Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation....

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Microstructure-Properties: IIElastic Effects, Interfaces

27-302

Lecture 7

Fall, 2002

Prof. A. D. Rollett

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Materials Tetrahedron

Microstructure Properties

ProcessingPerformance

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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 (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.

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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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Notation

• D dislocation spacing in interface • 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• misfit parameter• constrained misfit parameter• V volume [of precipitate]• GP-zone, “, ’ metastable precipitates in Al-

Cu system• stable precipitate in Al-Cu system

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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 (< ~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.

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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, 3 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

Dislocation Structure

Disordered 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 by:

1/D = 2sin(/2)/b /b for small angles.

b

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):

gb = E0 ln, where

µb/4π(1-); A0 = 1 + ln(b/2πr0)

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LAGB experimental results

• Experimental results on copper.

[Gjostein & Rhines, Acta metall. 7, 319 (1959)]

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[001]

[101][111]

0.30

0.26

0.23

0.33

Low Angle Grain Boundary Energy Yang et al. High

Low

[335]

[323][727]

[203]

[205]

[105]

[215]

[117]

[113]

[8411]

Measurements of low angle grainboundary energy in 99.98%Al

Scripta Materiala (2001) 44: 2735-2740

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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 1, 3, 5, 7….

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Exptl. vs. Calculated G.B. energies for 99.89% Al

Hasson & Goux

<100>Tilts

<110>Tilts

Twin

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Dislocation models of HAGBs• Boundaries near CSL points expected to exhibit

dislocation networks, which is observed.

<100> twist boundariesin gold. 5 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 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 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.

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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, is given by the following simple formula:

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/ .

For small enough misfits, this can be written as:D = d/

• The Burgers vector, b, of the interface dislocations is given by

b = (dA+dB)/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, see figure 3.36 in Porter & Easterling.

• The limit to dislocation-based structures is at ~ 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/ . The result is, Gb2/2d. 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.

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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 {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.

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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 {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.

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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 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.

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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

• Widmanstätten’s name is associated with platy precipitates that possess a definite crystallographic relationship with their parent phase.

• Examples:- ferrite in austenite (iron-rich meteors!)- ’ precipitates in Al-Ag (see fig. 3.42)- hcp Ti in bcc Ti (two-phase Ti alloys, slow cooled)- ’ precipitates in Al-Cu

• The latter example is based on the orientation relationship (001)’//{001}Al, [100]’//<100>Al. See fig. 3.41 for a diagram of the tetragonal structure of ’ 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 (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, is given by the following simple formula:

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 :

a’B - 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,

• For typical variations in moduli, the range of values

observed is .• The elastic energy associated with the dilatational

strains is of order V, where V is the volume. For the simplest case of isotropic matrix and precipitate, the elastic energy is independent of shape:

∆Gs = 4GV - G 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 <100>, hence elastic energy considerations favor discs perpendicular to <100>.

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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 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.

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Al-Cu precipitation sequence

• The sequence is: 0 1 + GP-zones 2 + “ 3 + ’ 4 +

• The phase are: n - fcc aluminum; nth subscript denotes each equilibriumGP 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).

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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).

• 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.

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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.• “ most likely nucleates on GP zones by adding

additional layers of Cu atoms.• Similarly, ’ nucleates on “ by in-situ transformation.• However, ’ 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.

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Coherency loss

• The growth of the penultimate precipitate, ’, 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 ’ ppts commonly have dislocations, see P&E 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 (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

= 4µ2 * 4πr3/3 + 4πr2coherent

∆Gnon-coherent = ∆Gelastic + ∆Ginterface

= 0 + 4πr2non-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 P&E write as st = non-coherent - coherent), one can estimate a critical radius, rcrit.:

rcrit = 3∆/4G2.• 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, , the minimum stress required to nucleate a dislocation (fig. 3.54a) is of order 3G and the minimum value of misfit to exceed the theoretical shear strength of a matrix is = 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, P&E 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).

Objective Coherency Homo- vs Hetero- phase Grain Bounds. Strained Interfaces Orient. Relation....

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