ENME 665
Advanced Materials Engineering
Sept. 09, 2022
Course Outline
▪
Objectives
❖ Understand the fundamentals of dislocation
theory and its application in strengthening,
deformation and fracture of metals.
❖ Understand various materials failure
mechanisms, including stress corrosion cracking,
hydrogen embrittlement, fatigue, creep, strain
ageing and corrosion.
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Course Outline (cont.)
▪ Instructor:
▪ Dr. Frank Cheng
▪ E-mail: fcheng@ucalgary.ca
▪ Teaching Assistant:
▪ Mr. Qing Hu, ying.hu1@ucalgary.ca
▪ Mr. Yinghao Sun, yinghao.sun1@ucalgary.ca
▪ Day, time and location
▪ Monday and Friday, 8:00-9:15 am
▪ ENA 103
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Course outline (cont.)
The following textbook(s) is recommended for this course:
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Course Outline (cont.)
▪ Examination (All in class)
▪ Quiz 1: Sept. 19th
▪ Mid-term 1: Oct. 3rd
▪ Quiz 2: Oct. 24th
▪ Mid-term 2: Nov. 4th
▪ Quiz 3: 25th
Reading week is Nov. 7-11.
▪ Mid-term 3: Dec. 5th
No class.
▪ Grading
Mid-term 1
25%
Mid-term 2
30%
Mid-term 3
15%
Quiz 1
10%
Quiz 2
10%
Quiz 3
10%
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Topic 1 – Fundamentals of Materials
Science and Engineering
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1. Crystalline structure
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Microstructure of solid materials
Amorphous solids
– Do not possess long-range order of atom
positions
– Undercooled liquid, glass, glues, ceramics, …
Crystalline solids
– Atoms arranged in a regular pattern, extending in
all three dimensions.
– Metals
Almost all solids prefer the crystalline state
– Why?
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Crystalline structure
▪ Crystal lattice: a lattice with atoms or ions attached
to the lattice points.
▪ The smallest possible part of crystal lattice,
determining the structure, is called unit cell.
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BCC packing
This type of structure has
two atoms per unit cell.
Each atom has 8 nearest
neighbours (coordination
number = 8).
BCC is not ideally close-packed
– Closest-packed direction: <111>
– Closest-packed plane: {110}
Common in
– Alkali metals (K, Na, Cs)
– Transition metals (Fe, Cr, V, Mo, Nb)
What is the fraction of the volume occupied by spheres for
a bcc type structure (packing efficiency)?
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FCC packing
This type of structure has four atoms
per unit cell.
Each atom has 12 nearest neighbours
coordination number = 12).
Close-packed planes ({111})
<110> is close-packed direction
Found in natural and noble metals (Al, Cu, Ag, Au, Pt, Pb)
and Transition metals (Ni, Co, Pd, Ir).
What is the packing efficiency for a fcc type structure?
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Hexagonal Closest Packing (HCP)
Each atom in an HCP
lattice is surrounded
by and touches 12
nearest neighbors,
each at a distance of
2r. Coordination
number = 12.
Divalent solids (Be,
Mg, Zn, Cd) and
Transition metals and
rare earths (Ti, Zr,
Co, Gd, Hf, Rh, Os)
The packing
efficiency of a
hcp structure is
72%.
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Closed packing of atoms
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Atom packing in FCC and HCP
crystals
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(FCC)
Void locations in the stacking
▪ Two interstitial voids in the two close-packed
crystal structures:
▪ Tetragonal void: surrounded by four atoms.
▪ Octahedral void: surrounded by six atoms.
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Sizing the voids in close-packed
structure
Tetragonal: maximum value of r = 0.225R
Octahedral: maximum value of r = 0.414R
Which type of voids has a bigger space?
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Octahedral Voids in FCC
Located at {1/2,1/2,1/2} and {1/2,0,0}
There are 4 octahedral voids per FCC cell.
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Tetrahedral Voids in FCC
Located at {1/4,1/4,1/4}
There are 8 tetrahedral voids per FCC cell
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Octahedral and tetrahedral voids in BCC
▪ Maximum value of r
▪ Tetrahedral: r = 0.291R
▪ Octahedral: r = 0.154R
▪ Octahedral and tetrahedral voids in BCC are
asymmetric.
▪ Movement of an interstitial atoms causes a nonsymmetrical
straining of crystal.
Why do the point defects in bcc have a more significant effect on
mechanical properties of metal than those in FCC?
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Stacking faults
Twins
– A change in the stacking sequence over many atoms
spacings
– -A-B-C-A-B-C-A-C-B-A-C-B-A-
Stacking fault
– A change over a few atom spacings
– -A-B-C-A-B_A-B-C-A-B-C– Stacking faults by themselves are simple twodimensional defects. They carry a certain stacking fault
energy; very roughly around a few 100 mJ/m2.
– Usually occurs in close-packed structure.
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Binary compounds - Substitutional
▪ BCC substitutional:
AB type (CsCl)
▪ Small size difference
FCC substitutional:
A3B type
– A atoms on face center
– Intermetallic
compounds (Cu3Au)
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Binary compounds - Interstitial
FCC Octahedral
Interstitial: NaCl
FCC Tetrahedral
Interstitial: ß-ZnS
HCP Tetrahedral
Interstitial: α-ZnS
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2. Diffusion in solids
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What is diffusion?
Diffusion: The phenomenon of material transport by
atomic motion.
Many reactions and processes that are important in the material
treatment rely on the mass transfer:
– Either with a specific solid (at microscopic level )
– Or from a liquid, a gas, or another solid phase.
This topic covers:
– Atomic mechanism
– Mathematics of diffusion
– Influence of temperature and diffusing species of the
diffusion rate
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Diffusion is a RATE PROCESS IN SOLIDS
Probability of finding an atom with energy E*

 E* - E
 kT

Probabilit y  e
 


T = absolute temperature, K
k = Boltzmann ' s constant = 1.38x 10- 23 J/(atom * K)
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Fraction of atoms or molecules having energies greater
than E* which is itself much greater than the average
energy E.
n
N total
= Ce
 E* 
- 
 kT 
where n = number of atoms with energy greater than E*
N total = total number of atoms or molecules in system
T = absolute temperature, K
k = Boltzmann ' s constant = 8.62x 10 -5 eV/K
C = constant
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DIFFUSION
• Interdiffusion: In an alloy, atoms tend to
migrate from regions of large concentration.
After some time
Initially
100%
0
Concentration Profiles
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DIFFUSION (cont.)
• Self-diffusion: In an elemental solid,
atoms also migrate.
All atoms exchanging positions are of
same type.
Label some atoms
After some time
No compositional diffusion in pure metal
changes.
C
A
D
B
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Diffusion mechanism
Atoms in solids are in constant motion rapidly changing positions.
Diffusion is just the stepwise migration of atoms from a lattice site
to other lattice site.
Two conditions for movement:
1. There must be an empty adjacent site
2. Atom must have sufficient energy to break bonds with neighbor
atoms
Atomic vibration:
– Every atom is vibrating very rapidly about its lattice position
within the crystal
– At any instant, not all vibrate with same frequency and
amplitude.
– Not all atoms have same energy
– Same atom may have different level of energy at different time
– Energy increases with temperature
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Diffusion mechanism (cont.)
Several different models for atomic motion
– Two dominate for metallic diffusion
VACANCY DIFFUSION
– Involves interchange of an atom from a normal lattice
position to an adjacent vacant lattice site or vacancy
– Necessitates presence of vacancies
– Diffusing atoms and vacancies exchange positions ➔ they
move in opposite directions
– Both self- and inter-diffusion occurs by this mechanism.
– The activation energy for diffusion is the sum of the energy
required to form a vacancy and the energy to move the
vacancy.
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Vacancy Diffusion
• Applies to substitutional
impurities
• Atoms exchange with
vacancies
• Rate depends on:
-- number of vacancies
-- activation energy to
exchange.
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Diffusion Mechanism (cont.)
INTERSTITIAL DIFFUSION
– Atoms migrate from an interstitial position to a neighboring
one that is empty
– Found for interdiffusion of impurties such as hydrogen,
carbon, nitrogen, and oxygen ➔ atoms small enough to fit
into interstitial positions.
– Host or substitutional impurity atoms rarely have interstitial
diffusion
Interstitial atoms are smaller and thus more mobile ➔
interstitial diffusion occurs much more rapidly than by
vacancy mode.
There are more empty interstitial positions than vacancies ➔
interstitial atomic movement have greater probability.
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Interstitial diffusion
Smaller atoms can diffuse between atoms.
The activation energy for diffusion is the energy required for
these atoms to squeeze through the small openings between the
host lattice atoms.
More rapid than vacancy diffusion.
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Steady-state diffusion
The quantity of an element that is transported within
another is a function of time ➔ diffusion is a timedependent process.
Diffusion flux (J)
– Rate of diffusion or mass transfer
– Defined as “mass or number of atoms (M)
diffusing through and perpendicular to a unit
cross-sectional area of solid per unit time.
– Mathematically, in differential form: J = M = l dM
At
A dt
A: area across which diffusion is occurring
t: time
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Steady-state diffusion (cont.)
Rate of diffusion independent of time
Flux proportional to concentration gradient =
C 1 C1
dC
dx
Fick’s first law of diffusion
C2
x1
x
C2
dC
J = −D
dx
x2
D  diffusion coefficient
dC C C2 − C1
if linear

=
dx
x
x2 − x1
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Steady-state diffusion (cont.)
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Non-steady-State Diffusion
Most practical diffusion
situations are non-steady.
Non-steady
– Diffusion flux and the
concentration flux at some
particular point of solid vary
with time
– Net accumulation or
depletion of the diffusing
species
– Figure shown concentration
profile at three different times
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Non-steady-State Diffusion
• Concentration
profile,
C(x), changes
with time.
• To conserve matter:
• Fick's First Law:
• Governing Equation:
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Solution for Semi-infinite Solid with constant surface concentration
Assumptions
– Initial concentration C0
– X = 0 at the surface and increases with distance into the solid
– Initial time = 0
Boundary conditions
– For t = 0,
C = Co at 0  x  
– For t > 0,
C = Cs (Constant surface concentration) at x=0
C = C0 at x = 
Solution
– erf( ) : Gaussian error function
C x − C0
 x 
= 1 − erf 

C s − C0
 2 Dt 
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Non-steady state diffusion
• Copper diffuses into a bar of aluminum.
Cs
C(x,t)
t
Co o
t1
t
t2 3
position, x
• General solution:
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Error function
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Diffusivity -- the proportionality constant between
flux and concentration gradient depends on:
1. Diffusion mechanism: Substitutional vs. interstitial
2. Temperature
3. Type of crystal structure of the host lattice. Interstitial
diffusion easier in BCC than in FCC.
4. Type of crystal imperfections
(a) Diffusion takes place faster along grain boundaries
than elsewhere in a crystal.
(b) Diffusion is faster along dislocation lines than
through bulk crystal.
(c) Excess vacancies will enhance diffusion.
5. Concentration of diffusing species
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Temperature Dependence of the Diffusion
Coefficient
 Qd 
D = Do exp  
 RT 
Qd
ln D = ln Do RT
D is the Diffusivity or Diffusion Coefficient ( m2 / sec )
Do is the prexponential factor ( m2 / sec )
Qd is the activation energy for diffusion ( joules / mole )
R is the gas constant ( joules / (mole deg) )
T is the absolute temperature ( K )
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Non-steady state diffusion
Sample Problem: An FCC iron-carbon alloy initially
containing 0.20 wt% C is carburized at an elevated
temperature and in an atmosphere that gives a surface
carbon concentration constant at 1.0 wt%. If after
49.5 h the concentration of carbon is 0.35 wt% at a
position 4.0 mm below the surface, determine the
temperature at which the treatment was carried out.
Solution:
C( x, t ) − Co
 x 
= 1 − erf 

Cs − Co
 2 Dt 
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Solution (cont.):
– t = 49.5 h
– Cx = 0.35 wt%
– Co = 0.20 wt%
C( x , t ) − Co
 x 
= 1 − erf 

Cs − Co
 2 Dt 
x = 4 x 10-3 m
Cs = 1.0 wt%
C( x, t ) − Co 0.35 − 0.20
 x 
=
= 1 − erf 
 = 1 − erf ( z )
Cs − Co
1.0 − 0.20
 2 Dt 
 erf(z) = 0.8125
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Solution (cont.):
We must now determine from Table 5.1 the value of z for which the
error function is 0.8125. An interpolation is necessary as follows
z
erf(z)
0.90
z
0.95
0.7970
0.8125
0.8209
Now solve for D
z − 0.90
0.8125 − 0.7970
=
0.95 − 0.90 0.8209 − 0.7970
z = 0.93
x
z=
2 Dt
D=
x2
4 z 2t
−3
2
 x2 
(
4
x
10
m)
1h


D =
=
= 2.6 x 10 −11 m2 /s
 4z 2t  ( 4)(0.93 )2 ( 49 .5 h) 3600 s


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Solution (cont.):
To solve for the temperature at
which D has above value, we use
a rearranged form of Equation:
Qd
T=
R(lnDo − lnD )
from Table 5.2, for diffusion of C in FCC Fe
Do = 2.3 x 10-5 m2/s Qd = 148,000 J/mol
T=
148,000 J/mol
(8.314 J/mol - K)(ln 2.3x10 −5 m2 /s − ln 2.6 x10 −11 m2 /s)
T = 1300 K = 1027°C
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Summary:
Structure & Diffusion
Diffusion FASTER for...
Diffusion SLOWER for...
• open crystal structures
• close-packed structures
• lower melting T materials
• higher melting T materials
• materials w/secondary
bonding
• materials w/covalent
bonding
• smaller diffusing atoms
• larger diffusing atoms
• cations
• anions
• lower density materials
• higher density materials
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