EMA5001 Lecture 11
(Interphase) Interfaces &
Precipitates Shape
© 2016 by Zhe Cheng
Coherent Interfaces (1)
Two crystals match very well at the interface plane

Same lattice structure and same orientation

Same lattice structure but different orientation

Different lattice structure
EMA 5001 Physical Properties of Materials
Zhe Cheng (2016)
11 Interfaces & Precipitates Shape
2
Coherent Interfaces (2)
Geometric match at the interface


Match of 2D lattice structure (typically closed packed plane)
Match of close-packed direction
Interfacial energy

If perfect match
− Coherent interface energy comes only
from chemical bonding to the “wrong” atoms:
ch Chemical contribution

If small geometric mismatch (< ~5%)
− Strain energy term appears
EMA 5001 Physical Properties of Materials
Zhe Cheng (2016)
11 Interfaces & Precipitates Shape
3
Semicoherent Interfaces (1)
Larger mismatch in lattice parameter (e.g., > ~5%)

Energetically more favorable to have semi-coherent interface w/ dislocations
dα
than w/ huge strain energy
D
┴
Misfit between two phases defined by  
d   d
d
The distance between two dislocations D will satisfy
D   d
i.e.,
d
D

Therefore, we have
┴
dβ
D
 d   d   d 
d

EMA 5001 Physical Properties of Materials
Zhe Cheng (2016)
11 Interfaces & Precipitates Shape
4
Semicoherent Interfaces (2)
Continue from p.5
 Relationship between relative mismatch and dislocation spacing D 
Considering Burger’s vector
b
We have D 

d  d 
d

2
b

For mismatch in two different directions, D1 
b1
1
D2 
b2
2
Interfacial energy come from two contributions
 ch Chemical contribution
 semicoherent   ch   st
 st Structural misfit contribution
1
For small misfit  (5%<  <25%),  st   
D
For large misfit of  > 25%, D 
EMA 5001 Physical Properties of Materials
d

 4d  , too much misfit Incoherent interface
Zhe Cheng (2016)
11 Interfaces & Precipitates Shape
5
Incoherent Interfaces
Large mismatch of the two adjacent phases

Different lattice structure that don’t have
matching planes
Or
 Large mismatch of lattice constant
Interfacial energy for incoherent
interfaces


Large interfacial energy
Not sensitive to orientation
EMA 5001 Physical Properties of Materials
Zhe Cheng (2016)
11 Interfaces & Precipitates Shape
6
Interfacial Energy on Precipitate Shape (1)
Simplest cases
 β precipitates in α single crystal


Strain free
Shape & orientation determined by minimization
of
Ai i

Fully coherent precipitate / GP Zone
 β precipitate has the same crystal structure
and similar lattice constant as α;


Matching (parallel) orientation
Shape: spherical
Incoherent precipitate
 β precipitate has very different crystal structure

and no match of any crystal plane
Shape
− Spherical
− Other shape – does not indicate coherency
EMA 5001 Physical Properties of Materials
Zhe Cheng (2016)
11 Interfaces & Precipitates Shape
Interfacial Energy on Precipitate Shape (2)
Partially coherent precipitate
 β precipitate has different crystal structure from
Contour of γ plot
 but w/ good match of only one crystal plane

γc
Shape
− If no other match: disk-shaped precipitates
Define
r
h
Plate radius
γi
Equilibrium shape
Plate thickness
Geometry of the plate satisfy
’ precipitate in Al-Cu
001 ' // 001
001 ' //001
2r  i

h c
Al
− If other cups exist - Other shapes
Phase Transformations in Metals & Alloys, Porter, 3rd Ed, 2008, p. 155
EMA 5001 Physical Properties of Materials
Zhe Cheng (2016)
11 Interfaces & Precipitates Shape
8
Interfacial Energy on Precipitate Shape (3)
To prove the relationship for plate-like precipitate shape
Define total interfacial energy for the plate-like precipitate:
G   A j  j  2r 2 c  2rh i
Because total volume of the plate-like precipitate is fixed, i.e.,
 h  V / r 2
r 2 h  V
Therefore,
V
V
2
G  2r  c  2r 2  i  2r  c  2  i
r
r
dG
V
For total interfacial energy to reach minimum,
 4r c  2 2  i  0
dr
r
2
V
r h
We have 4r c  2 2  i  2 2  i
r
r
2
Therefore
2r c  h i
or
2r  i

h c
EMA 5001 Physical Properties of Materials
Zhe Cheng (2016)
11 Interfaces & Precipitates Shape
9
Precipitates on Grain Boundaries
Precipitates on grain boundaries with a coherent interface


Coherent interface
Planar (coherent) interface on one side
Curved (incoherent) interface on other sides
α
α
β
Precipitates on grain boundaries without any coherent interfaces
αβ
 αα Grain boundary energy in α matrix
α
 αβ Incoherent interface energy
αα
 β
  Interphase angle
α

αβ
We have
   2  Cos
2
    
    
   2 
  180o
α
α
β
α
α
  120o
β
α
EMA 5001 Physical Properties of Materials
α
α
β
  0o
α
β α
α
Zhe Cheng (2016)
α
α
β
11 Interfaces & Precipitates Shape
α
α
α
10
Misfit Strain on Precipitate Shape
General precipitates

Shape determined by minimization of
− GS Strain energy term
 A
i i
 Gs
Coherent precipitate


Conservation of lattice sites
Shape of precipitates depends on
− Matrix: isotropic vs. anisotropic
− Precipitates: hard vs. soft
− Misfit: <5% vs. >5%

For isotropic matrix and the same elastic modules
Gs  4  2  V
Incoherent precipitate



Lattice sites do not have to remain unchanged
No coherency strain
Strain due to volume misfit  may exist
EMA 5001 Physical Properties of Materials
Zhe Cheng (2016)
11 Interfaces & Precipitates Shape
11
Loss of Coherency during
Precipitate Growth
Continue from p.11
For isotropic matrix, if volume misfit exist:
Gs 
2 2
 V  f c / a 
3
f(c/a) shape effect factor
Incoherent precipitates usually oblate spheroid
Loss of coherency during precipitation
Phase Transformations in Metals & Alloys,
Porter, 3rd Ed, 2008, p. 163
For coherent spherical precipitate
4
Gcoherent  A ch  Gs  4r 2 ch  4  2  r 3
3
G
Gcoherent
For semi-coherent precipitate (assume no volume misfit)
Gsemi coherent  A ch   st   4r 2  ch   st 
We have
rcrti 
3 st
1

4  2  2
Gsemi coherent
rcrit
r
r < rcriti: coherent; r > rcriti, semi-coherent
EMA 5001 Physical Properties of Materials
Zhe Cheng (2016)
11 Interfaces & Precipitates Shape
12
Solid Liquid Interfaces
Two types of solid-liquid interfaces
Liquid
Liquid
Flat
Diffuse
Solid
Lf
Tm
Solid
Lf
~ 4 R
Tm
Si, Ge, and nonmetals
Most metals
Relationship between SV, SL and LV
  SL  0.45 b  b  0.3 SV
 LV
V

 SL  0.15 SV

For the same material in solid and liquid states,  SV
EMA 5001 Physical Properties of Materials
R
Zhe Cheng (2016)
L
α  SL
  SL   LV
11 Interfaces & Precipitates Shape
 SV
13
Expectations about Surface & Interfaces
Understand that surface or interface free energy are the excess energy
with respect to bulk
Be able to estimate the surface free energy for low index crystal
surface in cubic lattices
Understand why surface energy change with orientation
Understand the impact of surface energy on crystal shape
Be able to name types of low angle grain boundaries and explain how
grain boundary energy change with misorientation angle
Understand why certain special high angle grain boundaries (e.g., twin
boundary) have low energy
Understand what determines the shapes of grains and grain
boundaries and the driving forces for grain boundary movement
Understand the origin of grain boundary segregation and the
relationship between grain boundary and bulk molar fraction of solute
EMA 5001 Physical Properties of Materials
Zhe Cheng (2016)
11 Interfaces & Precipitates Shape
14
Expectations about Surface & Interfaces
Be able to qualitatively describe coherent, semi-coherent, and
incoherent interfaces
Understand that the precipitate shape is determined by minimization of
interface energy and strain energy and be able to derive geometric
relationship for simple case when strain energy is omitted
Understand how grain boundaries impact precipitate shape
Understand from energy point of view the origin of the loss of
coherency for a precipitate as it grows
EMA 5001 Physical Properties of Materials
Zhe Cheng (2016)
11 Interfaces & Precipitates Shape
15
Interface Control vs. Diffusion Control
T
L
α
T1
Gα
α+β
A Xe X0
B
X0
1
1
Low interface mobility
Relative fast diffusion
β
α
Interface control
X0
Diffusion controlled


Xe Xi
0
XB
interface controlled


 Bi
Gβ
Xi
Xe
High interface mobility
Local equilibrium at interface
Diffusion control
Mixed controlled
EMA 5001 Physical Properties of Materials
Zhe Cheng (2016)
11 Interfaces & Precipitates Shape
16
Porter exercise 3.3
Total “surface” free energy
Gsurface   Ai i  l1 1  l2 2
Shape of the 2D crystal determined by minimization of the total
“surface” energy
The total area is a constant
l1  l2  A
Therefore, for minimization of Gsurface   Ai i  l1 1  l2 2
A
d (l1 1   2 )
dGsurface
A
ll
l
l1

  1  2  2   1  1 22  2   1  2  2  0
dl1
dl1
l1
l1
l1
Therefore,
l1  2

l2  1
EMA 5001 Physical Properties of Materials
Zhe Cheng (2016)
11 Interfaces & Precipitates Shape
17