APRESENTAÇÃO_Recovery, recrystallization & grain growth.pdf

1. Recovery, recrystallization & grain growth NFPL139: Physics of Materials II Faculty of Mathematics and Physics, Charles University Prague, Czech Republic Jana Šmilauerová

2. Cold working I cold-working – plastic deformation of metals and alloys at low temperatures (relative to their melting point Tm) I typically below 0.3 – 0.4 Tm (in K) I typical cold-working techniques are e.g. rolling, forging, extrusion I most energy of the cold working is heat, but some fraction is stored in the material → strain energy of defects (dislocations, point defects) I amount of stored energy ↑ with ↑ severity of the deformation process, ↓ temperature and by changing pure metal to alloy Stored energy and the fraction of the total energy provided by cold working of high-purity Cu as a function of tensile elongation [1], adapted from Gordon et al. Trans. AIME 203, 1043 (1955)

3. Cold working I cold working introduces a large number of dislocations into the metal I annealed (soft) metal ρ ≈ 1010 − 1012 m−2 I heavily deformed metal ρ ≈ 1014 − 1016 m−2 I each dislocation associated with lattice strain → ↑ dislocation density ↑ strain energy I point defects are also responsible for a part of stored energy (e.g. vacancies or interstitials due to dragging jogs on screw dislocations) I significant changes in physical and mechanical properties → increase in strength, hardness, electrical resistance; decrease in ductility; broadening of X-ray diffraction peaks due to lattice distortion I see work (strain) hardening in Physics of materials I I percent cold work (% CW) sometimes more convenient than strain % CW = A0 − Ad A0 × 100, where A0 and Ad are the initial and deformed cross-sectional areas

4. Cold working Influence of cold working on a) yield strength, b) tensile strength and c) ductility of steel (blue), brass (orange) and copper (red) [2]

5. Cold working Effect of cold working on the character of the stress-strain curve for low-carbon steel [2]

6. Cold working techniques Common metalworking techniques: (a) forging (open and closed die), (b) rolling, (c) extrusion (direct and indirect), (d) wire drawing, (e) stamping/pressing [3]

7. Gibbs free energy of cold-worked materials I Gibbs free energy associated with the cold work: ∆G = ∆H − T∆S, where ∆H is the enthalpy (stored strain energy), T is the thermodynamic temperature, ∆S the increase in entropy I plastic deformation increases the entropy, but the effect is small compared to ∆H ∆G ≈ ∆H I G of cold-worked metal is higher than that of annealed condition → microstructurally metastable state → spontaneous softening possible in order to ↓ G I lowering of G not simple process due to complexity of the cold-worked microstructure I processes which ↓ G are often associated with motion of atoms or vacancies → temperature sensitive (exponential laws)

8. Release of stored energy I stored energy released during heating – can be measured using differential scanning calorimetry (DSC)∗ – note that the changes are not phase transformations I broad exothermic peak ∼ 100 − 280 ◦ C ⇒ recovery (RV) – rearrangement of defects in deformed grains I large exo peak ∼ 300 − 480 ◦ C ⇒ recrystallization (RX) – a completely new set of strain-free grains develops, growing at the expense of previous deformed grains I further heating ⇒ grain growth – some recrystallized grains grow at the expense of neighbouring grains ∗ 80% cold-rolled ultra-high-purity Fe [F. Scholz et al. Scripta Mater. 40, 949 (1999)]

9. Release of stored energy

10. Release of stored energy Schematic illustration of the effect of annealing on the microstructure of a cold-worked metal: (a) cold-worked condition, (b) after recovery, (c) after recrystallization, (d) after grain growth [3]

11. Release of stored energy Anisothermal anneal curve for cold-worked Ni and the effect on hardness and electrical resistivity of the material. Denoted by C is the peak associated with recrystallization [1], adapted from H. M. Clarebrough et al. Proc. R. Soc. London 232A, 252 (1955)

12. Recovery I rearrangement of defects to lower the stored energy I material still strong but less brittle than in the CW condition I driving force: strain energy stored in defects – point defects and dislocations Recovery of point defects I first phase when heating up I non-equilibrium concentration of point defects created during deformation decreases towards equilibrium I recombination of vacancies and interstitials I annihilation of point defects at traps (e.g. grain and phase boundaries, edge dislocations, surface) I weak impact on mechanical properties of the material

13. Recovery Recovery of dislocations I annihilation of excess dislocations (positive with negative edge d., right-hand with left-hand screw d.) – both slip and climb I polygonization I rearrangement of edge dislocations of the same sign into low-angle grain boundaries (tilt boundaries) or screw d. into twist boundaries I lower strain energy (strain fields of adjacent dislocations cancel out partially) I crystal parts between LAGB are relatively dislocation- and strain-free → subgrains divided by subboundaries, also known as mosaic structure Rearrangement of dislocations in deformed (bent) crystal into low-energy configurations [1]

14. Recovery Recovery of dislocations I annihilation of excess dislocations (positive with negative edge d., right-hand with left-hand screw d.) – both slip and climb I polygonization I rearrangement of edge dislocations of the same sign into low-angle grain boundaries (tilt boundaries) or screw d. into twist boundaries I lower strain energy (strain fields of adjacent dislocations cancel out partially) I crystal parts between LAGB are relatively dislocation- and strain-free → subgrains divided by subboundaries, also known as mosaic structure In Laue paterns, the diffraction spots from deformed (bent) crystal are asterated (elongated) (A). After recovery by polygonization, X-ray reflections break into series of separated spots (B) [1]

15. Dislocation movement during polygonization I both slip and climb required to rearrange dislocations I low temperatures – climb not possible (depends on vacancy motion which is a thermally activated process); slip also more difficult I rate of polygonization increases with temperature

16. Dislocation movement during polygonization Polygonization during annealing of a bent FeSi (bcc) single crystal (optical microscopy, 750×). All samples were annealed for one hour at the given temperature. The surface is perpendicular to the bending axis and also to the (011̄) slip plane and to the (111) plane along which subboundaries form, see the schematics. Intersections of dislocations with the surface are observed as dark dots – etch pits [1] and W. R. Hibbard Jr. et al. Acta Metall 4, 306 (1956)

17. Dislocation movement during polygonization I in materials deformed by processes more complex than bending – slip on multiple intersecting slip planes → polygonization results in more complex subgrain structures Polygonized structure (subboundaries) in a FeSi single crystal deformed 8% by cold rolling and annealed 1 h at 1100 ◦ C [1] and W. R. Hibbard Jr. et al. in Creep and recovery, ASM, 52 (1957)

18. Recrystallization I a completely new set of grains is formed → the previous deformed grains are replaced by defect-free crystals by nucleation and growth I driving force: stored energy of CW – strain energy of dislocations I reduction of strength and hardness, increase in ductility I kinetics of recrystallization similar to nucleation and growth × recovery rate decreases with time as the driving force (strain energy stored in defects) is exhausted ⇒ S-shaped curves – slow start and finish, maximum reaction rate in between Kinetics of recrystallization [Wikipedia] fraction recrystallized, i.e. the shape of the curve (Avrami equation): f = 1 − exp (−ktn )

19. Recrystallization I a completely new set of grains is formed → the previous deformed grains are replaced by defect-free crystals by nucleation and growth I driving force: stored energy of CW – strain energy of dislocations I reduction of strength and hardness, increase in ductility I kinetics of recrystallization similar to nucleation and growth × recovery rate decreases with time as the driving force (strain energy stored in defects) is exhausted ⇒ S-shaped curves – slow start and finish, maximum reaction rate in between Isothermal recrystallization curves for pure Cu (99.999%) cold-rolled 98% [1] and B. F. Decker et al. Trans. AIME 188, 887 (1950)

20. Recrystallization I τ – time interval after which the recrystallization at any given temperature reaches a constant fraction (e.g. 50% recrystallized) I 1/T vs. log τ is a straight line, which can be expressed as 1 T = K log τ + C, where K is the slope and C the intercept of the curve with the y-axis Reciprocal absolute temperature vs. time for half-recrystallization [1] and B. F. Decker et al. Trans. AIME 188, 887 (1950)

21. Recrystallization I the previous equation can be also expressed as 1 τ = A exp − Qrecryst RT , where 1 τ is the rate at which a given percentage (here 50%) of the structure is recrystallized, Qrecryst is the activation energy for recrystallization and R is the gas constant I NB: conceptual difference between Qrecryst and e.g. activation energy (enthalpy) for vacancy motion I vacancies – activation energy is the height of the energy barrier for atomic jump I recrystallization – not clear, probably several processes → Qrecryst considered as an empirical quantity I generally, Qrecryst changes continuously during recrystallization (the driving force – stored energy of cold work – is depleted) I the Arrhenius-type equation only empirical, but has been found to hold for a number of metals and alloys

22. Avrami equation – derivation fraction recrystallized: f = 1 − exp (−ktn ) I previously mentioned S-shaped curve typical for many transformation reactions – nucleation and growth processes I theory by Kolmogorov (1937), Johnson and Mehl (1939) and Avrami (1939) ⇒ Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation I assumptions: I nucleation is homogeneous and random I new grains nucleate at the rate Ṅ per unit volume I growth is isotropic (spherical) and stops only after impingement I the growth rate Ġ does not depend on the extent of transformation I recrystallization can only take place in the non-recrystallized volume, but for derivation it is convenient to introduce an extended volume (newly transformed volume if the whole sample was still untransformed)† † Alternative derivation of the Avrami equation without the concept of the extended volume: Siclen: Random nucleation and growth kinetics. Phys. Rev. B 54 (1996) 11845

23. Avrami equation – derivation I the number N of nuclei appearing in the sample volume V during a time interval from τ to τ + dτ, where 0 τ t, τ is the incubation time for nucleation N = ṄV dτ I the radius of formed nuclei in time t Ġ(t − τ) I the change of extended volume due to new nuclei in the time interval dVext = (volume of one nucleus) × (number of nuclei) = 4π 3 Ġ3 (t − τ)3 ṄV dτ I the total extended volume by integration Vext = Z dVext = 4π 3 Ġ3 ṄV Z t 0 (t − τ)3 dτ = π 3 Ġ3 ṄVt4

24. Avrami equation – derivation I a part of the total extended volume is not real – it lies on previously transformed material I the increment of the “real” part of extended volume is (due to random nucleation) proportional to the untransformed volume fraction dVreal = dVext 1 − Vreal V Z dVext = Z dVreal 1 − Vreal V Vext = −V ln 1 − Vreal V = −V ln(1 − f) I substitution for Vext ln(1 − f) = − π 3 Ġ3 Ṅt4 f = 1 − exp(− π 3 Ġ3 Ṅt4 ) f = 1 − exp(−ktn )

25. Avrami equation – derivation f = 1 − exp(−ktn ) k . . . . . depends on nucleation and growth rates Ṅ, Ġ → very sensitive to T n . . . . . numerical exponent ∼ 1 − 4, independent of T (if the nucleation mechanism does not change) Isotropic growth (3D) n Ṅ = const. 4 Ṅ decreases as a power function of t 3-4 Ṅ decreases rapidly → all nucleation at the beginning ⇒ site saturated nucleation 3 n Ṅ = const. saturated nucleation 2D 3 2 1D 2 1

26. Recrystallization temperature I temperature at which a given material completely recrystallizes in a finite time (e.g. usually 1 h) I not a fixed characteristic, depends on many factors: I annealing time (↑ time → ↓ Trecryst) I alloying elements (alloys ↑ Trecryst than pure metals) I percent cold work (↑ CW → ↓ Trecryst) I temperature of CW (↑ TCW → ↓ stored energy → ↑ Trecryst) Temperature and time relationships for a complete recrystallization of 13% and 51% CW zirconium metal. Recrystallization is promoted by increasing amount of cold work. Note that the slope of the curves is not the same → Qrecryst varies with the amount of CW [1] and R. M. Treco: Proc. of AIME Regional Conference on Reactive Materials (1956) 136.

27. Recrystallization temperature I temperature at which a given material completely recrystallizes in a finite time (e.g. usually 1 h) I not a fixed characteristic, depends on many factors: I annealing time (↑ time → ↓ Trecryst) I alloying elements (alloys ↑ Trecryst than pure metals) I percent cold work (↑ CW → ↓ Trecryst) I temperature of CW (↑ TCW → ↓ stored energy → ↑ Trecryst) I example: for a given sample Qrecryst = 200 kJ/mol and recrystallization is finished after 1 h at 600 K → from the Arrhenius-type equation we find that I 10 K lower, full recrystallization after 2 h; after 1 h only partial RX I 10 K higher, only about 30 min needed for complete recrystallization I 20 K higher, about 15 min for complete recrystallization ⇒ large sensitivity of the recrystallization process to small changes in temperature → in practice, recrystallization temperature is treated as a fixed property of the metal and the time factor is neglected

28. Formation of nuclei during recrystallization I nucleation of new grains at places of high lattice-strain energy, e.g. slip lines and intersections, deformation twins intersections, close to grain boundaries and triple junctions I a number of different models, but some common points I nuclei can grow only if they are above some critical size (∼ 15 nm) I nuclei must be surrounded (at least partly) by a high-angle grain boundary (low-angle GB is much less mobile) I polycrystals (Bailey, Hirsch‡ ) – difference of dislocation density across GB → less deformed grain migrates into the deformed one driven by lowering energy by destroying defects at the advancing GB Schematic illustration of three types of GB nuclei. The hexagonal networks are subgrains [1] and Bay et al. Metall. Trans. A15, 287 (1984) ‡ J. E. Bailey, P. B. Hirsch, Proc. R. Soc. A267, 11 (1962)

29. Recrystallized grain size I grain size right after recrystallization finishes (before grain growth) I ↑ severity of deformation ⇒ ↑ number of nuclei (at points of high lattice strain) ⇒ smaller recrystallized grain size I usually does not depend on the temperature of recrystallization (!) – when activation energies for nucleation and growth are approximately equal (exception e.g. Al, where Qnucl much larger) Effect of cold work on recrystallized grain size in deformed brass [1] and Smart et al. Trans. AIME 152, 103 (1943)

30. Recrystallized grain size I if deformation too low, RX does not happen → critical amount of cold work – minimum CW after which RX happens (in a reasonable time) I critical deformation not a metal/alloy property – depends on the type of deformation (torsion, rolling etc.) and on its mechanism (e.g. easy glide → few nucleation sites for new grains → higher critical amount of deformation) I critical deformation and recrystallized grain size technologically important – e.g. metal sheets which need to be further cold-formed into some shape: I small grains → surface unaffected during plastic deformation I large grains → rough surface due to anisotropy of deformation within individual grains – orange-peel effect → undesirable Surface of an Al tube (a) before and (b) after forming, showing the orange-peel effect [Chao et al. KnE Materials Science (2016) 24]

31. Other aspects affecting recrystallization Purity of the metal/composition of the alloy I pure metals – more rapid recrystallization I even small amounts of impurities (∝ 0,01%) can raise the recrystallization temperature by hundreds of degrees I depends also on the type of foreign element I impurity atoms can segregate to grain boundaries and retard their motion → solute drag (can be used to retain strength of materials at high temperatures) Inpurity effect on the recrystallization temperature (30 min annealing) of 80% cold-rolled Al [1]

32. Other aspects affecting recrystallization Second-phase particles I precipitates and other particles can also pin down the grain boundary and impede its migration Fine particles of (Fe,Al)2Zr block the motion of a grain boundary in Fe-17Al-4Cr-0.3Zr [P. Kratochvı́l et al. J. Mater. Eng. Perform. 21, 1932-1936 (2012)] Interaction of a mica grain with a migrating grain boundary in quartz; scale bar is 0.2 mm [J. I. Urai et al. Mineral and rock deformation 36, 161-199 (1978)]

33. Other aspects affecting recrystallization Degree of deformation I ↑ severity of deformation → ↑ stored energy → ↓ incubation time I small deformation → coarse recrystallized grains due to a small number of nucleated grains Effect of tensile deformation on recrystallization kinetics of Al annealed at 350 ◦ C [4] and Anderson et al. Trans. Metal. Soc. AIME 161, 14 (1945)

34. Other aspects affecting recrystallization Homogeneity of deformation I inhomogeneous stored deformation → inhomogeneous recrystallization → nonuniform grain size Small recrystallized grains in AA8006 (Al-Fe-Si-Mn) alloy, TEM Schematics of recrystallization in an inhomogeneously deformed material, dark areas are deformed more [4]

35. Other aspects affecting recrystallization Initial grain size I grain boundaries interfere with slip during cold working ⇒ larger strains in the lattice near GB ⇒ more nucleation sites for new set of grains I ↓ initial grain size ⇒ ↑ stored deformation ⇒ ↑ driving force ⇒ ↑ number of nuclei and ↓ recrystallized grain size I affects also the kinetics of recrystallization Influence of the initial grain size on the kinetics of recrystallization of Cu cold-rolled 93% and annealed at 225 ◦ C – fraction recrystallized and JMAK plot [4] and Hutchinson et al. Scr. Metall. 23, 671 (1989)

36. Other aspects affecting recrystallization Initial grain size I initially coarse-grained material – preferential nucleation at grain boundaries or in shear bands ⇒ less homogeneous RX Illustration of the effect of the initial grain size on the homogeneity of nucleation [4] Recrystallization at shear bands in Cu [4] and Adcock et al. J. Inst. Met. 27, 73 (1922)

37. Other aspects affecting recrystallization Stacking fault energy I low SFE (e.g. Cu, brass, austenitic steel) → larger spacing between the two partial dislocations → more difficult cross-slip → more likely to form Lomer-Cottrell barriers → ↑ work hardening → ↑ stored energy and ↑ driving force for RX which starts earlier and at lower T I high SFE → small spacing between partials → high mobility of dislocations, including cross-slip → easier annihilation → ↓ dislocation density → ↓ stored energy → ↑ incubation time and temperature of RX Illustration of the relationship between a perfect dislocation and partial dislocations Cross slip of an extended dislocation, constriction is necessary [5]

38. Other aspects affecting recrystallization Stacking fault energy I low SFE (e.g. Cu, brass, austenitic steel) → larger spacing between the two partial dislocations → more difficult cross-slip → more likely to form Lomer-Cottrell barriers → ↑ work hardening → ↑ stored energy and ↑ driving force for RX which starts earlier and at lower T I high SFE → small spacing between partials → high mobility of dislocations, including cross-slip → easier annihilation → ↓ dislocation density → ↓ stored energy → ↑ incubation time and temperature of RX Formation of a Lomer-Cotterll barrier [5]

39. Recrystallization and grain growth in 33% CW brass. (a) cold-worked grain structure, (b) initial stage of recrystallization – recrystallized grains are the very small ones, (c) deformed grains partially replaced by recrystallized ones, (d) complete recrystallization, (e-f) grain growth at different temperatures [2]

40. Cold, hot and warm working I cold working – plastic deformation below Trecryst I hot working – plastic deformation above Trecryst

41. Further points on recovery and recrystallization I recovery and recrystallization can occur after deformation (static RV, RX, cold working) or during deformation (dynamic RV, RX, hot working) I dynamic recovery and recrystallization may be utilized to deform the material to large strains – these processes partially compensate work hardening at temperatures above I ∼ 0.3 Tm for dynamic recovery I ∼ 0.6 − 0.7 Tm for dynamic recrystallization I recrystallization either discontinuous (nucleation and growth of distinct new grains) or continuous (gradual evolution of the deformed microstructure into a recrystallized one – grains with a lower dislocation density grow at the expense of more deformed grains, no nucleation) I primary recrystallization: static discontinuous

42. Continuous recrystallization I typically after deformation to large strains and at high temperatures → microstructure with predominantly HAGB I no recognizable nucleation and growth, the microstructure evolves relatively homogeneously throughout the material I retains the deformed texture EBSD maps of AA8006 (Al-Fe-Mn) annealed after cold rolling: (a) ε = 0.69, T = 250 ◦ C – discontinuous RX and (b) ε = 3.9, T = 300 ◦ C – continuous RX. HAGB and LAGB are denoted by black and white lines, respectively [4]

43. Dynamic recovery I similar mechanisms as in the static process I cross-slip and climb of dislocations I formation of subgrains (less developed, less regular LAGB structure) I rapid and extensive especially in materials of a high stacking fault energy – usually the only dynamic process which occurs I initial stages of deformation – increase of the flow stress as dislocations multiply and interact → ↑ driving force for recovery I at a certain strain, dynamic equilibrium between rates of work hardening and recovery → steady/state flow stress and a constant dislocation density Summary of microstructure changes ocurring during dynamic recovery [4]

44. Dynamic recovery I similar mechanisms as in the static process I cross-slip and climb of dislocations I formation of subgrains (less developed, less regular LAGB structure) I rapid and extensive especially in materials of a high stacking fault energy – usually the only dynamic process which occurs I microstructural evolution depends also on the deformation temperature and strain rate (in addition to the strain) – often combined in a single Zener-Hollomon parameter Z = ε̇ exp Q RT , where Q is the activation energy Stress-strain curves for Al-1Mg at 400 ◦ C and different strain rates [4] and Puchi et al. Proc. Int. Conf. on Thermomechanical Processing of Steels 2, 572 (1988)

45. Dynamic recrystallization I occurs in metals in which recovery processes are slow (low SFE) after a critical deformation is reached I microstructure evolution – new grains nucleate at old grain boundaries → further grains nucleate at the boundaries of the growing grains → a band of recrystallized grains is formed → eventually, a full recrystallization Microstructure development during DRX, original grain boundaries are shown by dotted lines. (a)-(d) large initial grain size, (e) small initial grain size [4] The mean size of the dynamically recrystallized grains does not change as recrystallization proceeds, unlike in static RX. Ni deformed at 800 ◦ C, ε̇ = 0.057 s−1 [4] and Sah et al. Metal Sci. 8, 325 (1974)

46. Dynamic recrystallization I occurs in metals in which recovery processes are slow (low SFE) after a critical deformation is reached I microstructure evolution – new grains nucleate at old grain boundaries → further grains nucleate at the boundaries of the growing grains → a band of recrystallized grains is formed → eventually, a full recrystallization DRX at prior grain boundaries in Cu at 400 ◦ C (ε̇ = 0.02 s−1 , ε = 0.7) [4] and Ardakani et al. Acta Metall. 42, 763 (1994)

47. Dynamic recrystallization I several models od dynamic recrystallization – growth of a dynamically recrystallized grain depends on the distribution and density of dislocations (both free and subgrain d.) I schematics – the boundary A moves to the right into unrecrystallized material with a high dislocation density ρm I passage of the GB reduces the dislocation density to almost zero I but the continued deformation raises the dislocation density in the new grain → ρ in the grain increases and eventually approaches ρm Schematic diagram of dislocation density at the dynamic recrystallization front [4]

48. Dynamic recrystallization I dislocation density inside the new grains increases as the deformation continues → limited growth, GB migration also impeded by nucleation of further grains → the process repeats I a critical deformation (εc) is needed to initiate dynamic recrystallization, this occurs slightly before the σmax I εc decreases monotonically with Z The effect of temperature on the stress-strain curves of a steel sample, ε̇ = 1.3 · 10−3 s−1 . The trend of the curves would be the same but reversed for different ε̇ at a constant temperature [4] and Petkovic et al. Can. Metall. Q. 14, 137 (1975)

49. Dynamic recrystallization I wavy character of stress-strain curves (low Z, i.e. rapid RX): material softens due to DRX processes → for further DRX, the deformation needs to be increased again – work hardening observed I slow RX (high Z) – the S-shaped RX curves overlap → dynamic equilibrium between work hardening and DRX → constant stress The relationship between stress-strain curves and the rate of recrystallization (S-shaped curves)

50. Dynamic recrystallization I dynamic recrystallization might not be complete after deformation is stopped I nuclei and small grains formed in the dynamic process can further grow and evolve through static recovery and recrystallization (when annealing continues or when cooling is slow) I very heterogeneous, partly dynamically recrystallized microstructure I this phenomenon known as metadynamic recrystallization

51. Grain growth I growth of recrystallized grains (or any grains in any polycrystalline material) I large grains grow at the expense of small ones I driving force – surface energy of the GBs I larger grains → smaller number of grains → lower GB area → lower surface energy I analogy with soap bubbles Growth of soap cells in a flat glass container. Numbers in the bottom right corners of each snapshot are times since the beginning of experiment [1] and C. S. Smith, ASM Seminar, Metal interfaces, 65 (1952)

52. Grain growth I bubbles – pressure difference across a curved surface due to surface tension γ ∆p = 4γ R , where R is the radius of curvature I larger pressure at the side concave toward the cell centre → net flow of atoms from high- to low-pressure region → decrease in size I e.g. small triangular cells – wall curvature in order to maintain the angle of 120◦ (i.e. the equilibrium angle for a junction of three walls with equal surface tension) I NB: cell geometry I less than six sides → concave walls → cell unstable I more than six walls → convex walls → cell tends to grow I six walls → straight sides (the only planar geometrical figure having the average internal angle between its straight sides of 120◦ is the hexagon) I NB: correlation between the size of the cells and the number of their walls

53. Grain growth I the number of sides of a cell may change – migration of the walls due to curvature, the boundaries try to reach the equilibrium angle for a junction of the given number of boundaries Illustration of a mechanism which changes the number of sides of a grain/cell during growth [1]

54. Grain growth I geometrical coalescence – encounter of two grains whose relative orientations are such that the boundary between them has much lower surface energy than that of an average boundary I higher probability in textured materials; large coalesced grain has a high number of sides → potential for rapid growth Two grains with similar orientation (A and B) meet as a result of disappearance of grain C. The resulting boundary ab is in principle a subboundary and the grains A and B may be regarded as a single grain [1]

55. Grain growth I generally, cells/grains must be considered in 3D Five types of basic 3D processes in grain growth. (B) - (C) and (D) - (E) are inverse processes [1]

56. Grain growth I in metals and other crystalline materials, the boundary migration occurs due to diffusion of atoms across the boundary from the concave side to the convex one, similar to soap bubbles I the reason for this diffusion is not widely agreed upon – a theory: tighter binding on the convex side due to the presence of more neighbours in the lattice Schematics of grain growth by boundary migration [2]

57. Grain growth law I assumption: metallic grain growth is the result of surface energy minimization and diffusion of atoms across the boundary → we can use the soap bubble analogy I the rate of growth is assumed to be proportional to the wall curvature c dD dt = K0 c, where D is the average grain diameter and K0 is a proportionality constant I assumption: the wall curvature is inversely proportional to the diameter of an average-sized grain dD dt = K D I after integration D2 = Kt + C I integration constant C from the initial condition D(t = 0) = D0 D2 − D2 0 = Kt

58. Grain growth law I diffusion of atoms across the boundary considered as an activated process → the constant K can be expressed as K = K0 exp − Q RT , where Q is the empirical activation energy of the process, T is the absolute temperature and R the gas constant I the grain growth law can be then rewritten as a function of both t and T D2 − D2 0 = K0t exp − Q RT I or in the logarithmic form log D2 − D2 0 t = − Q 2.3RT + log K0

59. Grain growth law log D2 − D2 0 t = − Q 2.3RT + log K0 I the quantity log D2 −D2 0 t is linearly proportional to the reciprocal of the absolute temperature, 1 T , with − Q 2.3R being the slope I for isothermal annealing at temperature T, D2 is a linear function of annealing time t The logarithms of the slopes of grain-growth isotherms are inversely proportional to the absolute temperature. Data for α-brass (Cu-10%Zn) [1] and P. Feltham et al. Acta Met. 6, 539 (1958)

60. Grain growth law I generally, most experimental isothermal grain-growth data correspond to the empirical law Dn − Dn 0 = Kt where n is either equal to 2 or greater, K and n are time independent and with increasing temperature, n decreases and approaches the value of 2 I NB: the grain growth law applies for average grain size, it does not tell us anything about the growth of individual grains I growth rate also impacted by impurity atoms (depending on their size and the degree of distortion they introduce into the lattice) or foreign particles which hinder the motion of grain boundaries → practical application – limiting grain growth, “pinning” of GBs

61. Secondary recrystallization I when normal grain growth is inhibited by particles/inclusions, secondary recrystallization often occurs after primary RX I local coarsening of the microstructure – abnormal growth of a small number of grains which grow at the expense of their smaller neighbours I exaggerated grain growth as a result of surface-energy considerations, not the strain energy of CW as in primary recrystallization I construction materials – secondary recrystallization undesirable as the large grains negatively impact the strength and ductility of the material

62. Schematic illustration of the main processes during annealing of deformed material: (a) deformed state, (b) recovered. (c) partially recrystallized, (d) fully recrystallized, (e) grain growth, (f) secondary recrystallization (abnormal grain growth) [4]

63. Questions 1. Explain why is it desirable in materials processing to cold work and then recrystallize the metallic materials. How do mechanical properties of the processed material change after each step and why? 2. Explain why strain hardening is not observed in some metals, such as tin and lead, during deformation at room temperature. 3. Would you expect ceramic materials to recrystallize? Why?

64. References [1] R. Abbaschian, L. Abbaschian, and R. E. Reed-Hill. Physical Metallurgy Principles. Cengage Learning, 2009. [2] W. D. Callister and D. G. Rethwisch. Materials Science and Engineering. Wiley, 2010. [3] D. R. Askeland, P. P. Fulay, and W. J. Wright. The Science and Engineering of Materials. Cengage Learning, 2010. [4] F. J. Humphreys and M. Hatherly. Recrystallization and related annealing phenomena. Elsevier, 2017. [5] D. Hull and D. Bacon. Introduction to Dislocations. Butterworth-Heinemann, 2001.

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