1. Microstructure From Processing: Evaluation and Modelling Grain size control: Lecture 3
Martin Strangwood,
Phase Transformations and Microstructural Modelling, School of Metallurgy and Materials

2. Introduction
Thermal exposure generally increases grain size and the control is by (i) transformation, (ii) solute drag or (iii) Zener drag
Added deformation leads to an increase in stored energy that can be used to form ‘new’ grains

3. Starting state
Annealing converts a high energy state (deformed) into a lower energy one and so the processes are driven by the energy of the deformed starting state
Deformation introduces a number of defects:
Point defects - vacancies and interstitials (latter only really for irradiated material); vacancies often arise from dislocation motion, e.g. jogs
Vacancies
Dislocations
Jog

4. Starting state
Annealing converts a high energy state (deformed) into a lower energy one and so the processes are driven by the energy of the deformed starting state
Deformation introduces a number of defects:
Line defects - dislocations
Planar defects - stacking faults and boundaries
Defects store excess energy in elastic strain fields around them and annealing removes them at a rate dependent on the type and number of defects

5. Defect behaviour Energy Number Mobility Overall Vacancies Dislocations
Stacking faults
Grain boundaries
Low
High
High
High
Low
Low

6. Defect behaviour Energy Number Mobility Overall Vacancies Small
Dislocations
Dominant
Stacking faults
Secondary
Grain boundaries
Tertiary
Low
High
High
High
Low
Low

7. Stages in annealing
The dominant nature of dislocations means that these determine the stages of annealing
Recovery - re-arrangement of dislocations, but little reduction in numbers

8. Stages in annealing
Primary recrystallisation - removal of excess dislocations reducing stored energy as new dislocation-free grains form
Uniform grain growth - reduction in grain boundary area
Secondary recrystallisation - non-uniform grain growth
Tertiary recrystallisation

9. Deformed structure - Cu
FCC structure with high stacking fault energy
Structures imaged on normal-transverse plane after cold deformation
Rolling direction

10. Deformed structure - Cu
Cells
Up to 10 % 
~ m diameter
{111}
Microbands ~ 40 x 0.2 m
> 10 % 

11. Deformed structure - Cu
Microbands rotate parallel to rolling plane
Increasing 
35°
Macroscopic shear bands form - thin sheets of elongated narrow cells
Higher 
These features are associated with nucleation of recrystallised grains

12. Deformed structure - 70:30 brass
FCC structure with low stacking fault energy
Initially just a random arrangement of dislocations with no clear defined cell structure
% cold reduction
Very thin deformation twins form on planes of high shear stress, ~ 0.02 m thick
% cold reduction
Twins rotate to become parallel to sheet surfaces

13. Deformed structure - 70:30 brass
% cold reduction
Macroscopic shear bands form at lower strains than in high SFE materials
Random arrangement of dislocations

14. Recrystallisation and softening
High SFE, e.g. Cu
Primary recrystallisation Hv
Work hardened
50 % softening = 50 % primary recrystallisation
Annealing temperature

15. Recrystallisation and softening
Low SFE, e.g. brass
Recovery
Primary recrystallisation Hv
Work hardened
Excessive recovery leads to retardation of recrystallisation
50 % softening
50 % primary recrystallisation
Annealing temperature

16. Laws of recrystallisation
Minimum deformation needed to initiate recrystallisation
Activation barrier
Smaller amounts of deformation need higher recrystallisation temperatures
Thermally activated
Increasing annealing times reduce temperature needed for recrystallisation
Diffusional growth

17. Laws of recrystallisation
Recrystallised grain size depends on on degree of deformation (chiefly) and annealing temperature (to a lesser extent); smaller grains result from larger degrees of deformation and lower annealing temperatures
Nucleation rate effects
As original grain size increases, greater amounts of deformation are required for equivalent recrystallisation time and temperature
Driving force and nucleation site reduction
The amount of ‘cold’ work needed for equivalent hardening increases as deformation temperature increases
Rearrangement of defects during deformation and reduced driving force

18. Laws of recrystallisation
New grains do not grow into deformed grains of similar orientation
Texture affects recrystallisation
Continued heating can cause the recrystallised grain size to increase (not always the case)
Grain boundaries can drive some annealing processes

19. Dislocation stored energy
Dislocations store the greatest amount of excess energy in a cold deformed metallic alloy and primary recrystallisation is driven by the removal of excess dislocations
The size of this driving force can be estimated in two ways

20. Stress-strain curve Hot working?
Al-based alloys can have a fairly flat work hardening curve
200
Work done
Stress
Strain
0.5
~10 % is stored; P1 ~ 107 J m-3
Hot working?

21. Dislocation density
Excess dislocations store energy is strain energy around the extra half plane
Dislocation energy per unit length can be multiplied by dislocation density (length / unit volume) to give total energy
 = shear modulus; b = Burger’s vector

22. Dislocation density in Al
For a dislocation density of N dislocations per unit area of slip plane, then:
Recovery reduces the value of 

23. Measuring stored energy?

24. Measuring stored energy
DSC
XRD
EBSD
TEM

25. SIBM
Grains with different Schmid factors will start yielding under different applied stresses
During deformation the Schmid factors change as the sample gets longer and thinner so that the slip planes and directions rotate
During cold rolling adjacent grains will undergo different levels of strain and so have different stored energies (dislocation densities) B A
Dislocation density of A < dislocation density of B

26. Boundary bulging
If the stabler A grain is allowed to grow into the less stable grain B by bulging of the A/B grain boundary then there will be a decrease in free energy (Gv) but an increase in interfacial energy
Original A/B boundary
Bulged A/B boundary B A L
Bulge is stable and can grow if:
Schmid factor can give up to 10 % variation, hence viable

27. Intragranular nucleation
Grain refinement would require greater nucleation and the formation of new recrystallised grains within the deformed grains would also block the growth of those from grain boundaries
During deformation (depending on SFE) sub-grains can form separated by boundaries (SGB) with a few degrees misorientation
Under the correct processing conditions transition bands can form across grains and act as intragranular nucleation sites for recrystallised grains

28. Transition bands
Transition bands consist of a large number density of fine sub-grains, which achieve a large orientation change rapidly
{111}
Transition bands
The high number density of fine sub-grains gives conditions for SIBM with the misorientation established allowing growth into the surrounding sub-grains

29. Transition band formation
As plastic deformation proceeds the slip planes and directions rotate towards the tensile axis (i.e. towards or away from <110> depending on sense of stress and crystal structure)
Rotation away from <110> can be either to <100> or <111> so that parts of a grain can rotate one way and other parts the other way; these two parts are separated by a transition band

30. Transition band formation
BCC
FCC
Tension
Transition bands
Compression

31. Dutta-Sellars Equations

32. Dutta-Sellars Equations
The D-S equations have a relatively simple form and are easy to apply with clearly identified parts, that have been shown to fit data well
Original fit data

33. Dutta-Sellars Equations
The original fitting data used was limited to 0.3 strain; 0.03 – 0.04 wt % Nb; and a mean prior austenite grain sizes in the range 12 – 100 μm
Application of the D-S equations to more recent datasets has revealed less good agreement
Improvement of the D-S approach is needed along with expansion to more realistic aspects
Subsequent data

34. Re-crystallisation Ranges
Initially a wt % Nb steel was used that had been homogenised at 1225 °C for four days
Re-heating at 1225 °C for one hour prior to Gleeble testing gave a uni-modal large grain size ( μm) for which the original D-S equations should apply

35. Grain Size Distributions
Relationships of the type
were replaced by a simple halving of the original grain size size on recrystallisation – this effectively limits each prior austenite grain to the formation of two recrystallised grains
This allowed each class in the starting grain size distribution to be treated separately to build up a recrystallised grain size distribution (a similar approach has been used for TSDR material by the CEIT group).

36. Grain Size Distributions
This approach gives good quantification with the percentage recrystallised and for grain size distributions both unimodal and bimodal

37. Grain Size Distributions

38. Literature Values
A grain class-based approach improves the fit for our experimental data and for literature data. The outliers generally have high (0.09 wt %) and low (0.02 wt %) [Nb] or higher [N].
Birmingham data
Literature data
The grain-size class approach expand the original D-S equations and should be used as standard for future applications

39. Segregation Effects during Hot Rolling
The presence of solute-rich and solute-depleted regions will alter:
Degree of dissolution during re-heating (already mentioned)
Driving force for re-precipitation of strain-induced precipitates during deformation and holding
Solute drag
The differing Nb levels (currently experimental but to be predicted) can be used as input for the D-S equations with the appropriate grain size distribution

40. Segregation Effects during Hot Rolling
For an average grain size this leads to a widening of the partial recrystallisation regime

41. Segregation Effects during Hot Rolling
For the use of a class-based prediction then assumptions need to be made regarding the fraction of the each class in solute-rich and solute-depleted material
The D-S equations are then applied for each class and composition

42. Recrystallisation Prediction
The use of different compositions for different regions gives good agreement with Gleeble simulations (single hit)

43. Grain Size Distributions
The class-based segregated approach gives reasonable agreement for grain size distribution
1225/1025
1225/975
The extent of recrystallisation for different grain size classes changes with deformation temperature

44. Grain size distributions
Grain size distributions were evaluated in terms of D5% (grain size class constituting the first 5 area% of the total area measured), Dmode (the mode grain size class) and Dmax (the largest grain size class in the distribution).
Dmode
Dmax
D D5% 5%

45. Effect of strain and initial grain size on recrystallised grain size
The recrystallised grain size decreases with an increase in strain for all initial grain sizes during single hot deformation – shown here for mode grain size:
Whilst the mode grain size decreases as the applied strain increases, the rate of refinement decreases and the initially different grain size values converge at high strain.

46. Predicting the recrystallised grain size distribution
The modified equation (exponents derived from the classical theory for rate of nucleation) was used to predict the grain size distributions
Grain boundary nucleation is assumed in deriving the equation
Original equation
Modified equation
D’ values were fitted to the measured grain size distribution in terms of D5%, Dmode and Dmax for Fe-30Ni using the original equation in order to establish the best fit D’ values

47. Best fitted D’ values for predicting the D5% Dmode & Dmax D5%, Dmode & Dmax in the µm, µm & µm distributions for different applied strains
D’ represents the ‘efficiency’ of nucleation of new grains and is dimensionless.
For D5% Dmode & Dmax D’ values increase steadily with increasing strain for the smallest initial grain size distribution
A flattish region is observed with low strain for the largest initial grain size distribution
A lower D’ value for the Dmode and D5% might suggest that fine grains recrystallise first due to higher stored energy.

48. Best fitted D’ values for predicting the D5% & Dmax in the µm, µm & µm distributions for different applied strains
D5% & Dmax have similar trends to those observed for the Dmode D’ values.
D’ values increase steadily with increasing strain for the smallest initial grain size distribution
A flattish region is observed with low strain for the largest initial grain size distribution

49. Variation of stored energy with initial grain size
The deformation curves shown are for the µm, µm and
µm samples deformed to a strain of 0.3 cold strain
The finest initial grain size distribution (50-60 µm) material shows a greater work hardening rate than the other samples, which is consistent with a higher stored energy for the same plastic strain

50. Example distribution fit - 110-120 micron mode, 0.7 strain
D’ = 0.83 reported in literature for predicting Carbon Manganese (C-Mn) steels does not give a good fit to the measured grain size distribution Carbon Manganese (C-Mn) steels
D’ = 1.1 reported in literature for predicting Nb - containing steels was used to show that the literature equations can predict the mode grain size
The variable D’ approach gives a good fit to the measured grain size distribution

51. Summary of RMS errors for predicted grain size distributions in highly alloyed steels
RMS error
Alloy type
Initial mode grain size / µm
Strain
D' variation
Constant D’ approach (1.1)
Strip steel
0.15
3.14
7.71
0.3
7.61
16.00
0.45
8.00
21.87
High Aluminium steel
25-30
0.7
27.92
11.70
7.00
9Cr forging steel
10.83
12.50
4.42
18.80
6.90
17.00
The fit between measured and predicted grain size distributions for the variable D’ approach deteriorates as the complexity of the alloy increases and at lower strain values, although in almost all cases the fit is better than for a constant D’ approach

52. Control of grain size
The coverage of distributions has been extended to consider
strain and grain size (including the order of recrystallisation)
rates of recrystallisation
growth rates (and particle distribution)