Crystal Imperfections in Solids [7]
¾ Agregation of atoms (temperature decreasing):
vapour/gas
liquid
solid
¾ Solidification: result of casting of molten material
• Step 1: Nuclei formation
• Step 2: Nuclei grow to form crystals – grain
structure
Schematic illustration of the stages during solidification of molten metal; each
small square represents a unit cell. (a) Nucleation of crystals at random sites in
the molten metal; note that the crystallographic orientation of each site is
different. (b) and (c) Growth of crystals as solidification continues. (d) Solidified
metal, showing individual grains and grain boundaries; note the different angles
at which neighboring grains meet each other.
1>
Strength of Materials
¾Based on the bond strength most materials
should be much stronger than they are.
¾The strength for a strong ionic bond should
be about 7 GPa …
… but more typical strength is 300 MPa !
¾Why do we have three orders of
magnitude difference?
Answer: Imperfections/defects in structure
2>
Crystal Imperfections in Solids
¾ The atomic arrangements in a crystalline
lattice is almost always not perfect.
¾ There are “defects” in the way atoms are
arranged in the crystalline solids.
¾ So we can say that in crystalline solids some
Lattice Irregularities are always present.
¾ These crystalline defects are not bad. Some
are intentionally introduced to improve the
material.
3>
Types of Crystalline Defects
¾ CRYSTALLINE DEFECTS: classified on the
basis of their geometry and dimensionallity.
¾ POINT DEFECTS (no dimensional)
¾ LINE DEFECTS (1D)
¾ INTERFACIAL/SURFACE DEFECTS (2D)
¾ VOLUME DEFECTS (3D)
4>
Classification of Defects Based on Dimensionality
0D
(Point defects)
1D
(Line defects)
2D
(Surface / Interface)
3D
(Volume defects)
Vacancy
Dislocation
Surface
Twins
Impurity
Interphase
boundary
Precipitate
Frenkel
defect
Grain
boundary
Voids /
Cracks
Schottky
defect
Twin
boundary
Stacking
faults
5>
Dimensional scale of defects
significant effect on mechanical properties
6>
Point Defects
¾ POINT DEFECT: Imperfect point-like regions in
the crystal about the size of 1-2 atomic diameters
Vacancy
Non-ionic
crystals
0D
(Point defects)
Ionic
crystals
Interstitial
Impurity
Substitutional
Frenkel defect
Schottky defect
7>
Point Defects
¾ Vacancy: missing atom from an atomic site
¾ Atoms around the vacancy displaced
¾ Tensile stress field produced in the vicinity
X
Tensile stress field of vacancy
8>
Vacancies
¾ Usually introduced during solidification, at high T
¾ The vacancy concentration in pure elements is very
low at low temperatures.
¾ The probability that an atomic site is vacancy ~ 10-6
at low temperature
¾ The probability that an atomic site is vacancy ~ 10-3
at melting temperature
¾ can affect physical and electronic structures around
them → influence properties like color, conductivity
¾ they play a critical role in diffusion: control the self
diffusion and substitutional diffusion rates
¾ Movements of atoms coupled with movements of
vacancies
9>
Vacancies Equilibrium of Point Defects
¾ Each lattice site is a potential vacancy site, so the
equilibrium number of vacancies formed as a result of
thermal vibrations may be calculated from thermodynamics.
¾ At equilibrium, the fraction of lattices that are vacant (or
vacancy concentration) at a given temperature is given
approximately by the equation:
n
(− G f / kT )
=e
N
¾ where n is the number of point defects (number of vacancy
sites) in N sites and Gf is free energy of formation of the
defects (vacancy), T is the absolute temperature, k is the
Boltzman constant (1.38 x 10-23 J/atom.K or 8.62 x 10-5 eV/atom.K).
¾ A large numbers of additional (non equilibrium) vacancies
can be introduced by thermomechanical processing.
10>
The Effect of Temperature on Vacancy Concentration
¾ Calculate the concentration of vacancies in FCC copper
at room temperature (25oC). Assume that 20,000 cal are
required to produce a mole of vacancies in copper.
Solution: The lattice parameter of FCC copper is 0.36151
nm. The number of copper atoms, or lattice points, per
cm3 is:
4 atoms/cell
22
3
n=
=
8
.
47
×
10
copper
atoms/cm
(3.6151×10 −8 cm) 3
At room temperature,
⎛ Qν ⎞
T = 25 + 273 = 298 K
nν = n exp⎜
⎟
⎝ RT ⎠
cal
⎛
⎞
20,000
−
⎜
⎟
⎛
22 atoms ⎞
mol
⎜
⎟
.
exp
= ⎜ 8.47 × 10
⎟
3
cal
cm ⎠
⎜ 1.987
⎟
⎝
×
298K
⎜
⎟
mol − K
⎝
⎠
8
3
= 1.815 × 10 vacancies/cm
11>
Impurities in Solids
¾ Pure metal, containing only one type of atoms,
is not possible.
¾ Impurity atoms are always present.
¾ These atoms exists as point defects.
¾ In alloys, impurity atoms (alloying element
atoms) are intentionally added.
¾ An alloy is usually a solid solution of two or
more types of atoms.
¾ Examples:
Fe + C = steel
Cu + Zn = brass
Ni + Cu = monel
Al + Cu = dural
12>
Types of Solid Solutions
¾ SUBSTITUTIONAL IMPURITY
• Foreign atom replacing the parent atom in the crystal
• E.g. Cu sitting in the lattice site of FCC-Ni
¾ INTERSTITIAL IMPURITY
• Foreign atom sitting in the void of a crystal
• E.g. C sitting in the octahedral void in HT FCC-Fe
SUBSTITUTIONAL
Tensile Stress Fields
INTERSTITIAL
Compressive Stress Fields
13>
Hume-Rothery Rules for Solid Solution
¾ Hume-Rothery (1899-1968) states 4 Conditions
that must be satisfied for obtaining large solubility
of the substitutional solute in a given solvent lattice:
1.
2.
3.
4.
¾
Atomic Size Factor: The atomic size difference between
the solute and solvent atoms must be less than ± 15%.
Crystal Structure: Crystal structure of both solute and
solvent must be same.
Electronegative: The difference must be small. If this
difference is large ionic compound will form instead of
solid solution.
Valence: Higher valence metals will dissolve easily than
low valence metals.
When one or more is violated, partial solubility is
obtained.
14>
Hume-Rothery Rules for Solid Solution
¾ Ni + Cu: will they have large Solid Solubility?
Checking 4 H-R rules …
rule
Atomic Size
Ni
Cu
0.125nm 0.128nm
Crystal structure
FCC
FCC
Electronegativity
1.8
1.9
Valence
+2
+1
Answer: Yes, they will total solubility.
15>
Hume-Rothery Rules for Solid Solution
¾ Zn + Cu: will they have large Solid Solubility?
rule
Zn
Checking 4 H-R rules …
Atomic Size
Cu
0.133nm 0.128nm
Crystal structure
HCP
FCC
Electronegativity
1.6
1.9
Valence
+2
+1
Answer: No, they will not (partial solubility).
16>
Carbon Solid Solution in Iron
In FCC iron, carbon atoms are located at octahedral
sites. In BCC iron, carbon atoms enter tetrahedral
sites which are larger than octahedral ones. What
would be the atomic percentage of carbon in each
type of iron if all the interstitial sites were filled?
SOLUTION: We can find a total of 12 tetrahedral interstitial sites for
BCC structure. If all sites of BCC cell are occupied atomic
percentage of carbon in BCC iron would be:
The number of octahedral interstitial sites is 4 interstitial sites per
FCC unit cell. So, the atomic percentage of carbon in FCC iron
would be:
17>
Carbon Solid Solution in Iron
Interstitial C sitting in the octahedral void in HT FCC-Fe
‰ rOctahedral void / rFCC atom = 0.414
‰ rFe-FCC = 1.29 Å
‰ rC = 0.71 Å
⇒
rOctahedral void = 0.414 x 1.29 = 0.53 Å
‰ ⇒ Compressive strains around the C atom
‰ Solubility limited to 2 wt% (9.3 at%)
Interstitial C sitting in the tetrahedral void in RT BCC-Fe
‰ rTetrahedral void / rBCC atom = 0.29 • rC = 0.71 Å
⇒
rTetrahedral void = 0.29 x 1.258 = 0.364 Å
‰ rFe-BCC = 1.258 Å
‰► But C sits in smaller octahedral void - displaces fewer atoms
‰ ⇒ Severe compressive strains around the C atom
‰ Solubility limited to 0.008 wt% (0.037 at%)
18>
Specification of Composition or Concentration
m1
x 100
– WEIGHT PERCENT C1 =
m1 + m2
(wt%)
n m1
x 100
C =
n m1 + n m 2
(at%)
– ATOM PERCENT
being
'
1
m1
nm1 =
A1
m = mass; n = moles (atoms) - Compositions are easily converted from one
type to the other by manipulating m to n, or vice versa, using the atomic weight,
“A”.
19>
Specification of Composition or Concentration
¾ Example: Determine the composition, in atom percent, of
an alloy that consists of 97 wt% aluminum and 3 wt% copper.
(number of Al atoms)
C Al =
(number of Al atoms) + (number of Cu atoms)
(mass of Al atoms)
(number of Al atoms) =
(atomic weight of Al)
(97g Al)
(number of Al atoms) =
= 3.595 mol Al
(26.98 g/mol Al)
(3g Cu)
(number of Cu atoms) =
= 0.047 mol Cu
(63.55 g/mol Cu)
3.595
C Al =
= 98.7%at Al
C Cu = 100 - C Al = 1.3%at Cu
3.595 + 0.047
20>
Solid Solution Strengthening in Steels
Interstitial solute
Substitutional
solute
21>
Point Defects in Ceramics
¾ Vacancies
-- vacancies exist in ceramics for both cations and anions
¾ Interstitials
-- interstitials exist for cations
-- interstitials are not normally observed for anions because
anions are large relative to the interstitial sites
Cation Interstitial
Cation Vacancy
Adapted from Fig. 5.2, Callister & Rethwisch 3e.
(Fig. 5.2 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.)
Anion Vacancy
22>
Point Defects in Ceramics
¾ 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.
23>
Point Defects in Ceramics
¾ Schottky Defect
To maintain the charge neutrality, remove 1 cation and 1 anion;
this creates 2 vacancies.
24>
Point Defects in Ceramics
¾ Since there are both anions and cations in ceramics, a substitutional
impurity will replace the host ion most similar in terms of charge.
¾ Charge balance must be maintained when impurities are present.
• Example: NaCl
Na +
Cl -
• Substitutional cation impurity
Ca2+
Na+
Na+
without impurity Ca2+ impurity
• Substitutional anion impurity
O2-
without impurity
ClClO2- impurity
cation
vacancy
Ca2+
with impurity
anion vacancy
with impurity
25>
Semiconductor Ceramics
¾ Intrinsic semiconductor - A semiconductor in
which properties are controlled by the element or
compound that makes the semiconductor and not
by dopants or impurities.
¾ Extrinsic semiconductor - A semiconductor
prepared by adding dopants, which determine the
number and type of charge carriers.
¾ Doping - The addition of a small percentage of
foreign atoms in the regular crystal lattice of
silicon or germanium produces dramatic changes
in their electrical properties, producing n-type and
p-type semiconductors.
26>
Extrinsic Semiconductor
¾ Semiconductor N-type: The addition of pentavalent impurities such as Antimonium, Arsenic or
Phosphorus contribute free electrons, greatly
increasing the conductivity
of the intrinsic semiconductor.
Donor impurities:
¾ Sb
¾ As
¾P
27>
Extrinsic Semiconductor
¾ Semiconductor P-type: The addition of trivalent
impurities such as Boron, Aluminium or Gallium to an
intrinsic semiconductor creates deficiencies of
valence electrons, called "holes".
Acceptor impurities:
¾B
¾ Al
¾ Ga
28>
Point Defects
29>
Linear Defects
¾ Dislocations are linear defects and represent a line
around which atoms in the crystalline lattice are
misaligned.
¾ Types of Dislocations:
¾ EDGE DISLOCATION: A dislocation introduced into
the crystal by adding an ‘‘extra half plane’’ of atoms.
¾ SCREW DISLOCATION: A dislocation produced by
skewing a crystal so that one atomic plane produces a
spiral ramp about the dislocation.
¾ “MIXED” DISLOCATION: A dislocation that contains
partly edge components and partly screw components
30>
Edge Dislocation
Burgers vector b ⊥ dislocation line
b
Shear stress
Compressive stress field is colored RED and tensile stress field is BLUE.
31>
Screw Dislocation
Burgers vector b || dislocation line
Shear stress
Dislocation
line
Burgers
vector b
b
Top view
There are not compressive or tensile stress fields associated to screw
dislocation, only pure shear around dislocation line.
32>
Edge, Screw, and Mixed Dislocations
Mixed
Edge
Screw
During sliding Burgers vector does not change.
33>
Burgers Vector
¾ Burgers Vector b represents the magnitude and
direction of lattice distortion created by the dislocation.
– EDGE DISLOCATION b is perpendicular to dislocation
– SCREW DISLOCATION b is parallel to dislocation
34>
Slip Systems
¾ Dislocations move more readily in some crystal planes and
directions than in others as we will see.
¾ The combination of both the slip plane and direction form the
slip system.
¾ Slip plane is generally taken as the closest packed plane in the
system. Slip direction is taken as the direction on the slip
plane with the highest linear density.
¾ The line direction of a screw dislocation is in the same
direction as its Burger’s vector.
¾ An edge dislocation has its Burger’s vector perpendicular to
the line direction of a dislocation
¾ A dislocation having a line direction not parallel or
perpendicular to the Burger’s vector is considered a mixed
dislocation.
35>
Schematic of Dislocation Slip
Edge dislocation
Screw dislocation
36>
Slip Systems for Dislocations
Examples of slip plane
and direction for BCC
37>
Burgers Vector Calculation
¾ Calculate the length of the Burgers vector in copper.
SOLUTION: Copper has an FCC
crystal structure. The lattice
parameter of copper (Cu) is
0.36151 nm. The close-packed
directions, or the directions of the
Burgers vector, are of the form
<110>. The repeat distance along
the directions is one-half the face
diagonal, since lattice points are
located at corners and centers of
faces.
The length of the Burgers vector, or the repeat distance, is:
b = 1/2(0.51125 nm) = 0.25563 nm
38>
Formation of Dislocations
¾ Frank-Read sources:
Dislocations propagating
on slip plane from a
Frank-Read source
Two atractive
Frank-Read
interacting
Two repulsive
Frank-Read
interacting
http://www.numodis.fr/tridis/
39>
http://www.numodis.fr/tridis/
Dislocation Motion and Interaction
dislocations moving through a forest of dislocations
40>
http://www.numodis.fr/tridis/
Dislocation Motion and Interaction
pile-up at a grain boundary and overcoming it.
41>
http://www.numodis.fr/tridis/
Dislocation Motion and Interaction
dislocation multiplication and transmission in a bicrystal
42>
http://www.numodis.fr/tridis/
Dislocation Motion and Interaction
dislocation bowing a spherical particle.
43>
Dislocation Motion and Interaction
Mobility of an edge dislocation with different kinds of solute atoms
44>
Observing Dislocations in Thin Foils
¾ Dislocations can be observed in thin foil samples,
a very thin piece of material (< 100nm), which
be able to diffract a high voltage accelerated electron
beam of a transmission electron microscope (TEM).
(c) 2003 Brooks/Cole Publishing / Thomson Learning
top view
side view
TEM micrograph
45>
Observing Dislocations in Thin Foils
Source: http://www.numodis.fr/tridis/TEM/index.html
46>
Interfacial Defects
¾ INTERFACIAL DEFECTS are boundaries that have 2 or 3
dimensions and normally separate regions of the materials
that have different crystal structures and/or crystallographic
orientations.
¾ EXTERNAL SURFACES: One of the most obvious
imperfection boundaries is the external surface because the
crystal structure terminates.
¾ Surface atoms are not bonded to the maximum number of
nearest neighbors so they have higher energy state than
interior atoms.
¾ INTERNAL SURFACES:
¾ grain boundaries
¾ twin boundaries
¾ Stacking faults
¾ Of all these, the grain boundaries are the most important
from the mechanical properties point of view.
47>
Grain Structure
Crystalline solids generally consist of millions of individual
grains separated by boundaries. Each grain is a single crystal.
Within each individual grain there is a systematic packing of
atoms. Therefore each grain has different orientation and is
separated from the neighboring grain by grain boundary.
grain
grain boundary
48>
Polycrystalline Materials
¾ At the grain boundary, there is a disturbance in the
atomic packing.
HAGB: θ > 15 degrees
49>
Polycrystalline Materials
HALL-PETCH:
Small grain size
equates
to more strength
Hall-Petch mechanism: dependence of strength on grain
size for a number of metals and alloys.
50>
Grain Growth: Simulations in 2D & 3D
https://www.youtube.com/watch?v=J_2FdkRqmCA
https://www.youtube.com/watch?v=Ac_ca_NeRnw
51>
Tilt Boundary
¾ The simplest grain boundary consists of
a configuration of edge dislocations
between two grains.
¾ The misfit in the orientation of the two
grains (one on each side of the boundary)
is accommodated by a small perturbation
of the regular arrangement of crystals in
the boundary region.
¾ Schematic shows some vertical atomic
planes termination in the boundary and
each termination is represented by an
edge dislocation.
52>
Twist Boundary
¾ When the angle of
misorientation is parallel to
the boundary due to an array
of screw dislocations.
53>
Twin Boundary
¾ A twin boundary is a special type of grain boundary across
which there is a specific mirror lattice symmetry. Twin results
from atomic displacement that are produced from applied
mechanical shear forces, and also during annealing heat
treatments following deformation (annealing twins).
Twinning (FCC): {111} <211>
http://practicalmaintenance.net/?p=1135
54>
Comparison
Slip
Twinning
orientation of atoms
remains the same
reorientation of atomic
direction across twin plane
displacements take place atomic displacement is less
in exact atomic spacings than interatomic spacing
55>
Stacking Faults
¾ Error in packing sequence:
SF
perfect
defect
56>
Stacking Faults
¾ Bright Field Image in TEM:
http://www.tf.uni-kiel.de/matwis/amat/def_en/kap_6/backbone/r6_3_3.html
57>
Bulk or Volume Defects
¾ Much larger defects than the previous ones,
usually introduced during processing and
fabrication steps
¾ Examples include:
– Pores
– Cracks
– Foreign Inclusions
– Other Phases
Stress-corrosion cracking of na Inconel heat exchanger tube.
58>
References
¾ CALLISTER JR, W. D. AND RETHWISCH, D. G.
Materials Science and Engineering: An Introduction, 9th edition.
John Wiley & Sons, Inc. 2014, 988p. ISBN: 978-1-118-32457-8.
¾ ASHBY, M. and JONES, D. R. H.
Engineering Materials 1: An Introduction to Properties, Applications and Design.
4th Edition. Elsevier Ltd. 2012, 472p. ISBN 978-0-08-096665-6.
¾ CALLISTER JR, W. D. AND RETHWISCH, D. G.
Fundamentals of Materials Science and Engineering: An Integrated Approach, 4th ed.
John Wiley & Sons, Inc. 2012, 910p. ISBN 978-1-118-06160-2.
¾ MITTEMEIJER, E. J.
Fundamentals of Materials Science: The Microstructure–Property Relationship Using
Metals as Model Systems. Springer-Verlag Berlin. 2010, 594p. ISBN 978-3-642-10499-2.
¾ ASKELAND, D. AND FULAY, P.
Essentials of Materials Science & Engineering, 2nd Edition.
Cengage Learning. 2009, 604p. ISBN 978-0-495-24446-2.
¾ ABBASCHIAN, R., ABBASCHIAN, L., AND REED-HILL, R. E.
Physical Metallurgy Principles, 4th Ed. Cengage. 2009, 750p. ISBN 978-0-495-08254-5.
¾ SMALLMAN, R. E. and NGAN, A.H.W.
Physical Metallurgy and Advanced Materials, 7th Edition.
Elsevier Ltd. 2007, 650p. ISBN 978-0-7506-6906-1.
¾ http://www.tf.uni-kiel.de/matwis/amat/def_en/
Nota de aula preparada pelo Prof. Juno Gallego para a disciplina Ciência dos Materiais de Engenharia.
® 2015. Permitida a impressão e divulgação.
http://www.feis.unesp.br/#!/departamentos/engenharia-mecanica/grupos/maprotec/educacional/
59