Mt 610 phasetransformationsinsolids_i

430 views

Published on

0 Comments
2 Likes
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total views
430
On SlideShare
0
From Embeds
0
Number of Embeds
2
Actions
Shares
0
Downloads
0
Comments
0
Likes
2
Embeds 0
No embeds

No notes for slide

Mt 610 phasetransformationsinsolids_i

  1. 1. MT 610Advanced Physical Metallurgy Session : Phase Transformations in Solids I Materials Technology School of Energy and Materials
  2. 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. 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. 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. 5. Diffusional transformations Long-rang diffusion  Precipitatereaction  Eutectoid transformation Short-rang diffusion  Ordering reaction  Massive transformation  Polymorphic transformation 5
  6. 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. 7. Variation of precipitate reaction 7
  8. 8. Supersaturated solid solutionα 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. 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. 10. β precipitates2 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. 11. Homogeneous nucleation of βα solid solution passes solvus line B atoms in α matrix diffuse to form a small volume with β composition Nucleation processB atoms rearrange themselves to form β crystal structure 11
  12. 12. Homogeneous nucleation of β During nucleation process  α/β interfaces create → leading to an activation energy barrier α’ α 12 β
  13. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 27. Real homogeneous nucleationβ 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. 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. 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. 30. Heterogeneous nucleation of β ∆Ghet = ∆Ghom – ∆Gd ∆Ghet = (–V∆GV) + Aγ + V∆GS – ∆Gd 30
  31. 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. 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. 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. 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. 35. Interface 35
  36. 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. 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. 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. 39. CuAl2 precipitates in Al grainsand at grain boundaries ASM Handbook, Vol. 9 39
  40. 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. 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. 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. 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. 44. Growth of planar incoherentinterfacesβ 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. 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. 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. 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. 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. 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. 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. 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. 52. Growth of plates and needlesβ precipitates have a constant thickness and a cylindrically curved incoherent edge of radius r Growth governed by volume diffusion- controlled process 52
  53. 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. 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. 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. 56. Plate-like precipitate Widmanstätten precipitation  Plates lie along {111} matrix planes. ASM Handbook 56
  57. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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

×