1 Objective Coherency Homo vs Heterophase Grain Bounds

1 Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt Microstructure-Properties: II Elastic Effects, Interfaces 27 -302 Lecture 7 Fall, 2002 Prof. A. D. Rollett

2 Materials Tetrahedron Processing Performance Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt Microstructure Properties

3 Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • The objective of this lecture is to show 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.

4 Objective Coherency Homo- vs Heterophase 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.

5 Notation Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • • • D b d. B d. A a. B a. A dislocation spacing in interface Burgers vector of interface dislocations interplanar spacing of {hkl} in phase B interplanar spacing of {hkl} in phase A lattice parameter in phase B lattice parameter in phase A d misfit parameter e constrained misfit parameter V volume [of precipitate] GP-zone, q“, q’ metastable precipitates in Al-Cu system • q stable precipitate in Al-Cu system

6 Elasticity and Interface Structure Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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.

7 Basic results Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • For small precipitates (< ~5 nm), 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.

8 Coherence at interfaces Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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.

9 Homophase vs. Heterophase Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces • There is a useful comparison that can be made between grain boundaries (homophase) and interphase boundaries (heterophase). Structure G. B. Interface atoms no boundary coherent match (or, S 3 coherent twin in fcc) interface dislocations low angle g. b. semicoherent 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 Elasticity same structure and in the same (rotational) position because Al-Cu ppt of the chemical difference. Orient. Relation.

10 LAGB to HAGB Transition Objective Coherency • LAGB: steep rise with angle. HAGB: plateau Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt Dislocation Structure Disordered Structure

11 Read-Shockley model Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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 = 2 sin(q/2)/b q/b for small angles. b D

12 Read-Shockley, contd. Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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 (r 0 sets the core energy of the dislocation): ggb = E 0 q(A 0 - lnq), where E 0 = µb/4π(1 -n); A 0 = 1 + ln(b/2πr 0)

13 LAGB experimental results • Experimental results on copper. Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt [Gjostein & Rhines, Acta metall. 7, 319 (1959)]

14 [001] Low Angle Grain Boundary Energy High Yang et al. Objective [117] Coherency [105] Homo- vs Hetero[205] phase Grain Bounds. Strained Interfaces 0. 33 0. 30 0. 26 [113] 0. 23 [215] [335] [203] Low [8411] Orient. Relation. Elasticity Al-Cu ppt Scripta Materiala (2001) 44: 2735 -2740 [101] vs. [727] [323] [111] Measurements of low angle grain boundary energy in 99. 98%Al

15 High angle g. b. structure Objective Coherency Homo- vs Heterophase Grain Bounds. • High angle boundaries have a disordered structure. • Bubble rafts provide a useful example. • Disordered structure results in a high energy. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt Low angle boundary with dislocation structure

16 Energy of High Angle Boundaries Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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 nonzero 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 S 1, S 3, S 5, S 7….

17 Exptl. vs. Calculated G. B. energies for 99. 89% Al <100> Tilts Objective Coherency Homo- vs Heterophase Grain Bounds. Twin <110> Tilts Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt Hasson & Goux

18 Dislocation models of HAGBs • Boundaries near CSL points expected to exhibit dislocation networks, which is observed. Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt <100> twist boundaries in gold. S 5 twist Boundary in gold, showing dislocation structure

19 Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt 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 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 m. J. m-2. • Even in the case of perfect atomic matching, there is always a chemical contribution to the interface energy.

20 Strained interfaces Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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, d. A and d. B, the misfit parameter, d is given by the following simple formula: d = (d. B - d. A)/ d. A • See figs. 3. 34, 3. 35. • We shall see later, that the misfit that can be accommodated by elastic strain is limited.

21 Semi-coherent interfaces Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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 1 D is given by: D = d B/ d. For small enough misfits, this can be written as: D = d/ d. • The Burgers vector, b, of the interface dislocations is given by b = (d. A+d. B)/2

22 Semi-coherent interfaces, contd. Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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 m. J. m-2 range; these can be estimated in the usual manner where we divide the energy per unit length of the dislocations, Gb 2/2, by the dislocation spacing, d/ d. The result is, Gb 2 d/2 d. So for a shear modulus of, say 50 GPa, a spacing of 4 b and a misfit of 0. 25, the interface energy is 50*0. 3/4/8 ~ 470 m. J. m-3.

23 Complex semi-coherent interfaces Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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

24 Orientation relationships in iron Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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.

25 Orientations from KS OR Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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.

26 Incoherent Interfaces Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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).

27 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 Objective general, the same over the entire surface. Coherency • Analogy with grain boundaries: the boundary of an island grain (fully enclosed within another grain) varies from pure twist at Homo- vs Heteroopposite poles, to pure tilt around its equator. phase • Thus, some precipitates possess a mixture of interface types Grain around their perimeter. Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt Misorientation axis Tilt boundary on the equator Twist boundaries on the poles

28 Fully coherent precipitates Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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-4 Ag. • In all cases, the crystal structure is the same in parent and product; also the lattice parameters are similar.

29 Partially coherent precipitates Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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.

30 Widmanstätten morphology Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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!) - 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.

31 Incoherent precipitates Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • Al alloys provide many examples of incoherent precipitates that lack orientation relationships. • Cu. Al 2 (q) in Al - fig. 3. 44 Al 6 Mn in Al Al 3 Fe 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.

32 Elastic Effects Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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.

33 Misfit Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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. 47 c.

34 Misfit definitions Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • Given lattice parameters specified as a. A and a. B, the misfit parameter, d is given by the following simple formula: d = (a. B - a. A)/ a. A • 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 - a. A)/ a. A

35 Elastic strain energy Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • The constrained misfit and the unconstrained misfit are related to each other. For the simplest case of identical moduli in parent and product, e = 2 d/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 d 2 V, where V is the volume. For the simplest case of isotropic matrix and precipitate, the elastic energy is independent of shape: ∆Gs = 4 Gd 2 V - G is the shear modulus.

36 Strain energy: anisotropy Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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>.

37 Elasticity vs coherency Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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.

38 Al-Cu precipitation sequence Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • The sequence is: a 0 a 1 + GP-zones a 2 + q“ a 3 + q’ a 4 + q • The phase are: an - fcc aluminum; nth subscript denotes each equilibrium GP 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 bodycentered tetragonal (bct).

39 Al-Cu driving forces Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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.

40 Al-Cu ppt structures Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt GP zone structure

41 Nucleation sites, reversion Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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. 31 a. • 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.

42 Coherency loss Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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. 30 c. • Why? Again, a competition exists between volumetric elastic energy, and interfacial energy.

43 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 Objective case, not for the non-coherent case. Also, the energy of noncoherent interface is significantly larger than that of the Coherency coherent interface. Thus: Homo- vs ∆Gcoherent = ∆Gelastic + ∆Ginterface Hetero= 4µd 2 * 4πr 3/3 + 4πr 2 gcoherent phase ∆Gnon-coherent = ∆Gelastic + ∆Ginterface Grain = 0 + 4πr 2 gnon-coherent Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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.

44 Coherency loss, estimates Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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 gst = gnon-coherent - gcoherent), one can estimate a critical radius, rcrit. : rcrit = 3∆g/4 Gd 2. • 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. 54 a) is of order 3 Ge and the minimum value of misfit to exceed 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.

45 Impact on Properties Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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.

46 Summary Objective Coherency Homo- vs Heterophase Grain Bounds. Strained Interfaces Orient. Relation. Elasticity Al-Cu ppt • 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).