notes 01 spring2014
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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.
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