Cellular Automata Modeling of Grain Coarsening and Refinement

during the Dynamic Recrystallization of Pure Copper

Ho Won Lee and Yong-Taek Im*

National Research Laboratory for Computer Aided Materials Processing, Department of Mechanical Engineering, KAIST,373-1 Gusongdong, Yusonggu, Daejeon 305-701, Korea

In this study, a cellular automata technique was developed to simulate a dynamic recrystallization process of pure copper. Moore’sneighboring rule was applied with partial fraction and time step control in the current approach to represent the grain growth kinetics moreaccurately. The cellular automata model developed in this study was applied to a simulation of the dynamic recrystallization of pure copperduring hot deformation and compared with the experimental flow stresses and grain sizes determined from hot compression tests for validation.The predicted results were in reasonably good agreement with the experimental results. The grain coarsening and refinement phenomena werealso investigated in detail. Finally, the effects of the process parameters on the microstructure and flow stress were investigated from varioussimulation results. [doi:10.2320/matertrans.M2010116]

(Received April 7, 2010; Accepted June 28, 2010; Published August 11, 2010)

Keywords: cellular automata, dynamic recrystallization, copper, grain coarsening, grain refinement, partial fraction

1. Introduction

In the automobile industry, there is currently a strongdriving force to produce high-strength metals owing toenvironmental and energy concerns. The basic strengtheningmechanisms include substructural strengthening, solid solu-tion strengthening, precipitation strengthening, grain bound-ary strengthening, and phase transformation strengthening.In particular, grain boundary strengthening through grainrefinement is known to enhance both the strength andtoughness of a material.1)

Abundant metallurgical phenomena are related to theevolution of the grain size. However, dynamic recrystalliza-tion (DRX) is one of the key processes that govern the grainsize of materials during hot deformation. DRX generallytakes place in metals such as copper and nickel whichhave low or medium levels of stacking fault energy. Duringthe DRX process, new grains originate at high-angle grainboundaries such as prior grain boundaries, the boundariesof recrystallized grain, and deformation bands and twins.The new grains grow to the high dislocation density of theoriginal grain due to the driving force, which is driven by thedifference in the dislocation density. However, recrystallizedgrains will cease to grow as the material deforms further,which has the effect of reducing the dislocation densitydifference. The resulting microstructure, especially the grainsize, varies with the deformation condition. Therefore,understanding the DRX is essential when attempting tocontrol the mechanical properties of a material.

Various models have been proposed thus far to predictmicrostructural changes during recrystallization. Typically,the Johnson-Mehl-Avrami-Kolmogrov (JMAK) theory2) iswidely used to model homogeneous recrystallization kinet-ics. However, the JMAK equation is not adequate for actualapplications due to the heterogeneous nature of the recrys-tallization process. Recently, many modeling methods havebeen proposed to solve this problem, such as the vertex

model,3) the Monte Carlo model,4,5) and the phase fieldmodel.6) Although these models successfully describe micro-structural evolution during recrystallization, the CA model isused most often due to its straightforward time and lengthscale calibrations. Cellular automata model for recrystalliza-tion was firstly introduced by Hesselbarth and Gobel.7) Theysuccessfully described recrystallization kinetics that was notpredicted by the JMAK theory by introducing inhomoge-neous characteristics of recrystallization. However, moststudies have focused on static recrystallization and DRX withgrain refinement phenomena.7–10) Grain coarsening phenom-enon has received less attention, although it occurs during hotdeformation processes such as forging and rolling becauseof local irregularities in the process parameters.

Therefore, in the present investigation, CA modeling ofgrain coarsening and refinement during the DRX of purecopper was conducted. To ensure isotropic grain growth,Moore’s neighboring rule was applied with a partial fractionand controlled time step in the CA model. To verify thedeveloped model, simulated microstructures and flowstresses were compared to the compression results. Graincoarsening and refinement phenomena related to the flowcurves were subsequently investigated in detail. Finally,to investigate the effect of process parameters such as thetemperature, strain, and strain rate on the microstructuralevolution and flow stress, numerous numerical simulationswere carried out under different conditions.

2. Experimental

A set of compression tests was conducted to investigate themicrostructural and flow stress changes during the DRXprocess. Commercially available pure copper was used in theexperiments. Its chemical composition is given in Table 1.To ensure a homogeneous initial microstructure, the rawmaterial was vacuum-annealed at 700�C for one hour andfurnace-cooled to room temperature. The initial microstruc-ture obtained from this process is given in Fig. 1, and theinitial grain size was measured as approximately 64 mm.*Corresponding author, E-mail: ytim@kaist.ac.kr

Materials Transactions, Vol. 51, No. 9 (2010) pp. 1614 to 1620#2010 The Japan Institute of Metals

Cylindrical specimens with a height of 15mm and a diameterof 10mm were prepared from the processed material bymachining. Hot compression tests were conducted using aGleeble machine at a constant temperature and strain rate.The compression temperatures varied from 500 to 700�Cand the strain rates varied from 0.002 to 0.01 s�1. First, thespecimen was heated to the working temperature at a heatingrate of 10�C/s. After a holding time of 30 s, the specimen wasdeformed up to a strain of 0.8. The compressed specimenwas water-quenched immediately after the deformation forfurther microstructural investigation. The quenched speci-men was cut in parallel along the compression axis and waspolished using SiC and diamond papers. Finally, electro-chemical polishing was conducted to prepare the electronbackscattered diffraction (EBSD) specimen.

An EBSD system (EDAX-TSL/Hikari) attached to fieldemission scanning electron microscope (FE-SEM, FEI/Nova230) was used in the current study to investigatethe microstructural changes that occurred during the DRXprocess. The acceleration voltage and working distance was20 kV and 11mm, respectively. Due to the different grainsizes, the scanned area was varied from 0:3� 0:3 to1� 1mm2 and the step size was varied from 1.2 to 5.0 mm.Finally, while neglecting twin boundaries, an equivalentcircle diameter (ECD)2) was calculated using an orientationimaging microscopy (OIM) map.

3. CA Modeling of the DRX

In the current investigation, two-dimensional square cellswere employed for the CA modeling. Every cell had fivestate variables: a grain number variable that represented

different grains; a variable representing recrystallized frac-tion; and three state variables representing whether or notthe current cell is recrystallized, nucleated, and/or locatedon the boundary, respectively. Conventional Moore’s andvon Neumann’s neighboring rules11) with a deterministictransformation rule cannot demonstrate isotropic growthkinetics. Whereas the von Neumann’s neighborhood containsbesides the central cell itself only the four cells directlyabove, below and besides the central cell, the Moore’sneighborhood contains the central cell and all eight cellsadjacent to it. As a result, the final shape of the grains iseither diagonal or square, both of which are unrealistic.Therefore, in the current study, Moore’s neighboring rulewas implemented by allowing a partial recrystallized fractionin a cell.

The fraction increment �F during the time increment �t

was calculated by summing the fraction from the interfacecell during each time increment, as follows:

�F ¼X

vi ��t=lengi ð1Þ

where vi is the incoming velocity from the i-th neighboringcell, �t is the time increment, and lengi represents thedistance between the current and i-th neighboring cell. Thegrain shape using Moore’s neighboring rule with a partialfraction and time step control can be nearly circular aftera certain time step compared to those using Moore’s andvon Neumann’s neighboring rules, as shown in Fig. 2.In addition, the time step was determined by finding theminimum time step to complete the grain growth of onepartial cell.

Cellular automata simulations were conducted with thefollowing steps. The process is also shown in Fig. 3.Calculation of the minimum time step to ensure betterresults; calculation of dislocation density changes by workhardening at each time step; nucleation of the recrystalliza-tion embryo by comparing the dislocation densities in thegrain boundaries with the critical dislocation density;simulation of the growth of the nucleus to the highdislocation density area.

Fig. 1 EBSD OIM map of prior microstructure after annealing.

Fig. 2 The growth aspects of grain using Moore’s neighboring rule with

partial fraction and time step control.

Table 1 Chemical composition of pure copper used in the current study.

Cu O Se S Pb Ag Sn Fe Ni

mass

%99.410 0.442 0.009 0.031 0.030 0.010 0.034 0.032 0.002

Cellular Automata Modeling of Grain Coarsening and Refinement during the Dynamic Recrystallization of Pure Copper 1615

To model the dislocation density changes by work hard-ening, the ‘one-parameter’ model by Kocks and Mecking12)

was used in the current study due to its simplicity andapplicability in a hot deformation process. The Kocks-Mecking (KM) model is based on the assumption that theaverage dislocation density determines the kinetics of theplastic flow. In the kinetic equation for hot deformation,the flow stress is proportional to the square root of thedislocation density, as follows:

� ¼ �Gbffiffiffi�

pð2Þ

In this equation, � is a numerical constant, G denotes theshear modulus, b is the magnitude of Burger’s vector ofdislocation, and � is the average dislocation density. Thechange in the dislocation density of a coarse-grained orsingle-phase material may be considered to consist of twocomponents, as follows:

d�=d" ¼ k1ffiffiffi�

p� k2� ð3Þ

Here, k1 ¼ �s � k2=�Gb, k2 ¼ f ðT ; _""Þ, and �s ¼ fA1 _"" expðQdef=RTÞg1=n1 . The initial dislocation density was assumedto be 109 for all of the initial grains, and the increment of thedislocation density was calculated for every time step usingeq. (3).

Nucleation in the dynamic recrystallization process occurswhen the dislocation density reaches a critical value. In thepresent investigation, the critical dislocation density pro-posed by Roberts and Ahlblom13) was used.

�c ¼ ð20� _""=3blM�2Þ1=3 ð4Þ

In this equation, l is the dislocation mean free path, M

represents the grain boundary mobility, � is the dislocation

line energy (expressed as � ¼ 0:5Gb2), and � is the grainboundary energy. In eq. (5), the grain boundary mobility (M)is given, as shown below.14)

M ¼ �D0bb=kT � expð�Qb=RTÞ ð5Þ

Here, � is the characteristic grain boundary thickness, D0b

denotes the boundary self-diffusion coefficient, Qb is theactivation energy for boundary diffusion, k represents theBoltzmann’s constant, and R is the gas constant. In eq. (4),the dislocation mean free path (l) was calculated from thefollowing equation, as originally formulated by Takeuchiand Argon:15)

ð�=GÞ � ðl=bÞn2 ¼ K1 ð6Þ

In eq. (6), n2 and K1 are material constants. Nucleation inthe dynamic recrystallization process occurs on the grainboundaries of prior grains. This assumption corresponds tothe grain bulging nucleation mechanism of dynamic recrys-tallization. However, if the dislocation density of therecrystallized grain reaches a critical value, the grainboundaries of the recrystallized grain can also be consideredas a possible position for nucleation. The dislocation densityof the recrystallized grain was set close to zero and theorientation was randomly selected.

The nucleation rate model suggested by Ding and Guo10)

was selected in the current study of nucleation. The modelconsiders that the nucleation rate for dynamic recrystalliza-tion is a function of both the temperature and strain rate, asfollows:

_nn ¼ C _""m1 expð�Qnucl=RTÞ ð7Þ

where C, m1, and Qnucl are constants. The driving force forthe growth of the nucleus in dynamically recrystallizedgrains is the stored strain energy difference between therecrystallized grain and the prior grains. A grain boundarymoves with a velocity (v) in response to the net pressure (P)on the boundary. It is generally assumed that the velocity isdirectly proportional to the pressure and that the constantof proportionality is the mobility (M) of the boundary, asfollows:2)

v ¼ MP ¼ M�ð�m � �dÞ ð8Þ

where �d and �m are the dislocation density of thedynamically recrystallized grain and that of the prior grainmatrix, respectively.

In the current investigation, the dynamic recrystallizationof pure copper under hot compression was simulated withvarious temperatures and strain rates. The initial micro-structure was created by the normal grain growth of the site-saturated nucleus with randomly selected orientations, andthe initial grain size was set as 64 mm, as initially measured.The simulation area was 1� 1mm2, and 62,500 squarecells with a periodic boundary condition were used. Theparameters of the CA model simulated are given in Table 2.

Start

End

Calculate time step dt

Calculate dislocation increment of every grain for dt

ρ ρi > c

nucleation

Nucleus growth

t > tend

Y

N

Y

N

Fig. 3 Schematics of the numerical procedure of cellular automata for

calculating the DRX.

Table 2 Parameters used in the CA model.

G

(N/m2)

b

(m)

�

(J/m2)

�D0b

(m3/s)

Qb

(kJ/mol)A1 n1

Qdef

(kJ/mol)

4:21� 1010 2:56� 10�10 0.625 1:09� 10�11 162 3:51� 1043 0.131 294

1616 H. W. Lee and Y.-T. Im

4. Results and Discussion

4.1 Experimental result and validation of the CA modelThe flow stress curves obtained by the hot compression

tests are shown in Fig. 4. The flow stress was higher forthe large Zener-Hollomon parameter (Z) described as Z ¼_"" expðQact=RTÞ, in other words, at a high strain rate and lowtemperature. The critical strain, necessary to initiate theDRX, decreased as the Z value decreased. The flow stresscurve at high Z values showed a broad single-peak (e.g.,500�C and 0.1 s�1). On the other hand, the flow stress curvesat low Z values showed oscillating multi-peak curves underlow strain and steady curves under high strain. It is knownthat single- and multi-peak curves are related to the prior andfinal grain sizes. If the prior grain size is greater than twotimes the recrystallized grain size, a single-peak curve mayoccur.16)

The relationship between the measured average grain sizeand Z is represented in Fig. 5. It was compared with thevalues from Blaz et al.17) Considering different measurementtechniques and materials, the current experimental resultshowed the general characteristics of the DRX process well.

To verify the developed CA model, flow stress curves andfinal grain sizes at various temperatures and strain rates arecompared with the experimental results, as shown in Fig. 6and Fig. 7, respectively. The proposed CA model predictedthe flow stress generally well, although it overpredicted thesteady state stress at 500�C and 0.01 s�1 and underpredicted

the flow stress at 600�C and 0.002 s�1. This occurred becausethe characteristics of the single- and multi-peak curves wererather different from each other. To represent both curvesaccurately, the CA model should be considered differentlyin some manner (e.g., nucleation).

In spite of the errors that resulted, the CA resultsrepresented the single- and multi-peak characteristics well.The flow stress curve at 500�C and 0.01 s�1 showed only asingle-peak curve because the ratio of the prior grain size andthe recrystallized grain was greater than two, as shown in

0.0 0.2 0.4 0.6 0.80

20

40

60

80

100

120 500 °C 600 °C 700 °C 800 °C

Strain,

0.01 s-1

0.0 0.2 0.4 0.6 0.80

10

20

30

40

50

60

70

80

Strain,

0.1 s-1

0.01 s-1

0.005 s-1

0.002 s-1600 °C

ε

Str

ess,

σ /

Mpa

Str

ess,

/M

pa

ε

σ

Fig. 4 Experimental flow stress curves for various deformation conditions.

1E11 1E12 1E13 1E14 1E15 1E16 1E17 1E18 1E190

10

20

30

40

50

60

70

80

90

100

Current [11]

Ave

rage

Gra

in D

iam

eter

, Drx

/µm

Zener-Holomonn Parameter, Z

Fig. 5 The relationship between the experimental grain size and Z.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

10

20

30

40

50

60

70

80

90

100

110

120

Experimental 500°C-0.01s-1

600°C-0.01s-1

600°C-0.005s-1

600°C-0.002s-1

700°C-0.01s-1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

10

20

30

40

50

60

70

80

90

100

110

120

500°C-0.01s-1

600°C-0.01s-1

600°C-0.005s-1

600°C-0.002s-1

700°C-0.01s-1

CA analysis

Str

ess,

σ/M

paS

tres

s, σ

/Mpa

εStrain,

Strain, ε

Fig. 6 The comparison of flow stress curves between the experimental and

CA model.

Cellular Automata Modeling of Grain Coarsening and Refinement during the Dynamic Recrystallization of Pure Copper 1617

Fig. 7. The predicted grain sizes were generally in goodagreement with the experimental results, although someerrors existed in the results with small grains.

4.2 Grain coarsening and refinementAs explained earlier, the shape of the flow stress curves is

related to the ratio of the prior and recrystallized grain size. Inother words, it is related to microstructural phenomena suchas grain coarsening and refinement during the DRX process.Therefore, the grain coarsening, refinement phenomena, andthe characteristics of the flow curves are investigated in detailin this study.

The simulated microstructure during grain coarsening bythe DRX process is given in Fig. 8, and the changes in theflow stress and average grain size are represented in Fig. 9. InFig. 8, similar to the single-peak DRX, the nuclei originatedat the prior grain boundary past the critical strain. However,most of the prior grain boundaries were consumed rapidlycompared to the single-peak DRX due to the rapid grain

boundary velocity and the larger recrystallized grain size. Asa result, no more nucleation occurred after a certain strain(about 0.13), and the grain structure was maintained untilnew nuclei originated at the recrystallized grain site. There-fore, the grain structures at the strains of 0.14 and 0.16 werevery similar. The flow stress also showed a curve similar tothat of the single-peak DRX up to a strain of 0.1, as shown inFig. 9. However, the flow stress increased again past a strainvalue of 0.12 owing to the lack of nucleation between theDRX cycles. The next DRX cycle was then initiated past acertain deformation, and the stress was decreased again byfurther nucleation. The average grain size was also oscillat-ing owing to the several cycles of the DRX that were run. Theaverage grain size initially decreased immediately after theinitiation of the current cycle of the DRX as a consequenceof the nucleation stage. It then increased rapidly due to thegrowth and remained constant until the next DRX cycle. Thisphenomenon was repeated for nine cycles for the currentcondition.

T500-

0.01

T600-

0.01

T600-

0.00

5

T600-

0.00

2

T700-

0.01

0

10

20

30

40

50

60

70

80

90

Experimental CA model

Prior grain size: 64µm

Multi peaks

Single peak

Ave

rage

Gra

in D

iam

eter

, Drx

/µm

Fig. 7 The comparison of final grain diameter between the CA model and

experimental observation.

0 0.1 0.12

0.14 0.16 0.18

Fig. 8 Changes of the microstructure during grain coarsening by the DRX.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

10

20

30

Ave

rage

Gra

in D

iam

eter

, Drx

/µm

Strain, ε

Str

ess,

σ/M

Pa

σ

60

70

80

90

100Drx

.T=700°C, ε=0.01 s-1

1st 2nd 3rd 4th 5th 6th 7th 8th 9th

Fig. 9 Changes of the flow stress and average grain size during grain

coarsening by the DRX.

1618 H. W. Lee and Y.-T. Im

The simulated microstructure during the grain refinementprocess with the single-peak DRX is given in Fig. 10, andthe changes in the flow stress and average grain size arerepresented in Fig. 11. In Fig. 10, the nuclei for the DRXprocess originated at the prior grain boundaries (the blackline in the figure) past the critical strain. The recrystallizedgrain then grew into the prior grain structure (the white areain the figure). The recrystallized grain underwent furtherdeformation and reached the critical dislocation densityagain. Therefore, the nuclei also originated at the boundariesof the recrystallized grains before consuming all of the priorgrain boundaries. This implies that more than one DRX cyclewas taking place simultaneously in the grain for the single-peak DRX. In Fig. 11, the slope of the flow stress waschanged past the critical strain (about 0.2) and reached itspeak at a strain of about 0.25. The flow curves then softenedand reached a steady state at a strain value of 0.4. Theaverage grain size curve changed rapidly twice due to thedifferent cycles of the DRX process. The average grain sizewas held constant up to the critical strain, decreased rapidly

directly past the critical strain, and then became saturated.However, it decreased rapidly again at a strain of 0.4 whenthe second DRX cycle occurred at the recrystallized grainboundaries.

The grain refinement flow stress curve by the DRX processcan also show a multi-peak result (e.g., 600�C and 0.005 s�1

in Fig. 6). In this case, the mechanism is similar to that ofgrain coarsening. Although the recrystallized grain size issmaller than the prior grain size, it is large enough toconsume the prior grain boundaries before the initiation ofthe next DRX cycle. The change in the average grain sizeis also observable in Fig. 12.

4.3 Effect of process parameters on the DRXTo investigate the effect of the process parameters on the

microstructure, a histogram of the grain size is representedin Fig. 13. The final grain size decreased when the strain rateincreased and the temperature decreased. This was due to therelationships among the nucleation rate, mobility, and theresulting grain size. If the strain rate increases, the nucleationrate will be increased. Therefore, the grain size decreases as

0 0.3 0.4 0.5

0.6 0.7 0.8

Fig. 10 Changes of the microstructure during grain refinement by the single-peak DRX (500�C and 0.01 s�1).

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

10

203040

50

60

7080

90

100110

σ

Str

ess,

σ/M

Pa

Strain, ε

.T=500°C, ε=1 s-1

25

30

35

40

45

50

55

60

65

Drx

1st cycle 1stand 2nd cycle

Fig. 11 Changes of the flow stress and average grain size during grain

refinement by the single-peak DRX.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

10

20

30

40

50

Str

ess,

σ/M

Pa

Strain, ε

σ

48

50

52

54

56

58

60

62

64

66.

Ave

rage

Gra

in D

iam

eter

, Drx

/µm

Drx

T=600°C, ε=0.005 s-1

Fig. 12 Changes of the flow stress and average grain size during grain

refinement by the multi-peaks DRX.

Cellular Automata Modeling of Grain Coarsening and Refinement during the Dynamic Recrystallization of Pure Copper 1619

the strain rate increases. If the temperature increases, thenucleation rate and mobility will increase together. However,the change in the growth rate is rather important compared tothe change in the nucleation rate in the current condition.Therefore, the result is that the grain size increases as thetemperature increases. In the upper three cases in the figure,the fraction of grains having smaller grain sizes than averageis larger. The deviation from the average grain size wasincreased when the temperature increased and the strain ratedecreased. The large deviation in the grain size can also beobserved in Fig. 8.

5. Conclusions

A cellular automata model using Moore’s neighboring rulewith a partial fraction and controlled time step for theanalysis of dynamic recrystallization was successfully de-veloped in the present study. To validate the developedmodel, flow stresses and recrystallized grain sizes weredetermined experimentally and compared with the CAresults. The predicted result was generally in good agreementwith the experimental results, although the former showedsome errors in particular conditions. The grain coarseningand refinement phenomena were also investigated. In termsof grain refinement, the flow stress curves showed both asingle-peak and multiple-peaks up to the recrystallized grainsize, although a multi-peak curve was noted during the graincoarsening phenomena. Finally, the effects of the processparameters were investigated in this study to derive resultsthat matched those in earlier findings.

Acknowledgements

The authors wish to acknowledge the grant of NationalResearch Laboratory program of the Ministry of Education,Science and Technology through National Research Foun-dation (No. R0A-2006-000-10240-0) and BK21.

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0 40 80 120 160 200 2400.00

0.02

0.04

0.06

0.08

0.10

0.12

500°C, 0.01s-1

0 40 80 120 160 200 2400.00

0.02

0.04

0.06

0.08

0.10

0.12 600°C, 0.01s-1

0 40 80 120 160 200 2400.00

0.02

0.04

0.06

0.08

0.10

0.12

600°C, 0.005s-1

0 40 80 120 160 200 2400.00

0.02

0.04

0.06

0.08

0.10

0.12 600°C, 0.002s-1

0 40 80 120 160 200 2400.00

0.02

0.04

0.06

0.08

0.10

0.12 700°C, 0.01s-1

Grain Diameter, D/µm Grain Diameter, D/µm Grain Diameter, D/µm

Grain Diameter, D/µm Grain Diameter, D/µm

Num

ber

frac

tion,

f

Num

ber

frac

tion,

f

Num

ber

frac

tion,

f

Num

ber

frac

tion,

f

Num

ber

frac

tion,

f

Fig. 13 Distributions of the grain size with different deformation conditions.

1620 H. W. Lee and Y.-T. Im