2 ESV= solid vapor surface energy

If plot actual or theoretical vs. orientation for 3-d crystals, canmake radius and obtain a plot

A 2-d section through

-plot can be used tofind the minimum

energy shape for a

crystal by using the

Wulff construction :

at each point on polar

-plot construct aplane (line of 2-d

plot) normal to the

radius. The inner

envelope of such

planes gives theminimum energy

shape. This shape will

nothave the

minimum surface

area for a given

volume.

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For a faceted shape, total surface energy is :

E= i Ai ; i surf energy of ith facet, Ai =area of ith facet

if cusps are shallow or absent, equilib. shape may not have flat facets

and total surf energy will be:

E= dA

Non-spherical equilib. shapes have greater area but less energy/unit

area and therefore lower total surface energy

bulk heat treated UO2 where pores

achieved a near equilibrium shape. The

length of the facets in the small void can

be used to determine the relative energiesof (100) and (110) planes.

e.g. quartz

Grain Boundaries

crystal defects include

0-D Point defects (vacancies

and interstitials)

1-D Line defects

(dislocations)

2-D Planar defects (stacking

faults)

Grain boundaries represent another kind of planar defect, the

region between two crystals of the same phase of different

orientation.

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Misorientation between 2

grains can be described by an

axis-angle pair: rotation of oabout specific [uvw] axis of

grain A produces unit cell

orientation of grain B.

Tilt : [uvw] in boundary

Twist: [uvw] normal to

boundary

Low angle

boundary (LAB):

misorientation

angle is small;

accommodated by

an array of

regularly spaced

dislocations

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The spacing of the dislocations determines the misorientation of

the boundary.

sin() = b/D (in the case of low angles = b/D) ; b=magnitude of burgers vector

for very small , D is large and disloc energy/length is ~ that ofindividual disloc in bulk crystal

E~ Gb2/2 energy/unit length;

G=shear modulus

is ~ E (length of disloc/unitarea of LAB)

so E/D

E decreases from Gb2/2 as increases as strain fields overlap andcancel, reducing elastic strain energy

Seeing low-angle GBsby (A) decoration (Agparticles on a GB inKCl) and (B) etch pits(LiF).

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Twist boundary:an array of crossed screw dislocs in boundary

The twist when the GB emerges at the

surface.

Array of screw

dislocations in the tilt

grain boundary in

molybdenum

vicinal to the 70.5o

[110], {112}.

Unsymmetric boundary: 2

sets of edge dislocs of

different bs; doesnt split

angle between lattices

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Two sets of orthogonalscrew dislocations in a lowangle (001) twist GB in Si

Above a misorientation angle of 5-10o,concept of individual

dislocs in LAB breaks down since disloc cores are too closetogether. High angle boundaries have constant and energy ismostly due to missing atoms.

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due to a) wrong bond lengths, b) gb vacancies (broken bonds)

Some bonds are too long, and we get gb

vacancies. The extra space at gbs contributes

to: 1)fast gb diffusivity

2) segregation of solute atoms to gb

concentration of structural vacs is indep. of T, so

contribute to pre-exponential constant Do gb

LABHAB

HAB

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More bubble rafts...

HRTEM images of two high-angle GBs in

ZnO: (a) near-symmetric; (b) asymmetric.

Low-angle tilt GB in spinel showing an array ofclimb-dissociated edge dislocations. (A) GB viewed at

an angle. (B) Dislocations viewed end-on at highermagnification.

Same phenomena in ceramics but more

complex (ions)

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Polyhedra that are not present in the perfect crystal can be present at a

high-angle GB, and they can accommodate larger impurity ions than

can the bulk.

GB in Al2O3 shows the creationof new polyhedra in the GB.Inset: tilted view of therepeating group of polyhedra.

A high angle boundary is an impediment to dislocatonmotion. Dislocation pile-ups will occur.

Hall-Petch equation, d is the

average grain diameter, and 0 andky are constants for a particular

material. H-P is not valid for very

large (i.e., coarse) grain and

extremely fine grain polycrystalline

materials.

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Quenched austenitic nitrogen steel Fe-18Cr-14Mn-0,6N.

The high-angle boundaries are practically impenetrable barriers to moving dislocations.The distance between the dislocations in the shown pileups when approaching the

boundary diminishes. The stress on the leading dislocations, which can be estimated by the

dark shadows at A, B, and C, is very high. When the stress reaches a critical level, the

dislocation sources in the boundary will be activated to produce dislocations in the grain

on the left.

If equilibrate specimen with a gb normal to the surface

Surface and gb tensions must balance at intersection of a gb with

free surface

2[SV

cos (/2)] = b

b ~ 1/3 SV , so above is exaggerated (if b=1/3 SV , =160.8o

can calc b from measured and SV

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b / SV ~ 1/3 ; gb has fewer missing bonds than a freesurface [0.23-0.40 in table]

but remember gb = 2 X-tal surfaces, so~ 5/6 of energy isrecovered! Even for random high-angle (high-) g.b.

shear

twin

parent

Twin Boundaries

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If twin boundary plane coincides with ideal twin plane (={111}

in fcc) twin boundary is perfectly coherent: no vacancies or

incorrect bond lengths for first nearest neighbors (secondnearest neighbor mistakes generate a small

Examples of twinned grains found in differentminerals. The twin plane is outlined in blue.

Ceramics

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Two parallel {111} twin boundaries in

spinel having different structures

(identified by the arrows). The insets

show regions A and B at highermagnifications.

A variety of twin boundaries can form in

Al2O3 by mirroring the structure across

low-index planes that are not mirror planes

in the perfect crystal;e.g. the (1104) plane

Curved twin boundaries in a

thin film of NiO on an Al2O3substrate. (The hexagonal

pattern is a moire

interference effect).

GBs that correspond to a special twin orientation are not necessarily

flat (in which case they are probably more similar to other high-angle boundaries). The twin boundaries in the NiO film below

occur because NiO can grow on a basal-alumina substrate in two

twin-related orientations. (will this happen in the bulk?) Can see the

twin boundaries because the density is lower at the GB.

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Twin boundaries, like other GBs, can

accommodate new ions so that the

chemistry changes along the twin

plane. The shearing acts to change thechemistry along that plane.

A periodic repetition of alternating

twin planes, each of which

accommodates ordered impurities, can

give rise to a new structure. This

process is known as chemical

twinning.

The structure of -alumina as a repetition of chemicaltwin boundaries. This particle of -alumina contains

a sheet of spinel which is like a sheet of a second

phase. The twin planes are not actually spinel twin

planes because the chemistry is different on the twin

plane. Sand F are a block of spinel and a stacking

fault.

Coincident Site Lattice (CSL)

If we rotate one lattice about a low index direction like [111], at 1 or 2 specific

rotations some of the lattice positions of lattice 1 coincide with those of lattice 2

exactly. The set of coincident sites is arranged on a periodic lattice with much

greater spacing than the parent latttice.

Count the number of lattice sites that

are shared by the crystals on either side

of a boundary and divide by the total

number of lattice sites and invert, gives

value. Boundaries with small

(3,5,7,9,27,etc.) value are considered

special boundaries

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Geometric model ofa36.87o[100] tilt bicrystal with

simple cubic lattice. The circles

represent the positions of

individual atoms in misoriented

crystals, the empty circles denote

the coincidence sites.

Two-dimensional section of a CSL with 5 36.9 [1 0 0]

twist orientation

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1000

104

105

106

107

0 10 20 30 40

CriticalCurrentA/cm2

Misorientation Angle o

YBCO Critical current density between grains is very dependent onmisorientation

Super-conducting transport properties

of grain boundaries in YBa2Cu3O7

bicrystals D. Dimos, P. Chaudhari and

J. Mannhart, Phys. Rev. B 41, 4108

(1990). 946 citations

Mg metal is burned in air and the

resulting MgO smoke particles are caught

on a grid. The relative orientations

between cube particles determined in

TEM

Bicrystal particles form with a strong

preference for certain orientations in

which the two crystals have a fraction

of their lattice sites in common.

frequency of occurrence,f(), of

the misorientation angle

MgO smoke experiment

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Example: Effect of CSL on Pb Electrodes in Lead-Acid Batteries

Palumbo et al. [Palumbo, G., E. M. Lehockey, and P. Lin (1998).Applications for grain boundary engineered materials. JOM 50(2): 40-

43.] have shown that the crystallographic nature of grain boundaries in Pb

have a strong effect on the resistance of Pb electrodes (in the form of

lattice-work grids) to failure via intergranular corrosion and creep-cracking.

More specifically, Pb that has been processed to have a high fraction of

special boundaries, i.e. coincidence site lattice boundaries with low sigma

numbers, exhibit significantly longer lifetimes.

The next slide illustrates the difference in performance for Pb-Ca-Sn-Ag lead-

acid positive battery grids following 40 charge-discharge cycles. The

image on the left is the as-cast material with 7% special boundaries (3

29); the image on the right is the grain boundary engineered material with67.6% special boundaries.

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LABLABLABLAB

Experimental data for symmetric tilt boundaries show low at LABorientations, twin orientations (70.5o rotation about and at 130 about

(may be a Gleiter boundary - repeating group with little distortion)

No vacancies; small bond length errors;

38.2o

misorientation

twin

Grain Growth

A grain structure is never stable since b >0. However, it can be metastable (atleast locally) due to force balance of surface tensions.

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A gb will experience a moment tending to rotate it towards lower orientation if = ()

Fx

= 0 ; pulls both ways on segment

Fy = 0 ; Fy at opposite ends

M = 0; moment due to Fy =Fyl is balanced by (d/dl

If 12 = 13 = 23 then 1= 2= 3

i.e, if no orientation dependence (soap bubbles!)

in general, 23 = 13 (cos-3 ) + 12 (cos-2 ) whichleads to

23 /sin 1= 13 /sin 2= 12 /sin 3

for isotropic boundary energy

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pressure due to surface curvature

If a surface isnt flat, then P0. The vapor pressure above a curved surface is

not the same as above a flat surface due to P.

V is the molar volume, P is the vapor pressure, Po is the vapor pressure over a

flat surface (YoungLaplace equation)

where M is the molecular weight, and is the density.

If r1=r2,

The vapor pressure of a spherical particle is a function of its radius. The vapor pressure at the surface of a particle is higher if r is small. Small particles or small voids have large surface energies. At high temperatures, small particles tend to dissolve as the large particles

grow (Ostwald ripening).

Small particles have lower melting temperatures than large particles.

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Local metastable gb configuration can occur if grain shapes of

1,2,3 give correct angles. Annealing will establish correct angles

by atoms crossing gbs, but this will lead to curvature.

Curved boundaries exert pressure P=2b/r, which raises energy

G=Vm P= 2bVm/r

Boundaries tend totry to become flat to

minimize area. If #

boundaries opposite.

Concave grain shrinks, giving net grain growth

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http://www.youtube.com/watch?v=J_2FdkRqmCA

http://www.youtube.com/watch?v=Ac_ca_NeRnw

Unequal activation barriers

leads to unequal jump rates

anything that leads to a

potential difference can

lead to conditions

favoring grain boundary

migration

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Recrystallization is driven by excess energy of disloc tangles

- unequal jump rates

Driving force for low-D grain to "eat"

high-D grainG=b2DVm

(Vm changes 1/volume to 1/mol)

a) 33% CW brassb) New crystals nucleate after 3

sec. at 580C.

c) After 4 seconds

d) After 8 seconds

New crystals are formed that:

--have a small disl. density

--are small

--consume cold-worked crystals.

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e) After 8 s, 580C

f) After 15 min, 580C

At longer times, larger grains consume smaller ones,

grain boundary area (and therefore energy) is reduced.

A process used to make large single crystals of pure metals, solid solutions, and

intermetallic phases in which a fine-grained polycrystal is strained uniformly in small

amounts and heated gradually so that one recrystallizing nucleus consumes the entire

specimen, producing crystal that can as large as 100 cm3

Strain Anneal Methods

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Recrystallization and TemperatureRecrystallization temperature: temperature at which recrystallization is completed

within one hour.

Typically recrystallization

temperature is 1/3 to 1/2 of the

melting temperature (oK), but

depends on impurity

concentration and prior CW

(can be as high as 0.7 Tm in

some alloys).

Re-xtal behavior of a

brass alloy

Recrystallization and Cold WorkRecrystallization depends on the amount of cold work performed. Below acritical limit of cold work, it is impossible to initiate recrystallization

(local strain energy insufficient as driving force for crystal nucleation).

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1. Strengthening by Grain Size Reduction

Restricted motion of dislocation since grain boundaries act as barriersa) slip systems are usually not aligned and structural disorder at grain boundary

adds to this misalignment. Low angle grain boundaries are less efficient in

blocking than high angle grain boundaries

b) deformation even in favorable slip systems can be limited by spatialrestrictions

c) A stress concentration at end of a slip plane may trigger new dislocations in an

adjacent grain.

d) Grain size reduction is the only strengthening mechanism that does notreduce

ductility

A reduction in grain size increases the concentration of grain boundaries, which

impede dislocation motion. The Hall-Petch relation describes yield strength as a

function of grain size:

ys 0 kyd0.5

While many materials obey this relation over limited grain size or stress, it is not

always true!

Grain size d can be controlled by the rate of solidification, by plastic deformation

and by appropriate heat treatment.

Strengthening by Grain Size Reduction

70 Cu - 30 Zn

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Impurities at grain boundaries

Decreasing solubility

is largely due to atom

size / strain

atoms that are too

large / too small will

have stronger gb

binding and lower

solubility in the bulk

decreasing

Little open space, smallerenrichment, less impurity

effect

Impurities retard gb

motion because they have

to diffuse to keep up with

gb. ( If g.b. moves ahead

of impurity, system free

energy is increased. )

The strain field

interaction is like amechanical attraction to

gb

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Segregation results in lower boundary energies. Only lowenergy configurations arent sensitive to segregation.

Energetics of Impurity Drag on GBs

The basic idea is that the segregation of impurities to a

boundary lowers the free energy of the system.

Therefore there is a potential well present at the boundary

and a driving pressure must be applied to pull the

boundary out of the well.

Potential Energy, U

Position, xDrag Force

Position, x

Boundary at x=0 Moving Boundary (x=0)

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GBs in Ceramics

There may be large local changes in density at the interface because the bondingis ionic, covalent, or mixed ionic/covalent; hence ceramic GBs have a spacecharge.

Because the unit cells of all but the simplest binary compounds are large, it islikely that a GB with a fixed misorientation angle and interface plane exists in

more than one (meta)stable configuration. However, there will still be only one

minimum energy.

Energy is dependent on the GB plane, just as it is for surfaces. Hence, steps andfacets on these GBs are also important, and actually necessary for the GB to

move.

Many ceramics are processed in the presence of a second or third phase far awayfrom conditions of thermodynamic equilibrium, so a remnant of this phase may

remain at the GB even if it is not the lowest-energy configuration. Even more so

than metals, impurities will segregate to GBs.

Metals try to make the density of atoms at GBs uniform due to the nature of theelectron gas. GBs in ceramics may be much more open. The density of atoms in

the GB can be very different from that in the bulk grains.

Sintering another curvature effect

Sintering is a coalescence mechanism involving particles (usually, and typically

polycrystalline) in contact.

A neck forms between two particles and thickens as atoms aretransported into the region.

From the Gibbs-Thomson effect :

The driving force for neck growth is to reduce the total surface

energy of the system.

Since atoms on the convex island surfaces have a greater activity

than atoms situated in the concave neck ; an effectiveconcentration gradient between these regions develops.

mass transport into the neck.

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Two radii to consider (x, neck radius and

radius of curvature).A variety ofpossible transport mechanisms to the neck.

With three spheres, they can also move

together until they form a pore.

Differences in bulk pressure , vacancy concentration and vapor pressure can induce

material transport.The material transport due to the difference in interface curvature occurs under the

parallel action of various mechanisms.

Material transport mechanisms during sintering

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material transport paths

The dominant mechanism can vary depending on particle size, neck radius,temperature and time for a given system.

The interparticle distance can be reduced only by bulk material flow via viscousflow or by material transport from the grain boundary via atom movement. If material comes to the neck from the particle surface, interparticle distance is

not reduced but the neck size is increased by redistribution of material.

Therefore, the grain boundary is the source of material transport for densificationand shrinkage in crystalline powder compacts.

Diffusional flow is the most important mechanism of material transport. It is

based on the concept that a certain concentration of vacancies exists in thecrystal lattice of a metal.

Vacancy concentration is a function of temperature and the chemical potential

(or stress) to which the surface is subjected. Consequently, a gradient of

vacancies exists between a highly curved convex surface, which has a higher

vacancy concentration, and an adjacent flat surface, which has a lower vacancy

concentration.

The difference in vacancy concentration under surfaces with different radii of

curvature causes a flux of vacancies away from the highly curved surface to the

flat surface, which is equivalent to a diffusional flow of atoms in the opposite

direction.

Assuming the two particles are single crystals, with different orientations, a

grain boundary is formed at the neck. The difference in curvature at the neck and

the adjacent flat surface causes a difference in stress and chemical potential

between the two points, which in turn produces a gradient in the concentration of

vacancies between the highly curved neck surface, which has a high vacancy

concentration, and the adjacent flat surface, which has a lower concentration.

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Stages during Sintering

Theoretically, sintering kinetics show that

radiusnecktheisxwheretTAm

n

x )(

r

Can develop transport mechanism dependent relationships between the neck

/ sphere radius ratio and T,t.

DL is the lattice self-diffusion

coefficient,DS is the surface

diffusion coefficient, k is the

Boltzmann constant, Vm is the

vacancy (atomic) volume,x is

the neck dimension, t is the

sintering time, T is the sintering

temperature, is the surfaceenergy, r is the particle size of

the powders, s is the diffusionthickness of the surface

diffusion.

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Sintering diagram of an

aggregate of 38 m radius Ag

spheres. Contours of constanttime show the neck sizes found

after sintering for those periods

of time at various temperatures.

Ashby diagram identifies the dominant sintering mechanism under various experimental

conditions and shows the rate of sintering that results from all the mechanisms acting

together. Boundary lines between regions of dominant mechanisms indicate the experimental conditions

under which the contributions of two different mechanisms to sintering (neck growth) are the same.

Sintering is divided into three regions: stage 0 where adhesion between particles occurs at thebeginning of sintering, stage 1 where the driving force for sintering decreases as the neck grows,and stages 2 and 3 where the driving force increases with neck growth (spherical pore stage).

e.g. if sintered at 0.8 Tm, the diagram shows that after particle adhesion at the very beginning, theneck growth occurs dominantly by surface diffusion, grain boundary diffusion and lattice diffusion

in temporal sequence.