interfaces gb

8/18/2019 Interfaces GB

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University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Interfaces: Grain boundaries and interphase interfaces

(not tested )

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


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Grain boundaries

2 special cases can be distinguished:

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 andthe orientation of the boundary plane. 5 independent variables (degrees of freedom) are neededto 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.

pure tilt boundary - axis of rotation is parallel to the plane of the boundary

pure twist boundary - axis of rotation is perpendicular to the plane of the boundary

axis of tilt boundary

axis of tiltsymmetry plane

twist axis


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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

θ≈

θ=

bb D

)2/sin(2

D - dislocation spacing

θ - misorientation angle

22sin

b D =

θ

for small θ

y

x

recall our discussion of dislocation walls

h

low-angle symmetrical tilt boundary in asimple cubic lattice


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In asymmetric tilt boundary, the second set ofdislocations appears so that the boundary planemoves off the plane of reflectional symmetry

low-angle asymmetric tilt boundary in a simplecubic lattice

φ - is the angle of inclination of the boundary plane withrespect to the symmetric orientation

φθ=⊥

cos

b D

φθ=

−

−sin

||

b D


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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 perfectlyto the adjoining crystals, with the distorted regions

localized along the dislocation cores

low-angle twist boundary in a simple cubic latticeatoms 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

TEM image of a small angle

tilt boundary in Si

for small θ, the distance between dislocations is large and the

energy of the grain boundary, γGB, is proportional to thedislocation density:

θγ ~1

~ D

GB

as θ increases, the strain fields of dislocationsincreasingly cancel out and γGB tend to saturate

xxσ

when θ approaches ~15º, core regions of thedislocations start to overlap and the description of

GB in terms of dislocation wall is no longer useful

GBγ

θ

low-angle

random high-angle GB

10-15º


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Energy of grain boundaries

energy of random high angle GB:SV

GB γ≈γ 3

1examples: GBγ

SV

γ

Pt

Ag

Au

Cu

1140

14101670

2340 660

375

378625

][mJ/m2

energy of symmetrical tilt boundary in Al

(open disordered structure)

there are specific combinations of GBmisorientations and boundary planesthat 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

For a given misorientation, the energy of GBwill depend on the orientation of the GB plane

In general, GB energy is a function of at least5 parameters needed to describe the boundary

Twin boundary - special case of low angle,high symmetry grain boundary. Mostcommonly, twinning corresponds to mirrorsymmetry 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

twin boundary


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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 lowenergies (or large areas on low-energy planes + small areas of connecting high-energy planes)

faceting : even though the total GBarea increases, the energy decreases

somewhat similar todislocations adopting low-

energy configurations inPeierls 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) forcoherent twins and is smaller for other low-energy GBs - lookcurved in a microscope.

metal carbide precipitation on GBs (1),incoherent twin boundaries (2) & coherenttwin 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 twocrystals. 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 tolow-energy GBs


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Special high-angle grain boundaries: Coincidence site lattice

CSL is characterized by Σ that is defined as

Σ = volume ratio of the unit cell of the CSL to that of the original crystal latticeΣ = 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)

tilt boundary

(GD plane ⊥ paper)

Σ5, 36.95, 36.9°° in cubic lattice in cubic lattice twist boundary

(GD plane || paper)

GB that contains a high density of lattice pointsin CSL is expected to have low energy becauseof 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


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Energy of high-angle grain boundaries

deviations from the ideal CSLorientation may be accommodated by local atomic relaxation or theinclusion of dislocations into the boundary

experimentatomistic modeling

(a, b) and (c,d) symmetric tilt boundaries

low Σ boundaries tend to havelower energies than average

the correlation with Σ is not simple- there is no monotonous energydecrease with increasing Σ


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Energy of high-angle grain boundaries

• 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

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

disclination and disclination-structural unit modelsLi, Surf. Sci. 31,12, 1972Gertsman 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 differentfrom the bulk of the grains

no universal theory exists to describe high-angle GBs


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Interphase boundaries

coherent (commensurate) interface: two crystals match perfectly atthe interface plane (small lattice mismatch can be accommodated byelastic strain in the adjacent crystals)

semicoherent (discommensurate) interface: lattice mismatch isaccommodated by periodic array of misfit dislocations

incoherent (incommensurate) interface: disordered atomic structureof the interface

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

semicoherent

incoherent

1

12

a

aa −=δlattice misfit at the interface:

even in the case of perfect atomic matching, there is always a chemical contribution to the interface energy


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strain-free coherent (commensurate) interfaces:

• two crystals match perfectly at the interface plane

• interfacial plane has the same atomic configuration in both phases

same crystal structure

different crystal structure

Coherent interphase boundaries

Example: interface between α and κ phases in Cu-Si

(111) plane of α phasematches almost perfectly(0001) plane in κ phase

hcp κ phasefcc α phase

The indices of the planes comprising the boundary do not have to be thesame in each phase but orientation relationship between the two phasesshould be satisfied. This relationship is specified in terms of a pair of parallel planes and directions, i.e., {hkl }α//{hkl }β and α//β

orientation relationship:

{ } { }κα

κα 0211//101and0001//111


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This coherency strain reduces the interfacial energy at the expense of increasing energy of thetwo 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)

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 lowerhalf uniformly compressed, the crystals match perfectly.

12 aa >

01

12 >−

=δa

aa

Smith and Shiflet, Mater. Sci. Eng. 86, 67, 1987

While the structure of the interface is perfect, the interfacial energy is due to the bonding betweenatoms from different phase (has only chemical contribution): 2

mJ/m2001−≈γ=γ chemcoh


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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 withlocalized strain due to an array of dislocations that do not create long-range strain fields

δ and/or d are toolarge to maintaincoherent interface

Semicoherent interphase boundaries

dislocation spacing in 1D:

d

δ≈

δ=

ba D 2

2

21 aab +

= - Burgers vector of misfit dislocations

1

12

a

aa −=δ

In two dimensions, a network involving more than one Burgersvector may be required to accommodate the misfit

1

11

δ

=b

D

2

22

δ=

b D


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Semicoherent interphase boundaries

coherency strain is partially relieved by misfit dislocations,with residual compressive strain present in the film

ab

a D D 44.53.621 =δ>≈=

13.0=−

=δCu

Cu Ag

a

aa 1

2

1 [110]21

[110]2

b

b−

=

=

r

r

2ab =

A63.3=Cua

A09.4= Ag a

Example: lattice-mismatched Ag film - Cu-substrate interface

Wu, Thomas, Lin, Zhigilei, Appl. Phys. A 104, 781-792, 2011.

the limit to dislocation-based structures is at δ ~ 0.25 ⇒ D = 4b ⇒ cores start to overlap

γstr can be estimated similarly to low-angle GBs, by dividing the energy per unit length of thedislocations, 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

Energies of semicoherent interfaces have both chemical and structural(distortions due to the misfit dislocations) contributions:

2mJ/m500200 −≈γ+γ=γ str chem semicoh

similarly to low-angle GB, γstr ~ δ (proportional to density of dislocations) for large D butfollowing a logarithmic dependence and saturate as D decreases


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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

2mJ/m1000500 −≈γ+γ=γ str chemincoh

large interfacial energy largely dominated by the structural contribution:

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 α phase matrix

the interface around a precipitate is, in general, not the same over theentire surface - precipitates possess a mixture of interface types alongtheir surface

min1

→γ=γ ∑=

N

i

ii

tot Aremember our discussionof the equilibrium shapeof crystallites and Wulff

construction

minimum free energy of this system corresponds to the orientationrelationship and shape optimized to give the lowest

α

β

Examples:

precipitation of fcc Co in Cu matrix, fcc Ag in Al matrix

fully-coherent precipitates are called Guinier-Preston zones

coherent precipitates

small precipitates can form low-energy coherent

interfaces on all sides if α and β phases have thesame crystal structure and similar lattice parameter

3D reconstruction of GP zones in Al-2.7at.% Ag alloy (TEM and atom probe tips)Marquis, Bartelt, Leonard,Microsc. Microanal. 12, 1724 CD, 2006


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Examples:hcp Ti in bcc Ti (slowly cooled two-phase Ti alloys)

tetragonal θ’ phase precipitates in Al-Cu

hcp γ’ precipitates in Al-Ag

partially coherent precipitates

when α and β phases have different crystal structures, orientation relationship leading to low-energy 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 determinedsimilarly to the equilibrium shape of crystallites (γ-plot and Wulff construction) ⇒ large coherent facets terminated by incoherent edges

cohγ iγ

orientation relationship:

{ } { }κακα

0211//101and0001//111

Moore and Howe,

Acta Materialia 48, 4083, 2000 γ’ precipitate inAl–4.2 at.% Ag alloy


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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 placeduring very slow cooling that takes several million years ⇒ the presence of large-scaleWidmanstätten patterns proves extraterrestrial origin of the material


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incoherent precipitates

very different crystal structures or random orientation ⇒ absence of coherent or semi-coherentinterfaces ⇒ γ-plot and Wulff construction predict roughly spherical shapes of precipitates

Examples of incoherent precipitates in Al: CuAl2, Al6Mn, Al3Fe

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

heterogeneous nucleation at GB can give rise to precipitates that are incoherent on one side, andsemi-coherent on the other side

precipitation on GB

shapes of precipitates are defined by minimization of theinterfacial energy and balance of interfacial tensions at junctions of the interfaces and GB

Cu-In alloy

α precipitate and GB triple point of α-β Cu-In alloy

interfaces A and B are incoherent, C is semicoherent


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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 asfor the interfacial energy. The two effects can compete: this is one reason for changes duringgrowth, such as the loss of coherency.

min1 →Δ+γ∑= s N

iii G A

coherency strain should be accounted

for in minimization of the free energy:

elastic strain energy

α

αβ −=δ

a

aa

the elastic energy associated with the dilatational strains is of order δ 2 V , where V is the volumeof precipitate

for isotropic matrix and precipitate, the elastic energy is independent of shape: ∆G s = 4G V

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 and harder along ⇒ elastic energy considerations favor discs parallel to {100}


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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 whichhas a larger nucleation barrier

Example: in Al alloys with 5% Cu (maximum solid solubility of Cu in Al at T e

The sequence is α0 → α1 + GP-zones → α2 + θ“→ α3 + θ’→ α4 + θ

αn - fcc aluminum; nth subscript denotes each equilibrium

GP zones - mono-atomic layers of Cu on (001)Al

θ“ - thin discs, fully coherent with matrixθ’ - disc-shaped, semi-coherent on (001)θ’ bct

θ - 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 forincoherent or semicoherent ones, the free energies of crystals with coherent and non-coherent precipitates can be written as

232interface 4

344 r r GGGG chemelasticcoherent π×γ+π×δ=Δ+Δ=Δ

( ) 2interface 40 r GGG str chemelasticcoherent non π×γ+γ+=Δ+Δ=Δ −

at r > r cr , dislocations can be nucleated ⇒ the character of theinterface will change ⇒ coherency will be lost

r cr r

coherent GΔ

coherent nonG −Δ

GΔ

coherent noncoherent GG −Δ=Δ 24

3

δ

γ=

Gr st cr

δγ ~ st for semicoherent interfaces with large D:

δ

1~cr r

interfaces gb

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