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- 1. MT 610Advanced Physical Metallurgy Session : Phase Transformations in Solids I Materials Technology School of Energy and Materials
- 2. Contents Diffusional transformations Long-range diffusion Precipitate reaction Eutectoid transformation Short-range diffusion Ordering reaction Massive transformation Polymorphic transformation Diffusionless transformations Martensitic transformation 2
- 3. Phase transformations in solids Mostly occurred by thermally activated atomic movements Two behaviors of atomic movements Diffusion-controlled process of atoms (diffusional transformation) The other, the diffusion cannot take place because of some restrictions such as insufficient time for atomic diffusion, (diffusionless transformation) 3
- 4. Atomic movements Diffusion-controlled 1 interatomic spacing Atomic movements Diffusionless Atomic movements are less than one interatomic spacing 1 interatomic spacing Atomic movements 4
- 5. Diffusional transformations Long-rang diffusion Precipitatereaction Eutectoid transformation Short-rang diffusion Ordering reaction Massive transformation Polymorphic transformation 5
- 6. Precipitate reaction A metastable supersaturated solid solution of α’ transforms to two phases of More stable solid solution phase of α (same crystal structure as α’) and Either stable or metastable α’ → α + β precipitate phase of β 6
- 7. Variation of precipitate reaction 7
- 8. Supersaturated solid solutionα at To : A-8% B When reached T1 α : 100 % at A-8% B β : 0 % When reached T2 Att = 0, α at A-8% B becomes unstable and supersaturated with B solute atoms The unstable and supersaturated α is denoted as α’. 8
- 9. β precipitates Equilibrium At T22,, metastable supersaturated solid solution α’ phase Transforms to more stable α phase with composition of A-5% B (same crystal structure as α’ phase) and Allows precipitates of β phase to form with composition of A-96% B α’ → α +β α matrix β precipitates 9
- 10. β precipitates2 approaches of nucleation of solid β Homogeneous nucleation B atoms diffuse to form small volume of β composition with a critical nucleus size of precipitates β in a matrix of α. Heterogeneous nucleation Nucleation sites are non-equilibrium defects such as excess vacancies, dislocations, grain boundaries, stacking faults, inclusions, and free surfaces. 10
- 11. Homogeneous nucleation of βα solid solution passes solvus line B atoms in α matrix diffuse to form a small volume with β composition Nucleation processB atoms rearrange themselves to form β crystal structure 11
- 12. Homogeneous nucleation of β During nucleation process α/β interfaces create → leading to an activation energy barrier α’ α 12 β
- 13. Homogeneous nucleation of β During nucleation process, 3 components of ∆G Creationof β precipitates (driving force) - Volume free energy reduction of V∆GV Creation of α/β interfaces - Increase of Aγ Misfit strain energy between α and β - Increase of V∆GS α∆Ghom = (–V∆GV) + Aγ + V∆GS β 13
- 14. Homogeneous nucleation of β ∆Ghom = (–V∆GV) + Aγ + V∆GS Assuming a spherical nucleus with r V = (4/3) πr3 A = 4 πr2 Critical radius r* = 2γ / (∆GV – ∆GS) Necessary free energy change ∆G* = 16πγ3 / 3(∆GV – ∆GS)2 14
- 15. Homogeneous nucleation of β Critical radius r* = 2γ / (∆GV – ∆GS) Necessary free energy change ∆G* = 16πγ3 / 3(∆GV – ∆GS)2 For a given undercooling, If r < r*, the system will lower its free energy by dissolving the embryos back into solid solution. If r > r*, the system will lower its free energy by allowing the nuclei to grow. 15
- 16. Homogeneous nucleation of β Concentration of critical-sized nuclei per unit volume C* = Co exp(–∆G*/kT) cluster/m3 where Co : initial number of atoms/volume Homogeneous nucleation rate N = f C* nuclei/m3 s hom where f = ω exp (–∆Gm/kT) : how fast a critical nucleus can receive an atom from α matrix (atomic migration) and ∆Gm : activation energy for atomic migration 16
- 17. Homogeneous nucleation of β Homogeneous nucleation rate Nhom = ωCo exp(–∆Gm/kT) exp(–∆G*/kT) where ∆G* = 16πγ3 / 3(∆GV – ∆GS)2 Assumption ∆Gm is constant, and ∆GS is ignored. Consider ∆Gm and ∆GS 17
- 18. Homogeneous nucleation rate Nhom = ωCo exp(–∆Gm/kT) exp(–∆G*/kT) where ∆G* = 16πγ3 / 3(∆GV – ∆GS)2 Main factor controlling ∆G* is the driving force for precipitation ∆GV . 18
- 19. Driving force for transformation Initial composition Xo Solution treated at T1 Then, quench to T2 Alloy is supersaturated with B Alloy tries to precipitate β When α → α+β completed, free energy decreases by ∆Go ∆Go is a driving force for transformation. 19
- 20. β precipitation Initially First nuclei of β do not change α composition from Xo Small amount of materials with nucleus composition βXB (P) is removed from α phase Free energy of the system decreases by ∆G1 ∆G1 = αµA βXA + αµB βXB (per mol β removed) 20
- 21. β precipitation Rearranged into β crystals Freeenergy of the system decreases by ∆G2 (Q) ∆G2 = βµA βXA + βµB βXB (per mol β formed) Driving force for nucleation ∆G = ∆G2 – ∆G1 (per mol β) n 21
- 22. β precipitation Volume free energy decrease ∆G = ∆Gn/Vm V (per unit volume of β) For dilute solution ∆G ∝ ∆X V where ∆X = Xo – Xe Drivingforce for precipitation increases with increasing undercooling ∆T below Te. 22
- 23. When consider ∆Gm and ∆GS∆G* = 16πγ3 / 3(∆GV – ∆GS)2 Taking the misfit strain energy term into account, the effective driving force become (∆GV – ∆GS) Equilibrium temperature reduces from Te to Te’ (effective equilibrium temperature) 23
- 24. Homogeneous nucleation rateNhom = ωCo exp(–∆Gm/kT) exp(–∆G*/kT) Schematically plot 2 exponential terms Atomic mobility : exp(–∆Gm/kT) Potential concentration of nuclei exp(–∆G*/kT) Combination of 2 terms results in Nhom 24
- 25. NoticeNhom = ωCo exp(–∆Gm/kT) exp(–∆G*/kT) exp(–∆G*/kT) term is zero until ∆Tc (critical undercooling) is reached – above, β does not form . exp(–∆Gm/kT) term decreases rapidly as T decreases – atomic mobility is too small. 25
- 26. Undercooling ∆T If∆T < ∆Tc , N is negligible (∆GV is too small).N is maximum at intermediate ∆T. If ∆T >> ∆Tc , N is small or negligible (Diffusion becomes too slow). 26
- 27. Real homogeneous nucleationβ precipitate is not always spherical. The most effective way of minimizing ∆G* is to form nuclei with the smallest total interfacial energy by Form the same orientation relationship with the matrix Have coherent interfaces Example is formation of metastable GP zones in the Al-Cu alloys. 27
- 28. Al-Cu alloys The equilibrium consists of two solid phases : α4, θ Precipitate process α → α1 + GP Zone → α2 + θ” → α3 + θ’ → α4 + θ 28
- 29. Heterogeneous nucleation of β Usually precipitate in matrix α Nucleation sites of nonequilibrium defects Excess vacancies Nucleus creations Dislocations decrease some free Grain boundaries energy with an amount Stacking faults of ∆Gd Inclusions Therefore, help reducing Free surfaces activation energy barrier 29
- 30. Heterogeneous nucleation of β ∆Ghet = ∆Ghom – ∆Gd ∆Ghet = (–V∆GV) + Aγ + V∆GS – ∆Gd 30
- 31. Heterogeneous nucleation of β At α/α grain boundary Assumption : no misfit strain energy at a α/β interface Optimum embryo shape for nucleation is when total interfacial free energy is minimized. 2 spherical caps At the precipitate ∆Gd = Aααγαα r* = 2γαβ/ ∆GV 31
- 32. Heterogeneous nucleation of β Activationenergy barrier ∆G*het/∆G *hom = ∆V*het/∆V *hom = S(θ) S(θ)is a shape factor S(θ) = ½ (2 + cosθ) (1 – cosθ)2 1.0 ∆ G*het /∆ G*hom = S(θ) 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 cos θ 32
- 33. Heterogeneous nucleation of β ∆G*het/∆G *hom = ∆V*het/∆V *hom = S(θ) ∆G* and ∆V*het can be reduced further by het nucleation on Grain edge Grain corner 1.0 Grain boundaries ∆ G*het /∆ G*hom = S(θ) Grain edges 0.8 Grain corners 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 cos θ 0.8 1.0 33
- 34. Heterogeneous nucleation of β When matrix and precipitate are compatible and allow formation of lower energy coherent facets, nucleus Will form whenever possible Will have an orientation relationship with matrix 34
- 35. Interface 35
- 36. Heterogeneous nucleation of β Heterogeneous nucleation rate Nhet = ωC1 exp(–∆Gm/kT) exp(–∆G*/kT) where C1 is concentration of heterogeneous nucleation sites per unit volume. The rates can be obtained at very small driving forces. 36
- 37. Heterogeneous nucleation of β Heterogeneous nucleation rate Nhet = ωC1 exp(–∆Gm/kT) exp(–∆G*/kT) Homogeneous nucleation rate Nhom = ωCo exp(–∆Gm/kT) exp(–∆G*/kT) Differences in ω and ∆Gm are not significant Nhet/Nhom = C1/Co exp{(∆G*hom–∆G*het)/kT} where C1/Co is a ratio between the boundary thickness δ and the grain size D 37
- 38. Heterogeneous nucleation of βNhet/Nhom = C1/Co exp{(∆G*hom–∆G*het)/kT} Nucleation at grain boundary C1/Co C1/Co = δ/D = 10-5 Nucleation at grain edge C1/Co = (δ/D)2 = 10-10 Nucleation at grain corner C1/Co = (δ/D)3 = 10-15 δ = 0.5 nm D = 50 µm 38
- 39. CuAl2 precipitates in Al grainsand at grain boundaries ASM Handbook, Vol. 9 39
- 40. Growth of precipitate Solutes must migrate from deep within the parent phase. Solute from precipitate to precipitate/parent interfaces Long-range diffusion controlled Time-dependent velocity Interface controlled Constant velocity 40
- 41. Growth of precipitate Nuclei will grow on Direction of smallest nucleation barrier Smallest critical volume Minimum total interfacial free energy (in the case of strain-free) 41
- 42. Growth of precipitate Ifnuclei are enclosed by a combination of coherent and semicoherent facets, Incoherent interface will have higher mobility, so the interface can advance faster. 42
- 43. Growth of planar incoherentinterfaces Found at grain boundaries Depletion of solute in α from Co to Ce Form β slab precipitate Solute concentration is Cβ Grow from zero thickness at rate of v 43
- 44. Growth of planar incoherentinterfacesβ slab advances dx of a unit area on i/c Volume of β material (1·dx) must be converted from α (Ce) to β (Cβ) Materials required per unit area = 1·dx (Cβ – Ce) Solute B atoms migrate = D (dC/dx)i/c dt 44 where D is interdiff. coef. = XBDA + XADB
- 45. Growth of planar incoherentinterfaces Materials advancing= Solute atom migration 1·dx (Cβ – Ce) = D (dC/dx)i/c dt v = dx/dt = [D/(Cβ – Ce)] (dC/dx)i/c Approximation (dC/dx)i/c using “conservation of solute” α Area = β Area ½ L ∆Co = (Cβ – Co) x 45
- 46. Growth of planar incoherentinterfaces½ L ∆Co = (Cβ – Co) x∆Co/L = (∆Co)2/{2 x (Cβ – Co)}∆Co/L ≈ (dC/dx)i/c(dC/dx)i/c = (∆Co)2/{2 x (Cβ – Co)} v = dx/dt = [D/(Cβ – Ce)] (dC/dx)i/c 46 = [D/(C – C )] (∆C ) /{2 x (C – C )} 2
- 47. Growth of planar incoherentinterfacesv = dx/dt = D(∆Co)2/{2 x (Cβ – Ce) (Cβ – Co)} Assumption Cβ – Ce ≈ C β – Co Mole fraction : ∆X = ∆Cdx/dt = D(∆Xo)2/{2 x (Xβ – Xe)2}x dx = D(∆Xo)2/{2 x (Xβ – Xe)2} dtx2 = Dt(∆Xo)2/(Xβ – Xe)2x = ∆Xo/(Xβ – Xe) √(Dt) → precipitate thickening 47
- 48. Growth of planar incoherentinterfaces Precipitatethickening x = ∆Xo/(Xβ – Xe) √(Dt) x ∝ √(Dt)v = dx/dt v = ∆Xo/2(Xβ – Xe) √(D/t) Supersaturation ∆Xo before precipitation ∆Xo = Xo – Xe 48
- 49. Growth of planar incoherentinterfacesx = ∆Xo/(Xβ – Xe) √(Dt)v = ∆Xo/2(Xβ – Xe) √(D/t)∆Xo = Xo – Xe Growth rate is low when Small ∆T → small ∆X Large ∆ T → small diffusion 49
- 50. A sink for solute Grain boundaries can act as a collector plate of a sink for solute. Volume diffusion of solute to grain boundary Diffusion along the grain boundaries Diffusion along the α/β interfaces 50
- 51. End of precipitate growth Precipitates stop advancing/growing when the matrix composition reaches Xe everywhere – there are no longer excess solute supply for precipitation. 51
- 52. Growth of plates and needlesβ precipitates have a constant thickness and a cylindrically curved incoherent edge of radius r Growth governed by volume diffusion- controlled process 52
- 53. Growth of plates and needles Concentration gradient to drive diffusion through the edge is ∆C/L where L = kr, and k is a constant. v = D∆C/{kr(Cβ – Cr)} = D∆X/{kr(Xβ – Xr)} where ∆X = Xo – Xr = ∆Xo(1-r*/r) and ∆Xo = Xo – Xe v = D ∆Xo (1-r*/r) / {kr (Xβ – Xr)} For spherical tip, X = X (1+2γV /RTr) r e m For cylindrical tip, X = X (1+γV /RTr) 53 r e m
- 54. Growth of plates and needles For a constant thickness, Lengthening rate v is constant with time; therefore, x ∝ t. Lengthening rate v is varied with D and r. 54
- 55. Plate-like precipitate Observed by a ledge mechanism Broad faces are semicoherent Limit migration of atoms Atoms will migrate and attach at the ledges Their interfaces are incoherent. Growth requires lateral motion of ledges achieved by diffusion v = uh/λ = D∆Xo/{kλ(Xβ – Xe)} = constant 55
- 56. Plate-like precipitate Widmanstätten precipitation Plates lie along {111} matrix planes. ASM Handbook 56
- 57. Spinodal structure Homogenous precipitates of 2-phase mixtures resulting from a phase separation that occurs under certain conditions of temperature and composition 57
- 58. Spinodal structure Xois solution treated at To Then, aged at TA α tends to separate into 2- o phase mixture Initially, G o Xo becomes unstable and try to decreases its total free energy by producing small fluctuations in composition resulting in A-rich and B-rich regions 58
- 59. Spinodal structure Xo is unstable and try to decreases its total free energy Up-hill diffusion Down-hill diffusion until equilibrium phases of α1 and α2 are reached at compositions of X1 and X2 59
- 60. Spinodal structure α1 and α2 phase mixture occurs by continuous growth of initially small amplitude fluctuations Controlled by atomic migration and diffusion 60
- 61. Spinodal structure Xo will decay with time ∆X = ∆Xo exp(–t/τ) where τ is a relaxing time τ = λ2/(4π2D) where λ is wavelength of fluctuation and D is diffusion coefficient. 61
- 62. Spinodal structure TEM micrographs 2.5 – 10 nm in metallic system Contrast comes mainly from structure factor differencesFe-28.5 wt.% Cr-10.6 wt.% Co Aged at 600°C 4 hCu-33.5 at.% Ni-15 at.% Fe Aged at 775°C 15 min, λ ≈ 25 nm ASM Handbook 62
- 63. Cellular precipitate Precipitation of a second phase from a supersaturated solid solution May occur through a reaction involving the formation of colonies Consisting of a 2-phase mixture That grow and consume the matrix. The transformation is very similar to the eutectoid reaction. 63
- 64. Cellular precipitate Morphology Alternating lamellae of precipitate phase and depleted matrix Duplex cells Cooperative growth of 2 phases Originate from matrix grain boundaries 64
- 65. Cellular precipitate Cellular fronts advance into the supersaturated matrix and spatially partitions the structure into transformed and untransformed regions. Composition and orientation of α’ phase changes discontinuously from Cα’ to Cα for α phase colony. 65
- 66. Cellular precipitate Solutes to form β phase colony diffuse from the neighboring α colonies with a distance d = So/2 Solution redistribution occurs by diffusion along the interface at the composition distribution region Assumed Solute distribution is linear 66
- 67. Cellular precipitate Amount of solute rejected from α’ to form β plates with the rate of dm/dt dm/dt = Jdiff A = R(Cβ – Cα’) where R is the interface velocity Jdiff = DB(∂C/∂x) and A = λ (2 sides) = 2λ dm/dt = DB(∂C/∂x) 2λ = R(Cβ – Cα’) DB{(Cα’ – Cα)/d} 2λ = R(Cβ – Cα’) R = {2 DB λ/d} {(Cα’ – Cα)/(Cβ – Cα’)}where d = (So – Sβ)/2, for small distance 67 = SoSβ/2 d
- 68. Cellular precipitate Cellularor discontinuous precipitation growing out uniformly from the grain boundaries Fe-24.8Zn alloy Aged at 600°C 6 min (W.C. Leslie) 68

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