Kinetics of AlN precipitation in microalloyed steel

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Modelling Simul. Mater. Sci. Eng. 18 (2010) 055003 (16pp) doi:10.1088/0965-0393/18/5/055003

Kinetics of AlN precipitation in microalloyed steel

R Radis1,2,3 and E Kozeschnik1

1 Christian Doppler Laboratory ‘Early Stages of Precipitation’, Institute of Materials Science andTechnology, Vienna University of Technology, Favoritenstraße 9-11,A-1040 Vienna, Austria2 Institute for Materials Science and Welding, Graz University of Technology, Kopernikusgasse24, A-8010 Graz, Austria

E-mail: rene.radis@tugraz.at and ernst.kozeschnik@tuwien.ac.at

Received 14 November 2009, in final form 14 February 2010Published 29 April 2010Online at stacks.iop.org/MSMSE/18/055003

AbstractIn this work, the thermodynamic information on AlN formation in steel andexperimental data on AlN precipitation kinetics are reviewed. A revisedexpression for the Gibbs energy of AlN is presented with special emphasison microalloyed steel. Using the software package MatCalc, computersimulations of AlN precipitation are performed and compared with independentexperimental results from the literature. A new model for grain boundaryprecipitation is employed, which takes into account fast short-circuit diffusionalong grain boundaries as well as slower bulk diffusion inside the grain,together with the classical treatment for randomly distributed precipitates withspherical diffusion fields. It is demonstrated that the precipitation of AlNcan be modelled in a consistent way if precipitation at grain boundaries anddislocations is taken into account, dependent on chemical composition, grainsize and annealing temperature. It is also demonstrated that, for consistentsimulations, the influence of volumetric misfit stress must be taken intoaccount for homogeneous precipitation of AlN in the bulk and heterogeneousprecipitation at dislocations.

1. Introduction

The precipitation of second phase particles, e.g. nitrides, carbides or complex carbonitrides,plays an important role in the production process of high-strength low-alloy (HSLA) steel.Apart from Nb, V and Ti, Al plays a major role as a microalloying element. Importantmechanical and technological properties, such as hot ductility [1–3], impact toughness [3],deep drawability [4] or weldability [5], can be improved by microalloying additions of Al. Onthe other hand, precipitation of AlN can cause embrittlement and induce cracking phenomena,such as rock candy fracture [6–12]. Some authors [13] report that AlN without any other trace

3 Author to whom any correspondence should be addressed.

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Modelling Simul. Mater. Sci. Eng. 18 (2010) 055003 R Radis and E Kozeschnik

element has no influence on the hot ductility of high purity iron. The most significant effectof AlN in steel is in grain size control [3, 14–16], which directly influences these properties.A summary of the effects of AlN in steel can be found in the extensive review of Wilson andGladman [5].

The equilibrium crystallographic structure of AlN is the hexagonal wurtzite structure witha lattice constant of a = 0.311 nm and c = 0.4978 nm. In the early stages of precipitation,simple cubic structure with a = 0.412 nm is also observed [17–20]. A detailed investigationof the two crystallographic structures can be found in [21].

For a better understanding of the effects of AlN on the mechanical properties of steel, itis essential to understand its precipitation kinetics. Several authors [17, 22–29] have reportedabout the precipitation of AlN in undeformed ferrite. Fewer investigations have been madeinto the precipitation kinetics of AlN in undeformed austenite [29–33].

Because of considerable volumetric misfit between the AlN precipitate and the steelmatrix, the precipitation of AlN in austenite occurs predominantly at grain boundaries [34–38].In austenite, grain growth can be effectively controlled by AlN precipitation. For instance,Militzer et al [39] showed that, in the presence of AlN precipitates, there is very little graingrowth (40 µm after 10 min) during isothermal annealing at temperatures below 1100 ◦C,whereas fast grain growth is observed at 1150 ◦C (200 µm after 10 min). They suggested thatcomplete dissolution of AlN above 1100 ◦C is responsible for the large grain size in their steel.Gao and Baker [40] claimed that the presence of AlN precipitates is responsible for grain sizecontrol in Al–V–N steel. Speich et al [41] suggested that next to Ti additions, Al is may bethe most effective element in avoiding austenite grain coarsening at high temperatures. Apartfrom AlN precipitation at the austenite grain boundaries, AlN is also found at ferrite grainboundaries [37].

In addition to the numerous experimental studies, several theoretical treatments of AlNprecipitation are reported. Duit et al [24] carried out calculations of AlN precipitation in ferritebased on the Avrami equation. Cheng et al [37, 42, 43] modelled the precipitation of AlN inmicroalloyed steels. In [37, 42], these authors treated the coarsening and growth stages withoutaccounting for the nucleation process. In [43], they also considered the nucleation stage, butthey verified their model only on a single chemical composition.

Technology-oriented treatments are available for AlN precipitation during compact stripproduction (CSP) [44], thin slab casting and rolling (TSCR) [29] or coiling of hot stripsof microalloyed steels [45]. Zolotorevsky et al [25] as well as Kozeschnik et al [46]investigated the competing processes of AlN precipitation and recrystallization during batchannealing of low carbon steel. Biglari et al [20] modelled the precipitation kinetics ofAlN during internal nitridation with special emphasis on the crystallographic structure.Using thermodynamic calculations they demonstrated that, in recrystallized specimens, theprecipitation of incoherent, hexagonal AlN is favoured, while the precipitation of coherentcubic precipitates is preferred in cold rolled specimens.

Despite this large number of computational studies available in the literature, there isno comprehensive and rigorous description of the precipitation kinetics available for AlNprecipitation taking into account that AlN precipitation can occur simultaneously at grainboundaries and on dislocations, depending on chemical composition, grain size and annealingtemperature. In this work, the thermodynamic information on aluminium nitride in steelis investigated first. A modified expression for the Gibbs energy of AlN is presentedwith emphasis on the solubility of AlN in microalloyed steel. Using a new model forprecipitation at grain boundaries [47], computer simulations on AlN precipitation kinetics areperformed. The computed results are compared with independent experimental data from theliterature, spanning a wide range of chemical compositions and various isothermal annealing

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temperatures. The variety of experimental data is reproduced consistently with a single set ofsimulation parameters.

2. Computer simulations

For the present simulations, the thermo-kinetic software MatCalc (version 5.30 rel 1.075)[48–50] and the corresponding databases ‘mc steel’ [51] and ‘mc sample fe’ [52] are used. Inthis approach, the nucleation kinetics of precipitates are calculated from the classical nucleationtheory (CNT) [53, 54] extended for multi-component systems [48, 54, 55]. Accordingly, thetransient nucleation rate J is given as

J = N0Zβ∗ · exp

(−G∗

k · T

)· exp

(−τ

t

). (1)

Here J describes the rate at which nuclei are created per unit volume and time. N0 representsthe total number of potential nucleation sites. The Zeldovich factor Z takes into account thatthe nucleus is destabilized by thermal excitation as compared with the inactivated state. Theatomic attachment rate β∗ takes into account the long-range diffusive transport of atoms, whichis needed for nucleus formation if the chemical composition of the matrix is different from thechemical composition of the precipitate. T is the temperature, k the Boltzmann constant andt the time. The incubation time τ is given by [53]

τ = 1

2β∗Z2, (2)

and the critical energy for nucleus formation G∗ is

G∗ = 16π

3

γ 3

(�Gvol − �Gs)2, (3)

with the effective interfacial energy γ , the volume free energy change �Gvol and the misfitstrain energy �Gs. It is important to note that G∗ is the most essential quantity in nucleationtheory, when compared with the other quantities occurring in equation (1). G∗ contains thecube of the interfacial energy over the square of the effective driving force �Gvol − �Gs.Since G∗ appears in the exponent of the nucleation rate equation (1), small changes in γ

and/or �Gvol − �Gs can lead to huge variations in J , which is demonstrated in [56] in atreatment of the evolution of multimodal size distributions in a Ni-base superalloy.

The misfit strain energy is given by

�Gs = E

1 − νε2, (4)

where E is Young’s modulus and ν the Poisson constant of the matrix. ε is the linear misfitstrain, approximately given as

ε = 1

3

V Pmol − V M

mol

V Mmol

, (5)

where V Pmol and V M

mol are the molar volumes of the precipitate and the matrix, respectively.Once a precipitate is nucleated, its further growth is evaluated based on the evolution

equations for the radius and composition of the precipitate derived by Svoboda et al [48] in amean-field approach utilizing the thermodynamic extremal principle. Accordingly, the growthrate ρk and the rate of change of chemical composition cki of the precipitate with index k areobtained from solution of the linear system of equations

Aijyj = Bi, (6)

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Table 1. Solubility products of aluminium nitride in austenite.

Log [Al(wt%) × N(wt%)] Reference

1 4.382 − 11 085/T [60]2 0.528 − 5938/T [61]3 1.79 − 7184/T [62]4 4.5989 − 11 568/T [42]5 1.03 − 6770/T [31]6 1.95 − 7400/T [63]7 0.725 − 6180/T [3]8 0.309 − 6015/T [3]9 1.48 − 7500/T [14]

10 1.8 − 7750/T [32]11 3.577 − 10 020/T [8]12 2.923 − 9200/T [8]13 3.079 − 9295/T [8]14 6.4 − 14 356/T [44]15 0.18 − 5675/T + 2.4 × Al(wt%) [33]

where the variable yj represents the rates ρk and cki , as well as the Lagrange multipliers fromthe stoichiometry constraints (see [48]).

The system of equations (6) is solved for each precipitate k separately. The full expressionsfor the coefficients Aij and Bi , as used in this work, are given in [50]. The numerical timeintegration of ρk and cki is performed with the software MatCalc, based on the classicalnumerical Kampmann–Wagner approach [57], and it is described in detail in [49]. Theinterfacial energy is calculated from the generalized n-next nearest-neighbour broken-bondapproach [58], taking into account the influence of the precipitate size during nucleation [59].

In addition to the classical treatment of randomly distributed precipitates as developedin [48], the present calculations use a novel model [47] for the evolution of grain boundaryprecipitates. With this computational approach, a series of calculations for variousannealing temperatures is carried out and compared with independent experiments from theliterature.

2.1. Thermodynamics

A large number of experimental data are published on the solubility of AlN in austenite. Table 1and figure 1 summarize these data.

Unfortunately, the experimental data [3, 8, 31–33, 61–63] and the thermodynamicassessments [44, 60] show a very large scatter. Calculations with the commercial database‘TCFE3’ [64] deliver results which are in agreement with a group of solubility productspredicting very low solution temperatures, indicated by a dashed line. Furthermore, this curveseems to have a steeper slope compared with many of the other solubility products. The largevariations in the solubility products can be partially attributed to limitations of the variousexperimental quantification methods to the detection of AlN dissolution temperatures [26].Particularly, the Beeghly method [65], which is used by most of the authors, seems to becritical for a reliable detection of fine AlN precipitates. Furthermore, the solubility product ofAlN is reported to be influenced by additional alloying elements. Erasmus [3] demonstratedthat the AlN solution temperature in a 3.5%Ni steel is about 100 ◦C higher compared with aplain carbon steel due to the lower solubility of nitrogen in austenite. Honer and Baliktay [8]determined three different solubility products for three different alloys, see table 1. Another

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Figure 1. Solubility products for aluminium nitride in austenite.

possible reason can be attributed to the substitution of Al in the precipitate by other alloyingelements, e.g. Cr [17].

The large scatter in the available solubility products for AlN precipitation provided themotivation for detailed investigations already two decades ago. Wilson and Gladman [5]studied the available solubility products published by different authors. Based on theirassessment, they identified solubility product 5 [31] listed in table 1 as the ‘most likely’one. They found support in the work of Irvine et al [15], who reported that their results are inclose agreement with the results of Erasmus [3] and Konig et al [32]. In particular, the workof Konig et al [32] seems to be much substantiated, because they investigated the solubilityproduct of AlN using several different chemical compositions. In the present assessment, webase our analysis on the experiments reported in [32, 33, 37], where AlN precipitates havebeen clearly detected after different solution treatments, indicated by filled circles in figure 1,and have not been identified in treatments at higher temperatures. The chemical compositionsand different heat treatment temperatures are considered as the lower limit for the solubilityproduct of AlN in austenite. Accordingly, the following expression for the Gibbs energy ofthe AlN phase has been found most suitable for the simulations in this work:

GAlN − GSERAl − GSER

N = −262 982 + 63T . (7)

The corresponding curve is shown as a solid bold line in figure 1. The thermodynamicdescription is implemented in the database ‘mc steel’ [51]. The superscript ‘SER’ refersto the Standard Element Reference state defined in the CALPHAD technique [66].

2.2. Grain boundary diffusion versus bulk diffusion

An important ingredient for the treatment of precipitation of AlN precipitates with the newgrain boundary precipitation model [47] is the grain boundary diffusional mobility. Figure 2shows the temperature dependent self-diffusion coefficients of iron in the bulk and at the grainboundaries in bcc and fcc iron [67–69]. In ferrite, the ratio Dgi/Dbi is in the range 107–104

for temperatures between 500 and 800 ◦C. In austenite, this ratio ranges between 107 and 104

for temperatures between 800 and 1400 ◦C. The ratios of the diffusivities of all elements are

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Figure 2. Temperature dependent self-diffusion coefficients of iron in the bulk (Dbi ) and at grainboundaries (Dgi ) [67–69].

assumed to be similar to the ratio Dgi/Dbi of the self-diffusion of iron. These data are takeninto account in the present simulations.

2.3. Volumetric misfit between AlN precipitates and iron

For the treatment of precipitation of randomly distributed AlN particles at dislocations, it isnecessary to take into account the elastic strain energy �Gs caused by the volumetric misfitbetween the precipitates and the matrix. In the nucleation stage, this energy contribution cansubstantially reduce the available driving force for precipitation (see equation (3)) and thus thenucleation rate (equation (1)).

Figure 3 shows the volumetric misfit (solid line) along with the molar volumes of the bcc,fcc and AlN phases (dashed lines) versus temperature. Accordingly, the misfit is relativelyconstant over the entire temperature range and varies between 71% and 77%. However,these values represent the precipitate/matrix mismatch in an undisturbed crystal lattice andrepresent the so-called unconstrained misfit, which corresponds to the difference in latticeparameters of isolated bcc, fcc and AlN phases. In a precipitate/matrix composite, the matrixand precipitates will be elastically stressed, which leads to a so-called constrained misfit [70].This effect reduces the effective volumetric mismatch that must be inserted for the calculationof misfit energy.

Moreover, the misfit strain energy between the AlN nuclei and the matrix is stronglyinfluenced by lattice defects, such as dislocations and grain boundaries. For the former,precipitates can accommodate some of the volumetric misfit strain in the dislocation regionswith either compressive or tensile local stresses. For the latter, we assume that the grainboundary represents a heavily disturbed lattice region with an almost amorphous structure. Theprecipitate nuclei can form with incoherent phase boundary, where lattice vacancies can easilybe created or annihilated, and relaxation of misfit strains [73] can occur almost simultaneouslywith nucleus formation.

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Figure 3. Molar volumes of the phases bcc, fcc, AlN and the calculated volumetric misfit[71, 72].

The simulations have shown that a constant value of 19% effective volumetric misfit forprecipitates located at dislocations delivers good results for the calculation of precipitation ofAlN in ferrite as well as in austenite. No volumetric mismatch is considered for the precipitatesat grain boundaries due to the assumed immediate stress relaxation caused by vacancy creationor annihilation at the precipitate/matrix interface. Using a value of 70% mismatch for coherentbulk nucleation entirely suppresses the nucleation of AlN in all situations considered in thiswork, consistent with experimental evidence.

2.4. Young’s modulus

Considering misfit strain energies in precipitation kinetics calculations makes it necessary totake into account the temperature dependence of Young’s modulus (see equation (4)). Severalauthors have reported about this quantity in the ferrite phase field [74–82]. Considerably lessattention was paid to the austenite phase field [75, 83], see figure 4. Differences in the results areattributed mainly to different investigation methods [75], different chemical compositions [77]or different pre-treatments [78]. Most of the authors used the high frequency resonance method,e.g. [77, 78]. Fukuhara and Sanpei [75] analysed a plain low carbon steel using the ultrasonicpulse sing-around method. Both methods deliver accurate results, but Fukuhara and Sanpei [75]mention in their discussion that the ultrasonic pulse sing-around is the only technique for thedetermination of elastic parameters in the higher temperature range beyond the recrystallizationtemperature.

In this work, Young’s modulus of austenite is used according to the description of Fukuharaand Sanpei [75]. For the ferrite, the results of Peil and Wichers [74] are used, which aresimilar to the observations of Takeuti et al [80], in between the lower and upper limits ofDate [77] and a linear regression of the ultrasonic pulse sing-around investigations of Fukuharaand Sanpei [75]. The AlN particle is assumed to be a rigid body. This is manifested inequation (4) having its foundation in the Eshelby concept of elastic strains around a rigidellipsoidal inclusion. The temperature dependence of Young’s modulus of AlN is thereforeneglected.

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Figure 4. Temperature dependent Young’s modulus.

2.5. Simulation setup

Precipitation of AlN in ferrite and austenite is assumed to occur simultaneously on grainboundaries and dislocations. Therefore, two separate populations of AlN precipitates areconsidered in the simulations, different in the type of nucleation site and the value for thevolumetric misfit. Dislocation densities of 1012 m−2 and 1011 m−2 are assumed for annealedferrite and austenite [84], respectively. A constant value for the effective volumetric misfitof 19% is used for the precipitate/matrix mismatch at dislocations of AlN in ferrite andaustenite. For the calculation of the misfit strain energy, temperature dependent Young’smoduli are used as shown in figure 4. Furthermore, figure 2 shows the temperature dependentratios between diffusion at grain boundaries and in the bulk Dgi/Dbi , used for grain boundaryprecipitation [47]. The only variable input parameter in the simulations is the grain size, whichis unfortunately unknown for various experiments because the authors of these investigationsdo not provide these data. Therefore, we have decided to use values that conform toexperience for grain sizes in ferritic–pearlitic microstructure and austenite, respectively. Theslight variations applied in the simulations are considered as a legal means for fine-tuning,certainly justified by the complete lack of corresponding information and inherent variationsin chemical composition and experimental procedures. The grain sizes used in the calculationsare summarized in tables 2 and 3 and are between 5 and 15 µm for ferrite and 50 and 150 µmfor austenite. All simulations presented subsequently are otherwise performed with the sameidentical set of simulation parameters.

2.6. Materials

The numerical simulations are compared with experimental data from nine literature sources[17, 22, 24, 26, 28, 29, 31–33]. The Al and N contents are given in tables 2 and 3 for ferriteand austenite, respectively, and sorted according to the value of their solubility product.

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Table 2. Chemical composition of the alloys investigated in ferrite.

Log Grain sizeAlloy Al (wt%) N (wt%) [Al(wt%) × N(wt%)] Tsol (◦C) (µm) Reference

1 0.046 0.0067 −3.51 1187 15 [22]2 0.059 0.0051 −3.52 1187 10 [28]3 0.058 0.0050 −3.54 1181 7 [17]4 0.058 0.0038 −3.66 1150 7 [26]5 0.030 0.0070 −3.67 1143 15 [29]6 0.047 0.0043 −3.69 1142 10 [24]7 0.016 0.0065 −3.98 1072 5 [28]8 0.035 0.0029 −3.99 1072 10 [28]9 0.060 0.0017 −3.99 1074 15 [28]

10 0.019 0.0039 −4.13 1040 7 [28]

Table 3. Chemical composition of the alloys investigated in austenite.

Log Grain sizeAlloy Al (wt%) N (wt%) [Al(wt%) × N(wt%)] Tsol (◦C) (µm) Reference

11 0.055 0.0240 −2.88 1417 50 [32]12 0.120 0.0057 −3.16 1327 100 [33]13 0.079 0.0072 −3.25 1262 50 [31]14 0.058 0.0058 −3.47 1198 100 [31]15 0.030 0.0070 −3.67 1143 150 [29]

These alloys cover a wide range of different Al and N contents representative of conventionalmicroalloyed steel.

3. Results

Figures 5 and 6 show the calculated phase fraction versus time curves of AlN precipitated in15 different alloys at various temperatures. For ten alloys, precipitation occurred in the ferritephase field, see figure 5. The simulation of the precipitation kinetics in the austenite phasefield is shown in figure 6, using five different chemical compositions.

The grain sizes used for the calculations are given in tables 2 and 3. The dashed linesrepresent precipitates formed at dislocations, whereas the dotted lines represent precipitatesformed at grain boundaries. The sum of both phase fractions is displayed as solid lines. Again,the alloys are sorted according to decreasing Al and N contents.

Depending on chemical composition and annealing temperature, the calculations showonly precipitation at grain boundaries or simultaneous precipitation at grain boundaries anddislocations. Particularly in alloys with high Al and N contents, precipitation is observed atboth nucleation sites, which is shown in figures 5(a)–(m),(o),(p),(u),(x),(A) for precipitation inferrite, and in figures 6(a) and (b) for precipitation in austenite. Moreover, a strong dependenceof favoured nucleation site selection on the isothermal annealing temperature is observed.Whereas, at low temperatures and with higher driving forces, the majority of precipitates arelocated at dislocations, at higher annealing temperatures, precipitation at grain boundariesis predominant, see figures 5(a)–(n). For steels with low Al and N additions (see alloys6–10), which have solubility products smaller than approximately −3.7, a clear preference forprecipitation at ferrite grain boundaries is observed, see figures 5(p)–(F ).

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Figure 5. Calculated and experimental precipitation kinetics of AlN in ferrite for various alloysand temperatures; alloy composition in wt%.

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Figure 5. (Continued).

This observation is even more pronounced in the case of precipitation in austenite, inagreement with thermodynamic calculations of Cheng et al [43], who have also shown that�G∗

is much higher for homogeneous nucleation compared with nucleation at grain boundaries.These authors concluded that the kinetics of AlN are completely dominated by nucleation atgrain boundaries in the austenite phase field. Apart from alloy 11, this is also observed in ourcalculations, see figure 6. However, alloy 11 has a very high solubility product of −2.88. Thisleads to the predicted additional precipitation at dislocation, at least for annealing temperaturesup to 950 ◦C.

Figure 7 shows a comparison of the precipitation kinetics of the alloys with the lowest (alloy10) and highest (alloy 11) amounts of Al and N, depicted in time–temperature–precipitation(ttp) diagrams. In the simulations, the allotropic transformation from austenite to ferrite isassumed to occur at 800 ◦C. Since the precipitation kinetics simulations are carried out forrepresentative volume elements, we assume that there is only this sharp transition and nointercritical two-phase range is considered. The calculations are performed with a grain sizeof 10 µm for the ferritic matrix and 100 µm for the austenitic matrix, respectively. Similarto figures 5 and 6, the grain boundary precipitates are represented by dotted lines, whereas

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Modelling Simul. Mater. Sci. Eng. 18 (2010) 055003 R Radis and E Kozeschnik

Figure 6. Calculated and experimental precipitation kinetics of AlN in austenite for various alloysand temperatures; alloy composition in wt%.

Figure 7. Calculated ttp diagrams for alloy 10 (a) and alloy 11 (b).

dashed lines represent the precipitates formed at dislocations. For both populations, the timefor 5% and 95% precipitation (relative phase fraction) is displayed.

The two ttp diagrams in figure 7 reflect the features discussed for the previous figures.While, for alloy 10, the solution temperature of AlN is calculated with 1040 ◦C, the solutiontemperature for alloy 11 is calculated with 1417 ◦C. As a consequence, in alloy 10, the

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precipitation of AlN is observed only at temperatures below 900 ◦C. Particularly in austenite,precipitation occurs after long annealing times, e.g. t0.05 = 2 × 104 s with a nose temperatureof 825 ◦C. Moreover, above 600 ◦C, precipitation of AlN is observed only at grain boundaries(dotted lines). Nucleation at dislocations is predicted only for temperatures below 600 ◦C(dashed lines) with rather small volume fractions.

In alloy 11, the nose temperature for precipitation in austenite is calculated with 1150 ◦C.At this temperature, precipitation at grain boundaries occurs after 43 s. The discontinuityin the C-curve of the ttp diagram for precipitation at austenite grain boundaries is attributedto the simultaneous precipitation of AlN at dislocations below a temperature of 1000 ◦C.Due to the change from major precipitation at grain boundaries to favoured precipitation atdislocation, the total phase fraction of grain boundary precipitates is strongly reduced. Thus,the 5% and 95% phase fraction lines are shifted to shorter times. The fastest AlN precipitationkinetics are observed at a temperature of 800 ◦C for both alloys, which is the temperature ofthe assumed austenite to ferrite transformation. At this temperature, the precipitation of 5%of the equilibrium phase fraction is completed after 1.5 s and 0.15 s for alloy 10 and alloy11, respectively. Consequently, the present calculations suggest that AlN precipitation occursalmost instantaneously after austenite has transformed into ferrite.

4. Discussion

In order to get good agreement of our simulations with the available experimental databased on a single, identical set of input parameters, several physical mechanisms had to beconsidered. First, the thermodynamic basis for AlN formation has been reassessed and arevised Gibbs energy expression has been developed. A comparison of this expression withsolubility products reported in the literature shows that the new description delivers solubilityproducts, which are close to the ‘recommended’ solubility products of Wilson and Gladman [5].Although some discrepancies exist with the assessment of Hillert and Jonsson [60] in termsof the entropy of AlN (this possible discrepancy has already been admitted by Hillert andJonsson [60] in their discussion), the revised expression is capable of reproducing the necessarydriving forces for AlN precipitation in austenite and ferrite in a satisfying manner. Thisagreement could not be achieved with any of the alternative thermodynamic descriptions ofother databases. It is not clear whether any influence of the crystal structure (simple cubic orhexagonal) of the AlN nuclei exists on the precipitation kinetics.

The present analysis demonstrates that for the numerical description of AlN precipitationin microalloyed steel it is essential to account for the predominant precipitation of AlN at grainboundaries as well as at dislocations. The numerical simulations show that precipitation atgrain boundaries starts very quickly in the early stages of the precipitation process due to fastdiffusion of atoms along grain boundaries. However, the kinetics become significantly moresluggish in the later stages because of increasingly wide diffusion distances from the graininterior to the grain boundaries [47]. This effect is clearly reflected in the shape of the volumefraction versus time curves.

Particularly in alloys with high Al and N contents, simultaneous precipitation of AlNat grain boundaries and dislocations is observed in our simulations. The reason for this isfounded in the assumption that the misfit strain energy for the critical nucleus at dislocation ishigher compared with the values at grain boundaries. Thus, the activation barrier for dislocationnucleation is overcome only in alloys with higher Al and N contents. The preferred precipitationat dislocations at lower temperatures can be attributed to higher driving forces caused by higherundercooling and faster growth of dislocation precipitates characterized by approximatelyspherical diffusion fields. This issue has recently been discussed in an extensive parameter

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study when developing the new grain boundary precipitation model [47]. The sensitivity of therelevant input parameters for the present analysis is summarized in figure A1 in the appendix.

The present results correspond well with those of experimental TEM analyses, whereDogan et al [61] observed favoured precipitation of AlN along grain boundaries in a steel withlow Al and N contents and randomly distributed precipitates in a steel with higher amounts of Aland N. Moreover, the work of Wilcox and Honeycombe [36] shows that, at higher temperaturesin the austenite region, precipitation of AlN occurs predominantly at grain boundaries, whileMassardier et al [17] report about particles randomly distributed after annealing at 700 ◦C inthe ferrite phase field.

5. Summary

The present work deals with the description of the precipitation kinetics of AlN in microalloyedsteels. The thermodynamic information as well as experimental kinetic data are reviewedand a revised expression for the Gibbs energy of AlN is presented with special emphasis onmicroalloyed steel. Using the software package MatCalc, precipitation kinetics simulationsfor AlN are performed and compared with numerous independent experimental data from theliterature.

The simulations clearly demonstrate that AlN formation occurs by simultaneousprecipitation at grain boundaries and at dislocations. Depending on chemical composition,grain size and annealing temperature, predominant precipitation at grain boundaries and/orat dislocations is predicted. With the application of two models representing these twomechanisms, an excellent agreement between prediction and experimental data is achieved. Itis also shown that for a consistent description of AlN precipitation it is necessary to account forseveral physical mechanisms that are often neglected in these types of simulations, among themare the precipitate/matrix volumetric misfit and the temperature dependent Young’s modulus,composition-, temperature- and size-dependent interfacial energies, as well as the ratio betweenbulk and grain boundary diffusion.

Appendix

This section summarizes the sensitivity of the calculations on various input parameters inthe form of a parameter study. Therefore, a hypothetical alloy containing 0.05% Al and

Figure A1. Sensitivity of the calculations on various input parameters: (a) volumetric misfit,(b) Young’s modulus, (c) diffusion ratio Dgi /Dbi and (d) grain size.

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0.005% N is isothermally heat-treated at a temperature of 700 ◦C. The default input values are�V/V = 0.19%, E = 157 500 MPa, Dgi/Dbi = 105, and a grain size of 10 µm. Figure A1indicates the influence of each parameter on the simulation results. For a calculation ofprecipitation at dislocations, the volumetric misfit is varied between 0 and 0.2 in steps of0.05 (figure A1(a)). Young’s modulus is varied between 50 and 200 GPa in steps of 25 GPa(figure A1(b)). The main input parameters for precipitation at grain boundaries are the ratiobetween grain boundary diffusion and bulk diffusion Dgi/Dbi and grain size. The diffusionratio Dgi/Dbi is varied between 100 and 106 (figure A1(c)) and the grain sizes between 1 and100 µm (figure A1(d)).

All input parameters show a high influence on the precipitation kinetics calculations. Anincrease in the volumetric misfit and Young’s modulus shifts the precipitation to longer times.Additionally, a change in the slope of the phase fraction curves is observed by the variation ofthe diffusion ratio and grain size.

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