MicrostructureProperties: II Elastic Effects, Interfaces

1
Microstructure-Properties IIElastic Effects,
Interfaces

• 27-302
• Lecture 7
• Fall, 2002
• Prof. A. D. Rollett

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Materials Tetrahedron
Processing
Performance
Properties
Microstructure
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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 the misfit strain between
  two lattices at an interface that determines a
  dislocation density, and the (bulk) misfit
  parameter that determines the elastic energy
  associated with a precipitate.

4
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.

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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.

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Basic results

• For small precipitates (lt 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 (gt 1µm) are often spherical
  with incoherent interfaces in order to minimize
  volumetric free energy.

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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.

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Homophase vs. Heterophase

• There is a useful comparison that can be made
  between grain boundaries (homophase) and
  interphase boundaries (heterophase).Structure    
            G.B.                         Interface
• atoms no boundary coherentmatch (or, S3
  coherent twin in fcc) 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.

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LAGB to HAGB Transition

• LAGB steep risewith angle.HAGB plateau

Disordered Structure
Dislocation Structure
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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 by1/D
  2sin(q/2)/b ? q/b for small angles.

D
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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), whereE0 µb/4p(1-n) A0 1
  ln(b/2pr0)

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LAGB experimental results

• Experimental results on copper.

Gjostein Rhines, Acta metall. 7, 319 (1959)
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Low Angle Grain Boundary Energy Yang et al.
Scripta Materiala (2001) 44 2735-2740
High
117
105
113
205
215
335
203
Low
8411
323
727
Measurements of low angle grainboundary energy
in 99.98Al
? vs.
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High angle g.b. structure

• High angle boundaries have a disordered
  structure.
• Bubble rafts provide a useful example.
• Disordered structure results in a high energy.

Low angle boundarywith dislocation structure
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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.

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Exptl. vs. Calculated G.B. energies for 99.89 Al
lt100gtTilts
Twin
lt110gtTilts
Hasson Goux
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Dislocation models of HAGBs

• Boundaries near CSL points expected to exhibit
  dislocation networks, which is observed.

lt100gt twist boundariesin gold.
S5 twist Boundary in gold, showing dislocation
structure
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Heterophase boundaries coherent interfaces

• Coherent interfaces have perfect atomic matching
  at the boundary.
• See figures 3.32, 3.33 in PE.
• 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 hklA//hklB
  with ltuvwgtA//ltuvwgtB.
• For example, the hcp k phase precipitate in fcc
  copper has an almost perfect match through the
  orientation relationship, 111fcc//(0001)hcp and
  lt110gtfcc//lt11-20gthcp. 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.

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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.

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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 .For small enough misfits, this can be
  written as D d/ d.
• The Burgers vector, d, of the interface
  dislocations is given by d (dAdB)/2

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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.
• 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.

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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
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Orientation relationships in iron

• There are two well-known orientation
  relationships for fcc-bcc iron.
• The Nishiyama-Wasserman (NW) relationship is
  specified as 110bcc/111fcc,
  lt001gtbcc//lt101gtfcc.
• The NS relationship only gives good atomic fit in
  8 of the boundary area.
• The Kurdjumov-Sachs (KS) relationship is
  specified as 110bcc/111fcc,
  lt111gtbcc//lt101gtfcc.
• These two differ by only a 5.6 rotation in the
  interface plane.
• Better atomic matching is possible for irrational
  planes used.

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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 001lt100gt
  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.
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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).

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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
Tilt boundary on the equator
Twist boundaries on the poles
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Fully coherent precipitates

• One example of the Cu-Si system has been given.
• 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 PE 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.

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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.

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Widmanstätten morphology

• Widmanstättens name is associated with platy
  precipitates that possess a definite
  crystallographic relationship with their parent
  phase.
• Examples - ferrite in austenite (iron-rich
  meteors!) - 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//001Al, 100q//lt100gtAl.
  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.

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Incoherent precipitates

• Al alloys provide many examples of incoherent
  precipitates that lack orientation relationships.
• CuAl2 (q) in Al - fig. 3.44Al6Mn in AlAl3Fe 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.

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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.

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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.

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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, aB e (aB - aA)/ aA

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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 lt e lt 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 4µd2V - µ is
  the shear modulus.

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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 lt100gt, hence elastic energy
  considerations favor discs perpendicular to lt100gt.

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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 5Cu, 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.

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Al-Cu precipitation sequence

• The sequence is a0 ? a1 GP-zones ? a2 q?
  a3 q? a4 q
• The phase are an - fcc aluminum nth subscript
  denotes each equilibriumGP zones - mono-atomic
  layers of Cu on (001)Al 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).

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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).
• PE fig. 5.27 illustrates the sequence of
  successively greater free energy decreases and
  also successively greater ?G.
• PE 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.

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Al-Cu ppt structures
GP zone structure
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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 PE 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.

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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 PE fig. 5.30c.
• Why? Again, a competition exists between
  volumetric elastic energy, and interfacial energy.

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Coherency loss, analysis

• The assumptions of the simple analysis (PE
  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 4pr3/3
  4pr2gcoherent ?Gnon-coherent ?Gelastic
  ?Ginterface 0
  4pr2gnon-coherent
• At some size, the former becomes larger than the
  latter. Provided dislocations (or other defects)
  can be nucleated, the character of the interface
  will change, and coherency will be lost.

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Coherency loss, estimates

• It is possible to estimate the size, and type of
  precipitate for coherency loss.
• Given a difference in interfacial energy between
  coherent and non-coherent (which PE write as gst
  gnon-coherent - gcoherent), one can estimate a
  critical radius, rcrit. rcrit 3?g/4µd2.
• Fig. 3.53 illustrates the requirement for
  dislocation loops to be arranged on the perimeter
  of the precipitate (similar to a low angle grain
  boundary).
• Based on a constrained misfit, e, the minimum
  stress required to nucleate a dislocation (fig.
  3.54a) is of order 3µe and the minimum value of
  misfit to exceed the theoretical shear strength
  of a matrix is e 0.05. This implies that
  precipitates that have a small enough misfit will
  never lose coherency as they grow because they
  are unable to nucleate the required interfacial
  dislocations.

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Impact on Properties

• The most obvious impact of precipitation in
  metals is on mechanical strength.
• Precipitation was measured by hardness, long
  before the structure of the precipitates was
  known.
• Example age-hardening curves in Al-Cu alloys,
  PE fig. 5.37.

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Summary

• This lecture may be summarized by stating that
  both differences in elastic properties and
  interface structure exert a strong influence on
  precipitate morphology.
• Through their effect on free energy changes as a
  function of size, they also affect which
  precipitates actually nucleate under any given
  conditions.
• 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).