Computational Materials Science 107 (2015) 122–133

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Computational Materials Science

journal homepage: www.elsevier .com/locate /commatsci

Microstructure simulation on recrystallization of an as-cast nickel basedsingle crystal superalloy

http://dx.doi.org/10.1016/j.commatsci.2015.05.0200927-0256/� 2015 The Authors. Published by Elsevier B.V.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

⇑ Corresponding author. Tel.: +86 10 62795482; fax: +86 10 62773637.E-mail address: scjxqy@tsinghua.edu.cn (Q. Xu).

Li Zhonglin, Xu Qingyan ⇑, Liu BaichengKey Laboratory for Advanced Materials Processing Technology (Ministry of Education), School of Materials Science and Engineering, Tsinghua University, Beijing 100084, China

a r t i c l e i n f o

Article history:Received 28 March 2015Received in revised form 15 May 2015Accepted 16 May 2015Available online 5 June 2015

Keywords:RecrystallizationAs-castSingle crystal superalloysCellular automaton

a b s t r a c t

Recrystallization (RX) in an as-cast single crystal (SX) nickel-based superalloy was investigated usingsimulation and experiments. One cellular automaton (CA) method was proposed to predict RXmicrostructure of SX superalloys. The stored energy was obtained using one macroscopicphenomenon-based elastic–plastic model in this research, and the orthotropic mechanical propertieswere taken into account. Kinetics parameters were used on the basis of physical fundamentals, and dif-ferent values were employed in the dendritic arms and interdendritic regions. In order to validate thesimulation model, heat treatments under inert atmosphere were conducted on compressed cylinder sam-ples to induce RX. The RX microstructures on the middle section perpendicular to the cylinder axis wereobserved using EBSD technique. Both simulation and experiments show that the kinetics of recrystalliza-tion were significantly different in dendritic arms and interdendritic regions, and simulated microstruc-ture agrees well with experimental. The proposed model in this research can predict the kinetics,microstructural evolution during recrystallization of as-cast SX superalloys.

� 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Nowadays, nickel-based single crystal (SX) superalloys havebeen widely used for turbine blades. However, great care shouldbe taken to prevent defects such as stray grains [1,2], freckles [3–5], recrystallization [6–8], etc. Recrystallization (RX) can introducehigh-angle grain boundaries and degrade the creep [8,9] and fati-gue [10] properties significantly.

RX, which is intolerant in single crystal components, may ariseduring heat treatment and service term, and can be ascribed to theplastic deformation during manufacturing process. RX behaviors inwrought [11,12], powder metallurgy [13] and oxide dispersionstrengthened [14] superalloys have been studied by manyresearchers. Most of previous work on RX of SX superalloys focuseson the influence of annealing conditions [15,16] and orientationaldependence [17,18], microstructural features [19,20] and mechan-ical properties [8,21,22]. In addition, modeling and simulation havebeen employed to predict plastic strains and sites where RX canoccur [6,7,23,24]. However, microstructural simulations of recrys-tallization in SX nickel-based superalloys were rarely reported.

Till now, simulation on RX behavior has been conducted inmany materials [25–30], most on aluminium [31–35], magnesium

[36,37], and steel [38,39]. Nevertheless, little work has been doneon the simulation of RX in SX nickel-based superalloys. Manymethods, such as monte carlo [40,41], phase field [28,42] and cel-lular automaton [34,43,44], have been employed to predict themicrostructural evolution during RX process. Among these models,CA has gained the highest popularity for its simplicity and highercalculation efficiency. Difficulty of predicting RX microstructureof SX superalloys may result from two facts. First, this alloy consistat least two phases, and secondly little was known about the kinet-ics. The first problem can be solved by treating the gamma andgamma prime phases as only one phase, taking account into thehighly similar lattice parameters. The other one can hardly besolved in previous work, because the growth kinetics of RX in SXsuperalloys has been little researched and it is difficult to obtainthe kinetic parameters from experiments. One microstructuralsimulation on RX of SX superalloys was reported by Zambaldiet al. [18], and the activation energy for RX grain boundary motionwas set as high as 1290 kJ/mol with no physical basis given. Thisvalue violated the real condition obviously, though simulationand experimental results can agree with each other in his research.Furthermore, his model seems a little simple and many criticaldetails were not given.

In the present paper, one modified CA model was proposed toconduct the simulation of RX in as-cast SX superalloys.Driving force for RX was obtained using one macroscopic

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Z. Li et al. / Computational Materials Science 107 (2015) 122–133 123

phenomenon-based deformation model, considering the aniso-tropy of SX superalloys. Key simulation parameters were employedbased on physical fundamentals in this research. To make themodel more accurate, the influence of as-cast dendritic microstruc-ture on RX kinetics was also taken into account. Different growkinetics parameters were used in the dendritic arms and the inter-dendritic regions. In this research, experimental and simulated RXmicrostructures were compared, and this model can give us a bet-ter understanding of the influence of as-cast inhomogeneity on RXin SX superalloys.

2. Mathmatical models

2.1. Driving force

Deformed metal or alloys will experience three main stagesduring heating: recovery, recrystallization and grain growth, asshown in Fig. 1. The effect of recovery is strong for metals andalloys (such as Al) with high stacking fault energy (Al [45],166 mJ/m2). The stacking fault energy of pure Ni [46] is about125 mJ/m2, while most research indicates that the value for mostsingle crystal nickel-based superalloys is below 20 mJ/m2, as aresult of alloying elements Re, Mo, Nb, W, etc. [46–49]. Thus, therecovery of SX superalloy is very weak, which is also demonstratedby some experiments [16]. Therefore, recovery will be omitted inthis research.

Generally, deformed stored energy provides the driving forcefor recrystallization of deformed metals and alloys during heattreatment. In most previous work, the deformation stored energywas expressed in terms of dislocation density usingdislocation-based constitutive models [50,51]. However, thesemodels can hardly describe the deformation behaviors of SX super-alloys due to the large amount of the coherent precipitates.Another method is crystal plasticity finite element [18,25,52](CPFEM), which has gained a great popularity in describing theheterogeneous characteristics of deformation on mesoscale duringlast two decades. Nevertheless, it can hardly describe thestored-energy distribution on the entire component scale as aresult of its limitation of computing amount.

Hence, one macroscopic phenomenon-based deformationmodel is adopted in the present modeling to obtain the drivingforce for RX. This model can describe the mechanical behavior ofSX nickel-based superalloys, and gain popularity for its conciseequations as well as calculation speed. In this model,nickel-based single crystal superalloys can be treated as orthotro-pic materials, though inhomogeneity exists in single crystal com-ponents [6,7]. An elastic–plastic model was employed to describethe mechanical behavior, together with the elastic orthotropyand the Hill yield criterion. Fig. 2 shows Young’s modulus in all ori-entations at 980 �C and 1070 �C. High anisotropy of SX superalloyDD6 can be derived, and elastic modulus decreases with increasingtemperatures.

Most of the work expended in deforming a metal is given outand only a very small amount (�1%) remains as energy stored in

Fig. 1. General change of deformed metal during heat treatment.

the material [45]. Hence, the driving force (P) for recrystallizationcan be calculated using the following form:

P ¼ 0:01Upl ð1Þ

where Upl denotes plastic dissipated energy, and is obtained asfollows:

dUpl ¼Z epl

0r : depl ð2Þ

where r and epl are the stress and plastic strain vector. Hill’s flowfunction is introduced in the following form:

f ðrÞ¼ 1ffiffiffi2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðr22�r33Þ2þðr33�r11Þ2þðr11�r22Þ2þ2Kðr223þr231þr212Þ

qð3Þ

Here, K denotes anisotropic plastic parameter, and can beobtained by tensile or compression tests of the material in differentorientations. Isotropic hardening criterion was employed in thispaper. One commercial available FEM software (Abaqus) was usedfor the deformation calculation. More details and model parame-ters can be found in Ref. [6].

2.2. Nucleation model

Nucleation tends to occur in sites with high deformation storedenergy. In this research, nucleation model takes account into realphysical influences, e.g., annealing temperature and as-cast con-centration inhomogeneity. A continuous nucleation law is imple-mented. The nucleation rate _N is controlled by the followingequation:

N ¼ C0ðP � PcÞ exp �Q aRT

� �ð4Þ

with R the universal gas constant, T the absolute temperature, andQa the activation energy, and P the driving force, which can bedetermined using above deformation model. C0 is a scalingparameter (1.0 � 109 s�1 J�1 in this research). Pc is the criticalstored energy below which recrystallization will not occur. Qa hasdifferent values in dendritic arms (DAs) and interdendritic regions(IDRs).

The value of Pc was computed with the following relation [53]:

Pc ¼ 107ec

2:2ec þ 1:1 clagb ð5Þ

where ec is the critical plastic strain (here set to 2%) and clagb is thelow angle grain boundary energy, which is assumed as 0.6 J/m2 inthe present work.

2.3. Recrystallization and grain growth model

The movement of the recrystallization front is treated as astrain-introduced boundary migration process. The contributionof driving pressure arising from boundary curvature is neglectedfor its relative small amount in this stage. Thus, the main drivingforce in this stage is also the deformation stored energy. The veloc-ity of the RX front, V, moving into the deformed matrix can beexpressed as follows:

V ¼ MP ð6Þ

where M is the grain boundary mobility for the primary staticrecrystallization and P is driving pressure for the grain boundarymovement. The mobility can be estimated by

M ¼ M0 exp �QbRT

� �¼ D0b

2

kTexp �Q b

RT

� �ð7Þ

Fig. 2. Oritentational dependence of Young’s Modulus of SX superalloy DD6 at 980 �C (a) and 1070 �C (b).

124 Z. Li et al. / Computational Materials Science 107 (2015) 122–133

where D0 is the diffusion constant, b is the Burger’s vector, k is theBoltzmann constant and Qb is the activation energy for grain bound-ary motion. The grain boundary motion is controlled by the solutediffusion process from physical views. Temperature dependenceof the mobility is also incorporated. Different values for Qb are setfor DAs and IDRs to simulate the influence of as-castmicrostructure.

Grain growth is assumed to occur after the recrystallized frontsimpinge on each other. In this case, grain boundary curvature is thedriving force for boundary motion, which can be expressed by

P ¼ cj ð8Þ

where c is the grain boundary energy and j is the grain-boundarycurvature. In this model, the grain boundary energy is assumed tobe dependent on the misorientation angle h between two neighbor-ing grains. The grain boundary energy c can be calculated by theRead–Shockley equation [54]:

c ¼ cmhhm

� �1� ln h

hm

� �� �ð9Þ

with h the grain-boundary misorientation, cm the large angle grainboundary energy, and hm the large angle grain boundary misorien-tation (set to 15�). The grain boundary curvature can be calculatedusing one equivalent model [55], which considers the topologicalrelations between the current site and neighbor sites for a squarelattice.

In the present work, the grain orientation is represented usingEuler angle (u1 U u2). The misorientation h can be calculated by

h ¼min cos�1 trðO432DgÞ � 12

� � ð10Þ

where O432 is the symmetry operator. For FCC materials, there are24 equivalent orientations for each orientation [56]. Therefore,there are 24 transforming matrixes in the symmetry operator. Dgdenotes the transformation matrix between two neighboring grains(grain A and B), and can be expressed as

Dg ¼ gBg�1A � gAg�1B ð11Þ

The transformation matrix g of a grain can be calculated as

g ¼cos u1 cos u2 � sinu1 sin u2 cos U sinu1 cos u2 þ cos u1 sin� cos u1 sin u2 � sinu1 cos u2 cos U � sinu1 sinu2 þ cos u1 co

sinu1 sin U � cos u1 sin U

264

2.4. Influence of as-cast dendritic microstructure

Inhomogeneity always exists in as-cast nickel-based SX super-alloys, as shown in Fig. 3(a)–(c). Dendrites present in as-castmicrostructure, with fine and regular cubic gamma prime phasein dendritic core regions, coarse and irregular cubic gamma primephase in interdendritic regions. Bulky eutectic structure appears ininterdendritic regions.

Dendritic microstructure has a significant influence on RX dur-ing solution treatment [57]. Doherty suggested that coherent par-ticles will be at least four times effective than incoherent particlesin restraining grain-boundary motion [58]. Thus, solution behaviorof c0 phase is significant for the grain boundary motion during RXin SX nickel-based superalloys. It has been proven that RX initiallynucleates in the dendritic arms and grows rapidly at low tempera-tures, and it is difficult for grain boundary migration in the inter-dendritic regions (as shown in Fig. 3(d)). This can be ascribed totwo aspects: the morphology of the c0 phase (Fig. 3(a)–(c)) andsolute microsegregation. Finer c0 phase particles in dendritic armscan facilitate the solution. In addition, the c0 forming elements areamong the most strongly partitioning elements, and they are richin the interdendritic regions, causing the full solutioning tempera-ture to be much higher in IDRs than in the DAs.

Therefore, the influence of as-cast dendritic microstructure istaken into account in this research to make simulation more accu-rate. Different kinetics parameters are chosen for DAs and IDRs toachieve this. All cells are divided into two groups: DAs-Cell andIDRs-Cell. DAs-Cell represents the region where c0 phases are fullysolved at heating temperature, while IDRs-Cell represents wherethere are remaining c0 phases. Clearly, the fraction of DAs-Cell willincrease with temperatures, as shown in Fig. 4. In this research, thecell state of DAs of IDRs was determined based on the image pro-cessing technology, which is based on optical microstructure.Another possible method is based on the solution segregation.The cell state can be determined using the thermo-dynamic calcu-lation, based on as-cast dendritic results.

Activation energy for nucleation and recrystallization (graingrowth) in these two kinds of cells are given in Table. 1. It shouldbe noted that eutectics in IDRs are ignored. In addition, annealing

u2 sin U sinu2 sin Us u2 cos U cos u2 sin U

cos U

375 ð12Þ

Fig. 3. (a) Optical as-cast dendritic microstructures; (b) SEM showing finer c0 particles in DAs; (c) SEM showing coarse and irregular c0 particles in IDRs; (d) EBSDmicrostructures of recrystallized samples annealed at 1280 �C for 5 min.

Fig. 4. The distribution of dendritic state cells at the same region and different temperatures: (a) 1280 �C; (b) 1300 �C.

Z. Li et al. / Computational Materials Science 107 (2015) 122–133 125

Table 1Key parameters in the deformation model and CA model.

Parameter Value Unit

Activation energy for nucleation in DAs, Qa1 285 kJ/molActivation energy for nucleation in IDRs, Qa2 265 kJ/molActivation energy for recrystallization in DAs,

Qm1

250 kJ/mol

Activation energy for recrystallization in IDRs,Qm2

340 kJ/mol

Time step 0.01 sLarge angle boundary energy, cm 0.9 J/m2

Low angle boundary energy, clagb 0.6 J/m2

Critical plastic strain for recrystallization, ec 0.02Burgers vector, b 0.36 nmDiffusion constant, D0 7.5 � 10�4 m2/sBoltzman constant, K 1.38 � 10�23 J/KUniversal gas constant, R 8.3144 J/

(mol K)Cell size, dx or dy 0.0025 or

0.005mm

Transformation fraction of driving force 0.01

126 Z. Li et al. / Computational Materials Science 107 (2015) 122–133

twinning is also not taken into account in this model though itplays a crucial role in RX of SX nickel based superalloys, becauseit0s difficult to describe twinning on the micro-scale.

3. Cellular automaton algorithm

In this paper, a deterministic CA model is utilized to simulateRX of SX nickel-based superalloys during heat treatment, and thealgorithm is presented in detail in Fig. 5. A 2D square lattice (cell

Fig. 5. The cellular automata static re

size 2.5 lm or 5 lm) is employed. A von Neumann’s neighbor rule,which consider the nearest four neighbors cells, is used. The mate-rial consists of an ordered gamma-prime (c0, Ni3Al) precipitateswith L12 structure coherently embedded in a gamma (c) austeniticphase. Thus, gamma and gamma prime phases were treated as onephase in the simulation. The state of each cell site is characterizedby the following variables: the dendritic state variable indicatingwhether the cell belong to DAs or IDRs, the grain number variablerepresenting the different grains, the stored energy variable repre-senting the driving force, the order parameter variable indicatingwhether it belongs to RX boundaries and the fraction variable rep-resenting the recrystallized fraction. Some key parameters for thisCA model are shown in Table. 1.

The essentials of this CA model are summarized, as follows:

(a) Update the dendritic state variable, and obtain the storedenergy from FEM modeling results using linear interpolationmapping algorithm.

(b) Apply nucleation model according to Eqs. (4) and (5) if thereexists the deformed matrix, and the stored energy in nucle-ated cells is set to zero.

(c) Obtain the mobility M of the grain boundary in Eq. (7), anddetermine the driving force: stored energy or grain bound-ary energy via Eqs. (8)–(12) using a Von Neumannneighborhood.

(d) Calculate the velocity of a boundary cell and determining thedirection of the motion.

(e) Calculate the fraction of a recrystallization front cell or aboundary cell as follows:

crystalli

zation algorithm schema.

Table 2Nomina

Elem

wt.%

Z. Li et al. / Computational Materials Science 107 (2015) 122–133 127

f ¼ Vxdtdxþ Vydt

dy� VxVydt

2

dxdyð13Þ

where Vx and Vy are the velocity in X and Y directions, dx anddy are the sizes of a cell along the X and Y axes, and dt is thelength of the time step.

(f) Reassign the cell state variable of a recrystallization frontcell or a boundary cell if the boundary migrates through it(f P 1).

4. Experimental details

Experiments were designed and conducted to evaluate theaccuracy of the model. The second generation single-crystal super-alloy DD6 in as-cast condition was used. The chemical compositionis given in Table. 2.

In this research, hot compression tests were chosen to obtain10–12% plastic strain, as driving force for RX in single crystalsuperalloys. Cylinder test pieces was cut from as-cast test barsusing EDM (electrical discharge machining). The diameter is6 mm and the length is 10 mm (Fig. 6). Compressive tests wereconducted on Gleeble1500D (thermal physical simulator) at980 �C at a strain rate of 3 � 10�3 s�1. Only test samples within15� of h001i along the axis were employed. Test pieces were heldfor 1 min at testing temperatures before compression. Every com-pression sample was cut into two same smaller cylinders usingEDM, and tubed in silica glass under inert argon atmosphere toavoid oxidation. The samples were annealed at 1280 �C and1300 �C, and cooled in the air. The annealing times were 5 min,10 min, 30 min, 1 h, 2 h and 4 h.

In order to investigate the microstructure evolution duringrecrystallization, the middle section of cylinder pieces wasmechanically ground and electropolished using perchloric acid(10%) and dehydrated alcohol (90%) to provide adeformation-free flat surface. The samples were examined by anOxford detector within a MIRA3 LMH field emission gun scanningelectron microscope. EBSD data were collected using an accelerat-ing voltage of 20 kV and a magnification of 100 times with a stepsize of 10 lm was used to achieve a good combination of accuracyand analysis time. The data were then further analyzed using the

l chemical composition of DD6 alloy.

ent Cr Co Mo W Ta Re Nb Al Hf Ni

4.3 9 2 8 7.5 2 0.5 5.6 0.1 Balance

Fig. 6. The schematic of test pieces.

HKL CHANNEL5 suite of programs, assuming a FCC Ni-superalloystructure with a lattice parameter of 0.357 nm. In this experiment,c and c0 were detected as the same phase due to their highly sim-ilar lattice parameters. RX microstructure on the middle sectionwas obtained to compare with the simulation results.

5. Results

5.1. Deformation simulation

Hot compression modeling was conducted along [001] orienta-tion. A one-eighth-cylindrical model was used (shown in Fig. 7(a)),taking account into the four fold crystallographic symmetry andgeometrical symmetry along the compression axis. Hexahedralelement was chosen to conduct the FEM analysis, and 34,154nodes and 31,200 elements were used. The distribution of dissipa-tion plastic energy is nonuniform when considering the frictionbetween the compression head and the test piece, and the distribu-tion on the middle cross section is relatively uniform, as shown inFigs. 7(b) and 8. The energy distribution is relatively uniform onthe chosen section, which was chosen to provide stored energyand predict microstructure. As stated above, 1% of the dissipationplastic energy in FEM was transformed into CA cell results usingmapping Eq. (14). Fig. 8 shows that the interpolated values are ingood agreement with the original values. It should be noted thatthe coefficients below are different in every hexahedron element.

f ðx; yÞ ¼ a0 þ a1xþ a2yþ a3xy ð14Þ

5.2. Comparison between simulation and experimental results

Simulated and experimental recrystallized microstructure withdifferent solution time are shown in Figs. 9 and 10. As stated above,c and c0 phases were detected as one phase in the EBSD observa-tion. Colors in the orientation map (with three Euler angles) onlyrepresent crystal orientations instead of phases.

Clearly, RX initially nucleates in DAs and grows rapidly at1280 �C and 1300 �C. However, different RX behaviors appear atthese two temperatures. At 1280 �C, RX grains tend to exhibit den-dritic shape in the early stage, and recrystallization fraction is obvi-ously smaller than that at 1300 �C. This is related to the solutesegregation and different solution behavior of c0 at 1280 �C and1300 �C. The fraction of remaining c0 phase is much smaller at1300 �C than that at 1280 �C. After RX is completed in DAs, a largeamount of small grains present in IDRs and grows very slowly, asshown in Fig. 9(a, c, e, g and i). Small RX grains will disappear withtime through curvature-driven force, as shown in Fig. 9(g, i and k)and Fig. 10(g, i and k). The model in this paper can also predict theinfluence of as-cast dendritic microstructure. Relative frequencydistributions of grain area at different times are compared betweensimulation and experiments in Fig. 11. It is obvious that the area ofmost grains are below 2000 lm2 (about 50 lm in diameter) evenafter heated at 1300 �C for 4 h. At lower temperature (1280 �C), itis difficult for small grains to be merged.

In addition, it should be also noted that the twin grains cannotbe simulated in this research. Twinning effect was ignored in thismodel though a large amount of twinning grains can present dur-ing RX as a result of low stacking fault energy. In this research, alarge fraction of twining grain boundaries appeared after annealingfor about 2 h, as shown in Figs. 9(i) and 10(i). There is a small gapbetween experiments and simulation results at 2 h. Thus, the pro-posed models need to be further improved. However, in general,the simulated results can be acceptable from engineeringperspectives.

Fig. 7. The finite element mesh (a) and the simulated distribution (980 �C, compression head moving �0.5 mm) of plastic dissipation energy (b) assuming the frictioncoefficient between head and test piece as 0.1.

Fig. 8. Distribution of plastic dissipation energy (a) and stored energy (b): mapping from FEM results to CA square lattice results (section face with 0.5 mm to the middleface).

128 Z. Li et al. / Computational Materials Science 107 (2015) 122–133

5.3. Recrystallization kinetics

Porter and Ralph reported a activation energy of 790 kJ/mol forRX of polycrystalline Nimonic 115 alloy [12]. However, no experi-mental data for the activation energy Qb of the grain boundary atdifferent temperatures are available for SX superalloys till now.Zambaldi [18] set Qb as 1290 kJ/mol in his model. However, thesevalues are obviously high. Thus, in this research, the activationenergy Qb was estimated according to literatures.

The grain motion process is controlled by the elemental diffu-sion and precipitate solution behaviors. Therefore, the activationenergy for grain boundary motion can be estimated according tothe diffusion values. For Ni–Al binary system, the activation energyfor diffusion of Al in pure Ni is 284 kJ/mol [59], and activationenergy for diffusion of other elements (Ti, Co, Ru, Re, etc.) innickel-based superalloys are from 230 to 350 kJ/mol [60–62]. Inthis work, different values for DAs (250 kJ/mol) and IDRs(340 kJ/mol) were chosen to take account into the influence of

Fig. 9. Experimental and simulated recrystallization microstructure evolution (annealed at 1280 �C) with different solution time (a, b) 5 min; (c, d) 10 min; (e, f) 30 min; (g, h)60 min; (i, j) 120 min; (k, l)240 min; (a, c, e, g, i, k) experimental; (b, d, f, h, j, l) simulation.

Z. Li et al. / Computational Materials Science 107 (2015) 122–133 129

as-cast dendritic microstructure. Fig. 12 shows the simulated andexperimental recrystallized microstructures for the sampledeformed at 980 �C and heated at 1280 �C for 5 min. Both simula-tion and experiments show about 60% of deformed matrix hasrecrystallized.

Fig. 13 shows the variation of recrystallization fraction withtime at two temperatures. JMAK model [63] (contributed byJohnson, Mehl, Avrami and Kolmogorov) was usually used todescribe the kinetics of recrystallization. This model can give us abetter understanding of the recrystallization process. In general,the JMAK equation can be expressed as:

XðtÞ ¼ 1� exp � ts

� �q� �ð15Þ

where X is the recrystallization fraction, s is the characteristic time(X(s) = 1 � e�1) for recrystallization, q is the Avrami exponent and tis the time. The above equation can be changed into another form:

lnð� lnð1� XðtÞÞÞ ¼ q lnðt=sÞ ð16Þ

This means that the two terms ln(�ln(1 � X(t))) and ln(t) havethe linear relation, and JMAK model assumes homogeneous nucle-ation and spatially and temporally constant growth rate. The plotof ln(�ln(1�X(t))) vs ln(t) (Fig. 14) can still give much information

Fig. 10. Experimental and simulated recrystallization microstructure evolution (annealed at 1300 �C) with different solution time (a, b) 5 min; (c, d) 10 min; (e, f) 30 min;(g, h) 60 min; (i, j) 120 min; (k, l) 240 min; (a, c, e, g, i, k) experimental; (b, d, f, h, j, l) simulation.

130 Z. Li et al. / Computational Materials Science 107 (2015) 122–133

though the as-cast microstructure is inhomogeneous. RX process ofSX superalloys can be divided into three stages. At the first stage,RX grains grow rapidly in DAs; at the second stage, RX grains growvery slowly in IDRs as a result of the drag of second coherentphase; at final stage, recrystallized grains collide with each other,and recrystallization rate becomes slower. Thus, characteristictime increases at the three stages, as shown in Fig. 14. The charac-teristic time of the three stages at 1280 �C is 48.3 s, 344.5 s and601 s respectively, while 37 s, 50.6 s and 166 s respectively at

1300 �C. The characteristic time at 1300 �C is obviously shorterthan that at 1280 �C, showing the influence of annealing tempera-ture on RX behaviors.

6. Discussion

Comparison of experimental and simulated microstructures onthe whole cross section is shown in Fig. 15. In the simulatedresults, the RX grain boundaries are irregular and small grains still

Fig. 11. Relative frequency distribution of grain area:(a, b) 1280 �C; (c, d) 1300 �C; (a, c) 2 h; (b, d) 4 h.

Fig. 12. Simulated (a) and experimental (b) recrystallized microstructure for the sample heated at 1280 �C for 5 min (blue region standing for recrystallization, remaining fordeformed matrix). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Z. Li et al. / Computational Materials Science 107 (2015) 122–133 131

remain, showing that simulation agree well with the experiments.This is a result of taking account into the influence of as-castmicrostructure. It should be noted that the formation of annealingtwins was not taken into account in this research, though many

twin grains appear after heated for about 2 h (seen inFigs. 9(i) and 10(i)). If homogeneous kinetics parameters wereemployed, most grains will appear as polygons with no remainingsmall grains, shown in Fig. 16(b). This is very different from the

Fig. 13. Variation of recrystallization fraction with time for the samples annealed atdifferent temperatures.

Fig. 14. Recrystallization kinetics for the samples anneale

Fig. 15. Experimental (a) and simulated (b) recrystallization microstructure

132 Z. Li et al. / Computational Materials Science 107 (2015) 122–133

simulated results in Fig. 16(a), which uses different kinetic param-eters in DAs and IDRs. It’s evident that the method in this researchis more accurate. Both experiments and simulation results showthat the grain growth in IDRs is very slow and grain boundariesare rather irregulars. It can also be inferred that the homogenousassumption can be adopted at above solvus.

Kinetic parameters in both DAs and IDRs are based on the diffu-sion parameters in this research. Simulation results show that thisassumption conforms to the real situation. It should be noted thatthe activation energy, necessary to fit the experimental data, wasmuch higher compared to the values used for single phase puremetal. The activation energy in this research should, therefore, beconsidered as a phenomenological parameter, which includes notonly the diffusion effect, but also the phase dissolution effect.Additionally, it should be noted that Zambaldi set the activationenergy to be 1290 kJ/mol in his simulation, one order of magnitudehigher than that in this research. The main reason may be that thepre-exponential factor M0 in his research was set as high as1031 m3/Ns, which eliminated the diffusion effect. Furthermore,as-solutioned samples were used in his research.

d at different temperatures: (a) 1280 �C; (b) 1300 �C.

on the whole cross section of the sample annealed at 1300 �C for 4 h.

Fig. 16. Comparison of simulation results after 1300 �C for 4 h (a) taking account into the influence of as-cast dendritic microstructure; (b) homogeneous assumption.

Z. Li et al. / Computational Materials Science 107 (2015) 122–133 133

7. Conclusions

One modified CA method was developed for the characteriza-tion of primary recrystallization of an as-cast single crystal super-alloy. One phenomenological macroscopic elastic–plastic modelwas used to obtain the driving force for nucleation and recrystal-lization. A deterministic CA approach was utilized to predictrecrystallization microstructure. Simulated and experimentalresults agree well with each other, though some aspects of thismodel need to be further improved. The following conclusionscan be drawn:

(1) This model integrate CA method with one macroscopicphenomenon-based deformation model. This deformationmodel was used for obtaining the driving force forrecrystallization.

(2) RX microstructural evolution is observed using EBSD tech-nique to validate the simulation model. The activationenergy in IDRs is much higher than in DAs, which is alsotaken into account in this model.

(3) The activation energy for RX grain boundary motion is a phe-nomenological parameter, which includes not only the diffu-sion effect, but also the phase dissolution effect.

(4) Irregular grain boundaries after recrystallization below sol-vus are ascribed to the as-cast microstructure. Many smallgrains will be left in IDRs.

Acknowledgements

This research was funded by the National Basic ResearchProgram of China (No. 2011CB706801) and National NaturalScience Foundation of China (Nos. 51171089 and 51374137).

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Microstructure simulation on recrystallization of an as-cast nickel based single crystal superalloy1 Introduction2 Mathmatical models2.1 Driving force2.2 Nucleation model2.3 Recrystallization and grain growth model2.4 Influence of as-cast dendritic microstructure

3 Cellular automaton algorithm4 Experimental details5 Results5.1 Deformation simulation5.2 Comparison between simulation and experimental results5.3 Recrystallization kinetics

6 Discussion7 ConclusionsAcknowledgementsReferences