CHAPTER 5:
IMPERFECTIONS IN SOLIDS
ISSUES TO ADDRESS...
• What types of defects arise in solids?
• Can the number and type of defects be varied
and controlled?
• How do defects affect material properties?
• Are defects undesirable?
• How do point defects in ceramics differ from those
in metals?
• In ceramics, how are impurities accommodated
in the lattice and how do they affect properties?
1
Imperfections in Solids
• Introduction and Motivation
– Why study this?
• What I have shown you so far are “idealized” structures
• You would get the impression that solids (particularly crystalline
solids) have “perfect” structures
– This is clearly not the case
• All solids have imperfections, and these have profound influences on
mechanical (e.g. alloys of Ag/Cu v pure silver), electrical (e.g. doping
of silicon with boron) properties
• “Crystalline defect” – a lattice irregularity having one or more of its
dimensions on the order of an atomic distance
• We will also organize this discussion around whether the defect
involves a single atom (0 D), rows of atoms (1D), or interfacial (2D)
defects
TYPES OF IMPERFECTIONS
• Vacancy atoms
• Interstitial atoms
• Substitutional atoms
Point defects
• Dislocations
Line defects
• Grain Boundaries
Area defects
2
Imperfections in Solids
• Point defects in metals
– Start simple: what happens when you have a crystal structure
and one of the atoms is missing  vacancy
– Every crystal has vacancies
– Why? From a thermodynamic viewpoint, crystalline solids have
very low entropy (why?)
– The introduction of vacancies increases the entropy of the crystal
– Can estimate the number of vacancies in a crystal using a
Boltzmann type description
  Qv 

N v  N exp 
 k BT 
Qv is the energy required for forming a vacancy
For a “typical” metal Nv/N slightly below the melting
temperature is ~ 10-4
POINT DEFECTS
• Vacancies:
-vacant atomic sites in a structure.
distortion
of planes
Vacancy
• Self-Interstitials:
-"extra" atoms positioned between atomic sites.
distortion
of planes
selfinterstitial
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EQUIL. CONCENTRATION:
POINT DEFECTS
• Equilibrium concentration varies with temperature!
No. of defects
Activation energy
Q 
ND 
exp 
 D 

 kT 
N
No. of potential
Temperature
defect sites.
Boltzmann's constant
(1.38 x 10 -23 J/atom K)
(8.62 x 10 -5 eV/atom K)
Each lattice site
is a potential
vacancy site
4
MEASURING ACTIVATION
ENERGY
• We can get Q from
an experiment.
• Measure this...
• Replot it...
ND
ln
N
1
slope
-QD/k
1/T
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ESTIMATING VACANCY CONC.
• Find the equil. # of vacancies in 1m3 of Cu at 1000C.
• Given:
0.9eV/atom
Q 
ND
 exp 
 D 
= 2.7 · 10-4
 kT 
N
For 1m 3, N =
• Answer:
1273K
8.62 x 10-5 eV/atom-K
NA
x 1m3 = 8.0 x 1028 sites
 x
ACu
6
OBSERVING EQUIL. VACANCY CONC.
• Low energy electron
microscope view of
a (110) surface of NiAl.
• Increasing T causes
surface island of
atoms to grow.
• Why? The equil. vacancy
conc. increases via atom
motion from the crystal
to the surface, where
they join the island.
Movie m3
Reprinted with permission from Nature (K.F.
McCarty, J.A. Nobel, and N.C. Bartelt, "Vacancies in
Solids and the Stability of Surface Morphology",
Nature, Vol. 412, pp. 622-625 (2001). Image is
5.75 mm by 5.75 mm.) Copyright (2001) Macmillan
Publishers, Ltd.
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Imperfections in Solids
• Point defects in ceramics
– Point defects such as vacancies are also present in ceramics
– Here it is a bit more involved, since ceramics have both cations
and anions
– You might imagine that since the anions are (much?) larger than
the cations there are very few anion interstitial defects
– “Defect structure” is often used to denote defects in ceramics –
again need to deal with charge neutrality
Imperfections in Solids
• Point defects in metals
– A void left by an atom in the crystal structure is called a vacancy
– An additional atom is called a self-interstitial
• This is an atom that is located in one of the interstitial voids in the
crystal structure (remember those) that are usually not occupied
– These occur much less frequently than defects (why?)
Imperfections in Solids
• Point defects in ceramics
– Because of charge neutrality, defects in ceramics must occur in
pairs
– Frenkel defect -- cation-vacancy and cation-interstitial pair
(cation moves from regular site to interstitial site)
– Schottky defect – cation-vacancy and anion-vacancy pair (move
one cation and one anion from the lattice to the external surface)
DEFECTS IN CERAMIC
STRUCTURES
• Frenkel Defect
--a cation is out of place.
• Shottky Defect
--a paired set of cation and anion vacancies.
Adapted from Fig. 13.20,
Callister 5e. (Fig. 13.20 is from
W.G. Moffatt, G.W. Pearsall,
and J. Wulff, The Structure and
Properties of Materials, Vol. 1,
Structure, John Wiley and
Sons, Inc., p. 78.) See Fig.
12.21, Callister 6e.
• Equilibrium concentration of defects ~ e
QD / kT
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Imperfections in Solids
• Stoichiometry
– Stoichiometric materials are those where the exact cation:anion
ratio is predicted from the chemical formula (i.e. NaCl); deviation
from this leads to nonstoichiometric materials
– How would you get this?
• Ions with multiple oxidation states (e.g. Fe+2, Fe+3)
• Impurities
These types of issues can lead to
defects
Fe1-xO
x<1
Non-stoichiometry
Deficiency of Fe
Imperfections in Solids
• Impurities in solids (metals)
– It is very hard to make ultra-high purity metals
– Often turns out you don’t want to!
– Mixtures of metals (or alloys) often have better properties than
the pure compounds – e.g. alloying silver with copper
– Some definitions:
• Solid solution – single “solid phase” containing two (or more) metals
– Formed upon addition of impurity
• For binary metal solid solutions – solvent/solute nomenclature used
POINT DEFECTS IN ALLOYS
Two outcomes if impurity (B) added to host (A):
• Solid solution of B in A (i.e., random dist. of point defects)
OR
Substitutional alloy
(e.g., Cu in Ni)
Interstitial alloy
(e.g., C in Fe)
• Solid solution of B in A plus particles of a new
phase (usually for a larger amount of B)
Second phase particle
--different composition
--often different structure.
9
Imperfections in Solids
• Solid solutions
– In the case of a solid solution, upon addition of the impurity to the
host material, the crystal structure is maintained
– Analogy to liquids
• Add two miscible liquids together (water & methanol)
– The composition is homogeneous throughout
– Same idea with solid solutions – impurity atoms are randomly and
uniformly dispersed within the solid
ALLOYING A SURFACE
• Low energy electron
microscope view of
a (111) surface of Cu.
• Sn islands move along
the surface and "alloy"
the Cu with Sn atoms,
to make "bronze".
• The islands continually
move into "unalloyed"
regions and leave tiny
bronze particles in
their wake.
• Eventually, the islands
disappear.
Movie m4
Reprinted with permission from: A.K. Schmid,
N.C. Bartelt, and R.Q. Hwang, "Alloying at
Surfaces by the Migration of Reactive TwoDimensional Islands", Science, Vol. 290, No.
5496, pp. 1561-64 (2000). Field of view is 1.5
mm and the temperature is 290K.
10
Imperfections in Solids
•
Substitutional solid solutions
–
Regarding substitutional defects, there are several features
that determine how well the impurity dissolves in the host
material
1.
Atomic size factor – close in size, the better; if the difference is more than
about +/-15%, form new solid phases
Crystal structure – crystal structures for both species should be the same (i.e.
hard to dissolve an FCC metal in a BCC metal)
Electronegativity – closer the better; if the difference between the two is large,
they will likely form an intermetallic compound instead of a substitutional solid
solution
Valence – Metal will have a tendency to dissolve a metal of higher valency
versus lower valency
2.
3.
4.
–
Example: Cu and Ni are “model” substitutional solid solution
system
•
Size similar, both FCC, similar electronegativities, Ni+2, Cu+1
Imperfections in Solids
• Interstitial solid solutions
– Impurity atoms fill voids or interstices among the host lattice
– Given the high packing factors in metal structure (bcc, fcc, hcp),
the size of these positions is small
– So, dinterstitial atom < dhost atom
– Also very few of these defects (why?)
– Example – Carbon forms an interstitial solid with iron (what is
this?) – however can only add 2% carbon
Imperfections in Solids
• Impurities in Ceramics
– Can also have impurities in ceramics (again, issue of anions v
cations, electroneutrality)
IMPURITIES
• Impurities must also satisfy charge balance
• Ex: NaCl
• Substitutional cation impurity
• Substitutional anion impurity
O2-
initial geometry
ClClO2- impurity
anion vacancy
resulting geometry
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Imperfections in Solids
• Sections 5.5, 5.6 – Read
– 5.5 – can have point defects in polymers
– 5.6 – composition units conversion, you should know all of this already
from 204(?)
COMPOSITION
Definition: Amount of impurity (B) and host (A)
in the system.
Two descriptions:
• Weight %
• Atom %
• Conversion between wt % and at% in an A-B alloy:
CB =
C'BAB
C'AAA + C'BAB
x 100
• Basis for conversion:
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Imperfections in Solids
• Dislocations/linear defects
– A dislocation is a linear (one-dimensional) defect around which
atoms are misaligned
– Simplest type of dislocation – the presence of an extra portion of
a plane of atoms, which terminates inside the crystal
• This is called an edge dislocation
This is an extra plane of atoms
Note lattice distortion around plane
Atoms above dislocation line are
squeezed together, those below
are pushed apart
Magnitude and direction of
lattice distortion is expressed
in terms of the Burgers vector
Imperfections in Solids
• Edge dislocation
– Few other points
– Where the plane ends is often called the “edge dislocation line”
denoted by the ┴ symbol
– The symbol points up if the extra plane is in the “top” of the
crystal; points down if it is the “bottom” of the crystal
Also note the Burgers vector b
This is perpendicular to the edge
dislocation
LINE DEFECTS
Dislocations:
• are line defects,
• cause slip between crystal plane when they move,
• produce permanent (plastic) deformation.
Schematic of a Zinc (HCP):
• before deformation
• after tensile elongation
slip steps
13
Imperfections in Solids
• Screw dislocation
– Can think of this as being due to a
shear stress applied to the crystal
– The upper front region of the crystal
is shifted one atomic distance to the
right
– This also can be thought of as a
linear dislocation along the line AB
in the bottom Figure
Imperfections in Solids
• Screw dislocation
– Atomic positions above the “slip
plane” are open circles, those below
are solid circles
– Here the Burgers vector b is parallel
to the screw dislocation
In reality, most defects have
components of both types … these
are referred to as mixed dislocations
INCREMENTAL SLIP
• Dislocations slip planes incrementally...
• The dislocation line (the moving red dot)...
...separates slipped material on the left
from unslipped material on the right.
Simulation of dislocation
motion from left to right
as a crystal is sheared.
Click on image to animate
(Courtesy P.M. Anderson)
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BOND BREAKING AND
REMAKING
• 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)
Click on image to animate
15
Imperfections in Solids
• Interfacial defects
– We have now talked about zero-dimensional (i.e. point) and onedimensional (linear) defects. Can also have two-dimensional
defects
– These are boundaries that normally separate regions of the
material with different crystal structures and/or crystallographic
orientation (remember about grains?)
– The most common – external surfaces, grain boundaries,
stacking faults, twin boundaries, and phase boundaries
Imperfections in Solids
• Interfacial defects
– External surfaces
• Why defects? Surface atoms are not bonded to the same number
of atoms as the “bulk” atoms
• They are therefore in a higher energy state
• Nature (thermodynamics) says minimize your free energy (here
dominated by surface free energy effects)
• In liquids – typically get spherical droplets (why?)
– Solids can’t do this
• However, it is common to see structural rearrangement at surfaces
to minimize energy
Imperfections in Solids
• Interfacial defects
– Grain boundaries
• Boundaries between domains (or grains) of a polycrystalline
material that possesses different crystallographic orientations
• Within the boundary there is mismatch of the atomic arrangements
as compared to the crystallographic domains
• Orientation mismatch can be minor (small- or low-angle grain
boundaries) or major (large- or high-angle grain boundaries)
AREA DEFECTS: GRAIN
BOUNDARIES
Grain boundaries:
•
•
•
•
are boundaries between crystals.
are produced by the solidification process, for example.
have a change in crystal orientation across them.
impede dislocation motion.
Metal Ingot
Schematic
~ 8cm
grain
boundaries
Adapted from Fig. 4.7, Callister 6e.
heat
flow
Adapted from Fig. 4.10, Callister 6e. (Fig.
4.10 is from Metals Handbook, Vol. 9, 9th edition,
Metallography and Microstructures, Am. Society for
Metals, Metals Park, OH, 1985.)
16
Imperfections in Solids
• Interfacial defects
– Grain boundaries
• Tilt boundary – series of edge dislocations leading to a small-angle
grain boundary
Key point here – the edge dislocations are aligned
with respect to one another
Imperfections in Solids
• Interfacial defects
– Grain boundaries
• Turns out that atoms along the grain boundary have less regular
bonding as compared to that of the “bulk”
– This is analogous to surface atoms
•
•
•
•
As a result they have a higher energy
Tend to be more reactive
Often impurity atoms will segregate into/along the grain boundaries
Grow at elevated temperatures to reduce the total boundary energy
Imperfections in Solids
• Interfacial defects
– Twin boundaries
• Special type of grain boundary – across the grain boundary there is
a mirror symmetry
The region between the boundary
is called a twin
These are formed due to applied
shear forces (mechanical twins) or
during high temperature heat
treatments (annealing twins)
These twins occur on a definite
crystallographic plane and in a
specific direction
Imperfections in Solids
• Interfacial defects
– Stacking faults
• Exactly what it sounds like … the layers in a structure are stacked
incorrectly
• For example, in the FCC structure an interruption of the ABCABC..
stacking sequence of close-packed planes
– Phase boundaries – domains between solids that phase
separated (multiphase materials)
Imperfections in Solids
• Bulk/volume defects
–
–
–
–
–
–
These are “3-D” defects
Pores – holes in the material
Cracks
“Foreign inclusions” – a large impurity domain?
Usually result from processing/fabrication
All effect material properties
• Atomic vibrations – read
• Microscopic examination – read
– Key point – microscopy allows you to generate real space
images of solids
– Resolution of the image (i.e. size of the objects/features you can
see) depends on the wavelength of the radiation used
Cu atomic shells for (Cu0.25Ni0.75)343 at various
temperatures during the heating process
(MD simulations, S-P Huang and PB Balbuena, J. Phys. Chem. B, 2002)
0K
400 K
200 K
700 K
300 K
1300 K
Cu0.25Ni0.75 nanocluster (343 atoms) deposited on graphite (not shown)
(S-P Huang, D. S. Mainardi and P. B. Balbuena, Surf. Sci., 2003)
700 K
900 K
800 K
1200 K
OPTICAL MICROSCOPY (1)
• Useful up to 2000X magnification.
• Polishing removes surface features (e.g., scratches)
• Etching changes reflectance, depending on crystal
orientation.
close-packed planes
Adapted from Fig. 4.11(b) and (c),
Callister 6e. (Fig. 4.11(c) is courtesy
of J.E. Burke, General Electric Co.
micrograph of
Brass (Cu and Zn)
0.75mm
17
OPTICAL MICROSCOPY (2)
Grain boundaries...
• are imperfections,
• are more susceptible
to etching,
• may be revealed as
dark lines,
• change direction in a
polycrystal.
Adapted from Fig. 4.12(a)
and (b), Callister 6e.
(Fig. 4.12(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].)
18
Electron Microscopy
• Instead of light radiation, uses beams of
electrons
• A high-velocity electron becomes wave-like
(QM), and its wavelength is inversely
proportional to its velocity. If accelerated
(high V), wavelengths could be of ~0.003
nm (3pm). => high magnification and high
resolution
Electron microscopy
• Transmission electron microscope (TEM):
an electron beam goes through an specimen
(usually prepared as a thin foil) =>
transmission of a high % of the incident
beam.
• Transmitted beam is projected for
visualization. Magnifications of 1,000,000X
or more. (useful for dislocations)
Electron microscopy
• Scanning electron microscope (SEM): the
surface of a specimen is scanned with an
electron beam, and the reflected beam of
electrons is collected on a cathode ray tube
(similar to a TV screen). The surface must
be electrically conductive. Magnifications
from 10 to 50,000X.
SCANNING TUNNELING
MICROSCOPY
• Atoms can be arranged and imaged!
Photos produced from
the work of C.P. Lutz,
Zeppenfeld, and D.M.
Eigler. Reprinted with
permission from
International Business
Machines Corporation,
copyright 1995.
Carbon monoxide
molecules arranged
on a platinum (111)
surface.
Iron atoms
arranged on a
copper (111)
surface. These
Kanji characters
represent the word
“atom”.
Uses a tiny probe with a very sharp tip brought in very close proximity of the specimen. The probe
19
movements are monitored, and a 3-D image of the surface is generated.
SUMMARY
• Point, Line, and Area defects arise in solids.
• The number and type of defects can be varied
and controlled (e.g., T controls vacancy conc.)
• 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.)
20
ANNOUNCEMENTS
Reading: Chapter 5
HW # 4: Due Monday, February 19th
5.2; 5.7; 5.9; 5.11; 5.27; 5.31; 5.32; 5.34; 5D2
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