APRENSENTAÇÃO SOBRE O PROCESSO DE RECRISTALIZAÇÃO.

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.