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Influence of bainite reaction on the kinetics of carbon redistribution during the Quenchingand Partitioning process

Nishikawa, Arthur S.; Santofimia Navarro, Maria; Sietsma, Jilt; Goldenstein, Hélio

DOI10.1016/j.actamat.2017.09.048Publication date2018Document VersionAccepted author manuscriptPublished inActa Materialia

Citation (APA)Nishikawa, A. S., Santofimia Navarro, M., Sietsma, J., & Goldenstein, H. (2018). Influence of bainitereaction on the kinetics of carbon redistribution during the Quenching and Partitioning process. ActaMaterialia, 142, 142-151. https://doi.org/10.1016/j.actamat.2017.09.048

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Influence of bainite reaction on the kinetics of carbon redistribution during the Quenchingand Partitioning process

Arthur S. Nishikawaa,∗, Maria J. Santofimiab, Jilt Sietsmab, Hélio Goldensteina

aDepartment of Metallurgical and Materials Engineering, University of São Paulo, Av. Prof. Mello Moraes, 2463, 05508-030, São Paulo, BrazilbDepartment of Materials Science and Engineering, Delft University of Technology, Mekelweg 2, Delft, 2628 CD, The Netherlands

Abstract

In the present study the microstructural evolution and kinetics of carbon redistribution during the partitioning step of the Quenchingand Partitioning process are investigated by means of a modeling approach that simultaneously considers the martensite-austenitecarbon partitioning and the decomposition of austenite into bainitic ferrite. The development of the phase fractions, interfaceposition, and carbon compositions are analyzed for two different binary Fe-C alloys with distinct initial carbon compositionsand simulation geometries. The composition dependence of carbon diffusivity in austenite is taken into account for solving thediffusion field equations. Simulations indicate that kinetics of carbon partitioning from martensite to austenite is controlled bycarbon diffusion in martensite and it is little affected by simultaneous occurrence of bainite reaction. On the other hand martensite-austenite carbon partitioning strongly influences the bainite reaction by inhibiting the growth of bainitic ferrite.

Keywords: Quenching and Partitioning; Bainite; Modeling; Mixed-mode model

1. Introduction

The Quenching and Partitioning (Q&P) process was pro-posed by Speer and coworkers in 2003 [1] as a new heat treat-ment route to obtain bcc + fcc multiphase microstructures withoptimum properties [1, 2]. The Q&P process involves quench-5

ing of austenite between the martensite-start (Ms) and roomtemperatures to produce a controlled mixture of martensite (α’)+ austenite (γ). Then, in the so called “partitioning” step, thematerial is isothermally held at a higher temperature to promotethe diffusion of the carbon in the supersaturated martensite to10

the untransformed austenite. Thus the partitioning step is in-tended to carbon-enrich the austenite, allowing its stabilizationat room temperature. The success of this process depends on thesuppression of competitive reactions that act as sinks for car-bon, particularly carbides precipitation. Cementite suppression15

can be achieved by controlled additions of alloying elementsthat delay its formation, e.g. silicon and aluminum. In the re-sulting Q&P microstructure martensite confers high strength,whereas stabilized austenite favors good ductility due to occur-rence of TRIP effect [3].20

The essential mechanism of the Q&P process is the carbonredistribution from martensite to austenite. Speer et al. [1] haveproposed that the endpoint of carbon partitioning can be mod-eled assuming a “constrained carbon equilibrium” (CCE). TheCCE hypothesis assumes that during the partitioning step the25

mobility of iron and the substitutional species can be neglected,competitive reactions (e.g., carbides precipitation, decompo-sition of austenite) are fully suppressed, and the martensite-austenite interface can be assumed immobile or stationary. The

∗Corresponding author. Email: arthur.nishikawa@usp.br

constrained equilibrium is therefore reached by equilibration of30

carbon’s chemical potential only (µα’C = µ

γC). Initially, an infi-

nite set of compositions in martensite and austenite can fulfillthis relation. However, once this condition is coupled with theconservation equations derived from the other assumptions, agiven composition of austenite leads to an unique correspond-35

ing composition in martensite.Currently, theoretical considerations and experimental ob-

servations question the assumptions of the CCE model. Mi-gration of martensite-austenite interface has been directly ob-served by in-situ transmission electron microscopy [4] and its40

relevance discussed by computational models [5, 6]. Besides,competitive reactions are not always avoided during the Q&Pprocess, even in careful designed alloys. It has been reportedthat transition carbides can tolerate much more Si than cemen-tite and might form in different stages of the heat treatment, de-45

spite high alloying addition [7, 8]. Yet, formation of fine transi-tion carbides by martensite tempering is usually not considereddetrimental, whereas cementite can be of more concern [9]. De-composition of austenite may also take place during Q&P. Thenormally employed range of partitioning temperatures enables50

the bainite formation from austenite. Nevertheless, because ofhigh silicon content, bainite formation in Q&P steels is reportedto happen with absence of carbides (i.e., only bainitic ferrite)[10, 7]. Thus, similarly to TRIP-assisted steels, bainite reactionin Q&P steels contributes to carbon enrichment of the surround-55

ing austenite. However, decomposition of austenite decreasesthe fraction of austenite in the final product, affecting the prop-erties of the final product. Modern Q&P alloys have been de-signed with higher manganese and nickel content to improvethe hardenability of the austenite, decreasing the kinetics of bai-60

nite formation [2, 11]. However, employment of large fractions

Preprint submitted to Acta Materialia January 16, 2018

Link to formal publication “Acta Materialia” (Elsevier): https://doi.org/10.1016/j.actamat.2017.09.048

of alloying elements increases the production costs and may im-ply processing difficulties due to, for example, development ofsegregation bands. This is even more critical when consideringthe Q&P applied to cast products, such as cast iron, where the65

microsegregation formed during solidification is not mitigatedby further thermomechanical processing [12].

It is clear that a better view of how the competitive reac-tions affect the microstructural evolution during the Q&P pro-cess is necessary, as it strongly influences the final properties70

of the material. A physically-based model that accounts forthe involved metallurgical processes can provide a fundamen-tal understanding of how the interactions between the differentphenomena affect the final microstructure.

Different models are available in the literature for describ-75

ing the carbon partitioning from a supersaturated bcc phase toan fcc phase, either considering interface migration or not. Theproblem was first addressed by Mujahid and Bhadeshia [13]and by Hillert et al. [14] in the context of the discussion of themechanism of bainite reaction. In both papers the authors per-80

formed one-dimensional (1D) simulations to solve Fick’s sec-ond law for the bulk phases and evaluated the kinetics of decar-burization of a carbon-supersaturated ferrite plate. As bound-ary conditions, Mujahid and Bhadeshia assumed fixed com-positions determined by paraequilibrium at the α’/γ interface.85

Hillert argued that this assumption would lead to a disconti-nuity in the chemical potential of carbon at the interface. Al-ternatively, they proposed a condition of “local equilibrium forcarbon” at the interface, in which the equality of chemical po-tentials is obeyed, albeit only for the carbon. This hypothesis is90

essentially the same used by Speer et al. [1] in the formulationof the CCE theory. Most recently, the carbon partitioning underthe influence of a mobile α’/γ interface has been approached bycalculating the diffusion fields of carbon in a 1D grid [5, 6] andby phase-field modeling in two and three dimensions [15, 16].95

These studies have relied on Hillert’s local equilibrium for car-bon as thermodynamical condition at the interfaces.

The local kinetics of diffusional growth of ferrite from austen-ite at high temperatures has been extensively reported in theliterature. However, modeling the formation of Widmanstät-100

ten/bainitic ferrite is particularly challenging, given that there isstill no agreement about the mechanism of its growth. The twoconcurrent schools dispute whether the bainite reaction takesplace with or without diffusion during the growth of subunits,implying different thermodynamical conditions for its growth105

[17, 18]. According to the diffusionless theory, bainitic ferritegrows by successive nucleation and growth of subunits by a dis-placive mechanism until the carbon content in austenite reachesthe T0 limit. When T0 is reached, the free energies of fccand bcc are equal and, consequently, there should be no driv-110

ing force for nucleation of new bainite subunits. Conversely,the diffusional theory states that bainite grows by a reconstruc-tive mechanism controlled by diffusion of carbon [19, 20, 21].The thermodynamical limit for the transformation in these con-ditions is assumed to be the meta-stable equilibrium between115

ferrite and austenite. In turn, since mobilities of substitutionalelements at the range of temperatures for bainite formation arevery low, paraequilibrium condition is normally invoked to de-

scribe this limit.None of the proposed mechanisms for bainite reaction seems120

to describe the experimental results perfectly well. Experimen-tal points often lie beyond the T0 line for low temperatures andbelow it for high temperatures [22]. Similarly, the paraequilib-rium lines also fails to accurately describe the thermodynamicallimit for bainite reaction. Efforts have been made on both sides125

of the dispute to obtain a better description of the phenomenon.Among the researchers that defend the diffusional view it is ofgeneral agreement that the transformation stasis is caused bydissipation of the chemical driving force by solute drag at theinterfaces [23]. A few studies have achieved some success in130

predicting the bainite stasis using a so called GEB (Gibbs en-ergy balance) theory concurrently with an analytical solution ofthe mixed-mode model [24, 25]. However, it is reported thateven in similar conditions of temperature and composition thesolute drag theory can encounter problems for supporting the135

experimental data [26].An alternative theory for describing the thermodynamical

limits of the bainite reaction was proposed by Hillert and cowork-ers [22]. Hillert postulated the existence of a thermodynami-cal barrier for the growth of bainitic ferrite. Using experimen-140

tal data, the authors determined that the additional energies forbainite formation at 700, 450, and 300 °C are 107, 1283, and2329 J/mol, respectively. These values were then interpolatedusing a spline to get a smooth curve describing the variation ofthe extra energy with the temperature. Using the modified free145

energies of ferrite it was possible to obtain the thermodynamicallimit for the growth of Widmanstätten and bainitic ferrite, a socalled WBs limit. The WBs theory was able to predict reason-ably well the phase fractions and the austenite carbon contentfor several alloys, even when the effects of different alloying150

elements were considered.In the present study the microstructural evolution and ki-

netics of carbon redistribution during the partitioning step of theQ&P process are investigated by means of a modeling approachthat simultaneously considers the martensite-austenite carbon155

partitioning and the decomposition of austenite into bainitic fer-rite (henceforth, coupled model). The evolution of the phasefractions, interface position, and carbon compositions are ana-lyzed for two different binary Fe-C alloys with distinct initialcarbon compositions and simulation geometries. A more accu-160

rate assumption which considers the composition dependenceof carbon diffusivity in austenite is taken into account for solv-ing the diffusion field and the results are compared to the gen-erally adopted constant diffusivity approach.

2. Model165

The kinetic model for carbon diffusion has been implementedby numerically solving the laws of diffusion by means of a 1Dfinite-difference method (FDM). For modeling purposes, it isassumed that martensite has a bcc structure supersaturated incarbon. The main assumptions of the CCE theory are con-170

sidered for modeling the carbon diffusion from martensite toaustenite. Thus, during the partitioning step, the carbon in marten-site is redistributed to austenite without movement of the inter-

face and with no partitioning of substitutional elements. At themartensite-austenite interface the chemical potentials of carbon175

in both phases are set equal (local equilibrium for carbon). Thisis a necessary assumption in order to guarantee the continuity ofthe chemical potential along the interface. These assumptionsimply that for a given composition of austenite at the interfacethere is a unique corresponding composition in martensite.180

The growth of bainitic ferrite (αb) from the austenite hasbeen modeled using a mixed-mode approach for describing thesidewise movement of the ferrite-austenite interfaces (plate-thick-ening). Only the plate-thickening kinetics has been consideredbecause both the thickening and lengthening cannot be simul-185

taneously represented in the 1D case. The plate-thickening sce-nario represents the growth of a bainitic ferrite lath oriented inparallel to the martensite plate. In turn, the foundation of themixed-mode model is based on an energy balance between theavailable chemical driving force for movement of the interface190

(∆Gchem) and the dissipated energy due to the interface migra-tion (∆G f ric). These two terms can be expressed as follows:

∆Gchem =∑

xαi(µγi − µ

∆G f ric = VmvM

where xαi is the concentration of the component i in the grow-ing phase α, µp

i is the chemical potential of i in the phase p,Vm is the molar volume of the growing phase, M the interface195

mobility, and v the interface velocity, which assumes positivevalues for the ferrite growth. At the interface the local equi-librium of carbon is assumed, so that ∆Gchem arises from thedifferences of chemical potentials of iron and the substitutionalspecies, while ∆G f ric is function of the interface velocity and200

the interface mobility. By equating these two terms an expres-sion for the interface velocity in Fe-C system is obtained, asfollows:

v =MVm

(µγFe − µ

The interface mobility M is expressed in the familiar Arrhe-nius form, M = M0 exp[−Qa/(RT )]. In this work the parame-205

ters suggested by Gamsjäger et al. [27, 28], M0 = 2 × 10−4 m4J−1s−1

and Qa = 140 000 J mol−1, were used.The diffusion of carbon in martensite, austenite, and during

the growth of bainitic ferrite is modeled by numerically solvingFick’s second law of diffusion. The numerical solution for in-210

creasing partitioning times is determined by a finite-differencemethod (FDM) assuming that the diffusion coefficient of car-bon in austenite varies as a function of the composition. Thisassumption is more physically accurate than the traditional con-stant diffusivity approach, as the carbon diffusivity in austenite215

is strongly sensitive to the carbon content [14]. Fick’s secondlaw was discretized implicitly for the derivatives of the compo-sition and explicitly for the gradient of the diffusion coefficientfollowing the equations:

cti = − (ri + gi) ct+1

i+1 + (1 + 2ri) ct+1i − (ri − gi) ct+1

i−1 (4)

ri = Di∆t∆z2 (5)

gi =Di+1 − Di−1

4∆t∆z2 (6)

where the indices i and t refer respectively to the node posi-220

tion and the time step, cti is the mole fraction of carbon, Di is

the diffusion coefficient of carbon, ∆t is the time step and ∆z isthe step size. The diffusion coefficients of carbon in austeniteand ferrite/martensite phases were calculated using the equa-tions determined by Ågren [29, 30]:225

DγC = 4.53 × 10−7

[1 + yγC

(1 − yγC

) 8339.9T

]× exp

(1T− 2.221 × 10−4

) (17767 − 26436yγC

DαC = 2 × 10−6 exp

10115T

)× exp

{0.5898

arctan(1.4985 −

15309T

)] }(8)

both evaluated in m2/s, where T is the temperature in K and yγCis the site fraction of carbon in austenite, which is related to themolar fraction of carbon xγC by the relation yγC = xγC/(1 − xγC).

The FDM algorithm was applied in independent grids cor-responding to the different phases considered in the problem.230

In order to couple all grids without violating the conservationlaws, the boundary value problem at the interfaces was evalu-ated by means of the following equation:

− DαC

∣∣∣∣∣∣int

+ v(cγint − cαint

)= −Dγ

∣∣∣∣∣∣int

At the immobile martensite-austenite interface, evaluationof equation 9 for v = 0 implies that the flux of carbon must235

be equal in both phases. In turn, for the migrating bainitic fer-rite/austenite interface the boundary condition becomes a Ste-fan problem considering the interface velocity determined – inthe mixed-mode model – by equation 3. Neumann boundarycondition representing zero flux of carbon was applied to the240

extremities of the calculation domain in order to keep the do-main as a closed system.

3. Simulation conditions

Two Fe-C alloys with different initial carbon compositionswere chosen for the calculations, a low carbon steel Fe-0.25 wt %C245

and a high carbon alloy Fe-0.8 wt %C. Following Andrews equa-tion [31], martensite start temperatures (Ms) were estimated tobe 433 and 201 °C, respectively. Alloying elements commonlyused in Q&P steels (e.g., Mn and Si) were neglected, as addi-tional variables would be introduced. Moreover, it was found250

by previous thermodynamical calculations that such elementsdo not considerably affect the compositions of equilibrium andconstrained carbon equilibrium in a way the results could leadto misleading interpretations.

Different initial martensite-austenite morphologies were em-255

ployed for each alloy, reflecting the effect of the carbon in themorphology of martensite. For the Fe-0.25 wt %C alloy a filmmorphology was assumed, as previously used by Santofimiaet al. [6], consisting of martensite and austenite films with0.20 µm and 0.06 µm of thickness (23 vol % of austenite), as260

represented in Figure 1a. Nucleation of a single bainitic ferrite

0.20 μm 0.06 μm

αb nucleus

1.0 μm 1.0 μm

αb nuclei

1.0 μm 1.0 μm

c)0.8 wt.%C

0.25 wt.%C

0.8 wt.%C

α'/γ inteface

α' γ

Calculation domain

Figure 1: Schematic illustration of the morphologies considered in the simula-tions for a) Fe-0.25 wt %C alloy, b) Fe-0.8 wt % alloy with one initial nucleusof αb, and c) Fe-0.8 wt % alloys with three nuclei of αb.

plate is assumed to occur in the middle of the austenite film att = 0, when the partitioning step begins. All simulations werecarried out assuming the partitioning temperature of 350 °C, aspreviously considered by Santofimia and coworkers.265

Martensite in high carbon alloys presents plate morphologyconsiderably coarser than in low carbon steels and, since the Mstemperature is lower, a greater fraction of austenite is expectedat the same quenching temperature. Thus, for the Fe-0.8 wt %Calloy a microstructure consisting of a martensite plate 1.0 µm270

thick adjacent to a block of austenite 1.0 µm thick (50 vol % ofaustenite) has been considered. Such dimensions are consis-tent with martensite morphologies reported in works on Q&Papplied to alloys with similar carbon compositions [8, 12]. Re-garding the bainite reaction, two scenarios have been consid-275

ered for this alloy. In the first, a single bainitic ferrite plate isnucleated at instant t = 0 in the middle of the austenite block(Figure 1b). In the second scenario, three bainitic ferrite platesare nucleated equally spaced in the austenite block at t = 0(Figure 1c). Despite the fact that nucleation usually occurs het-280

erogeneously at defects, the homogeneous nucleation assump-tion is necessary in the 1D simulations, since the calculationdomain does not provide an extra dimension into which the par-ticle could grow.

In order to build a gradual understanding of the studied phe-285

nomena, the martensite-austenite carbon partitioning and thebainite reaction were first approached separately for the lowcarbon alloy. In addition, as most of the works in the literatureassume the simplification of constant carbon diffusivity, the ef-fect of adopting this simplification was tested by comparing the290

cases where constant DγC and composition dependent Dγ

C are as-sumed. The carbon partitioning was simulated using the samegeometry employed in the coupled model. Simulations of thebainitic ferrite growth were conducted considering an austenitefilm with the same dimensions as the untransformed austenite295

in the coupled model.The chemical potentials of carbon and iron at the partition-

ing temperature were determined by thermodynamical calcula-tions using the TCFE8 database with the Thermo-Calc® soft-ware. Additionally, in order to account for the discrepancy be-300

tween the experimental and computationally determined ther-modynamical limits for the bainite reaction, the chemical po-tentials of the bainitic ferrite were modified by considering theWBs theory. Following the extrapolation proposed by Hillert,the extra energy for the bainite reaction at 350 °C is 1965 J/mol.305

This energy was summed up to the chemical potentials of bainiticferrite, leading to a carbon content in austenite of approximately1.75 wt % at the metastable equilibrium, as is visualized in Fig-ure 2.

4. Results and discussion310

4.1. Effect of carbon diffusivity on kinetics of carbon partition-ing

Figure 3a shows the evolution of carbon profiles during thepartitioning process simulated assuming a constant value of Dγ

Cequals to 1.72 × 10−17 m2/s; Figure 3b shows the carbon pro-315

files obtained assuming composition dependence of DγC . In

both cases the carbon content of austenite at interface is veryhigh for short partitioning times and gradually decreases as thecarbon partitioning proceeds. Similar results have been pre-viously reported in the literature [5, 6]. It is remarkable that320

when DγC is assumed constant, the carbon composition at the

interface reaches a peak value of 6.99 wt %, which is higherthan the fraction of carbon in cementite. On the other hand,

0 1 2 3 4 5Carbon content (wt.%)

Ae3WBsT0

1.75 4.23

350 °C

Figure 2: Fe-C phase diagram showing lines proposed to be the critical limitfor growth of bainitic ferrite (Ae3, WBs, and T0).

when the composition dependence on the diffusivity is consid-ered the maximum composition is 3.11 wt %. Furthermore, car-325

bon in martensite at the interface quickly decreases to valuesnear zero when the effect of carbon concentration on the diffu-sivity is considered (< 0.01 wt % after 0.01 s). It is also noticedthat the gradient of carbon in austenite is different for each con-dition. The carbon profiles in martensite are relatively flat when330

DγC is constant. In turn, a steeper gradient is observed in marten-

site at the interface when the composition dependence of DγC is

considered.These observations can be explained by how the carbon dif-

fusivity affects the mass balance of carbon in the system. It335

can be proven [14] that, as long as soft impingement has notoccurred (i.e., for short times), the following relation must befulfilled in order to satisfy the mass conservation at the inter-face:

Dα’C int

DγCint

cγint − c0

c0 − cαint

where c0 is the initial carbon content, and cpint and Dp

Cint are340

the fraction of carbon and diffusion coefficient of carbon in thephase p at the interface. Evaluation of the diffusion coefficientof austenite (eq. 7) implies that an increase of 1 wt % in the car-bon increases the carbon diffusivity by about one order of mag-nitude. Thus, the high concentration of carbon at the interface345

leads to a local increase of the carbon diffusivity, decreasing theDα’

C intDγ

C intratio. Therefore, when the composition dependence of Dγ

is considered, cγint and cα’int must be lower in order to equation 10

be satisfied. For times longer than 0.1 s soft impingement oc-curs in martensite and interfacial compositions decrease, devi-350

ating from the values predicted by equation 10. It is worth men-tioning that the values of cγint and cα’

int are restrained by the localequilibrium of carbon and, therefore, cannot be freely changed.

The kinetics of carbon partitioning can be quantitatively vi-sualized by means of the evolution of the average carbon con-355

tent of martensite and austenite with time (Figure 3c). For con-

stant DγC the martensite is completely decarburized after about

2 s, whereas the same state is reached after only 0.1 s when thecomposition dependence of Dγ

C is considered. In both cases,once the martensite plate is completely decarburized the com-360

position of austenite reaches the value predicted by the CCEmodel (1.12 wt %). The different kinetics are explained on basisof which process controls the carbon partitioning. For composi-tion dependent Dγ

C the carbon partitioning is controlled by car-bon diffusion in martensite due to the rapid decrease of carbon365

in martensite at the interface for short times. Conversely, forconstant Dγ

C the process is controlled by diffusion in austenite.It is noteworthy that despite the decarburization of martensitebeing completed after short times (< 10 s), the homogeniza-tion of carbon in austenite takes substantially longer. From the370

carbon profiles, the homogenization of carbon in austenite isreached after 100 s when Dγ

C is assumed constant and 10 s whenit varies with composition. This happens because the homog-enization depends solely on the carbon diffusion in austenite,which occurs at a slower pace than diffusion in martensite.375

The carbon profiles during the growth of a single bainiticferrite plate are shown in figures 4a (constant Dγ

C) and 4b (DγC

varies with composition). In the mixed-mode model the com-positions at the interface gradually evolve from the initial com-position to the composition of (metastable) equilibrium. This380

composition is dictated by the WBs limit, corresponding to acarbon concentration of 1.75 wt %. At the beginning, the sys-tem is in a non-equilibrium state where the chemical potentialof iron is higher in austenite than in ferrite, leading to a posi-tive chemical driving force for the growth of the ferrite into the385

austenite. As the growth of bainitic ferrite proceeds, the carbonrejected from ferrite progressively builds up in austenite, the in-terfacial composition approaching the composition of equilib-rium. When the equilibrium is reached the chemical potentialof iron is equalized across the interface and, consequently, there390

is no more chemical driving force to be dissipated and the inter-face stops. The final transformed fraction and the final positionof the interface, determined by the lever rule, are 85 wt % and0.0045 µm, respectively.

In the carbon profiles the dotted lines represent the carbon395

content of austenite at the interface plotted as a function the po-sition of the interface. In both cases these lines show inflectionsas the interfaces advance, representing a decrease of the inter-face velocity. In Figure 4b the carbon profile at 1.5 s shows thatat this stage the diffusion field of carbon ahead of the interface400

interferes with the diffusion field coming from the other half ofthe austenite, i.e., soft impingement occurs. Soft impingementleads to a faster accumulation of carbon in the untransformedportion of austenite, therefore leading to a faster decrease ofthe chemical potential of carbon in austenite and the chemical405

driving force for the interface migration (eq. 1). Consequentlythe interface slows down at a higher rate, causing the observedinflections.

The comparison of the evolution of the interface positionswith time is shown in Figure 4c. The results are also com-410

pared with an analytical solution assuming constant diffusiv-ity in austenite and equilibrium compositions at the interface(dashed line). In the simulated curves the interface moves faster

10-4 10-3 10-2 10-1 100 101

Time (s)

. C inα0 a

) CCE (1.12 wt.%)

0.10 0.08 0.06 0.04 0.02 0.00 0.02Position (µm)

6.99 wt.%0.001 s

0.01 s

0.10 0.08 0.06 0.04 0.02 0.00 0.02Position (µm)

3.11 wt.%0.001 s

0.01 s

α 0, D γC constant

γ , D γC contant

α 0, D γC = f (c

γ , D γC = f (c

α' γ

Figure 3: Simulation of α’/γ carbon partitioning. a) Carbon profiles determined assuming DγC constant. b) Carbon profiles obtained assuming the dependence

of the carbon content in DγC . c) Comparison of the evolution of the average carbon content in α’ and γ assuming Dγ

C constant (black lines) and varying with thecomposition (Dγ

C = f (cγi ), red lines).

0.000 0.005 0.010 0.015 0.020 0.025 0.030Position (µm)

WBs (1.75 wt.%)

0.01 s

a) b) c)

0 2 4 6 8 10Time (s)

Metastable equilibrium

DγC = f(c

DγC constant

Analytical solution

0.000 0.005 0.010 0.015 0.020 0.025 0.030Position (µm)

WBs (1.75 wt.%)

0.01 s

10 s1.5 s

Figure 4: Simulation of growth of single plate of αb a) Carbon profiles determined assuming DγC constant. b) Carbon profiles obtained assuming the dependence of

the carbon content in DγC . c) Comparison of the evolution of the αb/γ interface position.

in the beginning and slows down by soft impingement whenthe position of equilibrium (0.0045 µm) is approached. The415

analytical solution does not account for soft impingement andtherefore equilibrium state is never reached in this case. As aconsequence of the higher diffusivity in austenite, the kineticsis faster when Dγ

C varies with the composition than when DγC

is constant (for both simulated case and analytical solution).420

However, the growth rate according to the analytical solution isslightly higher than in the simulated constant Dγ

C case, despitethe similar assumptions. This difference is observed becausethe interface migration in the mixed-mode model is limited byits mobility M, while the analytical solution effectively consists425

of an extreme case where M is infinite and the interface velocitydepends only on the diffusion of carbon in austenite.

From the above results, it is clear that the composition de-pendence of diffusivity has an important effect on the kineticsof the phase transformations taking place during the Q&P pro-430

cess. In the next section, the simulations considering the simul-taneous occurrence of the bainite reaction and the martensite-austenite carbon partitioning are presented considering this ef-fect.

4.2. Coupling martensite-austenite carbon partitioning and bai-435

nite reaction4.2.1. Low carbon alloy

Figure 5a shows the carbon profiles obtained with the cou-pled model for the low carbon alloy (0.25 wt %C) partitionedat 350 °C. The vertical lines represent the position of the inter-440

faces at the end of the simulation. As in the case of martensite-austenite carbon partitioning, for short times the carbon profilesin austenite are very sharp close to the martensite-austenite in-terface and attenuate for longer times. The carbon content ofmartensite at the interface quickly drops to a value near zero,445

leading to a process controlled by the carbon diffusion in marten-site. A flat carbon profile in austenite, denoting its homogeniza-tion, is obtained after 10 s.

The development of the carbon profile during the growth ofthe bainitic ferrite plate can be analyzed by means of Figure 5b,450

which shows a detail of the austenite-bainitic ferrite interface.The dotted line represents the austenite carbon content at theinterface plotted as function of the position of the interface. Aninflection in the dotted line is observed at about 0.3 s, associatedwith a decrease of the interface velocity due to soft impinge-455

ment, as discussed before. In this case the diffusion fields thatinteract are those resulting from the carbon partitioning fromthe martensite and due to the rejection of carbon during to the

0.10 0.08 0.06 0.04 0.02 0.00 0.02Position (µm)

3.12 wt.%

WBs (1.75 wt.%)

CCE (1.12 wt.%)

0.001 s

0.01 s

α' αbγ

0.000 0.005 0.010 0.015 0.020 0.025 0.030Position (µm)

0.001 s

0.01 s

10 s 0.3 s

Figure 5: Carbon profiles for the coupled model applied to the low carbon alloy(0.25 wt %). a) Full carbon profiles. b) Detail of carbon profiles close to thebainitic ferrite plate.

growth of bainite.Evolution of the average carbon content in martensite and460

austenite is shown in Figure 6 and compared to the results ofthe martensite-austenite carbon partitioning case. Evolution ofaverage carbon in martensite (black and red solid lines) showsno different behavior from one case to another. This occurs be-cause the decarburization of martensite is essentially controlled465

by carbon diffusion in martensite, so that its kinetics dependsonly on the size and composition of the martensite plate, beingvirtually insensitive to what occurs in the neighboring austen-ite. On the other hand, kinetics of carbon enrichment of austen-ite (dashed lines) shows some important differences. The two470

curves follow the same trend for very short times (< 0.03 s)but, as the rejection of carbon due to growth of bainitic ferritetakes place, the carbon enrichment of austenite becomes fasterin the coupled model, reaching the CCE composition earlierthan in the α’/γ case. Moreover, once the austenite-bainitic475

ferrite interface is free to move, the carbon content of austen-ite is no longer limited by the CCE composition determinedfrom the initial microstructure and keeps increasing until theequilibrium composition between bainitic ferrite and austenite(i.e., the WBs limit) is reached. In the end, the carbon diffu-480

sion achieves the stationary state and the chemical potential of

10-4 10-3 10-2 10-1 100 101

Time (s)

. C in

CCE (1.12 wt.%)

WBs (1.75 wt.%)

α 0, coup. model

α 0, α 0/ γ

γ , coup. model

γ , α 0/ γ

Figure 6: Comparison of the evolution of the average composition of α’ and γin the coupled model (black lines) and in the α’/γ model (red lines).

carbon becomes homogeneous in the system.A related point to consider is that the system does not estab-

lish a true equilibrium when the carbon profiles achieve the sta-tionary state. Unlike the bainitic ferrite, the chemical potentials485

of martensite (carbon supersaturated ferrite) were not modifiedto account for an extra energy and, consequently, the equilib-rium compositions for the martensite-austenite pair are givenby the extensions of the Ae1 and Ae3 lines. Hence, marten-site establishes a constrained carbon equilibrium in relation to490

a microstructure containing the phase fraction of austenite de-termined by the WBs limit. In the final state, metastable equi-librium between bainitic ferrite and austenite and CCE betweenmartensite and austenite are simultaneously established. Suchsituation is visualized in the schematic free energies curves shown495

in Figure 7.

Fe < µγFe

µα bFe = µγFe

1965 J/mol(WBs)

α bγ

x α bC

C = µα bC

= µ γC

Figure 7: Schematic diagram illustrating the free energies and compositions ofthe phases in the coupled model when the stationary state is reached.

4.2.2. High carbon alloyThe carbon profiles obtained from the simulations for the

high carbon case considering the nucleation of only one bainiteplate are shown in Figure 8a. As with all the other discussed500

0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50Position (µm)

1000 s

0.4 0.2 0.0 0.2 0.4Position (µm)

4.18 wt.%

WBs (1.75 wt.%)CCE (1.65 wt.%)

1000 s

α' γ

γ αb

Figure 8: Carbon profiles for the coupled model applied to the high carbonalloy (0.8 wt %) with only one nucleated bainite plate. a) Full carbon profiles.b) Detail of carbon profiles close to the bainitic ferrite plate.

cases, the compositions at the interface experience a rapid risein austenite and a fast drop in martensite. The higher composi-tion peak at the interface (4.18 wt%) when compared to the lowcarbon case can be explained on basis of equation 10. When theinitial composition c0 is raised, the right term of eq. 10 is eval-505

uated to a lower value. The equation is balanced when either

the ratioDα’

C intDγ

C int(left term) decreases or when the interfacial com-

positions increases. Both requisites are simultaneously fulfilledwhen the interfacial composition of austenite cγint increases, ex-plaining the result. The carbon partitioning from martensite510

to austenite takes longer than in the low carbon case, mainlyreflecting the coarser simulated microstructure. The marten-site plate is completely decarburized after 10 s and the carbonin austenite is homogenized at the WBs composition at about1000 s.515

Carbon profiles in austenite near the bainitic ferrite plateare shown in Figure 8b. At the beginning the interface movesforward consuming austenite, but at 50 s the direction of the in-terface changes and the plate starts shrinking. Such behavioris more clearly illustrated in Figure 9a, in which the evolution520

of the interface position and the carbon content in austenite atthe interface are exposed. When the interface shifts its direction

0 200 400 600 800 1000Time (s)

WBs (1.75 wt.%)

a)cγin t

Interface position

Figure 9: a) Evolution of position of the αb/γ interface with time (solid line)superimposed with the composition of austenite at the same interface (dashedline). b) Evolution of the chemical driving force available for αb growth.

of motion, the interfacial composition of austenite (cγint) reachesthe WBs composition and continues further increasing due tothe carbon partitioned from the martensite. Once the carbon525

concentration in austenite is higher than the equilibrium value,the chemical potential of iron becomes higher in austenite thanin ferrite. This causes the chemical driving force available forbainite growth to become negative (as shown in Figure 9b),which in turn induces the interface moving back towards the530

bainitic ferrite plate. Additionally, the change of the interfacemotion causes the development of a negative composition gra-dient in austenite in order to keep the mass balance preserved.At the moment the interface shifts its motion this gradient iszero, as can be seen in the carbon profile at 50 s (dashed line).535

Carbon profiles for the case assuming nucleation of threebainite nuclei in austenite are presented in Figure 10. Becauseof the symmetry condition at the extremities of the simulateddomain, only two (one and a half) bainitic ferrite plates, αb1and αb2, are represented in the figure. Regarding the carbon540

partitioning from martensite, this simulation represents all thepreviously reported phenomena. However, in contrast to thelast case, the present scenario exhibits two austenite regions (γ1and γ2) separated by the bainitic ferrite plates. The three mobilebainitic ferrite-austenite interfaces are named i1, i2, and i3, as545

0.4 0.2 0.0 0.2 0.40.00.51.01.52.02.53.03.54.04.5

0.4 0.2 0.0 0.2 0.40.0

0.4 0.2 0.0 0.2 0.4Position (µm)

0.4 0.2 0.0 0.2 0.40.0

1000 s

γ1α' γ2 γ2α' γ1

α' γ

α' γ1

αb1 αb2

i1 i2 i3

αb1 αb2

Figure 10: Carbon profiles for the coupled model applied to the high carbon alloy (0.8 wt %) with three nucleated bainite plates.

indicated in the figure.Due to the smaller dimensions of the austenite blocks, soft

impingement occurs after a shorter time than in the case assum-ing only one bainite plate. The carbon content in the austenitegrain closer to the martensite (γ1) rapidly increases, surpassing550

the WBs composition and reversing the movement of the inter-face i1 after only 4 s. The other austenite grain (γ2) experiencessoft impingement of the diffusion fields originated from the car-bon rejected during the growth of the bainite plates. After 50 sthe carbon in γ2 is completely homogenized at the WBs com-555

position, which is accompanied by the full stop of interfaces i2and i3. However, interface i1 keeps moving backwards as longas the equilibrium state in γ1 is not achieved. As this state isnot achieved before αb1 is completely dissolved, at about 90 s,a single austenite block, γ, is formed. This situation is illus-560

trated by the carbon profile at the imminence of dissolution ofthe plate, shown in Figure 10. Afterwards, the excess of carbonin austenite causes the reversal of the movement of the inter-face i3, establishing a situation similar to the one-bainite platescenario. The remaining bainitic ferrite plate keeps dissolving565

until the carbon in austenite is homogenized at the WBs com-position, which happens at about 1000 s.

In this work the martensite/austenite interface migration hasbeen neglected in order to avoid too many variables being si-multaneously evaluated in the model, in spite of such mobility570

being experimentally observed in some alloys [4, 32]. Indeed,carbon partitioning results in a difference of chemical poten-tials of iron (and substitutional elements) across the interface,leading to a driving force for martensite/austenite interface mi-gration [33]. However, whether interface migration will occur575

or not depends on various factors, such as composition, orien-

tation relationship, martensite morphology [32], and pinning ofinterface dislocations by segregation [34, 35]. The influenceof the martensite/austenite interface migration on carbon parti-tioning was modeled by Santofimia et al. considering different580

interface mobilities [5, 6]. In accordance to the results of thepresent model, Santofimia’s model showed that kinetics of de-carburization of martensite is very fast and it is little affected byinterface migration. On the other hand, evolution of carbon pro-files in austenite and interface position is very sensitive to inter-585

face mobility. In the case of a incoherent interface, Santofimiashowed that carbon partitioning induces interface migration to-wards austenite for short times and then back to martensite forlonger times. For the case of a semicoherent interface, car-bon profiles for short times are similar to those obtained in the590

immobile interface case, while for longer times the interfacemoves towards the martensite plate. In both cases equilibriumbetween the bcc and fcc phases is approached for longer parti-tioning times, such as in the present model.

The effect of mobility of the austenite/bainitic ferrite inter-595

face on carbon partitioning can be interpreted in a similar wayto Santofimia’s conclusions for martensite/austenite interfacemigration. In the case of adopting a mobility value infinitelylarge, the plate thickening of bainite would be controlled bycarbon diffusion. Since the martensite/austenite carbon par-600

titioning is governed by the carbon diffusion in martensite –much faster than carbon diffusion in austenite – the kineticsof martensite decarburization would be little changed from theprediction of the current model. On the other hand, a lowervalue of the interface mobility would imply a longer time nec-605

essary for the homogenization of carbon in austenite.The effect of varying the quenching temperature is also worth

discussing. As shown in the simulations, dissolution of bainiticferrite occurs by excess of carbon in austenite at the vicinity ofthe αb/γ interface. A higher fraction of martensite is produced610

at a lower quenching temperature. In this situation, the amountof carbon available to partition to austenite will be larger, en-hancing the dissolution process of bainitic ferrite. There willbe a critical quenching temperature where any bainitic ferriteprecipitate will be completely dissolved at the end of the parti-615

tioning step.In summary, results show that carbon partitioning from marten-

site to austenite significantly affects the kinetics of the bainitereaction. Partial or total dissolution of bainitic ferrite plates oc-curs due to accumulation of carbon coming from martensite to620

the adjacent austenite. The kinetics of these phenomena canbe partly explained by the 1D approach used in the currentwork. When the problem is approached in two or three dimen-sions, carbon is able to diffusive through other routes not rep-resented in the 1D case. Consequently, carbon is more prone to625

be trapped in isolated regions in 1D simulations than in 2D/3Dsimulations. This behavior explains the development of a moreheterogeneous distribution of carbon along austenite grains in2D/3D simulations, as pointed out by Takahama et al. [15].

Moreover, the kinetics of carbon partitioning is little af-630

fected by simultaneous occurrence of the bainite reaction. Thethermodynamical conditions in the martensite-austenite inter-face imply that carbon in martensite at the interface decreasesto zero at very short times, staying at a nearly constant valueuntil the end of the partitioning step. Consequently, the rate635

of carbon partitioning becomes controlled by carbon diffusionin martensite. The decarburization time will be essentially af-fected only by the dimensions of the martensite plate and theinitial carbon content. On the other hand, presence of carbidesmight play a substantial role in defining the kinetics of carbon640

partitioning. Once the carbides are formed in martensite, car-bon partitioning will be only possible if this is accompaniedby dissolution of carbides [7], which is presumably a processmuch slower than diffusion of carbon in martensite. The effectof carbides precipitation on the kinetics of carbon partitioning645

is currently being pursued and will be presented in future work.

5. Conclusions

The redistribution of carbon during the partitioning step ofthe Q&P process has been assessed by means of a modelingapproach that considers the simultaneous occurrence of parti-650

tioning of carbon between martensite and austenite and forma-tion of bainitic ferrite. Two alloy compositions (low and highcarbon) and different configurations for the bainite nuclei wereconsidered in the simulations. The main conclusions obtainedfrom this work are summarized as follows:655

1. Composition dependence of the diffusion coefficient ofcarbon in austenite substantially affects the results andshould not be disregarded in the simulations. When Dγ

Cis assumed constant the martensite-austenite carbon par-titioning is governed by diffusion in austenite and it is660

about 10 times slower than when DγC varies with compo-

sition.2. When the bainite reaction is suppressed the system reaches

the constrained carbon equilibrium (CCE) at the endpoint.When bainite reaction is considered the system estab-665

lishes a metastable equilibrium between bainitic ferriteand austenite and the CCE between martensite and austen-ite.

3. The interfacial composition of martensite decreases tovalues near zero for short partitioning times, causing the670

kinetics of martensite-austenite carbon partitioning to becontrolled by diffusion of carbon in martensite. The rateof decarburization of the martensite plates is not affectedby the simultaneous occurrence of the bainite reaction.However, rate of carbon-enrichment of austenite is sped675

up when formation of bainitic ferrite is considered.4. The kinetics of the bainite reaction is strongly influenced

by martensite-austenite carbon partitioning. The carbonpartitioned from martensite accumulates in austenite andslows down the growth of bainitic ferrite, eventually caus-680

ing the reversal of the movement of the interface. Ifthe bainitic ferrite plate is nucleated close enough to themartensite plate it can be completely dissolved by the de-scribed phenomena.

Acknowledgments685

Arthur Nishikawa acknowledges financial support by Capes(grant 7409/2015-00 PDSE). Hélio Goldenstein acknowledgespartial funding by CNPq.

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