Interfaces: Grain boundaries and interphase interfaces
 Structure and energy of grain boundaries
 Low-angle and high-angle grain boundaries
 Special low-energy high-angle grain boundaries
 Interphase interfaces
 Coherent, semicoherent and incoherent interphase boundaries
 Shape of precipitates: Effects of misfit strain and interfacial energy
 Loss of coherency
References:
Porter and Easterling, Ch. 3.3.1-3.3.3, 3.4
Allen and Thomas, Ch. 5.3
Jim Howe, Interfaces in Materials
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Grain boundaries
Single-phase polycrystalline material consist many crystals or grains that have different
crystallographic orientation. There exist atomic mismatch within the regions where grains meet.
These regions are grain boundaries.
Structure and energy of a grain boundary is defined by the misorientation of the two grains and
the orientation of the boundary plane. 5 independent variables (degrees of freedom) are needed
to define the rotation axis, rotation angle and the plane of the boundary. Rigid-body translation of
two grains with respect to each other add 3 more variables.
2 special cases can be distinguished:
pure tilt boundary - axis of rotation is
parallel to the plane of the boundary
axis of tilt
pure twist boundary - axis of rotation is
perpendicular to the plane of the boundary
boundary
twist axis
axis of tilt
symmetry plane
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Low-angle grain boundaries
Low-angle grain boundaries (misorientation ≤ 15) can be represented by an array of dislocations
In particular, low-angle tilt boundaries can be represented by an array of edge dislocations
D sin
 b

2 2
D
D - dislocation spacing
θ - misorientation angle
y
low-angle symmetrical tilt boundary in a
simple cubic lattice
b
b

2 sin( / 2) 
for small θ
recall our discussion of dislocation walls
h
x
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Low-angle grain boundaries
In asymmetric tilt boundary, the second set of
dislocations appears so that the boundary plane
moves off the plane of reflectional symmetry
b
D 
 cos 
D| 
b|
 sin 
φ - is the angle of inclination of the boundary plane with
respect to the symmetric orientation
low-angle asymmetric tilt boundary in a simple
cubic lattice
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Low-angle grain boundaries
Low-angle grain boundaries (misorientation ≤ 15) can be represented by an array of dislocations
Low-angle twist boundary is a cross-grid of two sets of screw dislocations
atoms between the dislocations fit almost perfectly
to the adjoining crystals, with the distorted regions
localized along the dislocation cores
low-angle twist boundary in a simple cubic lattice
atoms in crystal below boundary are shown by circles,
atoms above boundary are shown by dots
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Energy of low-angle grain boundaries
for small θ, the distance between dislocations is large and the
energy of the grain boundary, γGB, is proportional to the
dislocation density:
 GB
1
~ ~
D
as θ increases, the strain fields of dislocations
increasingly cancel out and γGB tend to saturate
when θ approaches ~15º, core regions of the
dislocations start to overlap and the description of
GB in terms of dislocation wall is no longer useful
 GB
TEM image of a small angle
tilt boundary in Si
10-15º
lowangle
random highangle GB

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 xx
Energy of grain boundaries
1 SV



energy of random high angle GB: GB
3
(open disordered structure)
examples:
Pt
 GB
2340
660
1410
378
Ag
1140
Cu
1670
Au
energy of symmetrical <110> tilt boundary in Al
 SV
[mJ/m 2 ]
375
625
there are specific combinations of GB
misorientations and boundary planes
that correspond to low energies
special high-angle grain boundaries
from Porter and Easterling
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Special high-angle grain boundaries
special boundary with good atomic fit  low grain boundary energy
twin boundary
In general, GB energy is a function of at least
5 parameters needed to describe the boundary
For a given misorientation, the energy of GB
will depend on the orientation of the GB plane
Twin boundary - special case of low angle,
high symmetry grain boundary. Most
commonly, twinning corresponds to mirror
symmetry around twinning plane.
coherent twin
boundary
incoherent twin boundary (much higher energy)
good atomic fit at coherent twin boundary  low energy comparable to that of a stacking fault
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Special high-angle grain boundaries: Faceting
strong dependence of the GB energy on the orientation of the boundary plane  optimization of
grain boundary - faceting, i.e., decomposition of the grain boundary plane into planes with low
energies (or large areas on low-energy planes + small areas of connecting high-energy planes)
faceting: even though the total GB
area increases, the energy decreases
somewhat similar to
dislocations adopting lowenergy configurations in
Peierls energy landscape
Faceting readily occurs and can reduce energy of the boundary misorientation of grains is more important than the orientation of
boundary planes.
The size facets can be large (observed in optical microscope) for
coherent twins and is smaller for other low-energy GBs - look
curved in a microscope.
metal carbide precipitation on GBs (1),
incoherent twin boundaries (2) & coherent
twin boundaries (3) in Fe-20Cr-25Ni (wt.%)
stainless steel
Sourmail & Bhadeshia,
Metall. Mater. Trans. A 36A, 23, 2005
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Special high-angle grain boundaries: Coincidence site lattice
Let’s consider rotation of two overlapping crystals with respect to each other about a certain
rotation axis. At certain misorientations one can get perfect overlap of the lattice sites in the two
crystals. The overlapping lattice sites create a new lattice called coincidence site lattice (CSL)
53.1º rotation of a cubic lattice about [100] cases 1/5 of the lattice sites to coincide
The (100) twist and (210) tilt GB shown above are high-density planes of CSL correspond to
low-energy GBs
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Special high-angle grain boundaries: Coincidence site lattice
CSL is characterized by  that is defined as
 = volume ratio of the unit cell of the CSL to that of the original crystal lattice
 = reciprocal density of coinciding sites
Σ1: perfect crystal of small deviations from
perfect crystal (low-angle GB)
Σ3: twin boundary - largest number of
coinciding lattice points (Σ is always odd)
GB that contains a high density of lattice points
in CSL is expected to have low energy because
of good atomic fit
high density of CSL lattice points requires both
special misorientation and the boundary plane 
pure tilt or twist boundaries are good candidates
Σ5, 36.9° <100> in cubic lattice
tilt boundary
(GD plane  paper)
twist boundary
(GD plane || paper)
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Energy of high-angle grain boundaries
low boundaries tend to have
lower energies than average
atomistic modeling
experiment
the correlation with Σ is not simple
- there is no monotonous energy
decrease with increasing Σ
deviations from the ideal CSL
orientation may be accommodated
by local atomic relaxation or the
inclusion of dislocations into the
boundary
<100> (a, b) and <110> (c,d) symmetric tilt boundaries
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Energy of high-angle grain boundaries
other models attempting to describe energies of GB include
structural (polyhedral) unit model proposed by Sutton and Vitek
Philos. Trans. R. Soc. London, Ser. A 309, 37, 1983
• for a given tilt axis there are short-period grain-boundary
structures consisting of a single type of structural unit
• GBs at intermediate misorientation angles can be
constructed by combining this units
• the minority units are considered to be dislocation cores
disclination and disclination-structural unit models
Li, Surf. Sci. 31,12, 1972
Gertsman et al., Phil. Mag. A 59, 1113, 1989
grain boundary regions can be disordered/amorphous, in
particular in polymers and ceramic materials
chemical composition of grain boundary regions can be different
from the bulk of the grains
no universal theory exists to describe high-angle GBs
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Interphase boundaries
interphase boundary separates two different phases which may have different composition,
crystal structure and/or lattice parameter  limited (if any) options for perfect matching of
planes and directions in the two crystals
depending on atomic structure, 3 types of interphase boundaries can
be distinguished: coherent, semicoherent, and incoherent
coherent (commensurate) interface: two crystals match perfectly at
the interface plane (small lattice mismatch can be accommodated by
elastic strain in the adjacent crystals)
semicoherent (discommensurate) interface: lattice mismatch is
accommodated by periodic array of misfit dislocations
incoherent
semicoherent
incoherent (incommensurate) interface: disordered atomic structure
of the interface
a  a1
lattice misfit at the interface:   2
a1
even in the case of perfect atomic matching, there is always a chemical
contribution to the interface energy
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coherent
Coherent interphase boundaries
strain-free coherent (commensurate) interfaces:
• two crystals match perfectly at the interface plane
• interfacial plane has the same atomic configuration in both phases
The indices of the planes comprising the boundary do not have to be the
same in each phase but orientation relationship between the two phases
should be satisfied. This relationship is specified in terms of a pair of
parallel planes and directions, i.e., {hkl}//{hkl}β and <uvw>//<uvw>β
Example: interface between  and κ phases in Cu-Si
same crystal structure
(111) plane of  phase
matches almost perfectly
(0001) plane in κ phase
hcp κ phase
fcc  phase
different crystal structure
orientation relationship:
111 //0001 and
1 10

// 1120

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Strained coherent interphase boundaries
Small differences in lattice parameter can be accommodated by elastic strain and coherent
interface can be maintained. If the upper crystal is uniformly strained in tension and the lower
half uniformly compressed, the crystals match perfectly.
a2  a1

a2  a1
0
a1
Smith and Shiflet, Mater. Sci. Eng. 86, 67, 1987
This coherency strain reduces the interfacial energy at the expense of increasing energy of the
two phases adjoining the interface  coherent interfaces are favored when
(1) interface is strong,
(2) misfit is small (few percent),
(3) the size of one of the crystals is small (thin overlayer or small precipitate)
While the structure of the interface is perfect, the interfacial energy is due to the bonding between
atoms from different phase (has only chemical contribution):   
 1 200 mJ/m 2
coh
chem
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Semicoherent interphase boundaries
When the energy due to the coherency strain becomes too large, formation of a semicoherent
interface can become energetically favorable  uniform elastic strains are replaced with
localized strain due to an array of dislocations that do not create long-range strain fields

a2  a1
a1
δ and/or d are too
large to maintain
coherent interface
d
dislocation spacing in 1D: D 
b
a1  a2
2
a2 b

 
- Burgers vector of misfit dislocations
In two dimensions, a network involving more than one Burgers
vector may be required to accommodate the misfit
D2 
b2
2
b1
D1 
1
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Semicoherent interphase boundaries
Example: lattice-mismatched Ag film - Cu-substrate interface

1
b

[1 1 0 ]
1
aCu  3.63 A
a Ag  aCu
2

 0.13

1 
aCu
a Ag  4.09 A
b 2  [11 0 ]
2
D1  D2  6.3a 
b
 5.44a

b
a
2
coherency strain is partially relieved by misfit dislocations,
with residual compressive strain present in the film
Wu, Thomas, Lin, Zhigilei,
Appl. Phys. A 104, 781-792, 2011.
Energies of semicoherent interfaces have both chemical and structural
(distortions due to the misfit dislocations) contributions:  semicoh   chem   str  200  500 mJ/m 2
γstr can be estimated similarly to low-angle GBs, by dividing the energy per unit length of the
dislocations, Gb2/2, by the dislocation spacing, b/ δ  γstr ≈ Gbδ/2. For G = 50 GPa, b = 3 Å,
and δ = 0.01, γstr = 500.30.01/2 = 75 mJ/m2
similarly to low-angle GB, γstr ~ δ (proportional to density of dislocations) for large D but
following a logarithmic dependence and saturate as D decreases
the limit to dislocation-based structures is at δ ~ 0.25  D = 4b  cores start to overlap
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Incoherent interphase boundaries
very different (incompatible) structure of the two phases or large lattice mismatch (δ ≥ 0.25)
prevents good matching across the interface  incoherent interface with disordered structure,
similar to random large-angle GB
large interfacial energy largely dominated by the structural contribution:
 incoh   chem   str  500  1000 mJ/m 2
table in Howe, Interfaces in Materials
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Shape of precipitates: Dependence on interfacial energy
let’s consider a strain-free precipitate of β phase in an  phase matrix
the interface around a precipitate is, in general, not the same over the
entire surface - precipitates possess a mixture of interface types along
their surface


minimum free energy of this system corresponds to the orientation
N
relationship and shape optimized to give the lowest tot
remember our discussion
   Ai  i  min of the equilibrium shape
coherent precipitates
small precipitates can form low-energy coherent
interfaces on all sides if  and β phases have the
same crystal structure and similar lattice parameter
i 1
of crystallites and Wulff
construction
Examples:
precipitation of fcc Co in Cu matrix, fcc Ag in Al matrix
fully-coherent precipitates are called Guinier-Preston zones
3D reconstruction of GP zones in Al-2.7
at.% Ag alloy (TEM and atom probe tips)
Marquis, Bartelt, Leonard,
Microsc. Microanal. 12, 1724 CD, 2006
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Shape of precipitates: Dependence on interfacial energy
partially coherent precipitates
when  and β phases have different crystal structures, orientation relationship leading to lowenergy coherent or semi-coherent interface may be found only for one habitat plane
other planes will be incoherent and will have higher interfacial energies
the equilibrium shape of the precipitate can then be determined
similarly to the equilibrium shape of crystallites (γ-plot and Wulff
construction)  large coherent facets terminated by incoherent edges
 coh
i
Examples:
hcp Ti in bcc Ti (slowly cooled two-phase Ti alloys)
tetragonal ’ phase precipitates in Al-Cu
hcp γ’ precipitates in Al-Ag
orientation relationship:
111 //0001 and
1 10

// 112 0
Moore and Howe,
Acta Materialia 48, 4083, 2000

γ’ precipitate in
at.% Ag alloy
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Zhigilei
Shape of precipitates: Dependence on interfacial energy
partially coherent precipitates - Widmanstätten pattern
cubic symmetry of the matrix  many possible orientations for the precipitate plates
Al–4 at.% Ag alloy
Widmanstätten pattern in iron meteorites: precipitation and growth of Ni-poor kamacite (bcc)
plates in the taenite (fcc) crystals  proceeds by diffusion of Ni at 450-700°C, and take place
during very slow cooling that takes several million years  the presence of large-scale
Widmanstätten patterns proves extraterrestrial origin of the material
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Shape of precipitates: Dependence on interfacial energy
incoherent precipitates
very different crystal structures or random orientation  absence of coherent or semi-coherent
interfaces  γ-plot and Wulff construction predict roughly spherical shapes of precipitates
some cusps on γ-plot may appear for certain crystallographic planes of the precipitate  faceting
that does not reflect the existence of coherent and semi-coherent interfaces
Examples of incoherent precipitates in Al: CuAl2, Al6Mn, Al3Fe
precipitation on GB
heterogeneous nucleation at GB can give rise to precipitates that are incoherent on one side, and
semi-coherent on the other side
shapes of precipitates are defined by minimization of the
interfacial energy and balance of interfacial tensions at
junctions of the interfaces and GB
 precipitate and GB triple point of -β Cu-In alloy
interfaces A and B are incoherent, C is semicoherent
Cu-In alloy
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Shape of precipitates: Effect of misfit strain
coherent precipitates
The effects of elastic interactions between the matrix and the precipitate can be as important as
for the interfacial energy. The two effects can compete: this is one reason for changes during
growth, such as the loss of coherency.
coherency strain should be accounted
for in minimization of the free energy:
 A
N
i 1
i i
 Gs  min
elastic strain energy

a  a
a
the elastic energy associated with the dilatational strains is of order 2 V, where V is the volume
of precipitate
for isotropic matrix and precipitate, the elastic energy is independent of shape: ∆Gs = 4G2V
effect of difference in elastic properties:
Precipitate stiffer than matrix: minimum elastic energy occurs for a sphere
Precipitate more compliant than matrix: minimum elastic energy occurs for a disc
Anisotropic matrix: most cubic metals are more compliant along <100> and harder along <111>
 elastic energy
considerations
discs
to {100}
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of Virginia,
MSE 6020: favor
Defects
and parallel
Microstructure
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Shape of precipitates: Strain energy vs. interfacial energy
competition between elastic energy and interfacial energy can result is a sequence of
precipitation reactions  appearance of successively more stable precipitates, each of which
has a larger nucleation barrier
Example: in Al alloys with 5% Cu (maximum solid solubility of Cu in Al at Te
The sequence is 0  1 + GP-zones  2 + “ 3 + ’ 4 + 
n - fcc aluminum; nth subscript denotes each equilibrium
GP zones - mono-atomic layers of Cu on (001)Al
“ - thin discs, fully coherent with matrix
’ - disc-shaped, semi-coherent on (001)’ bct
 - incoherent interface, ~spherical, complex body-centered tetragonal (bct)
The precipitate with the smallest nucleation barrier (generally) appears first. Small nucleation
barriers are associated with coherent interfaces (small interfacial energy) and similar lattices
(small elastic energies from misfit).
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Loss of coherency
competition of volumetric elastic strain energy and interfacial energy  precipitate may start as
fully coherent but nucleate interfacial dislocations once it reaches a critical size
Assuming that elastic strain energy is significant for the fully coherent precipitate but not for
incoherent or semicoherent ones, the free energies of crystals with coherent and non-coherent
precipitates can be written as
G
4
Gcoherent  Gelastic  Ginterface  4G 2  r 3   chem  4r 2
3
Gcoherent
Gnoncoherent  Gelastic  Ginterface  0   chem   str  4r 2
Gnoncoherent
rcr
r
Gcoherent  Gnoncoherent
for semicoherent interfaces with large D:
rcr 
3 st
4G 2
 st ~ 
at r > rcr, dislocations can be nucleated  the character of the
interface will change  coherency will be lost
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
rcr ~
1
