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Imperfections in Solids
Imperfections in Solids
• Solidification- result of casting of molten material• 2 steps
• Nuclei form
• Nuclei grow to form crystals – grain structure
• Start with a molten material – all liquid
• Crystals grow until they meet each other
nuclei crystals growing grain structureliquid
Polycrystalline Materials
Grain Boundaries
• regions between crystals
• transition from lattice of one region to that of the other
• ‘slightly’ disordered
• low density in grain boundaries
• high mobility
• high diffusivity
• high chemical reactivity
SolidificationGrains can be - equiaxed (roughly same size in all directions)
- columnar (elongated grains)
Columnar in
area with less
undercooling
Shell of
equiaxed grains
due to rapid
cooling (greater
T) near wall
Grain Refiner - added to make smaller, more uniform, equiaxed grains.
heat
flow
~ 8 cm
Point defects
Point defects, as the name implies, are imperfect point-
like regions in the crystal. Typical size of a point defect is
about 1-2 atomic diameters.
A vacancy is a vacant lattice position from where the atom
is missing. It is usually created when the solid is formed
by cooling the liquid. There are other ways of making a
vacancy, but they also occur naturally as a result of
thermal excitation, and these are thermodynamically
stable at temperatures greater than zero.
• Vacancies:-vacant atomic sites in a structure.
• Self-Interstitials:-"extra" atoms positioned between atomic sites.
Point Defects
Vacancy
distortion
of planes
self-interstitial
distortion of planes
7
Point Defects: Frenkel and Schottky
• Frenkel Defect
To maintain the charge neutrality, a cation vacancy-cation interstitial
pair occur together. The cation leaves its normal position and moves to the interstitial site.
• Schottky Defect
To maintain the charge neutrality, remove 1 cation and 1 anion;
this creates 2 vacancies.
Schottky
Defect
Frenkel
Defect
SELF-INTERSTITIAL: very rare occurrence
An interstitial atom or interstitialcy is an atom that
occupies a place outside the normal lattice position. It may
be the same type of atom as the rest surrounding it (self
interstitial) or a foreign impurity atom. Interstitialcy is most
probable if the atomic packing factor is low.
Another way an impurity atom can be fitted into a crystal
lattice is by substitution. A substitutional atom is a foreign
atom occupying original lattice position by displacing the
parent atom.
In the case of vacancies and foreign atoms (both
interstitial and substitutional), there is a change in the
coordination of atoms around the defect. This means that
the forces are not balanced in the same way as for other
atoms in the solid, which results in lattice distortion around
the defect.
SELF-INTERSTITIAL: very rare occurrence
• This defect occurs when an atom from the crystal occupies
the small void space (interstitial site) that under
ordinary circumstances is not occupied.
• In metals, a self-interstitial introduces relatively (very!)
large distortions in the surrounding lattice.
POINT DEFECTS
• The simplest of the point defect is a vacancy, or vacant lattice site.
•All crystalline solids contain vacancies.
• Principles of thermodynamics is used explain the necessity of the
existence of vacancies in crystalline solids.
• The presence of vacancies increases the entropy (randomness) of
the crystal.
• The equilibrium number of vacancies for a given quantity of
material depends on and increases with temperature as
follows:
Nv= N exp(-Qv/kT)
Equilibrium no. of vacanciesTotal no. of atomic sites Energy required to form vacancy
T = absolute temperature in Kelvin
k = gas or Boltzmann’s constant
Example Problem 4.1
Calculate the equilibrium number of vacancies per cubic meter for copper at
1000°C. The energy for vacancy formation is 0.9 eV/atom; the atomic weight and
density (at 1000 ° C) for copper are 63.5 g/mol and 8.4 g/cm3, respectively.
Solution.
Use equation 4.1. Find the value of N, number of atomic sites per cubic meter for
copper, from its atomic weight Acu, its density, and Avogadro’s number NA.
toequal ie )1273( 1000at vacanciesofnumber theThus,
/8.0x10
/5.63
)/10)(/4.8)(/10023.6(
3 28
336323
KC
matoms
molg
mcmcmgmolatomsx
A
NN
Cu
A
325
5
328
/m vacancies2.2x10
)1273)(/1062.8(
9.0(exp)/(8.0x10
exp
KKeVx
eVmatoms
kT
QNN v
v
Continuing:
And Note: for MOST MATERIALS just below
Tm Nv/N = 10-4
Here: 0.0022/8 = .000275 = 2.75*10-4
Two outcomes if impurity (B) added to host (A):• Solid solution of B in A (i.e., random dist. of point defects)
• Solid solution of B in A plus particles of a new
phase (usually for a larger amount of B)
OR
Substitutional solid soln.
(e.g., Cu in Ni)
Interstitial solid soln.
(e.g., C in Fe)
Second phase particle
--different composition
--often different structure.
Point Defects in Alloys
And:
• slip between crystal planes result when dislocations move,
• this motion produces permanent (plastic) deformation.
Are called Dislocations:
Schematic of Zinc (HCP):
• before deformation • after tensile elongation
slip steps which are
the physical
evidence of large
numbers of
dislocations
slipping along the
close packed plane
{0001}
Line Defects
Linear Defects (Dislocations)• Are one-dimensional defects around which atoms are misaligned
• Edge dislocation:• extra half-plane of atoms inserted in a crystal structure
• b (the berger’s vector) is (perpendicular) to dislocation line
• Screw dislocation:• spiral planar ramp resulting from shear deformation
• b is (parallel) to dislocation line
Burger’s vector, b: is a measure of lattice distortion and is
measured as a distance along the close packed directions in
the lattice
Edge DislocationEdge Dislocation
Fig. 4.3, Callister 7e.
• Dislocation motion requires the successive bumping
of a half plane of atoms (from left to right here).
• Bonds across the slipping planes are broken and
remade in succession.
Atomic view of edge
dislocation motion from
left to right as a crystal
is sheared.
(Courtesy P.M. Anderson)
Motion of Edge Dislocation
Screw Dislocation
MICROSCOPIC EXAMINATION
Applications
• To Examine the structural elements and defects that influence the
properties of materials.
• Ensure that the associations between the properties and structure (and
defects) are properly understood.
• Predict the properties of materials once these relationships have been
established.
Structural elements exist in ‘macroscopic’
and ‘microscopic’ dimensions
MACROSCOPIC EXAMINATION: The shape and average size or
diameter of the grains for a polycrystalline specimen are large
enough to observe with the unaided eye.
• Useful up to 2000X magnification (?).
• Polishing removes surface features (e.g., scratches)
• Etching changes reflectance, depending on crystal
orientation since different Xtal planes have different
reactivity.
Micrograph of
brass (a Cu-Zn alloy)
0.75mm
Optical Microscopy
Adapted from Fig. 4.13(b) and (c), Callister 7e. (Fig. 4.13(c) is courtesy
of J.E. Burke, General Electric Co.
crystallographic planes
Grain boundaries...
• are planer imperfections,
• are more susceptible
to etching,
• may be revealed as
dark lines,
• relate change in crystal
orientation across
boundary.Adapted from Fig. 4.14(a)
and (b), Callister 7e.(Fig. 4.14(b) is courtesy
of L.C. Smith and C. Brady,
the National Bureau of
Standards, Washington, DC
[now the National Institute of
Standards and Technology,
Gaithersburg, MD].)
Optical Microscopy
ASTM grain size number
N = 2n-1
number of grains/in2
at 100x magnification
Fe-Cr alloy(b)
grain boundary
surface groove
polished surface
(a)
GRAIN SIZE DETERMINATION
The grain size is often determined when the properties of
a polycrystalline material are under consideration. The
grain size has a significant impact of strength and
response to further processing
Linear Intercept method
• Straight lines are drawn through several
photomicrographs that show the grain
structure.
• The grains intersected by each line segment are
counted
• The line length is then divided by an average
number of grains intersected.
•The average grain diameter is found by dividing this
result by the linear magnification of the
photomicrographs.
ASTM (American Society for testing and Materials)
VISUAL CHARTS (@100x) each with a number
Quick and easy – used for steel
ASTM has prepared several standard comparison charts, all having different
average grain sizes. To each is assigned a number from 1 to 10, which is termed
the grain size number; the larger this number, the smaller the grains.
N = 2 n-1No. of grains/square inch
Grain size no.
NOTE: The ASTM grain size is related (or relates)
a grain area AT 100x MAGNIFICATION
Determining Grain Size, using a micrograph taken
at 300x
• We count 14 grains in a 1 in2 area on the image
• The report ASTM grain size we need N at 100x not 300x
• We need a conversion method!
2
12100
M is mag. of image
N is measured grain count at M
now solve for n:
log( ) 2 log log 100 1 log 2
log 2log 41
log 2
log 14 2log 300 41 7.98 8
0.301
n
M
M
M
m
MN
N M n
N Mn
n
For this same material, how many Grains would I expect /in2 at 100x?
1 8 1 2
2 2
8 1
2 2
2 2 128 grains/in
Now, how many grain would I expect at 50x?
100 100N 2 128*
50
N 128*2 512 grains/in
n
M
M
N
M
Electron Microscopes
beam of electrons of
shorter wave-length
(0.003nm) (when
accelerated across large
voltage drop)
Image formed with
Magnetic lenses
High resolutions and
magnification (up to
50,000x SEM); (TEM up
to 1,000,000x)
Summary
• Point, Line, and Area defects exist in solids.
• The number and type of defects can be varied and controlled
• T controls vacancy conc.• amount of plastic deformation controls # of dislocations• Weight of charge materials determine concentration of substitutional
or interstitial point ‘defects’
• Defects affect material properties (e.g., grain boundaries control crystal slip).
• Defects may be desirable or undesirable • e.g., dislocations may be good or bad, depending on whether plastic
deformation is desirable or not.• Inclusions can be intention for alloy development