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Microstructure-Properties: II Lecture 6: Elastic Effects, InterfacesObjective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Microstructure-Properties: II
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Materials Tetrahedron
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Objective
The objective of this lecture is to show how important (a) elastic effects are in controlling precipitation, and (b) the variety of interface structures that occur, and their importance in precipitation.
More specifically, this lecture examines the role of the interface (coherent vs. incoherent) in precipitate morphology and growth.
The main concepts are (a) the misfit strain between two lattices at an interface that determines a dislocation density, and (b) the (bulk) misfit parameter that determines the elastic energy associated with a precipitate.
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
References
Phase transformations in metals and alloys, D.A. Porter, & K.E. Easterling, Chapman & Hall. Chapter 3 is most relevant to this lecture.
Interfaces in Materials (1997), James M. Howe, Wiley Interscience.
Materials Principles & Practice, Butterworth Heinemann, Edited by C. Newey & G. Weaver.
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Notation
b Burgers vector of interface dislocations
dB interplanar spacing of {hkl} in phase B
dA interplanar spacing of {hkl} in phase A
aB lattice parameter in phase B
aA lattice parameter in phase A
d misfit parameter
q stable precipitate in Al-Cu system
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Elasticity and Interface Structure
Why consider elasticity and interface structure together? Both aspects exert a strong influence over not just the shape of the new phase that appears during a phase transformation but also which phase appears in the first place.
The fact that a new phase has a different composition means that its lattice has a different set of repeat distances (lattice parameters), even if the crystal structure is the same as the parent phase.
These differences mean (a) different elastic moduli between parent and product phase and (b) a mismatch in atomic positions at a parent-product interface.
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Basic results
For small precipitates (< ~5nm), precipitates are usually spherical with coherent interfaces in order to minimize surface energy.
For intermediate sizes, precipitates are often plates or needles in order to minimize surface energy in situations where one plane or direction is atomically similar between parent and product phases.
Large precipitates (> 1µm) are often spherical with incoherent interfaces in order to minimize volumetric free energy.
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Coherence at interfaces
Coherent/semi-coherent/incoherent interfaces: these terms are based on the degree of atomic matching across the interface.
Coherent interface means an interface in which the atoms match up on a 1-to-1 basis (even if some elastic strain is present).
Incoherent interface means an interface in which the atomic structure is disordered.
Semi-coherent interface means an interface in which the atoms match up, but only on a local basis, with defects (dislocations) in between.
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Homophase vs. Heterophase
There is a useful comparison that can be made between grain boundaries (homophase) and interphase boundaries (heterophase).
Structure              G.B.                         Interface
dislocations low angle g.b. semi-
coherent
disordered high angle g.b. incoherent
Remember: for a grain boundary to exist, there must be a difference in the lattice position (rotationally) between the two grains. An interface can exist even when the lattices are the same structure and in the same (rotational) position because of the chemical difference.
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
LAGB to HAGB Transition
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Read-Shockley model
Start with a symmetric tilt boundary composed of a wall of infinitely straight, parallel edge dislocations (e.g. based on a 100, 111 or 110 rotation axis with the planes symmetrically disposed).
Dislocation density (L-1) given by:
1/D = 2sin(q/2)/b q/b for small angles.
D
b
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Read-Shockley, contd.
For an infinite array of edge dislocations the long-range stress field depends on the spacing. Therefore given the dislocation density and the core energy of the dislocations, the energy of the wall (boundary) is estimated (r0 sets the core energy of the dislocation):
ggb = E0 q(A0 - lnq), where
E0 = µb/4π(1-n); A0 = 1 + ln(b/2πr0)
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
LAGB experimental results
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
vs.
Yang et al.
[001]
[101]
[111]
0.30
0.26
0.23
0.33
Chart1
1.5
0.0155991453
0.0155991453
1.5
3.5
0.0115902258
0.0115902258
3.5
4.5
0.0037749172
0.0037749172
4.5
5.5
0.0117473401
0.0117473401
5.5
6.5
0.0089953692
0.0089953692
6.5
7.5
0.0118427193
0.0118427193
7.5
8.5
0.0190700463
0.0190700463
8.5
9.5
0.0123558353
0.0123558353
9.5
10.5
0.0062383224
0.0062383224
10.5
11.5
0.0114309521
0.0114309521
11.5
12.5
0.0105316982
0.0105316982
12.5
13.5
0.0146173413
0.0146173413
13.5
0.0109658561
0.0109658561
15
0
0
0
0
0
0
0
0
0
0
0
0
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
High angle g.b. structure
Bubble rafts provide a useful example.
Disordered structure results in a high energy.
Low angle boundary
with dislocation structure
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Energy of High Angle Boundaries
No universal theory exists to describe the energy of HAGBs.
Abundant experimental evidence for special boundaries at (a small number) of certain orientations.
Good fit for special boundaries based on good fit of a certain fraction of the atoms at the interface.
Mathematically, special orientations are easier to characterize in terms of coincidence of lattice points between the two lattices (by imagining that they interpenetrate) leading to the Coincident Site Lattice (CSL).
Each special point (in misorientation space) expected to have a cusp in energy, similar to zero-boundary case but with non-zero energy at the bottom of the cusp.
Special boundaries defined by a “sigma number” which is the reciprocal of the fraction of lattice points (not atoms!) that coincide. For cubic materials, this number is always odd, so we have S1, S3, S5, S7….
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Exptl. vs. Calculated G.B. energies for 99.89% Al
Hasson & Goux
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Dislocation models of HAGBs
Boundaries near CSL points expected to exhibit dislocation networks, which is observed.
<100> twist boundaries
showing dislocation structure
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Heterophase boundaries:
coherent interfaces
See figures 3.32, 3.33 in P&E.
The indices of the planes comprising the boundary do not have to be the same in each phase.
In general, a coherent interface is based on an orientation relationship. This relationship is specified crystallographically in terms of a pair of planes and directions as in {hkl}A//{hkl}B with <uvw>A//<uvw>B.
For example, the hcp k phase precipitate in fcc copper has an almost perfect match through the orientation relationship, {111}fcc//(0001)hcp and <110>fcc//<11-20>hcp. The interfacial energy is estimated to be as low as 1 mJ.m-2.
Even in the case of perfect atomic matching, there is always a chemical contribution to the interface energy.
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Strained interfaces
Unlike the case of grain boundaries, there is an important elastic aspect of coherent interfaces.
Small differences in lattice parameter are accommodated by elastic strain.
Given lattice parameters specified as interplanar spacings, dA and dB, the misfit parameter, d is given by the following simple formula:
d = (dB - dA)/ dA
See figs. 3.34, 3.35.
We shall see later, that the misfit that can be accommodated by elastic strain is limited.
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Semi-coherent interfaces
The logical next step in typing interfaces is to note that too-large misfit strains can be accommodated (i.e. lower energy interfaces constructed) by replacing uniform elastic strains with dislocations (which localizes the strain into the dislocation cores), fig. 3.35.
The dislocation spacing in 1D is given by:
D = dB/ d .
D = d/ d.
The Burgers vector, b, of the interface dislocations is given by
b = (dA+dB)/2
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Semi-coherent interfaces, contd.
Just as for low angle grain boundaries, the interface energy is proportional to the dislocation density at small misfits and then following a logarithmic dependence at larger misfits.
In two dimensions, a network involving more than one Burgers vector may be required to accommodate the misfit, see figure 3.36 in Porter & Easterling.
The limit to dislocation-based structures is at
d ~ 0.25, corresponding to one dislocation every four plane spacings (where the cores start to overlap).
Interface energies are in the 200-500 mJ.m-2 range; these can be estimated in the usual manner where we divide the energy per unit length of the dislocations, Gb2/2, by the dislocation spacing, d/ d. The result is, Gb2d/2d. So for a shear modulus of, say 50 GPa, a spacing of 4b and a misfit of 0.25, the interface energy is 50*0.3/4/8 ~ 470 mJ.m-3.
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Complex semi-coherent interfaces
It can often happen that an orientation relationship exists despite the lack of an exact match.
Such is the case for the relationship between bcc and fcc iron (ferrite and austenite).
Note limited atomic match for the NS relationship
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Orientation relationships in iron
There are two well-known orientation relationships for fcc-bcc iron.
The Nishiyama-Wasserman (NW) relationship is specified as {110}bcc/{111}fcc, <001>bcc//<101>fcc.
The NS relationship only gives good atomic fit in 8% of the boundary area.
The Kurdjumov-Sachs (KS) relationship is specified as {110}bcc/{111}fcc, <111>bcc//<101>fcc.
These two differ by only a 5.6° rotation in the interface plane.
Better atomic matching is possible for irrational planes used.
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Orientations from KS OR
Based on a particular orientation relationship (OR), the orientations of new grains of the product phase can be predicted, as derived from a product phase.
Illustrated for the Kurdjumov-Sachs (KS) relationship for iron, with the {001}<100> starting orientation for the fcc (austenite) phase.
The different new orientations are called variants.
{001} pole figure, showing {001} poles in the eight variant positions of the ferrite phase for the KS OR, starting from a single austenite crystal in (001)[100] position.
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Incoherent Interfaces
Not surprisingly, incoherent interfaces have a disordered structure similar to high angle grain boundaries.
Their energies range up to 1 J.m-2.
Little is known about the detailed structure of such interfaces.
Large differences in crystal structure and lattice parameter between parent and product phases tend to mean that the interface must be incoherent.
Possibilities for partially coherent interfaces exist even under the latter circumstance, but better tools are need for prediction of interface structure and energy (current research topic, e.g. W. Reynolds).
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Interfaces in precipitates
In order to present examples of real systems, it is important to keep in mind that the interface around a precipitate is not, in general, the same over the entire surface.
Analogy with grain boundaries: the boundary of an island grain (fully enclosed within another grain) varies from pure twist at opposite poles, to pure tilt around its equator.
Thus, some precipitates possess a mixture of interface types around their perimeter.
Misorientation axis
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Fully coherent precipitates
Precipitation of Co from Cu is another example.
Guinier-Preston zones in the early stages of precipitation in Al alloys are another example.
See fig. 3.39 in P&E for Ag-rich zones in Al-4Ag.
In all cases, the crystal structure is the same in parent and product; also the lattice parameters are similar.
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Partially coherent precipitates
When only part of the surface of a precipitate can be coherent, it is said to be partially coherent.
Typically, one plane is coherent or semi-coherent.
As fig. 3.40 shows, the shape of the precipitate can be determined by the Wulff shape through the inverse ratio of the interfacial energies. Large coherent facets are terminated by incoherent edges.
Caution! An anisotropic shape can also be determined by either growth anisotropy or elastic anisotropy.
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Widmanstätten morphology
Widmanstätten’s name is associated with platy precipitates that possess a definite crystallographic relationship with their parent phase.
Examples:
- g’ precipitates in Al-Ag (see fig. 3.42)
- hcp Ti in bcc Ti (two-phase Ti alloys, slow cooled)
- q’ precipitates in Al-Cu
The latter example is based on the orientation relationship (001)q’//{001}Al, [100]q’//<100>Al. See fig. 3.41 for a diagram of the tetragonal structure of q’ whose a-b plane, i.e. (001), aligns with the (100) plane of the parent Al.
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Incoherent precipitates
Al alloys provide many examples of incoherent precipitates that lack orientation relationships.
CuAl2 (q) in Al - fig. 3.44
Al6Mn in Al
Al3Fe in Al
Note that heterogeneous nucleation at grain boundaries can give rise to precipitates that are incoherent on one side, and semi-coherent on the other side. This leads to significant differences in growth rate, fig. 3.45.
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Elastic Effects
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.
Elastic effects can influence precipitate shape.
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Misfit
Imagine that a certain volume of the parent phase (matrix) is removed and replaced by a different volume of product phase (precipitate). The difference in volume leads to a dilatational strain which is positive or negative, depending on the sign of the volume change.
In the case of identical crystal structures and a coherent interface, the parent and product have equal and opposite forces at the interface, see fig. 3.47c.
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Misfit definitions
Given lattice parameters specified as aA and aB, the misfit parameter, d is given by the following simple formula:
d = (aB - aA)/ aA
Note the similarity to the definition of misfit for coherent and semi-coherent interfaces.
In the case of dilatational/hydrostatic strains, one can define a constrained misfit or in situ misfit based on the strained precipitate lattice parameter, a’B :
e = (a’B - aA)/ aA
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Elastic strain energy
The constrained misfit and the unconstrained misfit are related to each other. For the simplest case of identical moduli in parent and product,
e = 2d/3
For typical variations in moduli, the range of values observed is d/2 < e < d.
The elastic energy associated with the dilatational strains is of order d2 V, where V is the volume. For the simplest case of isotropic matrix and precipitate, the elastic energy is independent of shape:
Gs = 4Gd2V
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Strain energy: anisotropy
The effect of modulus differences is interesting and asymmetric:
Precipitate stiffer than matrix: minimum elastic energy occurs for a sphere.
Precipitate more compliant than matrix: minimum elastic energy occurs for a disc (oblate ellipsoid).
Note that differences in elastic modulus can be synergistic or antagonistic to effects of interface structure.
Anisotropic matrix: most cubic metals are more compliant along <100>, hence elastic energy considerations favor discs perpendicular to <100>.
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Elasticity vs coherency
The competition between elastic energy and interfacial energy is illustrated by reference to specific examples in Al alloys.
Observation: a sequence of precipitation reactions is observed in Al-Cu alloys (containing up to, say 5%Cu, i.e. the maximum solid solubility of Cu in Al, at the eutectic temperature).
The sequence can be explained as the appearance of successively more stable precipitates, each of which has a larger nucleation barrier.
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Al-Cu precipitation sequence
The sequence is:
The phase are:
q“ - thin discs, fully coherent with matrix
q’ - disc-shaped, semi-coherent on (001)q’ bct.
q - incoherent interface, ~spherical, complex body-centered tetragonal (bct).
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Al-Cu driving forces
Each precipitate has a different free energy curve w.r.t composition. Exception is the GP zone, which may be regarded as continuous with the alloy (leading to the possibility of spinodal decomposition, discussed later).
P&E fig. 5.27 illustrates the sequence of successively greater free energy decreases and also successively greater G*.
P&E fig. 5.28 illustrates the point that the nucleation barriers are much smaller for each individual nucleation step when the next precipitate nucleates heterogeneously on the previous structure.
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Al-Cu ppt structures
GP zone structure
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Nucleation sites, reversion
The nucleation sites vary depending on circumstances.
q“ most likely nucleates on GP zones by adding additional layers of Cu atoms.
Similarly, q’ nucleates on q“ by in-situ transformation.
However, q’ can also nucleate on dislocations, see P&E fig. 5.31a.
The full sequence is only observable for annealing temperatures below the GP solvus. Any of the intermediate precipitates can be dissolved, reverted, by increasing the temperature above the relevant solvus, fig. 5.32.
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Coherency loss
The growth of the penultimate precipitate, q’, illustrates an important point about the loss of coherency that commonly occurs during growth.
A precipitate may start out fully coherent but nucleate interfacial dislocations once it reaches a critical size.
Illustration: large q’ ppts commonly have dislocations, see P&E fig. 5.30c.
Why? Again, a competition exists between volumetric elastic energy, and interfacial energy.
Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt
Coherency loss, analysis
The assumptions of the simple analysis (P&E section 3.4.4) are: elastic strain energy is significant for the fully coherent case, not for the non-coherent case. Also, the energy of non-coherent interface is significantly larger than that of the coherent interface. Thus:
Gcoherent = Gelastic + Ginterface
= 4µd2 * 4πr3/3…